Subgroups — #538×D₁₀₉ #28376366

Order 13952.

1336 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295616	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295615	C₂	1	2	yes	maximal, complemented
3	#20295614	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295612	C₂	1	2	yes	maximal, complemented
6	#20295611	C₂	1	2	yes	maximal, complemented
7	#20295610	C₂	1	2	yes	maximal, complemented
8	#20295609	C₂	1	2	yes	maximal, complemented
9	#20295608	C₂	1	2	yes	maximal, complemented
10	#126×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#124×D₁₀₉	C₂	1	2	yes	maximal, complemented
12	#123×D₁₀₉	C₂	1	2	yes	maximal, complemented
13	C₂²⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#121×D₁₀₉	C₂	1	2	yes	maximal, complemented
15	#121×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#120×D₁₀₉	C₂	1	2	yes	maximal, complemented
17	C₄×D₄×D₁₀₉	C₂	1	2	yes	maximal, complemented
18	C₂×C₂²⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
19	#538×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₄×D₁₀₉	C₂²	1	2	yes	—
23	C₂×D₄×D₁₀₉	C₂²	1	2	yes	—
24	C₂×D₄×D₁₀₉	C₂²	1	2	yes	—
25	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	complemented
29	#16812694	C₂²	1	2	yes	complemented
30	#16812694	C₂²	1	2	yes	complemented
31	#16812694	C₂²	1	2	yes	complemented
32	#16812693	C₂²	1	2	yes	complemented
33	#16812692	C₂²	1	2	yes	complemented
34	#16812692	C₂²	1	2	yes	complemented
35	#16812692	C₂²	1	2	yes	complemented
36	#16812688	C₂²	1	2	yes	complemented
37	#16812687	C₂²	1	2	yes	complemented
38	#16812686	C₂²	1	2	yes	complemented
39	#16812685	C₂²	1	2	yes	complemented
40	#16812685	C₂²	1	2	yes	complemented
41	#16812685	C₂²	1	2	yes	complemented
42	#16812685	C₂²	1	2	yes	complemented
43	#16812684	C₂²	1	2	yes	complemented
44	#16812684	C₂²	1	2	yes	complemented
45	#16812683	C₂²	1	2	yes	complemented
46	#16812682	C₂²	1	2	yes	complemented
47	#16812681	C₂²	1	2	yes	complemented
48	#16812681	C₂²	1	2	yes	complemented
49	#16812680	C₂²	1	2	yes	complemented
50	#16812679	C₂²	1	1	yes	characteristic, complemented
51	#16812678	C₂²	1	1	yes	characteristic, complemented
52	#16812677	C₂²	1	2	yes	complemented
53	#16812676	C₂²	1	2	yes	complemented
54	#16812675	C₂²	1	1	yes	characteristic, complemented
55	#16812674	C₂²	1	2	yes	complemented
56	#16812674	C₂²	1	1	yes	characteristic, complemented
57	#16812673	C₂²	1	1	yes	characteristic, complemented
58	#16812673	C₂²	1	2	yes	complemented
59	#16812672	C₂²	1	2	yes	complemented
60	#16812672	C₂²	1	2	yes	complemented
61	#16812672	C₂²	1	2	yes	complemented
62	#16812671	C₂²	1	2	yes	complemented
63	#16812671	C₂²	1	2	yes	complemented
64	#16812671	C₂²	1	2	yes	complemented
65	#16812670	C₂²	1	2	yes	complemented
66	#16812670	C₂²	1	1	yes	characteristic, complemented
67	#16812669	C₂²	1	2	yes	complemented
68	#16812669	C₂²	1	2	yes	complemented
69	#16812669	C₂²	1	2	yes	complemented
70	#16812668	C₂²	1	2	yes	complemented
71	#16812667	C₂²	1	2	yes	complemented
72	#16812667	C₂²	1	2	yes	complemented
73	#16812667	C₂²	1	2	yes	complemented
74	C₄×D₄₃₆	C₂²	1	2	yes	complemented
75	D₄×Dic₁₀₉	C₂²	1	2	yes	complemented
76	#16812661	C₂²	1	2	yes	complemented
77	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
78	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
79	#16812659	C₂²	1	2	yes	complemented
80	#16812659	C₂²	1	2	yes	complemented
81	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
82	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
83	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
84	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
85	C₂×C₂²⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
86	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
87	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
88	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
89	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
90	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
91	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
92	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
93	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
94	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
95	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
96	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
97	C₄²×D₁₀₉	C₂²	1	2	yes	complemented
98	#126×C₁₀₉	C₂²	1	2	yes	complemented
99	#124×C₁₀₉	C₂²	1	2	yes	complemented
100	#123×C₁₀₉	C₂²	1	2	yes	complemented
101	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
102	#121×C₁₀₉	C₂²	1	2	yes	complemented
103	#121×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
104	#120×C₁₀₉	C₂²	1	2	yes	complemented
105	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
106	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
107	C₂³×D₁₀₉	—	2	4	no	—
108	C₂³×D₁₀₉	—	4	2	no	—
109	C₂³×D₁₀₉	—	4	2	no	—
110	C₂³×D₁₀₉	C₂³	1	2	yes	—
111	C₂³×D₁₀₉	C₂³	1	2	yes	—
112	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
113	C₂³×D₁₀₉	C₂³	1	2	yes	—
114	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
115	C₂×Dic₂₁₈	C₂³	1	2	yes	complemented
116	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic, complemented
117	Q₈×D₁₀₉	—	4	2	no	—
118	C₂×D₄₃₆	C₂³	1	2	yes	—
119	C₂×D₄₃₆	C₂³	1	2	yes	—
120	C₂×D₄₃₆	C₂³	1	2	yes	—
121	C₂×D₄₃₆	C₂³	1	1	yes	characteristic, complemented
122	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
123	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
124	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
125	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
126	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
127	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
128	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	complemented
129	D₄×D₁₀₉	—	2	8	no	—
130	D₄×D₁₀₉	—	4	4	no	—
131	D₄×D₁₀₉	—	4	4	no	—
132	D₄×D₁₀₉	—	4	2	no	—
133	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
134	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
135	C₂²×Dic₁₀₉	C₂³	1	2	yes	complemented
136	C₂×C₄×D₁₀₉	—	4	2	no	—
137	C₂×C₄×D₁₀₉	—	4	2	no	—
138	C₂×C₄×D₁₀₉	—	4	4	no	—
139	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
140	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
141	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
142	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
145	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
146	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₁₀₉	C₂³	1	2	yes	complemented
149	C₂×C₄×D₁₀₉	C₂³	1	2	yes	complemented
150	C₄⋊Dic₁₀₉	C₂³	1	2	yes	complemented
151	C₄⋊Dic₁₀₉	C₂³	1	2	yes	complemented
152	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic, complemented
153	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic, complemented
154	#13182008	C₂³	1	2	yes	complemented
155	#13182008	C₂³	1	2	yes	complemented
156	#13182008	C₂³	1	2	yes	complemented
157	#13182008	C₂³	1	2	yes	complemented
158	#13182008	C₂³	1	2	yes	complemented
159	#13182008	C₂³	1	2	yes	complemented
160	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
161	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
162	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
163	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
164	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
165	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	complemented
166	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	complemented
167	#13182006	C₂³	1	2	yes	—
168	#13182006	C₂³	1	2	yes	—
169	#13182006	C₂³	1	2	yes	—
170	#13182006	C₂³	1	2	yes	—
171	#13182006	C₂³	1	2	yes	—
172	#13182006	C₂³	1	2	yes	—
173	#13182006	C₂³	1	2	yes	—
174	#13182006	C₂³	1	2	yes	—
175	#13182006	C₂³	1	2	yes	—
176	#13182006	C₂³	1	2	yes	—
177	#13182006	C₂³	1	2	yes	complemented
178	#13182006	C₂³	1	2	yes	complemented
179	#13182006	C₂³	1	2	yes	complemented
180	#13182006	C₂³	1	2	yes	complemented
181	C₄×Dic₁₀₉	C₂³	1	2	yes	complemented
182	C₄×Dic₁₀₉	C₂³	1	2	yes	complemented
183	C₄×Dic₁₀₉	C₂³	1	2	yes	complemented
184	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
185	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
186	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
187	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
188	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
189	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
190	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
191	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
192	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	complemented
193	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	complemented
194	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	complemented
195	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
196	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
197	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
198	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
199	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
200	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
201	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
202	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	complemented
203	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	complemented
204	C₄²×C₁₀₉	C₂³	1	2	yes	complemented
205	C₂³×C₁₀₉	—	2	2	no	—
206	C₂³×C₁₀₉	—	4	1	no	—
207	C₂³×C₁₀₉	—	4	1	no	—
208	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
209	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
210	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
211	C₂²×D₁₀₉	—	2	2	no	—
212	C₂²×D₁₀₉	—	2	2	no	—
213	C₂²×D₁₀₉	—	2	2	no	—
214	C₂²×D₁₀₉	—	2	4	no	—
215	C₂²×D₁₀₉	—	2	4	no	—
216	C₂²×D₁₀₉	—	2	4	no	—
217	C₂²×D₁₀₉	—	2	4	no	—
218	C₂²×D₁₀₉	—	2	4	no	—
219	C₂²×D₁₀₉	—	2	4	no	—
220	C₂²×D₁₀₉	—	2	4	no	—
221	C₂²×D₁₀₉	—	4	2	no	—
222	C₂²×D₁₀₉	—	4	2	no	—
223	C₂²×D₁₀₉	—	4	1	no	—
224	C₂²×D₁₀₉	—	4	1	no	—
225	C₂²×D₁₀₉	—	4	16	no	—
226	C₂²×D₁₀₉	#41	1	4	yes	—
227	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
228	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
229	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
230	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
231	Q₈×C₁₀₉	—	4	1	no	—
232	D₄×C₁₀₉	—	2	4	no	—
233	D₄×C₁₀₉	—	4	2	no	—
234	D₄×C₁₀₉	—	4	2	no	—
235	D₄×C₁₀₉	—	4	1	no	—
236	C₂×C₄×C₁₀₉	—	4	1	no	—
237	C₂×C₄×C₁₀₉	—	4	1	no	—
238	C₂×C₄×C₁₀₉	—	4	2	no	—
239	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
240	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
241	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
242	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
245	C₂²⋊D₁₀₉	—	2	4	no	—
246	C₂²⋊D₁₀₉	—	2	4	no	—
247	C₂²⋊D₁₀₉	—	4	2	no	—
248	C₂²⋊D₁₀₉	—	4	2	no	—
249	C₂²⋊D₁₀₉	—	4	2	no	—
250	C₂²⋊D₁₀₉	—	4	2	no	—
251	C₂²⋊D₁₀₉	—	4	2	no	—
252	C₂×Dic₁₀₉	—	4	1	no	—
253	C₂×Dic₁₀₉	—	4	1	no	—
254	C₂×Dic₁₀₉	—	4	2	no	—
255	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
256	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
257	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
258	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
259	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
260	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
261	D₄₃₆	—	2	4	no	—
262	D₄₃₆	—	4	2	no	—
263	D₄₃₆	—	4	2	no	—
264	D₄₃₆	—	4	1	no	—
265	C₄×D₁₀₉	—	2	4	no	—
266	C₄×D₁₀₉	—	2	4	no	—
267	C₄×D₁₀₉	—	2	2	no	—
268	C₄×D₁₀₉	—	2	4	no	—
269	C₄×D₁₀₉	—	2	2	no	—
270	C₄×D₁₀₉	—	2	4	no	—
271	C₄×D₁₀₉	—	4	1	no	—
272	C₄×D₁₀₉	—	4	2	no	—
273	C₄×D₁₀₉	—	4	1	no	—
274	C₄×D₁₀₉	—	4	2	no	—
275	C₄×D₁₀₉	—	4	2	no	—
276	Dic₂₁₈	—	4	2	no	—
277	Dic₂₁₈	—	4	1	no	—
278	C₂²×C₁₀₉	—	2	1	no	—
279	C₂²×C₁₀₉	—	2	1	no	—
280	C₂²×C₁₀₉	—	2	2	no	—
281	C₂²×C₁₀₉	—	2	2	no	—
282	C₂²×C₁₀₉	—	2	2	no	—
283	C₂²×C₁₀₉	—	2	2	no	—
284	C₂²×C₁₀₉	—	2	2	no	—
285	C₂²×C₁₀₉	—	2	2	no	—
286	C₂²×C₁₀₉	—	4	4	no	—
287	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
288	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
289	C₂×D₁₀₉	—	2	1	no	—
290	C₂×D₁₀₉	—	2	1	no	—
291	C₂×D₁₀₉	—	2	8	no	—
292	C₂×D₁₀₉	—	2	2	no	—
293	C₂×D₁₀₉	—	2	2	no	—
294	C₂×D₁₀₉	—	2	2	no	—
295	C₂×D₁₀₉	—	2	2	no	—
296	C₂×D₁₀₉	—	2	2	no	—
297	C₂×D₁₀₉	—	2	2	no	—
298	C₂×D₁₀₉	—	4	8	no	—
299	C₂×D₁₀₉	—	4	4	no	—
300	C₂×D₁₀₉	—	4	8	no	—
301	C₂×D₁₀₉	—	4	8	no	—
302	C₂×D₁₀₉	C₂×#41	1	2	yes	—
303	C₂×D₁₀₉	C₂×#41	1	2	yes	—
304	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
305	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
306	C₄×C₁₀₉	—	2	2	no	—
307	C₄×C₁₀₉	—	2	1	no	—
308	C₄×C₁₀₉	—	2	2	no	—
309	C₄×C₁₀₉	—	2	2	no	—
310	C₄×C₁₀₉	—	2	1	no	—
311	C₄×C₁₀₉	—	2	2	no	—
312	Dic₁₀₉	—	2	2	no	—
313	Dic₁₀₉	—	2	1	no	—
314	Dic₁₀₉	—	2	2	no	—
315	Dic₁₀₉	—	2	2	no	—
316	Dic₁₀₉	—	2	1	no	—
317	Dic₁₀₉	—	2	2	no	—
318	C₂×C₁₀₉	—	2	2	no	—
319	C₂×C₁₀₉	—	4	2	no	—
320	C₂×C₁₀₉	—	4	2	no	—
321	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
322	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
323	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
324	D₁₀₉	—	2	2	no	—
325	D₁₀₉	—	4	2	no	—
326	D₁₀₉	—	4	2	no	—
327	D₁₀₉	#538	1	4	yes	direct product factor
328	C₂×#538	—	109	1	no	maximal, Sylow
329	C₁₀₉	C₂×#538	1	1	yes	characteristic, complemented, Sylow
330	#538	—	109	2	no	—
331	#538	—	109	2	no	—
332	#538	—	109	2	no	—
333	#538	—	109	1	no	—
334	#538	—	109	2	no	—
335	#538	—	109	1	no	—
336	#538	—	109	2	no	—
337	#538	—	109	2	no	—
338	#538	—	109	1	no	—
339	#538	D₁₀₉	1	1	yes	characteristic, direct product factor
340	C₂×#126	—	109	2	no	—
341	C₂×#124	—	109	2	no	—
342	C₂×#123	—	109	2	no	—
343	C₂×C₂²⋊Q₈	—	109	1	no	—
344	C₂×#121	—	109	2	no	—
345	C₂×#121	—	109	1	no	—
346	C₂×#120	—	109	2	no	—
347	C₂×C₄×D₄	—	109	2	no	—
348	C₂²×C₂²⋊C₄	—	109	1	no	—
349	C₂⁵	—	109	1	no	—
350	C₂²×Q₈	—	109	1	no	—
351	C₂²×D₄	—	109	2	no	—
352	C₂²×D₄	—	109	2	no	—
353	C₂²×D₄	—	109	1	no	—
354	C₂²×D₄	—	109	2	no	—
355	C₂³×C₄	—	109	1	no	—
356	C₂³×C₄	—	109	1	no	—
357	C₂³×C₄	—	109	2	no	—
358	#126	—	109	2	no	—
359	#126	—	109	2	no	—
360	#126	—	109	2	no	—
361	#126	—	109	2	no	—
362	#126	—	109	2	no	—
363	#126	—	109	2	no	—
364	#126	—	109	2	no	—
365	#126	C₂×D₁₀₉	1	2	yes	complemented
366	#124	—	109	2	no	—
367	#124	—	109	2	no	—
368	#124	—	109	2	no	—
369	#124	—	109	2	no	—
370	#124	—	109	2	no	—
371	#124	—	109	2	no	—
372	#124	—	109	2	no	—
373	#124	C₂×D₁₀₉	1	2	yes	complemented
374	#123	—	109	2	no	—
375	#123	—	109	2	no	—
376	#123	—	109	2	no	—
377	#123	—	109	2	no	—
378	#123	—	109	2	no	—
379	#123	—	109	2	no	—
380	#123	—	109	2	no	—
381	#123	C₂×D₁₀₉	1	2	yes	complemented
382	C₂²⋊Q₈	—	109	1	no	—
383	C₂²⋊Q₈	—	109	1	no	—
384	C₂²⋊Q₈	—	109	2	no	—
385	C₂²⋊Q₈	—	109	1	no	—
386	C₂²⋊Q₈	—	109	2	no	—
387	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
388	#121	—	109	2	no	—
389	#121	—	109	2	no	—
390	#121	—	109	2	no	—
391	#121	—	109	1	no	—
392	#121	—	109	2	no	—
393	#121	—	109	1	no	—
394	#121	—	109	2	no	—
395	#121	—	109	2	no	—
396	#121	—	109	1	no	—
397	#121	—	109	2	no	—
398	#121	—	109	2	no	—
399	#121	—	109	2	no	—
400	#121	C₂×D₁₀₉	1	2	yes	complemented
401	#121	C₂×D₁₀₉	1	1	yes	characteristic, complemented
402	#120	—	109	2	no	—
403	#120	—	109	2	no	—
404	#120	—	109	2	no	—
405	#120	—	109	2	no	—
406	#120	—	109	2	no	—
407	#120	—	109	2	no	—
408	#120	—	109	2	no	—
409	#120	C₂×D₁₀₉	1	2	yes	complemented
410	C₄×D₄	—	109	2	no	—
411	C₄×D₄	—	109	2	no	—
412	C₄×D₄	—	109	2	no	—
413	C₄×D₄	—	109	2	no	—
414	C₄×D₄	—	109	2	no	—
415	C₄×D₄	—	109	2	no	—
416	C₄×D₄	—	109	2	no	—
417	C₄×D₄	C₂×D₁₀₉	1	2	yes	complemented
418	C₂×C₄⋊C₄	—	109	2	no	—
419	C₂×C₄⋊C₄	—	109	2	no	—
420	C₂×C₄⋊C₄	—	109	1	no	—
421	C₂×C₄⋊C₄	—	109	1	no	—
422	C₂×C₂²⋊C₄	—	109	1	no	—
423	C₂×C₂²⋊C₄	—	109	1	no	—
424	C₂×C₂²⋊C₄	—	109	1	no	—
425	C₂×C₂²⋊C₄	—	109	2	no	—
426	C₂×C₂²⋊C₄	—	109	2	no	—
427	C₂×C₂²⋊C₄	—	109	2	no	—
428	C₂×C₂²⋊C₄	—	109	2	no	—
429	C₂×C₂²⋊C₄	—	109	2	no	—
430	C₂×C₂²⋊C₄	—	109	2	no	—
431	C₂×C₂²⋊C₄	—	109	2	no	—
432	C₂×C₂²⋊C₄	—	109	2	no	—
433	C₂×C₂²⋊C₄	—	109	2	no	—
434	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
435	C₂×C₄²	—	109	2	no	—
436	C₂⁴	—	109	1	no	—
437	C₂⁴	—	109	2	no	—
438	C₂⁴	—	109	1	no	—
439	C₂⁴	—	109	2	no	—
440	C₂⁴	—	109	2	no	—
441	C₂⁴	—	436	2	no	—
442	C₂⁴	—	436	2	no	—
443	C₂⁴	—	218	4	no	—
444	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
445	C₂×Q₈	—	109	1	no	—
446	C₂×Q₈	—	109	2	no	—
447	C₂×Q₈	—	436	2	no	—
448	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
449	C₂×D₄	—	109	2	no	—
450	C₂×D₄	—	109	2	no	—
451	C₂×D₄	—	109	2	no	—
452	C₂×D₄	—	109	2	no	—
453	C₂×D₄	—	109	2	no	—
454	C₂×D₄	—	109	2	no	—
455	C₂×D₄	—	109	2	no	—
456	C₂×D₄	—	109	2	no	—
457	C₂×D₄	—	109	2	no	—
458	C₂×D₄	—	109	1	no	—
459	C₂×D₄	—	109	2	no	—
460	C₂×D₄	—	436	4	no	—
461	C₂×D₄	—	436	2	no	—
462	C₂×D₄	—	436	4	no	—
463	C₂×D₄	—	218	8	no	—
464	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
465	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
466	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
467	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
468	C₂²×C₄	—	109	1	no	—
469	C₂²×C₄	—	109	1	no	—
470	C₂²×C₄	—	109	2	no	—
471	C₂²×C₄	—	109	2	no	—
472	C₂²×C₄	—	109	1	no	—
473	C₂²×C₄	—	109	1	no	—
474	C₂²×C₄	—	109	2	no	—
475	C₂²×C₄	—	109	2	no	—
476	C₂²×C₄	—	109	2	no	—
477	C₂²×C₄	—	109	2	no	—
478	C₂²×C₄	—	109	2	no	—
479	C₂²×C₄	—	109	1	no	—
480	C₂²×C₄	—	109	1	no	—
481	C₂²×C₄	—	109	2	no	—
482	C₂²×C₄	—	436	2	no	—
483	C₂²×C₄	—	436	2	no	—
484	C₂²×C₄	—	436	4	no	—
485	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
486	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
487	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	complemented
488	C₄⋊C₄	—	109	2	no	—
489	C₄⋊C₄	—	109	2	no	—
490	C₄⋊C₄	—	109	2	no	—
491	C₄⋊C₄	—	109	2	no	—
492	C₄⋊C₄	—	109	2	no	—
493	C₄⋊C₄	—	109	2	no	—
494	C₄⋊C₄	—	109	2	no	—
495	C₄⋊C₄	—	109	2	no	—
496	C₄⋊C₄	—	109	1	no	—
497	C₄⋊C₄	—	109	1	no	—
498	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	complemented
499	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	complemented
500	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
501	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
502	C₂²⋊C₄	—	109	2	no	—
503	C₂²⋊C₄	—	109	2	no	—
504	C₂²⋊C₄	—	109	2	no	—
505	C₂²⋊C₄	—	109	2	no	—
506	C₂²⋊C₄	—	109	2	no	—
507	C₂²⋊C₄	—	109	2	no	—
508	C₂²⋊C₄	—	109	2	no	—
509	C₂²⋊C₄	—	109	2	no	—
510	C₂²⋊C₄	—	109	2	no	—
511	C₂²⋊C₄	—	109	2	no	—
512	C₂²⋊C₄	—	109	2	no	—
513	C₂²⋊C₄	—	109	2	no	—
514	C₂²⋊C₄	—	109	2	no	—
515	C₂²⋊C₄	—	109	2	no	—
516	C₂²⋊C₄	—	109	2	no	—
517	C₂²⋊C₄	—	109	2	no	—
518	C₂²⋊C₄	—	109	2	no	—
519	C₂²⋊C₄	—	109	2	no	—
520	C₂²⋊C₄	—	109	2	no	—
521	C₂²⋊C₄	—	109	2	no	—
522	C₂²⋊C₄	—	109	2	no	—
523	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
524	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
525	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
526	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
527	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
528	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	complemented
529	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	complemented
530	C₄²	—	109	2	no	—
531	C₄²	—	109	2	no	—
532	C₄²	—	109	2	no	—
533	C₄²	C₂²×D₁₀₉	1	2	yes	complemented
534	C₂³	—	2	2	no	—
535	C₂³	—	4	1	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	109	1	no	—
538	C₂³	—	109	4	no	—
539	C₂³	—	109	1	no	—
540	C₂³	—	109	2	no	—
541	C₂³	—	109	2	no	—
542	C₂³	—	436	1	no	—
543	C₂³	—	436	1	no	—
544	C₂³	—	436	2	no	—
545	C₂³	—	436	2	no	—
546	C₂³	—	436	16	no	—
547	C₂³	—	218	4	no	—
548	C₂³	—	218	4	no	—
549	C₂³	—	218	4	no	—
550	C₂³	—	218	4	no	—
551	C₂³	—	218	4	no	—
552	C₂³	—	218	4	no	—
553	C₂³	—	218	4	no	—
554	C₂³	—	218	2	no	—
555	C₂³	—	218	2	no	—
556	C₂³	—	218	2	no	—
557	C₂³	C₂³×D₁₀₉	1	2	yes	—
558	C₂³	C₂³×D₁₀₉	1	2	yes	—
559	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
560	Q₈	—	4	1	no	—
561	Q₈	—	436	1	no	—
562	Q₈	—	436	2	no	—
563	D₄	—	2	4	no	—
564	D₄	—	4	2	no	—
565	D₄	—	4	2	no	—
566	D₄	—	4	1	no	—
567	D₄	—	436	2	no	—
568	D₄	—	436	1	no	—
569	D₄	—	436	2	no	—
570	D₄	—	436	2	no	—
571	D₄	—	436	2	no	—
572	D₄	—	436	2	no	—
573	D₄	—	436	2	no	—
574	D₄	—	436	2	no	—
575	D₄	—	218	4	no	—
576	D₄	—	218	4	no	—
577	D₄	—	218	4	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	109	2	no	—
582	C₂×C₄	—	109	2	no	—
583	C₂×C₄	—	109	2	no	—
584	C₂×C₄	—	109	1	no	—
585	C₂×C₄	—	109	1	no	—
586	C₂×C₄	—	109	2	no	—
587	C₂×C₄	—	436	2	no	—
588	C₂×C₄	—	436	2	no	—
589	C₂×C₄	—	436	2	no	—
590	C₂×C₄	—	436	1	no	—
591	C₂×C₄	—	436	1	no	—
592	C₂×C₄	—	436	1	no	—
593	C₂×C₄	—	436	2	no	—
594	C₂×C₄	—	436	1	no	—
595	C₂×C₄	—	218	4	no	—
596	C₂×C₄	—	218	2	no	—
597	C₂×C₄	—	218	4	no	—
598	C₂×C₄	—	218	2	no	—
599	C₂×C₄	—	218	4	no	—
600	C₂×C₄	—	218	4	no	—
601	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
602	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
603	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
604	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
605	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
606	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	4	4	no	—
616	C₂²	—	109	2	no	—
617	C₂²	—	109	2	no	—
618	C₂²	—	109	2	no	—
619	C₂²	—	109	2	no	—
620	C₂²	—	436	4	no	—
621	C₂²	—	436	8	no	—
622	C₂²	—	436	8	no	—
623	C₂²	—	436	8	no	—
624	C₂²	—	218	8	no	—
625	C₂²	—	218	2	no	—
626	C₂²	—	218	2	no	—
627	C₂²	—	218	2	no	—
628	C₂²	—	218	2	no	—
629	C₂²	—	218	2	no	—
630	C₂²	—	218	2	no	—
631	C₂²	—	218	1	no	—
632	C₂²	—	218	1	no	—
633	C₂²	#41×D₁₀₉	1	2	yes	—
634	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
635	C₄	—	2	2	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	1	no	—
640	C₄	—	2	2	no	—
641	C₄	—	218	2	no	—
642	C₄	—	218	2	no	—
643	C₄	—	218	2	no	—
644	C₄	—	218	1	no	—
645	C₄	—	218	1	no	—
646	C₄	—	218	2	no	—
647	C₂	—	2	2	no	—
648	C₂	—	4	2	no	—
649	C₂	—	4	2	no	—
650	C₂	—	109	4	no	—
651	C₂	—	436	2	no	—
652	C₂	—	436	2	no	—
653	C₂	—	218	2	no	—
654	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
655	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic
656	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#538×C₁₀₉	C₂
hypercenter	#538	D₁₀₉
hypocenter	C₁₀₉	C₂×#538
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #538

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295616	C₂	semidirect
2. #20295615	C₂	semidirect
3. #20295614	C₂	semidirect
4. #20295613	C₂	semidirect
5. #20295612	C₂	semidirect
6. #20295611	C₂	semidirect
7. #20295610	C₂	semidirect
8. #20295609	C₂	semidirect
9. #20295608	C₂	semidirect
10. #126×D₁₀₉	C₂	semidirect
11. #124×D₁₀₉	C₂	semidirect
12. #123×D₁₀₉	C₂	semidirect
13. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
14. #121×D₁₀₉	C₂	semidirect
15. #121×D₁₀₉	C₂	semidirect
16. #120×D₁₀₉	C₂	semidirect
17. C₄×D₄×D₁₀₉	C₂	semidirect
18. C₂×C₂²⋊C₄×D₁₀₉	C₂	semidirect
19. #538×C₁₀₉	C₂	semidirect
21. C₂×Q₈×D₁₀₉	C₂²	semidirect
25. C₂×D₄×D₁₀₉	C₂²	semidirect
28. C₂²×C₄×D₁₀₉	C₂²	semidirect
29. #16812694	C₂²	semidirect
30. #16812694	C₂²	semidirect
31. #16812694	C₂²	semidirect
32. #16812693	C₂²	semidirect
33. #16812692	C₂²	semidirect
34. #16812692	C₂²	semidirect
35. #16812692	C₂²	semidirect
36. #16812688	C₂²	semidirect
37. #16812687	C₂²	semidirect
38. #16812686	C₂²	semidirect
39. #16812685	C₂²	semidirect
40. #16812685	C₂²	semidirect
41. #16812685	C₂²	semidirect
42. #16812685	C₂²	semidirect
43. #16812684	C₂²	semidirect
44. #16812684	C₂²	semidirect
45. #16812683	C₂²	semidirect
46. #16812682	C₂²	semidirect
47. #16812681	C₂²	semidirect
48. #16812681	C₂²	semidirect
49. #16812680	C₂²	semidirect
50. #16812679	C₂²	semidirect
51. #16812678	C₂²	semidirect
52. #16812677	C₂²	semidirect
53. #16812676	C₂²	semidirect
54. #16812675	C₂²	semidirect
55. #16812674	C₂²	semidirect
56. #16812674	C₂²	semidirect
57. #16812673	C₂²	semidirect
58. #16812673	C₂²	semidirect
59. #16812672	C₂²	semidirect
60. #16812672	C₂²	semidirect
61. #16812672	C₂²	semidirect
62. #16812671	C₂²	semidirect
63. #16812671	C₂²	semidirect
64. #16812671	C₂²	semidirect
65. #16812670	C₂²	semidirect
66. #16812670	C₂²	semidirect
67. #16812669	C₂²	semidirect
68. #16812669	C₂²	semidirect
69. #16812669	C₂²	semidirect
70. #16812668	C₂²	semidirect
71. #16812667	C₂²	semidirect
72. #16812667	C₂²	semidirect
73. #16812667	C₂²	semidirect
74. C₄×D₄₃₆	C₂²	semidirect
75. D₄×Dic₁₀₉	C₂²	semidirect
76. #16812661	C₂²	semidirect
77. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
78. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
79. #16812659	C₂²	semidirect
80. #16812659	C₂²	semidirect
81. C₄⋊C₄×D₁₀₉	C₂²	semidirect
82. C₄⋊C₄×D₁₀₉	C₂²	semidirect
83. C₄⋊C₄×D₁₀₉	C₂²	semidirect
84. C₄⋊C₄×D₁₀₉	C₂²	semidirect
85. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
86. C₂×#13182006	C₂²	semidirect
87. C₂×#13182006	C₂²	semidirect
93. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
94. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
95. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
96. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
97. C₄²×D₁₀₉	C₂²	semidirect
98. #126×C₁₀₉	C₂²	semidirect
99. #124×C₁₀₉	C₂²	semidirect
100. #123×C₁₀₉	C₂²	semidirect
101. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
102. #121×C₁₀₉	C₂²	semidirect
103. #121×C₁₀₉	C₂²	semidirect
104. #120×C₁₀₉	C₂²	semidirect
105. C₄×D₄×C₁₀₉	C₂²	semidirect
106. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
115. C₂×Dic₂₁₈	C₂³	semidirect
116. C₂×Dic₂₁₈	C₂³	semidirect
121. C₂×D₄₃₆	C₂³	semidirect
128. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
135. C₂²×Dic₁₀₉	C₂³	semidirect
148. C₂×C₄×D₁₀₉	C₂³	semidirect
149. C₂×C₄×D₁₀₉	C₂³	semidirect
150. C₄⋊Dic₁₀₉	C₂³	semidirect
151. C₄⋊Dic₁₀₉	C₂³	semidirect
152. C₄⋊Dic₁₀₉	C₂³	semidirect
153. C₄⋊Dic₁₀₉	C₂³	semidirect
154. #13182008	C₂³	semidirect
155. #13182008	C₂³	semidirect
156. #13182008	C₂³	semidirect
157. #13182008	C₂³	semidirect
158. #13182008	C₂³	semidirect
159. #13182008	C₂³	semidirect
165. C₂²⋊Dic₁₀₉	C₂³	semidirect
166. C₂²⋊Dic₁₀₉	C₂³	semidirect
177. #13182006	C₂³	semidirect
178. #13182006	C₂³	semidirect
179. #13182006	C₂³	semidirect
180. #13182006	C₂³	semidirect
181. C₄×Dic₁₀₉	C₂³	semidirect
182. C₄×Dic₁₀₉	C₂³	semidirect
183. C₄×Dic₁₀₉	C₂³	semidirect
185. C₂×Q₈×C₁₀₉	C₂³	semidirect
189. C₂×D₄×C₁₀₉	C₂³	semidirect
192. C₂²×C₄×C₁₀₉	C₂³	semidirect
193. C₄⋊C₄×C₁₀₉	C₂³	semidirect
194. C₄⋊C₄×C₁₀₉	C₂³	semidirect
195. C₄⋊C₄×C₁₀₉	C₂³	semidirect
196. C₄⋊C₄×C₁₀₉	C₂³	semidirect
202. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
203. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
204. C₄²×C₁₀₉	C₂³	semidirect
327. D₁₀₉	#538	direct (N × H)
329. C₁₀₉	C₂×#538	semidirect
339. #538	D₁₀₉	direct (N × H)
365. #126	C₂×D₁₀₉	semidirect
373. #124	C₂×D₁₀₉	semidirect
381. #123	C₂×D₁₀₉	semidirect
387. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
400. #121	C₂×D₁₀₉	semidirect
401. #121	C₂×D₁₀₉	semidirect
409. #120	C₂×D₁₀₉	semidirect
417. C₄×D₄	C₂×D₁₀₉	semidirect
434. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
448. C₂×Q₈	C₂²×D₁₀₉	semidirect
467. C₂×D₄	C₂²×D₁₀₉	semidirect
487. C₂²×C₄	C₂²×D₁₀₉	semidirect
498. C₄⋊C₄	C₂²×D₁₀₉	semidirect
499. C₄⋊C₄	C₂²×D₁₀₉	semidirect
500. C₄⋊C₄	C₂²×D₁₀₉	semidirect
501. C₄⋊C₄	C₂²×D₁₀₉	semidirect
528. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
529. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
533. C₄²	C₂²×D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295616	2
#20295615	2
#20295614	2
#20295613	2
#20295612	2
#20295611	2
#20295610	2
#20295609	2
#20295608	2
#126×D₁₀₉	2
#124×D₁₀₉	2
#123×D₁₀₉	2
C₂²⋊Q₈×D₁₀₉	2
#121×D₁₀₉	2
#120×D₁₀₉	2
C₄×D₄×D₁₀₉	2
C₂×C₂²⋊C₄×D₁₀₉	2
#538×C₁₀₉	2
C₂×#538	109

Subgroups — #538×D109 #28376366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #538×D₁₀₉ #28376366