Subgroups — #37583972

Order 24448.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 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	#27133271	C₂	1	1	yes	maximal, characteristic
3	#27132972	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132971	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132970	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132968	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132920	C₂	1	1	yes	maximal, characteristic
8	#27132918	C₂	1	1	yes	maximal, characteristic
9	#27132903	C₂	1	1	yes	maximal, characteristic
10	#27132901	C₂	1	1	yes	maximal, characteristic
11	#27132871	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132871	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132870	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132688	C₂	1	1	yes	maximal, characteristic
15	#462×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690277	C₂²	1	1	yes	characteristic
17	C₂×#15690277	C₂²	1	1	yes	characteristic
18	C₂×#15690270	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
20	#20259320	C₂²	1	1	yes	characteristic
21	#20259317	—	2	1	no	—
22	#20259317	—	2	1	no	—
23	#20259317	—	2	1	no	—
24	#20259316	—	2	1	no	—
25	#20259314	—	2	1	no	—
26	#20259314	C₂²	1	1	yes	characteristic
27	#20259313	C₂²	1	1	yes	characteristic
28	#20259312	—	2	1	no	—
29	#20259312	—	2	1	no	—
30	#20259312	—	2	1	no	—
31	#20259302	C₂²	1	1	yes	characteristic
32	#20259298	C₂²	1	1	yes	characteristic
33	#20259298	C₂²	1	1	yes	characteristic
34	#20259298	C₂²	1	1	yes	characteristic
35	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
36	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
38	#20259279	C₂²	1	1	yes	characteristic
39	#20259256	C₂²	1	1	yes	characteristic
40	#20259255	C₂²	1	1	yes	characteristic
41	#20259254	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	#20259247	C₂²	1	1	yes	characteristic
47	#20259247	C₂²	1	1	yes	characteristic
48	#20259246	C₂²	1	1	yes	characteristic
49	#20259246	C₂²	1	1	yes	characteristic
50	#20259246	C₂²	1	1	yes	characteristic
51	#20259246	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	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
61	#15690277	—	4	1	no	—
62	#15690277	—	4	1	no	—
63	#15690270	—	4	1	no	—
64	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₈×D₁₉₁	—	4	1	no	—
67	C₄⋊Dic₁₉₁	—	2	1	no	—
68	C₄⋊Dic₁₉₁	—	2	1	no	—
69	C₄⋊Dic₁₉₁	—	2	1	no	—
70	C₄⋊Dic₁₉₁	—	2	1	no	—
71	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
74	Dic₁₉₁⋊C₄	—	2	1	no	—
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	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
90	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
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	—	4	1	no	—
99	Q₁₆×C₁₉₁	—	4	1	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	1	no	—
110	Dic₃₈₂	—	2	1	no	—
111	C₂×Dic₁₉₁	—	2	1	no	—
112	C₂×Dic₁₉₁	—	2	1	no	—
113	C₂×Dic₁₉₁	—	2	1	no	—
114	C₂×Dic₁₉₁	—	2	1	no	—
115	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₉₁	—	2	1	no	—
119	C₄×D₁₉₁	—	2	1	no	—
120	C₁₉₁⋊C₈	—	2	1	no	—
121	C₁₉₁⋊C₈	—	2	1	no	—
122	Q₈×C₁₉₁	—	2	1	no	—
123	Q₈×C₁₉₁	—	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	C₂×C₄×C₁₉₁	—	2	1	no	—
129	C₂×C₄×C₁₉₁	—	2	1	no	—
130	C₂×C₄×C₁₉₁	—	2	1	no	—
131	C₂×C₄×C₁₉₁	—	2	1	no	—
132	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₉₁	—	2	1	no	—
136	C₈×C₁₉₁	—	2	1	no	—
137	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₉₁	—	2	1	no	—
139	C₂×D₁₉₁	—	2	1	no	—
140	C₂×D₁₉₁	—	2	1	no	—
141	C₄×C₁₉₁	—	2	1	no	—
142	C₄×C₁₉₁	—	2	1	no	—
143	C₄×C₁₉₁	—	4	1	no	—
144	C₄×C₁₉₁	—	4	1	no	—
145	C₄×C₁₉₁	—	4	1	no	—
146	C₄×C₁₉₁	—	4	1	no	—
147	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₉₁	—	2	1	no	—
150	Dic₁₉₁	—	2	1	no	—
151	Dic₁₉₁	—	2	1	no	—
152	Dic₁₉₁	—	4	1	no	—
153	Dic₁₉₁	—	4	1	no	—
154	Dic₁₉₁	—	4	1	no	—
155	Dic₁₉₁	—	4	1	no	—
156	C₂×C₁₉₁	#537	1	1	yes	characteristic
157	C₂×C₁₉₁	#578	1	1	yes	characteristic
158	C₂×C₁₉₁	#578	1	1	yes	characteristic
159	D₁₉₁	—	4	1	no	—
160	C₁₉₁	#3261	1	1	yes	characteristic, complemented, Sylow
161	#3261	—	191	1	no	maximal, Sylow
162	#541	—	191	1	no	—
163	#541	—	191	1	no	—
164	#479	—	191	1	no	—
165	#479	—	191	1	no	—
166	#479	—	191	1	no	—
167	#479	—	191	1	no	—
168	#470	—	191	1	no	—
169	#470	—	191	1	no	—
170	#467	—	191	1	no	—
171	#467	—	191	1	no	—
172	#462	—	191	1	no	—
173	#462	—	191	1	no	—
174	#462	—	191	1	no	—
175	#462	D₁₉₁	1	1	yes	characteristic, complemented
176	#424	—	191	1	no	—
177	C₂×Q₁₆	—	191	1	no	—
178	C₂×Q₁₆	—	191	1	no	—
179	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
180	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
181	C₂×M₁₆	—	191	1	no	—
182	C₂²×C₈	—	191	1	no	—
183	C₄⋊Q₈	—	191	1	no	—
184	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
185	#126	—	382	1	no	—
186	#126	—	382	1	no	—
187	#126	—	382	1	no	—
188	#126	—	382	1	no	—
189	#125	—	382	1	no	—
190	#125	—	382	1	no	—
191	#125	—	382	1	no	—
192	#125	—	382	1	no	—
193	#125	—	191	1	no	—
194	#125	—	191	1	no	—
195	C₂²⋊Q₈	—	191	1	no	—
196	C₂²⋊Q₈	—	191	1	no	—
197	C₂²⋊Q₈	—	191	1	no	—
198	C₂²⋊Q₈	—	191	1	no	—
199	C₄×Q₈	—	191	1	no	—
200	C₄×Q₈	—	191	1	no	—
201	C₄×Q₈	—	191	1	no	—
202	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
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	—	191	1	no	—
210	C₄⋊C₈	—	191	1	no	—
211	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
212	#103	—	191	1	no	—
213	#103	—	191	1	no	—
214	#103	—	191	1	no	—
215	#103	—	191	1	no	—
216	#103	—	191	1	no	—
217	#103	—	191	1	no	—
218	#103	C₂×D₁₉₁	1	1	yes	characteristic
219	#103	C₂×D₁₉₁	1	1	yes	characteristic
220	C₂×Q₈	—	191	1	no	—
221	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
224	C₂²×C₄	—	191	1	no	—
225	Q₁₆	—	4	1	no	—
226	Q₁₆	—	4	1	no	—
227	Q₁₆	—	764	1	no	—
228	Q₁₆	—	764	1	no	—
229	M₁₆	—	764	1	no	—
230	C₂×C₈	—	191	1	no	—
231	C₂×C₈	—	191	1	no	—
232	C₂×C₈	—	764	1	no	—
233	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	382	1	no	—
239	C₄⋊C₄	—	382	1	no	—
240	C₄⋊C₄	—	382	1	no	—
241	C₄⋊C₄	—	382	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	—	382	1	no	—
261	C₂²⋊C₄	—	191	1	no	—
262	C₂²⋊C₄	—	191	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	382	1	no	—
265	C₄²	—	382	1	no	—
266	C₄²	—	382	1	no	—
267	C₄²	—	191	1	no	—
268	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
269	C₂³	—	191	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	382	1	no	—
277	Q₈	—	382	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	382	1	no	—
283	C₂×C₄	—	382	1	no	—
284	C₂×C₄	—	382	1	no	—
285	C₂×C₄	—	382	1	no	—
286	C₂×C₄	—	382	1	no	—
287	C₂×C₄	—	382	1	no	—
288	C₂×C₄	—	191	1	no	—
289	C₂×C₄	—	191	1	no	—
290	C₂×C₄	—	191	1	no	—
291	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	382	1	no	—
297	C₈	—	382	1	no	—
298	C₂²	—	382	1	no	—
299	C₂²	—	382	1	no	—
300	C₂²	—	382	1	no	—
301	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	382	1	no	—
309	C₄	—	382	1	no	—
310	C₄	—	382	1	no	—
311	C₄	—	764	1	no	—
312	C₄	—	764	1	no	—
313	C₄	—	764	1	no	—
314	C₄	—	764	1	no	—
315	C₄	#20259367	1	1	yes	characteristic
316	C₄	#20259369	1	1	yes	characteristic
317	C₂	—	764	1	no	—
318	C₂	#27133242	1	1	yes	characteristic
319	C₂	#27133476	1	1	yes	characteristic
320	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	#3261
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
3. #27132972	C₂	semidirect
4. #27132971	C₂	semidirect
5. #27132970	C₂	semidirect
6. #27132968	C₂	semidirect
11. #27132871	C₂	semidirect
12. #27132871	C₂	semidirect
13. #27132870	C₂	semidirect
15. #462×C₁₉₁	C₂	semidirect
160. C₁₉₁	#3261	semidirect
175. #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
#27133271	2
#27132972	2
#27132971	2
#27132970	2
#27132968	2
#27132920	2
#27132918	2
#27132903	2
#27132901	2
#27132871	2
#27132870	2
#27132688	2
#462×C₁₉₁	2
#3261	191