Subgroups — #37582307

Order 24448.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133219	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15690288	C₂	1	1	yes	maximal, characteristic
3	#27132970	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132964	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132960	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132956	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132952	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132946	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132942	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132700	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132698	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132658	C₂	1	1	yes	maximal, characteristic
14	#27132658	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690288	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
19	#20259321	C₂²	1	1	yes	characteristic
20	#20259321	C₂²	1	1	yes	characteristic
21	#20259314	C₂²	1	1	yes	characteristic
22	#20259300	—	2	1	no	—
23	#20259300	—	2	1	no	—
24	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
25	#20259287	—	2	1	no	—
26	#20259287	—	2	1	no	—
27	C₄×D₇₆₄	—	2	1	no	—
28	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
29	#20259284	—	2	1	no	—
30	#20259284	C₂²	1	1	yes	characteristic
31	#20259280	—	2	1	no	—
32	#20259280	C₂²	1	1	yes	characteristic
33	#20259280	C₂²	1	1	yes	characteristic
34	#20259280	C₂²	1	1	yes	characteristic
35	C₄²×D₁₉₁	—	2	1	no	—
36	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
37	#20259256	C₂²	1	1	yes	characteristic
38	#20259255	C₂²	1	1	yes	characteristic
39	#20259255	C₂²	1	1	yes	characteristic
40	#20259254	C₂²	1	1	yes	characteristic
41	#20259253	C₂²	1	1	yes	characteristic
42	#20259253	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
46	#20259246	C₂²	1	1	yes	characteristic
47	#20259246	C₂²	1	1	yes	characteristic
48	#20259244	C₂²	1	1	yes	characteristic
49	#20259244	C₂²	1	1	yes	characteristic
50	#20259242	C₂²	1	1	yes	characteristic
51	#20259242	C₂²	1	1	yes	characteristic
52	#125×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	#107×C₁₉₁	C₂²	1	1	yes	characteristic
55	#106×C₁₉₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
57	#103×C₁₉₁	C₂²	1	1	yes	characteristic
58	#103×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690288	—	2	2	no	—
60	#15690288	Q₈	1	4	yes	complemented
61	C₂×Dic₃₈₂	—	2	1	no	—
62	C₂×Dic₃₈₂	—	2	1	no	—
63	C₂×D₇₆₄	—	2	1	no	—
64	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₉₁	—	2	1	no	—
66	C₂×C₄×D₁₉₁	—	2	1	no	—
67	C₂×C₄×D₁₉₁	—	2	1	no	—
68	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
71	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₁₉₁	—	2	2	no	—
74	C₈×D₁₉₁	—	2	2	no	—
75	C₄⋊Dic₁₉₁	—	2	1	no	—
76	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
79	Dic₁₉₁⋊C₄	—	2	1	no	—
80	Dic₁₉₁⋊C₄	—	2	1	no	—
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
84	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
88	C₄×Dic₁₉₁	—	2	1	no	—
89	C₄×Dic₁₉₁	—	2	1	no	—
90	C₄×Dic₁₉₁	—	2	1	no	—
91	C₄×Dic₁₉₁	—	2	1	no	—
92	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
93	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₉₁	—	2	1	no	—
99	C₄⋊C₄×C₁₉₁	—	2	1	no	—
100	C₄⋊C₄×C₁₉₁	—	2	1	no	—
101	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₉₁	—	2	1	no	—
105	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₉₁	—	2	1	no	—
107	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
108	Dic₃₈₂	—	4	2	no	—
109	Dic₃₈₂	—	4	2	no	—
110	D₇₆₄	—	2	4	no	—
111	D₇₆₄	—	2	1	no	—
112	D₇₆₄	C₂×Q₈	1	2	yes	—
113	C₂×Dic₁₉₁	—	2	1	no	—
114	C₂×Dic₁₉₁	—	2	1	no	—
115	C₂×Dic₁₉₁	—	2	1	no	—
116	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₉₁	—	2	4	no	—
120	C₄×D₁₉₁	—	2	2	no	—
121	C₄×D₁₉₁	—	4	1	no	—
122	C₄×D₁₉₁	—	4	2	no	—
123	C₄×D₁₉₁	—	4	2	no	—
124	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
125	C₄×D₁₉₁	#41	1	2	yes	—
126	C₁₉₁⋊C₈	—	2	1	no	—
127	C₁₉₁⋊C₈	—	2	1	no	—
128	Q₈×C₁₉₁	—	2	1	no	—
129	Q₈×C₁₉₁	C₂×Q₈	1	2	yes	—
130	C₂×C₄×C₁₉₁	—	2	1	no	—
131	C₂×C₄×C₁₉₁	—	2	1	no	—
132	C₂×C₄×C₁₉₁	—	2	1	no	—
133	C₂×C₄×C₁₉₁	—	2	1	no	—
134	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₉₁	—	2	1	no	—
138	C₈×C₁₉₁	—	2	1	no	—
139	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₉₁	—	2	1	no	—
141	C₂×D₁₉₁	—	2	1	no	—
142	C₂×D₁₉₁	—	2	2	no	—
143	C₂×D₁₉₁	—	4	1	no	—
144	C₂×D₁₉₁	—	4	1	no	—
145	C₂×D₁₉₁	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₉₁	—	2	2	no	—
147	C₄×C₁₉₁	—	2	1	no	—
148	C₄×C₁₉₁	—	2	1	no	—
149	C₄×C₁₉₁	—	4	1	no	—
150	C₄×C₁₉₁	—	4	1	no	—
151	C₄×C₁₉₁	—	4	1	no	—
152	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
154	Dic₁₉₁	—	2	1	no	—
155	Dic₁₉₁	—	4	1	no	—
156	Dic₁₉₁	—	4	1	no	—
157	Dic₁₉₁	—	4	1	no	—
158	Dic₁₉₁	C₂²⋊Q₈	1	2	yes	—
159	Dic₁₉₁	#135	1	2	yes	—
160	C₂×C₁₉₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₁₉₁	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₉₁	#575	1	1	yes	characteristic
163	D₁₉₁	—	2	2	no	—
164	D₁₉₁	—	4	2	no	—
165	C₁₉₁	#3076	1	1	yes	characteristic, complemented, Sylow
166	#3076	—	191	1	no	maximal, Sylow
167	#533	—	191	1	no	—
168	C₄×#41	—	191	1	no	—
169	#479	—	191	1	no	—
170	#479	D₁₉₁	1	1	yes	characteristic, complemented
171	#478	—	191	1	no	—
172	#478	—	191	1	no	—
173	#477	—	191	1	no	—
174	#476	—	191	1	no	—
175	#475	—	191	1	no	—
176	#474	—	191	1	no	—
177	#427	—	191	1	no	—
178	#427	—	191	1	no	—
179	C₂×C₄⋊C₈	—	191	1	no	—
180	#416	—	191	1	no	—
181	#416	—	191	1	no	—
182	C₂×#41	—	191	1	no	—
183	C₂²×C₈	—	191	1	no	—
184	C₂²×C₈	—	191	1	no	—
185	C₄⋊Q₈	—	191	1	no	—
186	C₄⋊Q₈	—	191	1	no	—
187	#125	—	191	1	no	—
188	#125	C₂×D₁₉₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	382	1	no	—
190	C₂²⋊Q₈	—	382	1	no	—
191	C₄×Q₈	—	382	1	no	—
192	C₄×Q₈	—	382	1	no	—
193	C₄×Q₈	—	191	1	no	—
194	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
195	C₄×D₄	—	382	1	no	—
196	C₄×D₄	—	382	1	no	—
197	C₄×D₄	—	191	1	no	—
198	C₄×D₄	—	191	1	no	—
199	#117	—	382	1	no	—
200	#117	—	191	1	no	—
201	#117	—	191	1	no	—
202	#117	—	191	1	no	—
203	C₂×C₄²	—	382	1	no	—
204	C₂×C₄²	—	191	1	no	—
205	#107	—	191	1	no	—
206	#107	—	191	1	no	—
207	#107	—	191	1	no	—
208	#107	C₂×D₁₉₁	1	1	yes	characteristic
209	#106	—	191	1	no	—
210	#106	—	191	1	no	—
211	#106	—	191	1	no	—
212	#106	C₂×D₁₉₁	1	1	yes	characteristic
213	C₄⋊C₈	—	191	1	no	—
214	C₄⋊C₈	—	191	1	no	—
215	C₄⋊C₈	—	191	1	no	—
216	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
217	#103	—	191	1	no	—
218	#103	—	191	1	no	—
219	#103	C₂×D₁₉₁	1	1	yes	characteristic
220	#103	C₂×D₁₉₁	1	1	yes	characteristic
221	#102	—	191	1	no	—
222	#102	—	191	1	no	—
223	#102	—	191	1	no	—
224	#102	—	191	1	no	—
225	#41	—	382	2	no	—
226	#41	—	191	4	no	—
227	C₂×Q₈	—	382	1	no	—
228	C₂×Q₈	—	382	1	no	—
229	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
230	C₂×D₄	—	382	1	no	—
231	C₂×D₄	—	191	1	no	—
232	C₂²×C₄	—	382	1	no	—
233	C₂²×C₄	—	382	1	no	—
234	C₂²×C₄	—	382	1	no	—
235	C₂²×C₄	—	191	1	no	—
236	C₂²×C₄	—	191	1	no	—
237	C₂²×C₄	—	191	1	no	—
238	C₂×C₈	—	382	2	no	—
239	C₂×C₈	—	382	2	no	—
240	C₂×C₈	—	191	1	no	—
241	C₂×C₈	—	191	1	no	—
242	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	382	1	no	—
248	C₄⋊C₄	—	382	1	no	—
249	C₄⋊C₄	—	382	1	no	—
250	C₄⋊C₄	—	382	1	no	—
251	C₄⋊C₄	—	382	1	no	—
252	C₄⋊C₄	—	191	1	no	—
253	C₄⋊C₄	—	191	1	no	—
254	C₄⋊C₄	—	191	1	no	—
255	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	382	1	no	—
259	C₂²⋊C₄	—	382	1	no	—
260	C₂²⋊C₄	—	382	1	no	—
261	C₂²⋊C₄	—	382	1	no	—
262	C₂²⋊C₄	—	382	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	382	1	no	—
265	C₄²	—	382	1	no	—
266	C₄²	—	382	1	no	—
267	C₄²	—	382	1	no	—
268	C₄²	—	191	1	no	—
269	C₄²	—	191	1	no	—
270	C₄²	—	191	1	no	—
271	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
272	C₂³	—	382	1	no	—
273	C₂³	—	191	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	764	2	no	—
276	Q₈	—	764	2	no	—
277	Q₈	Q₈×D₁₉₁	1	2	yes	—
278	D₄	—	382	4	no	—
279	D₄	—	382	1	no	—
280	D₄	—	191	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	382	4	no	—
286	C₂×C₄	—	382	2	no	—
287	C₂×C₄	—	382	1	no	—
288	C₂×C₄	—	382	1	no	—
289	C₂×C₄	—	382	1	no	—
290	C₂×C₄	—	191	1	no	—
291	C₂×C₄	—	191	1	no	—
292	C₂×C₄	—	191	1	no	—
293	C₂×C₄	—	191	2	no	—
294	C₂×C₄	—	191	2	no	—
295	C₂×C₄	—	764	1	no	—
296	C₂×C₄	—	764	2	no	—
297	C₂×C₄	—	764	2	no	—
298	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	382	1	no	—
304	C₈	—	382	1	no	—
305	C₂²	—	382	1	no	—
306	C₂²	—	382	1	no	—
307	C₂²	—	382	2	no	—
308	C₂²	—	191	2	no	—
309	C₂²	—	764	1	no	—
310	C₂²	—	764	1	no	—
311	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	382	1	no	—
319	C₄	—	191	2	no	—
320	C₄	—	191	2	no	—
321	C₄	—	764	1	no	—
322	C₄	—	764	1	no	—
323	C₄	—	764	1	no	—
324	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
325	C₄	#41×D₁₉₁	1	1	yes	characteristic
326	C₂	—	382	2	no	—
327	C₂	—	764	2	no	—
328	C₂	C₂²⋊Q₈×D₁₉₁	1	1	yes	characteristic
329	C₂	#135×D₁₉₁	1	1	yes	characteristic
330	C₂	#27133462	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#479×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#15690288	C₂²
hypercenter	#479	D₁₉₁
hypocenter	C₁₉₁	#3076
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #27133219	C₂	semidirect
3. #27132970	C₂	semidirect
4. #27132964	C₂	semidirect
5. #27132960	C₂	semidirect
6. #27132956	C₂	semidirect
7. #27132952	C₂	semidirect
8. #27132946	C₂	semidirect
9. #27132942	C₂	semidirect
10. #27132700	C₂	semidirect
11. #27132698	C₂	semidirect
12. C₄⋊C₈×D₁₉₁	C₂	semidirect
15. #479×C₁₉₁	C₂	semidirect
60. #15690288	Q₈	semidirect
165. C₁₉₁	#3076	semidirect
170. #479	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133219	2
C₄×#15690288	2
#27132970	2
#27132964	2
#27132960	2
#27132956	2
#27132952	2
#27132946	2
#27132942	2
#27132700	2
#27132698	2
C₄⋊C₈×D₁₉₁	2
#27132658	2
#479×C₁₉₁	2
#3076	191