Subgroups — #28377074

Order 13952.

1124 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295747	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295730	C₂	1	2	yes	maximal, complemented
3	#20295717	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295716	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295706	C₂	1	2	yes	maximal, complemented
6	#20295705	C₂	1	2	yes	maximal, complemented
7	Q₈×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295684	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295676	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295675	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295665	C₂	1	2	yes	maximal, complemented
12	#20295640	C₂	1	2	yes	maximal, complemented
13	#20295622	C₂	1	2	yes	maximal, complemented
14	#20295621	C₂	1	2	yes	maximal, complemented
15	#20295580	C₂	1	1	yes	maximal, characteristic, complemented
16	#20295555	C₂	1	1	yes	maximal, characteristic, complemented
17	#20295536	C₂	1	1	yes	maximal, characteristic, complemented
18	#20295533	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×#16812675	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂²⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
21	#20295505	C₂	1	2	yes	maximal, complemented
22	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
23	#548×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#13182028	C₂²	1	2	yes	—
25	C₂×#13182028	C₂²	1	1	yes	characteristic, complemented
26	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
30	#16812698	C₂²	1	1	yes	characteristic, complemented
31	#16812697	C₂²	1	2	yes	complemented
32	#16812697	C₂²	1	1	yes	characteristic, complemented
33	#16812694	C₂²	1	2	yes	complemented
34	#16812693	C₂²	1	2	yes	complemented
35	#16812692	C₂²	1	2	yes	complemented
36	#16812692	C₂²	1	2	yes	complemented
37	#16812692	C₂²	1	2	yes	complemented
38	#16812691	C₂²	1	1	yes	characteristic, complemented
39	#16812689	C₂²	1	2	yes	complemented
40	#16812689	C₂²	1	2	yes	complemented
41	#16812689	C₂²	1	2	yes	complemented
42	#16812685	C₂²	1	2	yes	—
43	#16812685	C₂²	1	2	yes	complemented
44	#16812684	C₂²	1	2	yes	—
45	#16812682	C₂²	1	2	yes	complemented
46	#16812681	C₂²	1	2	yes	—
47	#16812681	C₂²	1	2	yes	complemented
48	#16812681	C₂²	1	2	yes	complemented
49	#16812680	C₂²	1	2	yes	—
50	#16812680	C₂²	1	2	yes	complemented
51	#16812680	C₂²	1	2	yes	complemented
52	#16812680	C₂²	1	2	yes	complemented
53	#16812679	C₂²	1	1	yes	characteristic
54	#16812678	C₂²	1	2	yes	complemented
55	#16812678	C₂²	1	1	yes	characteristic, complemented
56	#16812678	C₂²	1	1	yes	characteristic, complemented
57	#16812677	C₂²	1	2	yes	—
58	#16812677	C₂²	1	2	yes	—
59	#16812676	C₂²	1	2	yes	complemented
60	#16812676	C₂²	1	2	yes	complemented
61	#16812676	C₂²	1	2	yes	complemented
62	#16812675	C₂²	1	2	yes	—
63	#16812675	C₂²	1	1	yes	characteristic
64	#16812675	C₂²	1	1	yes	characteristic
65	#16812675	C₂²	1	1	yes	characteristic, complemented
66	#16812675	C₂²	1	2	yes	complemented
67	#16812675	C₂²	1	1	yes	characteristic, complemented
68	#16812675	C₂²	1	1	yes	characteristic, complemented
69	#16812675	C₂²	1	2	yes	complemented
70	#16812672	C₂²	1	2	yes	—
71	#16812671	C₂²	1	2	yes	—
72	#16812670	C₂²	1	1	yes	characteristic
73	#16812670	C₂²	1	1	yes	characteristic, complemented
74	#16812667	C₂²	1	2	yes	—
75	#16812667	C₂²	1	2	yes	—
76	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
77	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
78	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
79	#16812664	C₂²	1	2	yes	complemented
80	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
81	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
82	D₄×Dic₁₀₉	C₂²	1	2	yes	complemented
83	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
84	#16812661	C₂²	1	1	yes	characteristic
85	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
86	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
87	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
88	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
89	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
90	#16812659	C₂²	1	2	yes	—
91	#16812659	C₂²	1	2	yes	—
92	#16812659	C₂²	1	1	yes	characteristic
93	#16812659	C₂²	1	1	yes	characteristic
94	#16812659	C₂²	1	2	yes	complemented
95	#16812658	C₂²	1	1	yes	characteristic, complemented
96	#16812657	C₂²	1	1	yes	characteristic, complemented
97	#16812656	C₂²	1	2	yes	—
98	#16812656	C₂²	1	2	yes	complemented
99	#16812655	C₂²	1	2	yes	complemented
100	#16812655	C₂²	1	2	yes	complemented
101	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
102	C₂×#13182008	C₂²	1	2	yes	complemented
103	C₂×#13182008	C₂²	1	2	yes	complemented
104	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
105	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
106	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
107	C₂×#13182006	C₂²	1	2	yes	—
108	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
109	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
110	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
111	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
112	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
113	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
114	#124×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
115	#124×C₁₀₉	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
118	#121×C₁₀₉	C₂²	1	2	yes	—
119	#121×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
120	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
122	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
123	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
124	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
125	#13182028	—	2	8	no	—
126	#13182028	—	4	2	no	—
127	C₂×Dic₂₁₈	C₂³	1	2	yes	—
128	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
129	C₂×Dic₂₁₈	C₂³	1	2	yes	—
130	Q₈×D₁₀₉	—	4	2	no	—
131	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
132	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
134	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
137	D₄×D₁₀₉	—	2	4	no	—
138	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
139	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
140	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
141	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
142	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₁₀₉	—	4	2	no	—
144	C₂×C₄×D₁₀₉	—	4	2	no	—
145	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
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	—
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	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
154	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
156	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
162	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
163	#13182008	C₂³	1	1	yes	characteristic
164	#13182008	C₂³	1	1	yes	characteristic
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	2	yes	—
173	#13182008	C₂³	1	2	yes	—
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	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	2	yes	—
184	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
185	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
186	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
187	#13182006	C₂³	1	2	yes	—
188	#13182006	C₂³	1	2	yes	—
189	#13182006	C₂³	1	1	yes	characteristic
190	#13182006	C₂³	1	1	yes	characteristic
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	#13182006	C₂³	1	2	yes	—
196	#13182006	C₂³	1	2	yes	—
197	#13182006	C₂³	1	2	yes	—
198	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
199	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
200	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
201	C₄×Dic₁₀₉	C₂³	1	2	yes	—
202	C₄×Dic₁₀₉	C₂³	1	2	yes	—
203	C₄×Dic₁₀₉	C₂³	1	2	yes	—
204	C₄×Dic₁₀₉	C₂³	1	2	yes	—
205	#41×C₁₀₉	—	4	2	no	—
206	C₂×Q₈×C₁₀₉	—	4	2	no	—
207	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
208	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
209	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
210	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
211	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
212	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
213	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
214	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
215	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
216	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
217	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
218	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
219	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
220	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
221	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
222	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
223	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
224	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
225	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
226	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
227	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
228	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
229	C₄²×C₁₀₉	C₂³	1	2	yes	—
230	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
231	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
232	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
233	C₂²×D₁₀₉	—	2	2	no	—
234	C₂²×D₁₀₉	—	2	2	no	—
235	C₂²×D₁₀₉	—	2	2	no	—
236	C₂²×D₁₀₉	—	2	2	no	—
237	C₂²×D₁₀₉	—	4	1	no	—
238	C₂²×D₁₀₉	—	4	1	no	—
239	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
240	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
241	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
242	Q₈×C₁₀₉	—	2	2	no	—
243	Q₈×C₁₀₉	—	2	2	no	—
244	Q₈×C₁₀₉	—	4	2	no	—
245	D₄×C₁₀₉	—	2	2	no	—
246	D₄×C₁₀₉	—	2	4	no	—
247	D₄×C₁₀₉	—	2	2	no	—
248	D₄×C₁₀₉	—	4	2	no	—
249	C₂×C₄×C₁₀₉	—	2	2	no	—
250	C₂×C₄×C₁₀₉	—	2	2	no	—
251	C₂×C₄×C₁₀₉	—	4	2	no	—
252	C₂×C₄×C₁₀₉	—	4	2	no	—
253	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
254	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
255	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
256	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
257	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
258	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
259	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
260	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
261	C₂²⋊D₁₀₉	—	2	4	no	—
262	C₂²⋊D₁₀₉	—	2	4	no	—
263	C₂²⋊D₁₀₉	—	2	2	no	—
264	C₂²⋊D₁₀₉	—	4	2	no	—
265	C₂²⋊D₁₀₉	#41	1	4	yes	complemented
266	C₂×Dic₁₀₉	—	2	4	no	—
267	C₂×Dic₁₀₉	—	2	4	no	—
268	C₂×Dic₁₀₉	—	4	1	no	—
269	C₂×Dic₁₀₉	—	4	1	no	—
270	C₂×Dic₁₀₉	—	4	2	no	—
271	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
272	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
273	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
274	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
275	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
276	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
277	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
278	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
279	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
280	D₄₃₆	—	2	2	no	—
281	C₄×D₁₀₉	—	2	4	no	—
282	C₄×D₁₀₉	—	2	2	no	—
283	C₄×D₁₀₉	—	2	2	no	—
284	C₄×D₁₀₉	—	2	2	no	—
285	C₄×D₁₀₉	—	2	2	no	—
286	C₄×D₁₀₉	—	2	2	no	—
287	C₄×D₁₀₉	—	4	1	no	—
288	C₄×D₁₀₉	—	4	1	no	—
289	C₄×D₁₀₉	—	4	1	no	—
290	C₄×D₁₀₉	—	4	1	no	—
291	C₄×D₁₀₉	—	4	2	no	—
292	Dic₂₁₈	—	2	2	no	—
293	Dic₂₁₈	—	2	4	no	—
294	Dic₂₁₈	—	4	2	no	—
295	C₂²×C₁₀₉	—	2	1	no	—
296	C₂²×C₁₀₉	—	2	1	no	—
297	C₂²×C₁₀₉	—	2	2	no	—
298	C₂²×C₁₀₉	—	2	1	no	—
299	C₂²×C₁₀₉	—	2	2	no	—
300	C₂²×C₁₀₉	—	2	1	no	—
301	C₂²×C₁₀₉	—	2	2	no	—
302	C₂²×C₁₀₉	—	2	1	no	—
303	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
304	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
305	C₂×D₁₀₉	—	2	1	no	—
306	C₂×D₁₀₉	—	2	1	no	—
307	C₂×D₁₀₉	—	2	1	no	—
308	C₂×D₁₀₉	—	2	1	no	—
309	C₂×D₁₀₉	—	2	1	no	—
310	C₂×D₁₀₉	—	2	1	no	—
311	C₂×D₁₀₉	—	2	1	no	—
312	C₂×D₁₀₉	—	4	4	no	—
313	C₂×D₁₀₉	—	4	4	no	—
314	C₂×D₁₀₉	C₂×#41	1	2	yes	—
315	C₂×D₁₀₉	2₋¹⁺⁴	1	2	yes	—
316	C₄×C₁₀₉	—	2	2	no	—
317	C₄×C₁₀₉	—	2	1	no	—
318	C₄×C₁₀₉	—	2	2	no	—
319	C₄×C₁₀₉	—	2	1	no	—
320	C₄×C₁₀₉	—	2	1	no	—
321	C₄×C₁₀₉	—	2	2	no	—
322	C₄×C₁₀₉	—	2	1	no	—
323	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
324	Dic₁₀₉	—	2	2	no	—
325	Dic₁₀₉	—	2	2	no	—
326	Dic₁₀₉	—	2	2	no	—
327	Dic₁₀₉	—	2	1	no	—
328	Dic₁₀₉	—	2	1	no	—
329	Dic₁₀₉	—	2	1	no	—
330	Dic₁₀₉	—	2	1	no	—
331	Dic₁₀₉	—	2	2	no	—
332	Dic₁₀₉	C₂×#41	1	2	yes	—
333	C₂×C₁₀₉	—	2	2	no	—
334	C₂×C₁₀₉	—	4	1	no	—
335	C₂×C₁₀₉	—	4	2	no	—
336	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
337	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
338	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
339	D₁₀₉	—	2	2	no	—
340	D₁₀₉	—	2	2	no	—
341	D₁₀₉	—	4	1	no	—
342	#3520	—	109	1	no	maximal, Sylow
343	C₁₀₉	#3520	1	1	yes	characteristic, complemented, Sylow
344	#556	—	109	1	no	—
345	#554	—	109	2	no	—
346	#552	—	109	1	no	—
347	#552	—	109	1	no	—
348	#551	—	109	2	no	—
349	#551	—	109	2	no	—
350	D₄×Q₈	—	109	1	no	—
351	#548	—	109	1	no	—
352	#548	D₁₀₉	1	1	yes	characteristic, complemented
353	#547	—	109	1	no	—
354	#547	—	109	1	no	—
355	#546	—	109	2	no	—
356	#542	—	109	2	no	—
357	#539	—	109	2	no	—
358	#539	—	109	2	no	—
359	#533	—	109	1	no	—
360	#529	—	109	1	no	—
361	#525	—	109	1	no	—
362	#525	—	109	1	no	—
363	C₂×C₂²⋊Q₈	—	109	1	no	—
364	C₂×C₂²⋊Q₈	—	109	1	no	—
365	#520	—	109	2	no	—
366	C₂×C₄×D₄	—	109	1	no	—
367	C₂×#41	—	109	1	no	—
368	C₂×#41	—	109	2	no	—
369	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic, complemented
370	C₂²×Q₈	—	109	1	no	—
371	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
372	C₂²×D₄	—	109	1	no	—
373	C₂³×C₄	—	109	1	no	—
374	C₂³×C₄	—	109	1	no	—
375	C₄⋊Q₈	—	109	1	no	—
376	C₄⋊Q₈	—	109	2	no	—
377	C₄⋊Q₈	—	109	1	no	—
378	#126	—	109	2	no	—
379	#126	—	109	2	no	—
380	#126	—	109	2	no	—
381	#126	—	109	2	no	—
382	#126	—	109	2	no	—
383	#125	—	109	2	no	—
384	#125	—	109	2	no	—
385	#125	—	109	2	no	—
386	#125	—	109	1	no	—
387	#124	—	109	2	no	—
388	#124	—	109	2	no	—
389	#124	C₂×D₁₀₉	1	1	yes	characteristic, complemented
390	#124	C₂×D₁₀₉	1	2	yes	complemented
391	#123	—	109	2	no	—
392	#123	—	109	2	no	—
393	#123	—	109	2	no	—
394	#123	—	109	2	no	—
395	#123	—	109	2	no	—
396	#123	—	109	2	no	—
397	#123	—	109	2	no	—
398	#123	—	109	2	no	—
399	#123	—	109	2	no	—
400	C₂²⋊Q₈	—	109	1	no	—
401	C₂²⋊Q₈	—	109	1	no	—
402	C₂²⋊Q₈	—	109	2	no	—
403	C₂²⋊Q₈	—	109	1	no	—
404	C₂²⋊Q₈	—	109	1	no	—
405	C₂²⋊Q₈	—	109	2	no	—
406	C₂²⋊Q₈	—	109	1	no	—
407	C₂²⋊Q₈	—	109	2	no	—
408	C₂²⋊Q₈	—	109	1	no	—
409	C₂²⋊Q₈	—	109	1	no	—
410	C₂²⋊Q₈	—	109	2	no	—
411	C₂²⋊Q₈	—	109	1	no	—
412	C₂²⋊Q₈	—	109	2	no	—
413	C₂²⋊Q₈	—	109	2	no	—
414	C₂²⋊Q₈	—	109	2	no	—
415	C₂²⋊Q₈	—	109	2	no	—
416	C₂²⋊Q₈	—	109	2	no	—
417	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
418	C₂²⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
419	#121	—	109	1	no	—
420	#121	—	109	1	no	—
421	#121	—	109	2	no	—
422	#121	—	109	2	no	—
423	#121	C₂×D₁₀₉	1	2	yes	—
424	#121	C₂×D₁₀₉	1	1	yes	characteristic, complemented
425	#120	—	109	2	no	—
426	#120	—	109	2	no	—
427	C₄×Q₈	—	109	1	no	—
428	C₄×Q₈	—	109	2	no	—
429	C₄×Q₈	—	109	2	no	—
430	C₄×Q₈	—	109	1	no	—
431	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
432	C₄×D₄	—	109	1	no	—
433	C₄×D₄	—	109	2	no	—
434	C₄×D₄	—	109	1	no	—
435	C₄×D₄	—	109	1	no	—
436	C₄×D₄	—	109	1	no	—
437	C₄×D₄	—	109	2	no	—
438	C₄×D₄	—	109	2	no	—
439	C₄×D₄	—	109	2	no	—
440	C₄×D₄	—	109	1	no	—
441	C₄×D₄	—	109	1	no	—
442	C₄×D₄	—	109	1	no	—
443	C₄×D₄	—	109	2	no	—
444	C₄×D₄	—	109	2	no	—
445	C₄×D₄	—	109	2	no	—
446	C₄×D₄	—	109	1	no	—
447	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
448	C₄×D₄	C₂×D₁₀₉	1	2	yes	complemented
449	#117	—	109	1	no	—
450	#117	—	109	1	no	—
451	#117	—	109	2	no	—
452	#117	—	109	2	no	—
453	#117	—	109	2	no	—
454	#117	—	109	2	no	—
455	C₂×C₄⋊C₄	—	109	2	no	—
456	C₂×C₄⋊C₄	—	109	2	no	—
457	C₂×C₄⋊C₄	—	109	1	no	—
458	C₂×C₄⋊C₄	—	109	1	no	—
459	C₂×C₄⋊C₄	—	109	2	no	—
460	C₂×C₄⋊C₄	—	109	1	no	—
461	C₂×C₂²⋊C₄	—	109	2	no	—
462	C₂×C₂²⋊C₄	—	109	1	no	—
463	C₂×C₂²⋊C₄	—	109	1	no	—
464	C₂×C₂²⋊C₄	—	109	2	no	—
465	C₂×C₄²	—	109	1	no	—
466	C₂⁴	—	109	1	no	—
467	C₂⁴	—	109	1	no	—
468	#41	—	4	2	no	—
469	#41	—	436	2	no	—
470	#41	—	218	8	no	—
471	C₂×Q₈	—	4	2	no	—
472	C₂×Q₈	—	109	2	no	—
473	C₂×Q₈	—	109	2	no	—
474	C₂×Q₈	—	109	1	no	—
475	C₂×Q₈	—	436	2	no	—
476	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
477	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
478	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
479	C₂×D₄	—	109	1	no	—
480	C₂×D₄	—	109	2	no	—
481	C₂×D₄	—	109	2	no	—
482	C₂×D₄	—	109	2	no	—
483	C₂×D₄	—	109	1	no	—
484	C₂×D₄	—	109	1	no	—
485	C₂×D₄	—	218	4	no	—
486	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
487	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
488	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
489	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
490	C₂²×C₄	—	109	1	no	—
491	C₂²×C₄	—	109	2	no	—
492	C₂²×C₄	—	109	1	no	—
493	C₂²×C₄	—	109	2	no	—
494	C₂²×C₄	—	109	2	no	—
495	C₂²×C₄	—	109	2	no	—
496	C₂²×C₄	—	109	1	no	—
497	C₂²×C₄	—	109	1	no	—
498	C₂²×C₄	—	109	1	no	—
499	C₂²×C₄	—	109	1	no	—
500	C₂²×C₄	—	109	1	no	—
501	C₂²×C₄	—	109	1	no	—
502	C₂²×C₄	—	109	1	no	—
503	C₂²×C₄	—	109	2	no	—
504	C₂²×C₄	—	109	1	no	—
505	C₂²×C₄	—	109	1	no	—
506	C₂²×C₄	—	436	2	no	—
507	C₂²×C₄	—	436	2	no	—
508	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
509	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
510	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
511	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
512	C₄⋊C₄	—	109	1	no	—
513	C₄⋊C₄	—	109	2	no	—
514	C₄⋊C₄	—	109	2	no	—
515	C₄⋊C₄	—	109	2	no	—
516	C₄⋊C₄	—	109	1	no	—
517	C₄⋊C₄	—	109	2	no	—
518	C₄⋊C₄	—	109	2	no	—
519	C₄⋊C₄	—	109	2	no	—
520	C₄⋊C₄	—	109	2	no	—
521	C₄⋊C₄	—	109	2	no	—
522	C₄⋊C₄	—	109	2	no	—
523	C₄⋊C₄	—	109	2	no	—
524	C₄⋊C₄	—	109	2	no	—
525	C₄⋊C₄	—	109	2	no	—
526	C₄⋊C₄	—	109	2	no	—
527	C₄⋊C₄	—	109	2	no	—
528	C₄⋊C₄	—	109	1	no	—
529	C₄⋊C₄	—	109	1	no	—
530	C₄⋊C₄	—	109	1	no	—
531	C₄⋊C₄	—	109	2	no	—
532	C₄⋊C₄	—	109	2	no	—
533	C₄⋊C₄	—	109	1	no	—
534	C₄⋊C₄	—	109	2	no	—
535	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
536	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
537	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
538	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
539	C₂²⋊C₄	—	109	1	no	—
540	C₂²⋊C₄	—	109	1	no	—
541	C₂²⋊C₄	—	109	2	no	—
542	C₂²⋊C₄	—	109	2	no	—
543	C₂²⋊C₄	—	109	2	no	—
544	C₂²⋊C₄	—	109	2	no	—
545	C₂²⋊C₄	—	109	2	no	—
546	C₂²⋊C₄	—	109	2	no	—
547	C₂²⋊C₄	—	109	2	no	—
548	C₂²⋊C₄	—	109	1	no	—
549	C₂²⋊C₄	—	109	2	no	—
550	C₂²⋊C₄	—	109	2	no	—
551	C₂²⋊C₄	—	109	1	no	—
552	C₂²⋊C₄	—	109	1	no	—
553	C₂²⋊C₄	—	109	1	no	—
554	C₂²⋊C₄	—	109	2	no	—
555	C₂²⋊C₄	—	109	2	no	—
556	C₂²⋊C₄	—	109	2	no	—
557	C₂²⋊C₄	—	109	2	no	—
558	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
559	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
560	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
561	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
562	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
563	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
564	C₄²	—	109	1	no	—
565	C₄²	—	109	1	no	—
566	C₄²	—	109	1	no	—
567	C₄²	—	109	2	no	—
568	C₄²	—	109	2	no	—
569	C₄²	—	109	2	no	—
570	C₄²	—	109	2	no	—
571	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
572	C₄²	C₂²×D₁₀₉	1	2	yes	—
573	C₂³	—	109	1	no	—
574	C₂³	—	109	1	no	—
575	C₂³	—	109	1	no	—
576	C₂³	—	436	1	no	—
577	C₂³	—	436	1	no	—
578	C₂³	—	218	2	no	—
579	C₂³	—	218	2	no	—
580	C₂³	—	218	2	no	—
581	C₂³	—	218	2	no	—
582	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
583	C₂³	C₂³×D₁₀₉	1	2	yes	—
584	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
585	Q₈	—	2	2	no	—
586	Q₈	—	2	2	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	436	2	no	—
589	Q₈	—	218	4	no	—
590	Q₈	—	218	2	no	—
591	D₄	—	2	2	no	—
592	D₄	—	2	4	no	—
593	D₄	—	2	2	no	—
594	D₄	—	4	2	no	—
595	D₄	—	109	4	no	—
596	D₄	—	436	2	no	—
597	D₄	—	218	2	no	—
598	D₄	—	218	2	no	—
599	D₄	—	218	4	no	—
600	D₄	—	218	4	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	109	1	no	—
606	C₂×C₄	—	109	2	no	—
607	C₂×C₄	—	109	1	no	—
608	C₂×C₄	—	109	2	no	—
609	C₂×C₄	—	109	1	no	—
610	C₂×C₄	—	109	1	no	—
611	C₂×C₄	—	109	2	no	—
612	C₂×C₄	—	109	2	no	—
613	C₂×C₄	—	109	1	no	—
614	C₂×C₄	—	436	1	no	—
615	C₂×C₄	—	436	1	no	—
616	C₂×C₄	—	436	1	no	—
617	C₂×C₄	—	436	1	no	—
618	C₂×C₄	—	436	2	no	—
619	C₂×C₄	—	436	1	no	—
620	C₂×C₄	—	436	1	no	—
621	C₂×C₄	—	436	2	no	—
622	C₂×C₄	—	218	2	no	—
623	C₂×C₄	—	218	4	no	—
624	C₂×C₄	—	218	2	no	—
625	C₂×C₄	—	218	4	no	—
626	C₂×C₄	—	218	4	no	—
627	C₂×C₄	—	218	2	no	—
628	C₂×C₄	—	218	2	no	—
629	C₂×C₄	—	218	2	no	—
630	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
631	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
632	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
633	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
634	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
635	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
636	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
637	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
638	C₂²	—	2	1	no	—
639	C₂²	—	2	1	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	1	no	—
646	C₂²	—	109	2	no	—
647	C₂²	—	109	2	no	—
648	C₂²	—	436	4	no	—
649	C₂²	—	436	4	no	—
650	C₂²	—	218	1	no	—
651	C₂²	—	218	1	no	—
652	C₂²	—	218	1	no	—
653	C₂²	—	218	1	no	—
654	C₂²	—	218	1	no	—
655	C₂²	—	218	1	no	—
656	C₂²	—	218	1	no	—
657	C₂²	#41×D₁₀₉	1	2	yes	—
658	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
659	C₄	—	2	1	no	—
660	C₄	—	2	2	no	—
661	C₄	—	2	1	no	—
662	C₄	—	2	1	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	2	1	no	—
666	C₄	—	109	2	no	—
667	C₄	—	218	2	no	—
668	C₄	—	218	2	no	—
669	C₄	—	218	1	no	—
670	C₄	—	218	1	no	—
671	C₄	—	218	1	no	—
672	C₄	—	218	2	no	—
673	C₄	—	218	1	no	—
674	C₄	—	218	2	no	—
675	C₄	#16812744	1	2	yes	—
676	C₂	—	2	2	no	—
677	C₂	—	4	1	no	—
678	C₂	—	4	2	no	—
679	C₂	—	436	1	no	—
680	C₂	—	218	2	no	—
681	C₂	—	218	2	no	—
682	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
683	C₂	C₂×#16812744	1	1	yes	characteristic
684	C₂	2₋¹⁺⁴×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#548×C₁₀₉	C₂
hypercenter	#548	D₁₀₉
hypocenter	C₁₀₉	#3520
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #548

Complements

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

N	H ≅ G/N	type
1. #20295747	C₂	semidirect
2. #20295730	C₂	semidirect
3. #20295717	C₂	semidirect
4. #20295716	C₂	semidirect
5. #20295706	C₂	semidirect
6. #20295705	C₂	semidirect
7. Q₈×C₂²⋊D₁₀₉	C₂	semidirect
8. #20295684	C₂	semidirect
9. #20295676	C₂	semidirect
10. #20295675	C₂	semidirect
11. #20295665	C₂	semidirect
12. #20295640	C₂	semidirect
13. #20295622	C₂	semidirect
14. #20295621	C₂	semidirect
15. #20295580	C₂	semidirect
16. #20295555	C₂	semidirect
17. #20295536	C₂	semidirect
18. #20295533	C₂	semidirect
19. C₂×#16812675	C₂	semidirect
20. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
21. #20295505	C₂	semidirect
22. C₄×D₄×D₁₀₉	C₂	semidirect
23. #548×C₁₀₉	C₂	semidirect
25. C₂×#13182028	C₂²	semidirect
26. C₂×Q₈×D₁₀₉	C₂²	semidirect
30. #16812698	C₂²	semidirect
31. #16812697	C₂²	semidirect
32. #16812697	C₂²	semidirect
33. #16812694	C₂²	semidirect
34. #16812693	C₂²	semidirect
35. #16812692	C₂²	semidirect
36. #16812692	C₂²	semidirect
37. #16812692	C₂²	semidirect
38. #16812691	C₂²	semidirect
39. #16812689	C₂²	semidirect
40. #16812689	C₂²	semidirect
41. #16812689	C₂²	semidirect
43. #16812685	C₂²	semidirect
45. #16812682	C₂²	semidirect
47. #16812681	C₂²	semidirect
48. #16812681	C₂²	semidirect
50. #16812680	C₂²	semidirect
51. #16812680	C₂²	semidirect
52. #16812680	C₂²	semidirect
54. #16812678	C₂²	semidirect
55. #16812678	C₂²	semidirect
56. #16812678	C₂²	semidirect
59. #16812676	C₂²	semidirect
60. #16812676	C₂²	semidirect
61. #16812676	C₂²	semidirect
65. #16812675	C₂²	semidirect
66. #16812675	C₂²	semidirect
67. #16812675	C₂²	semidirect
68. #16812675	C₂²	semidirect
69. #16812675	C₂²	semidirect
73. #16812670	C₂²	semidirect
76. Q₈×Dic₁₀₉	C₂²	semidirect
77. C₄×Dic₂₁₈	C₂²	semidirect
78. C₄×Dic₂₁₈	C₂²	semidirect
79. #16812664	C₂²	semidirect
82. D₄×Dic₁₀₉	C₂²	semidirect
83. D₄×Dic₁₀₉	C₂²	semidirect
89. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
94. #16812659	C₂²	semidirect
95. #16812658	C₂²	semidirect
96. #16812657	C₂²	semidirect
98. #16812656	C₂²	semidirect
99. #16812655	C₂²	semidirect
100. #16812655	C₂²	semidirect
101. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
102. C₂×#13182008	C₂²	semidirect
103. C₂×#13182008	C₂²	semidirect
105. C₄⋊C₄×D₁₀₉	C₂²	semidirect
106. C₄⋊C₄×D₁₀₉	C₂²	semidirect
112. C₂×#41×C₁₀₉	C₂²	semidirect
113. C₂²×Q₈×C₁₀₉	C₂²	semidirect
114. #124×C₁₀₉	C₂²	semidirect
115. #124×C₁₀₉	C₂²	semidirect
116. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
117. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
119. #121×C₁₀₉	C₂²	semidirect
120. C₄×Q₈×C₁₀₉	C₂²	semidirect
122. C₄×D₄×C₁₀₉	C₂²	semidirect
265. C₂²⋊D₁₀₉	#41	semidirect
343. C₁₀₉	#3520	semidirect
352. #548	D₁₀₉	semidirect
369. C₂×#41	C₂×D₁₀₉	semidirect
371. C₂²×Q₈	C₂×D₁₀₉	semidirect
389. #124	C₂×D₁₀₉	semidirect
390. #124	C₂×D₁₀₉	semidirect
417. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
418. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
424. #121	C₂×D₁₀₉	semidirect
431. C₄×Q₈	C₂×D₁₀₉	semidirect
448. C₄×D₄	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295747	2
#20295730	2
#20295717	2
#20295716	2
#20295706	2
#20295705	2
Q₈×C₂²⋊D₁₀₉	2
#20295684	2
#20295676	2
#20295675	2
#20295665	2
#20295640	2
#20295622	2
#20295621	2
#20295580	2
#20295555	2
#20295536	2
#20295533	2
C₂×#16812675	2
C₂²⋊Q₈×D₁₀₉	2
#20295505	2
C₄×D₄×D₁₀₉	2
#548×C₁₀₉	2
#3520	109