Subgroups — #552×D₁₀₉ #28376535

Order 13952.

1124 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	#20295718	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295717	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295716	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295715	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295714	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295713	C₂	1	2	yes	maximal, complemented
7	#20295712	C₂	1	2	yes	maximal, complemented
8	#20295711	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295710	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295709	C₂	1	2	yes	maximal, complemented
11	#20295708	C₂	1	2	yes	maximal, complemented
12	#126×D₁₀₉	C₂	1	2	yes	maximal, complemented
13	#125×D₁₀₉	C₂	1	2	yes	maximal, complemented
14	#125×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#123×D₁₀₉	C₂	1	2	yes	maximal
16	C₂²⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
19	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
20	#117×D₁₀₉	C₂	1	2	yes	maximal, complemented
21	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
23	#552×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
27	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
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	#16812691	C₂²	1	2	yes	complemented
37	#16812691	C₂²	1	1	yes	characteristic, complemented
38	#16812690	C₂²	1	1	yes	characteristic, complemented
39	#16812690	C₂²	1	2	yes	complemented
40	#16812690	C₂²	1	2	yes	complemented
41	#16812690	C₂²	1	1	yes	characteristic, complemented
42	#16812689	C₂²	1	2	yes	complemented
43	#16812689	C₂²	1	2	yes	complemented
44	#16812689	C₂²	1	2	yes	complemented
45	#16812689	C₂²	1	2	yes	complemented
46	#16812689	C₂²	1	2	yes	complemented
47	#16812689	C₂²	1	2	yes	complemented
48	#16812684	C₂²	1	2	yes	—
49	#16812684	C₂²	1	2	yes	—
50	#16812683	C₂²	1	2	yes	—
51	#16812682	C₂²	1	2	yes	—
52	#16812681	C₂²	1	2	yes	—
53	#16812681	C₂²	1	2	yes	—
54	#16812680	C₂²	1	2	yes	—
55	#16812679	C₂²	1	1	yes	characteristic, complemented
56	#16812679	C₂²	1	1	yes	characteristic, complemented
57	#16812678	C₂²	1	1	yes	characteristic, complemented
58	#16812678	C₂²	1	1	yes	characteristic, complemented
59	#16812677	C₂²	1	2	yes	complemented
60	#16812677	C₂²	1	2	yes	complemented
61	#16812676	C₂²	1	2	yes	complemented
62	#16812676	C₂²	1	2	yes	complemented
63	#16812675	C₂²	1	1	yes	characteristic, complemented
64	#16812675	C₂²	1	1	yes	characteristic, complemented
65	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
66	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
67	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
68	#16812664	C₂²	1	2	yes	complemented
69	#16812664	C₂²	1	1	yes	characteristic, complemented
70	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
71	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
72	#16812661	C₂²	1	1	yes	characteristic
73	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
74	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
75	#16812659	C₂²	1	1	yes	characteristic
76	#16812659	C₂²	1	1	yes	characteristic
77	#16812658	C₂²	1	1	yes	characteristic, complemented
78	#16812658	C₂²	1	2	yes	complemented
79	#16812657	C₂²	1	2	yes	complemented
80	#16812657	C₂²	1	1	yes	characteristic, complemented
81	#16812657	C₂²	1	2	yes	complemented
82	#16812657	C₂²	1	1	yes	characteristic, complemented
83	#16812656	C₂²	1	2	yes	complemented
84	#16812656	C₂²	1	2	yes	complemented
85	#16812656	C₂²	1	2	yes	complemented
86	#16812655	C₂²	1	2	yes	complemented
87	#16812655	C₂²	1	2	yes	complemented
88	#16812655	C₂²	1	2	yes	complemented
89	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
90	C₂×#13182008	C₂²	1	2	yes	complemented
91	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
92	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
93	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
94	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
95	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
96	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
97	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
98	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
99	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
100	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
101	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
102	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
103	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
104	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
105	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
106	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
107	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
108	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
109	C₄²×D₁₀₉	C₂²	1	2	yes	—
110	C₄²×D₁₀₉	C₂²	1	2	yes	—
111	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
112	#126×C₁₀₉	C₂²	1	2	yes	complemented
113	#125×C₁₀₉	C₂²	1	2	yes	complemented
114	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
115	#123×C₁₀₉	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
117	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
118	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
120	#117×C₁₀₉	C₂²	1	2	yes	complemented
121	#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
122	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
124	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
125	C₂×Dic₂₁₈	C₂³	1	2	yes	—
126	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
127	Q₈×D₁₀₉	—	2	4	no	—
128	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
131	D₄×D₁₀₉	—	2	4	no	—
132	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
133	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
134	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
135	C₂×C₄×D₁₀₉	—	2	8	no	—
136	C₂×C₄×D₁₀₉	—	2	4	no	—
137	C₂×C₄×D₁₀₉	—	2	4	no	—
138	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
139	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
140	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
141	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
142	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
143	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
146	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
147	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
148	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
149	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
153	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
154	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
155	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
156	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
158	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
159	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
160	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
161	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
162	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
163	#13182008	C₂³	1	2	yes	—
164	#13182008	C₂³	1	2	yes	—
165	#13182008	C₂³	1	2	yes	—
166	#13182008	C₂³	1	2	yes	—
167	#13182008	C₂³	1	2	yes	—
168	#13182008	C₂³	1	2	yes	—
169	#13182008	C₂³	1	2	yes	—
170	#13182008	C₂³	1	2	yes	—
171	#13182008	C₂³	1	2	yes	—
172	#13182008	C₂³	1	1	yes	characteristic
173	#13182008	C₂³	1	1	yes	characteristic
174	#13182008	C₂³	1	2	yes	—
175	#13182008	C₂³	1	2	yes	—
176	#13182008	C₂³	1	2	yes	—
177	#13182008	C₂³	1	2	yes	—
178	#13182008	C₂³	1	2	yes	—
179	#13182008	C₂³	1	2	yes	—
180	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
182	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
183	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
184	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
185	#13182006	C₂³	1	1	yes	characteristic
186	#13182006	C₂³	1	1	yes	characteristic
187	#13182006	C₂³	1	2	yes	—
188	#13182006	C₂³	1	2	yes	—
189	#13182006	C₂³	1	2	yes	—
190	#13182006	C₂³	1	2	yes	—
191	#13182006	C₂³	1	1	yes	characteristic
192	#13182006	C₂³	1	1	yes	characteristic
193	#13182006	C₂³	1	2	yes	—
194	#13182006	C₂³	1	2	yes	—
195	C₄×Dic₁₀₉	C₂³	1	2	yes	—
196	C₄×Dic₁₀₉	C₂³	1	2	yes	—
197	C₄×Dic₁₀₉	C₂³	1	2	yes	—
198	C₄×Dic₁₀₉	C₂³	1	2	yes	—
199	C₄×Dic₁₀₉	C₂³	1	2	yes	—
200	C₄×Dic₁₀₉	C₂³	1	2	yes	—
201	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
202	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
203	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
204	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
205	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
206	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
207	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
208	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
209	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
210	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
211	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
212	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
213	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
214	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
215	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
216	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
217	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
218	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
219	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
220	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
222	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
223	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
224	C₄²×C₁₀₉	C₂³	1	2	yes	—
225	C₄²×C₁₀₉	C₂³	1	2	yes	—
226	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
227	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
228	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
229	C₂²×D₁₀₉	—	2	2	no	—
230	C₂²×D₁₀₉	—	2	2	no	—
231	C₂²×D₁₀₉	—	2	2	no	—
232	C₂²×D₁₀₉	—	2	2	no	—
233	C₂²×D₁₀₉	—	2	2	no	—
234	C₂²×D₁₀₉	#41	1	4	yes	—
235	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
236	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
237	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
238	Q₈×C₁₀₉	—	2	2	no	—
239	D₄×C₁₀₉	—	2	2	no	—
240	C₂×C₄×C₁₀₉	—	2	4	no	—
241	C₂×C₄×C₁₀₉	—	2	2	no	—
242	C₂×C₄×C₁₀₉	—	2	2	no	—
243	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
246	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
247	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
248	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
251	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
252	C₂²⋊D₁₀₉	—	2	2	no	—
253	C₂²⋊D₁₀₉	—	2	2	no	—
254	C₂×Dic₁₀₉	—	2	4	no	—
255	C₂×Dic₁₀₉	—	2	2	no	—
256	C₂×Dic₁₀₉	—	2	2	no	—
257	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
258	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
259	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
260	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
261	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
262	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
263	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
265	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
266	D₄₃₆	—	2	2	no	—
267	C₄×D₁₀₉	—	2	4	no	—
268	C₄×D₁₀₉	—	2	4	no	—
269	C₄×D₁₀₉	—	2	4	no	—
270	C₄×D₁₀₉	—	2	4	no	—
271	C₄×D₁₀₉	—	2	2	no	—
272	C₄×D₁₀₉	—	2	2	no	—
273	C₄×D₁₀₉	—	2	2	no	—
274	C₄×D₁₀₉	—	2	4	no	—
275	C₄×D₁₀₉	—	2	2	no	—
276	C₄×D₁₀₉	—	2	4	no	—
277	C₄×D₁₀₉	—	2	2	no	—
278	C₄×D₁₀₉	—	2	2	no	—
279	C₄×D₁₀₉	—	2	2	no	—
280	C₄×D₁₀₉	—	2	2	no	—
281	C₄×D₁₀₉	#41	1	4	yes	—
282	Dic₂₁₈	—	2	2	no	—
283	Dic₂₁₈	—	2	4	no	—
284	C₂²×C₁₀₉	—	2	1	no	—
285	C₂²×C₁₀₉	—	2	1	no	—
286	C₂²×C₁₀₉	—	2	1	no	—
287	C₂²×C₁₀₉	—	2	1	no	—
288	C₂²×C₁₀₉	—	2	1	no	—
289	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
290	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
291	C₂×D₁₀₉	—	2	1	no	—
292	C₂×D₁₀₉	—	2	1	no	—
293	C₂×D₁₀₉	—	2	8	no	—
294	C₂×D₁₀₉	—	2	1	no	—
295	C₂×D₁₀₉	—	2	1	no	—
296	C₂×D₁₀₉	—	2	1	no	—
297	C₂×D₁₀₉	—	4	4	no	—
298	C₂×D₁₀₉	C₂×#41	1	2	yes	—
299	C₂×D₁₀₉	C₂×#41	1	2	yes	—
300	C₂×D₁₀₉	C₂×#41	1	2	yes	—
301	C₂×D₁₀₉	2₋¹⁺⁴	1	2	yes	—
302	C₄×C₁₀₉	—	2	2	no	—
303	C₄×C₁₀₉	—	2	2	no	—
304	C₄×C₁₀₉	—	2	2	no	—
305	C₄×C₁₀₉	—	2	1	no	—
306	C₄×C₁₀₉	—	2	1	no	—
307	C₄×C₁₀₉	—	2	1	no	—
308	C₄×C₁₀₉	—	2	1	no	—
309	C₄×C₁₀₉	—	2	2	no	—
310	C₄×C₁₀₉	C₂×#41	1	2	yes	—
311	Dic₁₀₉	—	2	2	no	—
312	Dic₁₀₉	—	2	2	no	—
313	Dic₁₀₉	—	2	2	no	—
314	Dic₁₀₉	—	2	1	no	—
315	Dic₁₀₉	—	2	1	no	—
316	Dic₁₀₉	—	2	1	no	—
317	Dic₁₀₉	—	2	1	no	—
318	Dic₁₀₉	—	2	2	no	—
319	Dic₁₀₉	C₂×#41	1	2	yes	—
320	C₂×C₁₀₉	—	2	2	no	—
321	C₂×C₁₀₉	—	4	1	no	—
322	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
323	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
324	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
325	D₁₀₉	—	2	2	no	—
326	D₁₀₉	—	4	1	no	—
327	D₁₀₉	#552	1	4	yes	direct product factor
328	C₂×#552	—	109	1	no	maximal, Sylow
329	C₁₀₉	C₂×#552	1	1	yes	characteristic, complemented, Sylow
330	#552	—	109	2	no	—
331	#552	—	109	2	no	—
332	#552	—	109	1	no	—
333	#552	—	109	2	no	—
334	#552	—	109	1	no	—
335	#552	—	109	1	no	—
336	#552	—	109	1	no	—
337	#552	—	109	1	no	—
338	#552	—	109	2	no	—
339	#552	—	109	1	no	—
340	#552	—	109	1	no	—
341	#552	D₁₀₉	1	1	yes	characteristic, direct product factor
342	C₂×#126	—	109	2	no	—
343	C₂×#125	—	109	2	no	—
344	C₂×#125	—	109	1	no	—
345	C₂×#123	—	109	2	no	—
346	C₂×C₂²⋊Q₈	—	109	1	no	—
347	C₂×C₂²⋊Q₈	—	109	1	no	—
348	C₂×C₄×Q₈	—	109	1	no	—
349	C₂×C₄×D₄	—	109	1	no	—
350	C₂×#117	—	109	1	no	—
351	C₂×#117	—	109	2	no	—
352	C₂²×C₄⋊C₄	—	109	1	no	—
353	C₂²×Q₈	—	109	1	no	—
354	C₂²×D₄	—	109	1	no	—
355	C₂³×C₄	—	109	2	no	—
356	C₂³×C₄	—	109	1	no	—
357	C₂³×C₄	—	109	1	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	#125	—	109	2	no	—
367	#125	—	109	2	no	—
368	#125	—	109	2	no	—
369	#125	—	109	2	no	—
370	#125	—	109	2	no	—
371	#125	—	109	2	no	—
372	#125	—	109	2	no	—
373	#125	—	109	2	no	—
374	#125	—	109	1	no	—
375	#125	—	109	1	no	—
376	#125	—	109	2	no	—
377	#125	—	109	1	no	—
378	#125	C₂×D₁₀₉	1	2	yes	complemented
379	#125	C₂×D₁₀₉	1	1	yes	characteristic, complemented
380	#123	—	109	2	no	—
381	#123	—	109	2	no	—
382	#123	—	109	2	no	—
383	#123	—	109	2	no	—
384	#123	—	109	2	no	—
385	#123	—	109	2	no	—
386	#123	—	109	2	no	—
387	#123	C₂×D₁₀₉	1	2	yes	—
388	C₂²⋊Q₈	—	109	1	no	—
389	C₂²⋊Q₈	—	109	1	no	—
390	C₂²⋊Q₈	—	109	1	no	—
391	C₂²⋊Q₈	—	109	1	no	—
392	C₂²⋊Q₈	—	109	2	no	—
393	C₂²⋊Q₈	—	109	2	no	—
394	C₂²⋊Q₈	—	109	1	no	—
395	C₂²⋊Q₈	—	109	1	no	—
396	C₂²⋊Q₈	—	109	2	no	—
397	C₂²⋊Q₈	—	109	2	no	—
398	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
399	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
400	C₄×Q₈	—	109	1	no	—
401	C₄×Q₈	—	109	2	no	—
402	C₄×Q₈	—	109	1	no	—
403	C₄×Q₈	—	109	1	no	—
404	C₄×Q₈	—	109	2	no	—
405	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
406	C₄×D₄	—	109	1	no	—
407	C₄×D₄	—	109	1	no	—
408	C₄×D₄	—	109	1	no	—
409	C₄×D₄	—	109	1	no	—
410	C₄×D₄	—	109	1	no	—
411	C₄×D₄	—	109	1	no	—
412	C₄×D₄	—	109	1	no	—
413	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
414	#117	—	109	1	no	—
415	#117	—	109	2	no	—
416	#117	—	109	1	no	—
417	#117	—	109	1	no	—
418	#117	—	109	2	no	—
419	#117	—	109	2	no	—
420	#117	—	109	2	no	—
421	#117	—	109	2	no	—
422	#117	—	109	2	no	—
423	#117	—	109	2	no	—
424	#117	—	109	2	no	—
425	#117	—	109	2	no	—
426	#117	C₂×D₁₀₉	1	1	yes	characteristic, complemented
427	#117	C₂×D₁₀₉	1	2	yes	complemented
428	C₂×C₄⋊C₄	—	109	2	no	—
429	C₂×C₄⋊C₄	—	109	1	no	—
430	C₂×C₄⋊C₄	—	109	1	no	—
431	C₂×C₄⋊C₄	—	109	1	no	—
432	C₂×C₄⋊C₄	—	109	2	no	—
433	C₂×C₄⋊C₄	—	109	2	no	—
434	C₂×C₄⋊C₄	—	109	1	no	—
435	C₂×C₄⋊C₄	—	109	1	no	—
436	C₂×C₄⋊C₄	—	109	1	no	—
437	C₂×C₄⋊C₄	—	109	2	no	—
438	C₂×C₄⋊C₄	—	109	2	no	—
439	C₂×C₄⋊C₄	—	109	1	no	—
440	C₂×C₄⋊C₄	—	109	2	no	—
441	C₂×C₄⋊C₄	—	109	2	no	—
442	C₂×C₄⋊C₄	—	109	2	no	—
443	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
444	C₂×C₂²⋊C₄	—	109	1	no	—
445	C₂×C₂²⋊C₄	—	109	2	no	—
446	C₂×C₂²⋊C₄	—	109	2	no	—
447	C₂×C₂²⋊C₄	—	109	1	no	—
448	C₂×C₂²⋊C₄	—	109	2	no	—
449	C₂×C₄²	—	109	2	no	—
450	C₂×C₄²	—	109	2	no	—
451	C₂×C₄²	—	109	1	no	—
452	C₂⁴	—	109	1	no	—
453	C₂⁴	—	109	1	no	—
454	C₂×Q₈	—	109	2	no	—
455	C₂×Q₈	—	109	1	no	—
456	C₂×Q₈	—	218	4	no	—
457	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
458	C₂×D₄	—	109	1	no	—
459	C₂×D₄	—	109	1	no	—
460	C₂×D₄	—	109	1	no	—
461	C₂×D₄	—	218	4	no	—
462	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
463	C₂²×C₄	—	109	1	no	—
464	C₂²×C₄	—	109	2	no	—
465	C₂²×C₄	—	109	1	no	—
466	C₂²×C₄	—	109	1	no	—
467	C₂²×C₄	—	109	1	no	—
468	C₂²×C₄	—	109	1	no	—
469	C₂²×C₄	—	109	2	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	1	no	—
478	C₂²×C₄	—	109	1	no	—
479	C₂²×C₄	—	109	1	no	—
480	C₂²×C₄	—	109	1	no	—
481	C₂²×C₄	—	218	4	no	—
482	C₂²×C₄	—	218	4	no	—
483	C₂²×C₄	—	218	8	no	—
484	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
485	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
486	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
487	C₄⋊C₄	—	109	2	no	—
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	—	109	2	no	—
499	C₄⋊C₄	—	109	2	no	—
500	C₄⋊C₄	—	109	2	no	—
501	C₄⋊C₄	—	109	2	no	—
502	C₄⋊C₄	—	109	2	no	—
503	C₄⋊C₄	—	109	2	no	—
504	C₄⋊C₄	—	109	1	no	—
505	C₄⋊C₄	—	109	1	no	—
506	C₄⋊C₄	—	109	2	no	—
507	C₄⋊C₄	—	109	2	no	—
508	C₄⋊C₄	—	109	1	no	—
509	C₄⋊C₄	—	109	2	no	—
510	C₄⋊C₄	—	109	1	no	—
511	C₄⋊C₄	—	109	2	no	—
512	C₄⋊C₄	—	109	2	no	—
513	C₄⋊C₄	—	109	2	no	—
514	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
517	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
518	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
520	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
522	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
523	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
524	C₂²⋊C₄	—	109	1	no	—
525	C₂²⋊C₄	—	109	2	no	—
526	C₂²⋊C₄	—	109	2	no	—
527	C₂²⋊C₄	—	109	1	no	—
528	C₂²⋊C₄	—	109	2	no	—
529	C₂²⋊C₄	—	109	1	no	—
530	C₂²⋊C₄	—	109	1	no	—
531	C₂²⋊C₄	—	109	2	no	—
532	C₂²⋊C₄	—	109	2	no	—
533	C₂²⋊C₄	—	109	2	no	—
534	C₂²⋊C₄	—	109	2	no	—
535	C₂²⋊C₄	—	109	1	no	—
536	C₂²⋊C₄	—	109	1	no	—
537	C₂²⋊C₄	—	109	2	no	—
538	C₂²⋊C₄	—	109	2	no	—
539	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
540	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
541	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
542	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
544	C₄²	—	109	2	no	—
545	C₄²	—	109	2	no	—
546	C₄²	—	109	2	no	—
547	C₄²	—	109	2	no	—
548	C₄²	—	109	2	no	—
549	C₄²	—	109	2	no	—
550	C₄²	—	109	1	no	—
551	C₄²	—	109	1	no	—
552	C₄²	—	109	1	no	—
553	C₄²	C₂²×D₁₀₉	1	2	yes	—
554	C₄²	C₂²×D₁₀₉	1	2	yes	—
555	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
556	C₂³	—	109	1	no	—
557	C₂³	—	109	4	no	—
558	C₂³	—	109	1	no	—
559	C₂³	—	109	1	no	—
560	C₂³	—	218	2	no	—
561	C₂³	—	218	2	no	—
562	C₂³	—	218	2	no	—
563	C₂³	—	218	2	no	—
564	C₂³	—	218	2	no	—
565	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
566	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
567	Q₈	—	2	2	no	—
568	Q₈	—	218	4	no	—
569	Q₈	—	218	2	no	—
570	D₄	—	2	2	no	—
571	D₄	—	218	2	no	—
572	D₄	—	218	2	no	—
573	D₄	—	218	2	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	109	1	no	—
578	C₂×C₄	—	109	4	no	—
579	C₂×C₄	—	109	1	no	—
580	C₂×C₄	—	109	1	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	—	218	2	no	—
588	C₂×C₄	—	218	2	no	—
589	C₂×C₄	—	218	2	no	—
590	C₂×C₄	—	218	2	no	—
591	C₂×C₄	—	218	4	no	—
592	C₂×C₄	—	218	4	no	—
593	C₂×C₄	—	218	4	no	—
594	C₂×C₄	—	218	4	no	—
595	C₂×C₄	—	218	2	no	—
596	C₂×C₄	—	218	4	no	—
597	C₂×C₄	—	218	2	no	—
598	C₂×C₄	—	218	4	no	—
599	C₂×C₄	—	218	2	no	—
600	C₂×C₄	—	218	2	no	—
601	C₂×C₄	—	218	4	no	—
602	C₂×C₄	—	218	2	no	—
603	C₂×C₄	—	218	2	no	—
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	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
608	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
609	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
611	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
612	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	109	2	no	—
619	C₂²	—	109	2	no	—
620	C₂²	—	109	2	no	—
621	C₂²	—	109	2	no	—
622	C₂²	—	436	4	no	—
623	C₂²	—	218	8	no	—
624	C₂²	—	218	1	no	—
625	C₂²	—	218	1	no	—
626	C₂²	—	218	1	no	—
627	C₂²	—	218	1	no	—
628	C₂²	—	218	1	no	—
629	C₂²	#41×D₁₀₉	1	2	yes	—
630	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	109	2	no	—
640	C₄	—	218	2	no	—
641	C₄	—	218	2	no	—
642	C₄	—	218	1	no	—
643	C₄	—	218	1	no	—
644	C₄	—	218	1	no	—
645	C₄	—	218	2	no	—
646	C₄	—	218	1	no	—
647	C₄	—	218	2	no	—
648	C₄	#41×D₁₀₉	1	2	yes	—
649	C₂	—	2	2	no	—
650	C₂	—	4	1	no	—
651	C₂	—	109	4	no	—
652	C₂	—	436	1	no	—
653	C₂	—	218	2	no	—
654	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
655	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
656	C₂	2₋¹⁺⁴×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#552×C₁₀₉	C₂
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#552	D₁₀₉
hypocenter	C₁₀₉	C₂×#552
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #552

Complements

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

N	H ≅ G/N	type
1. #20295718	C₂	semidirect
2. #20295717	C₂	semidirect
3. #20295716	C₂	semidirect
4. #20295715	C₂	semidirect
5. #20295714	C₂	semidirect
6. #20295713	C₂	semidirect
7. #20295712	C₂	semidirect
8. #20295711	C₂	semidirect
9. #20295710	C₂	semidirect
10. #20295709	C₂	semidirect
11. #20295708	C₂	semidirect
12. #126×D₁₀₉	C₂	semidirect
13. #125×D₁₀₉	C₂	semidirect
14. #125×D₁₀₉	C₂	semidirect
16. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
17. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
18. C₄×Q₈×D₁₀₉	C₂	semidirect
20. #117×D₁₀₉	C₂	semidirect
21. #117×D₁₀₉	C₂	semidirect
22. C₂×C₄⋊C₄×D₁₀₉	C₂	semidirect
23. #552×C₁₀₉	C₂	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. #16812691	C₂²	semidirect
37. #16812691	C₂²	semidirect
38. #16812690	C₂²	semidirect
39. #16812690	C₂²	semidirect
40. #16812690	C₂²	semidirect
41. #16812690	C₂²	semidirect
42. #16812689	C₂²	semidirect
43. #16812689	C₂²	semidirect
44. #16812689	C₂²	semidirect
45. #16812689	C₂²	semidirect
46. #16812689	C₂²	semidirect
47. #16812689	C₂²	semidirect
55. #16812679	C₂²	semidirect
56. #16812679	C₂²	semidirect
57. #16812678	C₂²	semidirect
58. #16812678	C₂²	semidirect
59. #16812677	C₂²	semidirect
60. #16812677	C₂²	semidirect
61. #16812676	C₂²	semidirect
62. #16812676	C₂²	semidirect
63. #16812675	C₂²	semidirect
64. #16812675	C₂²	semidirect
65. Q₈×Dic₁₀₉	C₂²	semidirect
66. C₄×Dic₂₁₈	C₂²	semidirect
67. C₄×Dic₂₁₈	C₂²	semidirect
68. #16812664	C₂²	semidirect
69. #16812664	C₂²	semidirect
77. #16812658	C₂²	semidirect
78. #16812658	C₂²	semidirect
79. #16812657	C₂²	semidirect
80. #16812657	C₂²	semidirect
81. #16812657	C₂²	semidirect
82. #16812657	C₂²	semidirect
83. #16812656	C₂²	semidirect
84. #16812656	C₂²	semidirect
85. #16812656	C₂²	semidirect
86. #16812655	C₂²	semidirect
87. #16812655	C₂²	semidirect
88. #16812655	C₂²	semidirect
89. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
90. C₂×#13182008	C₂²	semidirect
101. C₄⋊C₄×D₁₀₉	C₂²	semidirect
102. C₄⋊C₄×D₁₀₉	C₂²	semidirect
103. C₄⋊C₄×D₁₀₉	C₂²	semidirect
112. #126×C₁₀₉	C₂²	semidirect
113. #125×C₁₀₉	C₂²	semidirect
114. #125×C₁₀₉	C₂²	semidirect
116. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
117. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
118. C₄×Q₈×C₁₀₉	C₂²	semidirect
120. #117×C₁₀₉	C₂²	semidirect
121. #117×C₁₀₉	C₂²	semidirect
122. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
327. D₁₀₉	#552	direct (N × H)
329. C₁₀₉	C₂×#552	semidirect
341. #552	D₁₀₉	direct (N × H)
365. #126	C₂×D₁₀₉	semidirect
378. #125	C₂×D₁₀₉	semidirect
379. #125	C₂×D₁₀₉	semidirect
398. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
399. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
405. C₄×Q₈	C₂×D₁₀₉	semidirect
426. #117	C₂×D₁₀₉	semidirect
427. #117	C₂×D₁₀₉	semidirect
443. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295718	2
#20295717	2
#20295716	2
#20295715	2
#20295714	2
#20295713	2
#20295712	2
#20295711	2
#20295710	2
#20295709	2
#20295708	2
#126×D₁₀₉	2
#125×D₁₀₉	2
#123×D₁₀₉	2
C₂²⋊Q₈×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
C₄×D₄×D₁₀₉	2
#117×D₁₀₉	2
C₂×C₄⋊C₄×D₁₀₉	2
#552×C₁₀₉	2
C₂×#552	109

Subgroups — #552×D109 #28376535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #552×D₁₀₉ #28376535