Subgroups — #37582818

Order 24448.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133274	C₂	1	1	yes	maximal, characteristic
2	#27133139	C₂	1	1	yes	maximal, characteristic
3	#27133003	C₂	1	1	yes	maximal, characteristic
4	#27133002	C₂	1	1	yes	maximal, characteristic
5	#27132971	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132970	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132870	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132817	C₂	1	1	yes	maximal, characteristic
9	#27132817	C₂	1	1	yes	maximal, characteristic
10	#27132770	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132767	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132755	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132723	C₂	1	1	yes	maximal, characteristic
15	#462×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×#15690277	C₂²	1	1	yes	characteristic
18	C₂×#15690277	C₂²	1	1	yes	characteristic
19	#20259317	—	2	1	no	—
20	#20259317	—	2	1	no	—
21	#20259314	C₂²	1	1	yes	characteristic
22	#20259312	—	2	1	no	—
23	#20259312	—	2	1	no	—
24	#20259298	C₂²	1	1	yes	characteristic
25	#20259298	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₈₂	—	2	1	no	—
30	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
31	#20259287	—	2	1	no	—
32	#20259287	C₂²	1	1	yes	characteristic
33	#20259280	—	2	1	no	—
34	#20259280	C₂²	1	1	yes	characteristic
35	#20259279	—	2	1	no	—
36	#20259279	C₂²	1	1	yes	characteristic
37	#20259256	C₂²	1	1	yes	characteristic
38	#20259255	C₂²	1	1	yes	characteristic
39	#20259254	C₂²	1	1	yes	characteristic
40	#20259253	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#20259247	C₂²	1	1	yes	characteristic
43	#20259247	C₂²	1	1	yes	characteristic
44	#20259246	C₂²	1	1	yes	characteristic
45	#20259246	C₂²	1	1	yes	characteristic
46	#20259246	C₂²	1	1	yes	characteristic
47	#20259246	C₂²	1	1	yes	characteristic
48	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#20259232	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×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	C₂×Dic₃₈₂	—	2	1	no	—
60	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
61	Q₈×D₁₉₁	—	2	2	no	—
62	Q₈×D₁₉₁	—	2	2	no	—
63	C₂×C₄×D₁₉₁	—	2	1	no	—
64	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
65	#15690277	—	2	2	no	—
66	#15690277	—	2	2	no	—
67	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₉₁	—	2	1	no	—
70	C₄⋊Dic₁₉₁	—	2	1	no	—
71	C₄⋊Dic₁₉₁	—	2	1	no	—
72	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
75	Dic₁₉₁⋊C₄	—	2	1	no	—
76	Dic₁₉₁⋊C₄	—	2	1	no	—
77	Dic₁₉₁⋊C₄	—	2	1	no	—
78	Dic₁₉₁⋊C₄	—	2	1	no	—
79	Dic₁₉₁⋊C₄	—	2	1	no	—
80	Dic₁₉₁⋊C₄	—	2	1	no	—
81	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
82	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
87	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
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	—	2	1	no	—
93	C₄×Dic₁₉₁	—	2	1	no	—
94	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₉₁	—	2	2	no	—
99	Q₁₆×C₁₉₁	—	2	2	no	—
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	—	2	1	no	—
103	C₄⋊C₄×C₁₉₁	—	2	1	no	—
104	C₄⋊C₄×C₁₉₁	—	2	1	no	—
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₄²×C₁₉₁	—	2	1	no	—
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
109	Dic₃₈₂	—	2	4	no	—
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	#41	1	2	yes	—
112	C₂×Dic₁₉₁	—	2	1	no	—
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	1	no	—
121	C₄×D₁₉₁	—	2	1	no	—
122	C₁₉₁⋊C₈	—	2	1	no	—
123	C₁₉₁⋊C₈	—	2	1	no	—
124	Q₈×C₁₉₁	—	2	1	no	—
125	Q₈×C₁₉₁	—	2	1	no	—
126	Q₈×C₁₉₁	—	2	1	no	—
127	Q₈×C₁₉₁	—	2	1	no	—
128	Q₈×C₁₉₁	—	2	1	no	—
129	Q₈×C₁₉₁	#41	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	1	no	—
143	C₄×C₁₉₁	—	2	2	no	—
144	C₄×C₁₉₁	—	2	1	no	—
145	C₄×C₁₉₁	—	2	1	no	—
146	C₄×C₁₉₁	—	4	1	no	—
147	C₄×C₁₉₁	—	4	1	no	—
148	C₄×C₁₉₁	—	4	1	no	—
149	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
151	Dic₁₉₁	—	2	2	no	—
152	Dic₁₉₁	—	2	1	no	—
153	Dic₁₉₁	—	2	1	no	—
154	Dic₁₉₁	—	2	1	no	—
155	Dic₁₉₁	—	4	1	no	—
156	Dic₁₉₁	—	4	1	no	—
157	Dic₁₉₁	—	4	1	no	—
158	C₂×C₁₉₁	#525	1	1	yes	characteristic
159	C₂×C₁₉₁	#578	1	1	yes	characteristic
160	C₂×C₁₉₁	#578	1	1	yes	characteristic
161	D₁₉₁	—	4	1	no	—
162	C₁₉₁	#3127	1	1	yes	characteristic, complemented, Sylow
163	#3127	—	191	1	no	maximal, Sylow
164	#541	—	191	1	no	—
165	#519	—	191	1	no	—
166	#485	—	191	1	no	—
167	#485	—	191	1	no	—
168	#479	—	191	1	no	—
169	#479	—	191	1	no	—
170	#462	—	191	1	no	—
171	#462	D₁₉₁	1	1	yes	characteristic, complemented
172	#451	—	191	1	no	—
173	#451	—	191	1	no	—
174	#441	—	191	1	no	—
175	#441	—	191	1	no	—
176	C₄×Q₁₆	—	191	1	no	—
177	C₄×Q₁₆	—	191	1	no	—
178	#433	—	191	1	no	—
179	C₂²×Q₈	—	191	1	no	—
180	C₂×Q₁₆	—	191	1	no	—
181	C₂×Q₁₆	—	191	1	no	—
182	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
183	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
184	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
185	#126	—	382	1	no	—
186	#126	—	382	1	no	—
187	#125	—	382	1	no	—
188	#125	—	382	1	no	—
189	#125	—	191	1	no	—
190	C₂²⋊Q₈	—	191	1	no	—
191	C₂²⋊Q₈	—	191	1	no	—
192	C₄×Q₈	—	382	1	no	—
193	C₄×Q₈	—	382	1	no	—
194	C₄×Q₈	—	191	1	no	—
195	C₄×Q₈	—	191	1	no	—
196	C₄×Q₈	—	191	1	no	—
197	C₄×Q₈	—	191	1	no	—
198	C₄×Q₈	—	191	1	no	—
199	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
200	#117	—	382	1	no	—
201	#117	—	382	1	no	—
202	#117	—	191	1	no	—
203	#117	—	191	1	no	—
204	#107	—	191	1	no	—
205	#107	—	191	1	no	—
206	#106	—	191	1	no	—
207	#106	—	191	1	no	—
208	C₄⋊C₈	—	191	1	no	—
209	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
210	#103	—	191	1	no	—
211	#103	—	191	1	no	—
212	#103	—	191	1	no	—
213	#103	—	191	1	no	—
214	#103	—	191	1	no	—
215	#103	—	191	1	no	—
216	#103	C₂×D₁₉₁	1	1	yes	characteristic
217	#103	C₂×D₁₉₁	1	1	yes	characteristic
218	C₂²⋊C₈	—	191	1	no	—
219	C₂²⋊C₈	—	191	1	no	—
220	#97	—	191	1	no	—
221	C₄×C₈	—	191	1	no	—
222	C₂×Q₈	—	382	1	no	—
223	C₂×Q₈	—	382	2	no	—
224	C₂×Q₈	—	382	2	no	—
225	C₂×Q₈	—	191	1	no	—
226	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
229	C₂²×C₄	—	382	1	no	—
230	C₂²×C₄	—	191	1	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	382	2	no	—
234	Q₁₆	—	382	2	no	—
235	C₂×C₈	—	191	1	no	—
236	C₂×C₈	—	191	1	no	—
237	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	382	1	no	—
243	C₄⋊C₄	—	382	1	no	—
244	C₄⋊C₄	—	382	1	no	—
245	C₄⋊C₄	—	382	1	no	—
246	C₄⋊C₄	—	382	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	—	191	1	no	—
252	C₄⋊C₄	—	191	1	no	—
253	C₄⋊C₄	—	191	1	no	—
254	C₄⋊C₄	—	191	1	no	—
255	C₄⋊C₄	—	191	1	no	—
256	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	382	1	no	—
258	C₂²⋊C₄	—	382	1	no	—
259	C₂²⋊C₄	—	382	1	no	—
260	C₂²⋊C₄	—	191	1	no	—
261	C₂²⋊C₄	—	191	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	382	1	no	—
264	C₄²	—	382	1	no	—
265	C₄²	—	382	1	no	—
266	C₄²	—	382	1	no	—
267	C₄²	—	382	1	no	—
268	C₄²	—	382	1	no	—
269	C₄²	—	191	1	no	—
270	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
271	C₂³	—	191	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	382	4	no	—
278	Q₈	—	382	1	no	—
279	Q₈	—	191	2	no	—
280	Q₈	#15690286	1	2	yes	—
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	1	no	—
287	C₂×C₄	—	382	1	no	—
288	C₂×C₄	—	382	1	no	—
289	C₂×C₄	—	382	1	no	—
290	C₂×C₄	—	382	1	no	—
291	C₂×C₄	—	382	1	no	—
292	C₂×C₄	—	191	1	no	—
293	C₂×C₄	—	191	1	no	—
294	C₂×C₄	—	191	1	no	—
295	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
296	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	382	1	no	—
301	C₈	—	382	1	no	—
302	C₂²	—	382	1	no	—
303	C₂²	—	382	1	no	—
304	C₂²	—	382	1	no	—
305	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	382	1	no	—
313	C₄	—	382	1	no	—
314	C₄	—	382	2	no	—
315	C₄	—	382	1	no	—
316	C₄	—	764	1	no	—
317	C₄	—	764	1	no	—
318	C₄	—	764	1	no	—
319	C₄	#41×D₁₉₁	1	1	yes	characteristic
320	C₄	C₂×#15690286	1	1	yes	characteristic
321	C₂	—	764	1	no	—
322	C₂	#27133173	1	1	yes	characteristic
323	C₂	#27133476	1	1	yes	characteristic
324	C₂	#27133479	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#462×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂²×D₁₉₁	C₂×D₄
hypercenter	#462	D₁₉₁
hypocenter	C₁₉₁	#3127
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
5. #27132971	C₂	semidirect
6. #27132970	C₂	semidirect
7. #27132870	C₂	semidirect
10. #27132770	C₂	semidirect
11. #27132767	C₂	semidirect
12. Q₁₆×Dic₁₉₁	C₂	semidirect
13. #27132755	C₂	semidirect
15. #462×C₁₉₁	C₂	semidirect
162. C₁₉₁	#3127	semidirect
171. #462	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133274	2
#27133139	2
#27133003	2
#27133002	2
#27132971	2
#27132970	2
#27132870	2
#27132817	2
#27132770	2
#27132767	2
Q₁₆×Dic₁₉₁	2
#27132755	2
#27132723	2
#462×C₁₉₁	2
#3127	191