Subgroups — #37582070

Order 24448.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259352	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20259342	C₂	1	1	yes	maximal, characteristic
3	#27133239	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132885	C₂	1	2	yes	maximal, complemented
5	#27132878	C₂	1	2	yes	maximal, complemented
6	#27132873	C₂	1	2	yes	maximal, complemented
7	#27132868	C₂	1	2	yes	maximal, complemented
8	#27132690	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132662	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20259245	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15690287	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
14	#20259352	—	2	4	no	—
15	#20259342	—	2	4	no	—
16	C₂×#15690277	C₂²	1	1	yes	characteristic
17	C₂×#15690277	C₂²	1	2	yes	—
18	C₂×#15690275	C₂²	1	1	yes	characteristic
19	C₂×#15690275	C₂²	1	2	yes	—
20	C₂×#15690274	C₂²	1	1	yes	characteristic
21	C₂×#15690272	C₂²	1	1	yes	characteristic
22	C₂×#15690271	C₂²	1	1	yes	characteristic
23	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
24	#20259322	C₂²	1	2	yes	—
25	#20259310	C₂²	1	2	yes	—
26	#20259306	—	2	2	no	—
27	#20259302	—	2	1	no	—
28	#20259302	—	2	1	no	—
29	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
30	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
31	#20259245	C₂²	1	2	yes	—
32	#20259245	C₂²	1	2	yes	—
33	#20259245	C₂²	1	2	yes	—
34	#20259242	C₂²	1	2	yes	—
35	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₉₁	C₂²	1	2	yes	—
37	C₄×D₄×C₁₉₁	—	2	2	no	—
38	C₄×D₄×C₁₉₁	C₂²	1	2	yes	—
39	#117×C₁₉₁	—	2	1	no	—
40	#117×C₁₉₁	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
43	#15690287	—	2	2	no	—
44	#15690287	—	2	2	no	—
45	C₂×Dic₃₈₂	—	2	1	no	—
46	C₂×Dic₃₈₂	—	2	2	no	—
47	C₂×Dic₃₈₂	—	2	2	no	—
48	C₂×Dic₃₈₂	C₂³	1	2	yes	—
49	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
50	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
52	C₂²×Dic₁₉₁	—	2	1	no	—
53	C₂×C₄×D₁₉₁	—	2	1	no	—
54	#15690277	—	2	2	no	—
55	#15690277	—	2	4	no	—
56	#15690275	—	2	2	no	—
57	#15690275	—	2	4	no	—
58	#15690274	—	2	2	no	—
59	#15690272	—	2	2	no	—
60	#15690271	—	2	2	no	—
61	C₂×C₁₉₁⋊C₈	—	2	2	no	—
62	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
64	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
65	C₄⋊Dic₁₉₁	—	2	2	no	—
66	C₄⋊Dic₁₉₁	—	2	2	no	—
67	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
68	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
69	#41×C₁₉₁	—	2	2	no	—
70	#41×C₁₉₁	D₄	1	4	yes	—
71	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₉₁	—	2	1	no	—
73	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₉₁	—	2	1	no	—
75	C₂²×C₄×C₁₉₁	—	2	1	no	—
76	C₂²×C₄×C₁₉₁	—	2	2	no	—
77	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉₁	—	2	1	no	—
79	C₄⋊C₄×C₁₉₁	—	2	2	no	—
80	C₄⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉₁	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
84	C₂²⋊C₄×C₁₉₁	D₄	1	2	yes	—
85	C₄²×C₁₉₁	—	2	2	no	—
86	C₄²×C₁₉₁	C₂³	1	2	yes	—
87	C₂²×D₁₉₁	—	2	1	no	—
88	Dic₃₈₂	—	2	2	no	—
89	Dic₃₈₂	—	2	1	no	—
90	Dic₃₈₂	—	2	2	no	—
91	Dic₃₈₂	—	2	1	no	—
92	Dic₃₈₂	—	4	2	no	—
93	D₇₆₄	—	2	1	no	—
94	D₇₆₄	—	2	1	no	—
95	C₂²⋊D₁₉₁	—	4	2	no	—
96	C₂×Dic₁₉₁	—	2	2	no	—
97	C₂×Dic₁₉₁	—	2	1	no	—
98	C₂×Dic₁₉₁	—	4	2	no	—
99	C₄×D₁₉₁	—	4	2	no	—
100	C₁₉₁⋊C₈	—	2	1	no	—
101	C₁₉₁⋊C₈	—	2	1	no	—
102	C₁₉₁⋊C₈	—	2	2	no	—
103	C₂³×C₁₉₁	—	2	1	no	—
104	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₉₁	—	2	1	no	—
106	Q₈×C₁₉₁	C₂×D₄	1	2	yes	—
107	D₄×C₁₉₁	—	2	4	no	—
108	D₄×C₁₉₁	—	2	1	no	—
109	D₄×C₁₉₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉₁	—	2	1	no	—
111	C₂×C₄×C₁₉₁	—	2	4	no	—
112	C₂×C₄×C₁₉₁	—	2	2	no	—
113	C₂×C₄×C₁₉₁	—	8	2	no	—
114	C₂×C₄×C₁₉₁	—	8	1	no	—
115	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₉₁	#41	1	2	yes	—
120	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
121	C₂²×C₁₉₁	—	2	1	no	—
122	C₂²×C₁₉₁	—	2	1	no	—
123	C₂²×C₁₉₁	—	2	2	no	—
124	C₂²×C₁₉₁	—	4	1	no	—
125	C₂²×C₁₉₁	—	4	1	no	—
126	C₂²×C₁₉₁	#121	1	2	yes	—
127	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₉₁	—	4	1	no	—
129	C₂×D₁₉₁	—	4	1	no	—
130	C₂×D₁₉₁	—	4	1	no	—
131	C₄×C₁₉₁	—	2	2	no	—
132	C₄×C₁₉₁	—	2	2	no	—
133	C₄×C₁₉₁	—	2	2	no	—
134	C₄×C₁₉₁	—	4	2	no	—
135	C₄×C₁₉₁	#121	1	2	yes	—
136	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
138	Dic₁₉₁	—	4	1	no	—
139	Dic₁₉₁	—	4	2	no	—
140	C₂×C₁₉₁	—	2	2	no	—
141	C₂×C₁₉₁	—	4	2	no	—
142	C₂×C₁₉₁	C₂×#121	1	1	yes	characteristic
143	C₂×C₁₉₁	#577	1	1	yes	characteristic
144	C₂×C₁₉₁	#578	1	1	yes	characteristic
145	D₁₉₁	—	8	1	no	—
146	C₁₉₁	#3046	1	1	yes	characteristic, complemented, Sylow
147	#3046	—	191	1	no	maximal, Sylow
148	C₂×#137	—	191	1	no	—
149	C₂×#135	—	191	1	no	—
150	#536	—	191	1	no	—
151	#520	D₁₉₁	1	1	yes	characteristic, complemented
152	#464	—	191	2	no	—
153	#463	—	191	2	no	—
154	#462	—	191	2	no	—
155	#461	—	191	2	no	—
156	#424	—	191	1	no	—
157	#417	—	191	1	no	—
158	C₂×#103	—	191	1	no	—
159	C₂×#41	—	191	1	no	—
160	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
161	C₂²×Q₈	—	191	1	no	—
162	#137	—	382	4	no	—
163	#135	—	382	4	no	—
164	C₂×Q₁₆	—	191	1	no	—
165	C₂×Q₁₆	—	191	2	no	—
166	C₂×SD₁₆	—	191	1	no	—
167	C₂×SD₁₆	—	191	1	no	—
168	C₂×SD₁₆	—	191	2	no	—
169	C₂×D₈	—	191	1	no	—
170	C₂×M₁₆	—	191	1	no	—
171	C₂²×C₈	—	191	1	no	—
172	C₄⋊Q₈	—	191	2	no	—
173	#124	—	191	2	no	—
174	#123	—	382	2	no	—
175	C₂²⋊Q₈	—	382	1	no	—
176	C₂²⋊Q₈	—	382	1	no	—
177	C₄×Q₈	C₂×D₁₉₁	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×D₁₉₁	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	C₂×D₁₉₁	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
184	C₄⋊C₈	—	191	2	no	—
185	C₄⋊C₈	—	191	2	no	—
186	#103	—	191	2	no	—
187	#103	—	191	2	no	—
188	#103	—	191	2	no	—
189	#102	—	191	2	no	—
190	#41	—	2	2	no	—
191	#41	—	382	2	no	—
192	#41	—	382	2	no	—
193	#41	D₇₆₄	1	4	yes	—
194	C₂×Q₈	—	382	1	no	—
195	C₂×Q₈	—	382	2	no	—
196	C₂×Q₈	—	382	2	no	—
197	C₂×Q₈	—	191	2	no	—
198	C₂×Q₈	—	191	1	no	—
199	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	382	1	no	—
202	C₂×D₄	—	191	1	no	—
203	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	382	1	no	—
208	C₂²×C₄	—	382	1	no	—
209	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
210	Q₁₆	—	382	2	no	—
211	Q₁₆	—	382	4	no	—
212	SD₁₆	—	382	2	no	—
213	SD₁₆	—	382	2	no	—
214	SD₁₆	—	382	4	no	—
215	D₈	—	382	2	no	—
216	M₁₆	—	382	2	no	—
217	C₂×C₈	—	382	2	no	—
218	C₂×C₈	—	191	2	no	—
219	C₂×C₈	—	191	1	no	—
220	C₂×C₈	—	191	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	382	2	no	—
224	C₄⋊C₄	—	382	2	no	—
225	C₄⋊C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₉₁	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	382	2	no	—
230	C₂²⋊C₄	—	382	2	no	—
231	C₂²⋊C₄	C₂²⋊D₁₉₁	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂²×D₁₉₁	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	382	1	no	—
236	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	382	2	no	—
239	Q₈	—	382	1	no	—
240	Q₈	—	382	2	no	—
241	Q₈	—	382	1	no	—
242	Q₈	—	764	2	no	—
243	Q₈	C₂×D₇₆₄	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	382	1	no	—
247	D₄	—	382	1	no	—
248	D₄	—	764	2	no	—
249	D₄	C₂×D₇₆₄	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	382	2	no	—
256	C₂×C₄	—	382	1	no	—
257	C₂×C₄	—	764	2	no	—
258	C₂×C₄	—	764	2	no	—
259	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
262	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
263	C₂×C₄	#15690287	1	2	yes	—
264	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
265	C₈	—	382	1	no	—
266	C₈	—	382	1	no	—
267	C₈	—	382	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	764	1	no	—
274	C₂²	—	764	1	no	—
275	C₂²	—	764	1	no	—
276	C₂²	#20259296	1	2	yes	—
277	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	764	1	no	—
283	C₄	—	764	2	no	—
284	C₄	#20259296	1	2	yes	—
285	C₄	C₂²×D₇₆₄	1	1	yes	characteristic
286	C₄	C₂×#15690287	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	1528	1	no	—
290	C₂	C₂×#20259296	1	1	yes	characteristic
291	C₂	#27133474	1	1	yes	characteristic
292	C₂	#27133478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#520×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#20259342	C₂
hypercenter	#520	D₁₉₁
hypocenter	C₁₉₁	#3046
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#20259352	C₂	semidirect
3. #27133239	C₂	semidirect
4. #27132885	C₂	semidirect
5. #27132878	C₂	semidirect
6. #27132873	C₂	semidirect
7. #27132868	C₂	semidirect
8. #27132690	C₂	semidirect
9. #27132662	C₂	semidirect
10. C₂×#20259245	C₂	semidirect
11. #520×C₁₉₁	C₂	semidirect
146. C₁₉₁	#3046	semidirect
151. #520	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259352	2
C₂×#20259342	2
#27133239	2
#27132885	2
#27132878	2
#27132873	2
#27132868	2
#27132690	2
#27132662	2
C₂×#20259245	2
#520×C₁₉₁	2
#3046	191