Subgroups — #37575801

Order 24448.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₆₄	C₂	1	6	yes	maximal, complemented
2	#27132600	C₂	1	4	yes	maximal, complemented
3	#27132573	C₂	1	4	yes	maximal, complemented
4	#376×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₆₄	C₂²	1	6	yes	—
6	C₂²×C₄×D₁₉₁	C₂²	1	4	yes	—
7	C₄×D₇₆₄	—	2	24	no	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	6	yes	—
9	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	12	yes	—
10	C₂×C₄²×C₁₉₁	C₂²	1	3	yes	—
11	#95×C₁₉₁	C₂²	1	4	yes	—
12	C₂³×D₁₉₁	C₂³	1	4	yes	—
13	C₂×D₇₆₄	—	2	24	no	—
14	C₂×D₇₆₄	—	2	12	no	—
15	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
16	C₂×C₄×D₁₉₁	—	2	24	no	—
17	C₄⋊Dic₁₉₁	—	2	12	no	—
18	(C₂×D₁₉₁)⋊C₄	—	2	24	no	—
19	C₂²×C₄×C₁₉₁	C₂³	1	6	yes	—
20	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
21	C₄²×C₁₉₁	D₄	1	12	yes	complemented
22	C₂²×D₁₉₁	—	2	12	no	—
23	C₂²×D₁₉₁	—	2	12	no	—
24	C₂²×D₁₉₁	#41	1	8	yes	—
25	D₇₆₄	—	2	48	no	—
26	C₂×Dic₁₉₁	—	2	12	no	—
27	C₄×D₁₉₁	—	4	16	no	—
28	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₉₁	—	2	12	no	—
30	C₂×C₄×C₁₉₁	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₁₉₁	C₂×D₄	1	6	yes	—
32	C₂²×C₁₉₁	C₂²×D₄	1	3	yes	—
33	C₂²×C₁₉₁	C₂×#41	1	4	yes	—
34	C₂×D₁₉₁	—	2	24	no	—
35	C₂×D₁₉₁	—	4	12	no	—
36	C₂×D₁₉₁	—	4	4	no	—
37	C₄×C₁₉₁	—	2	12	no	—
38	C₄×C₁₉₁	#127	1	4	yes	—
39	Dic₁₉₁	—	4	4	no	—
40	C₂×C₁₉₁	#525	1	6	yes	—
41	C₂×C₁₉₁	C₂×#127	1	1	yes	characteristic
42	D₁₉₁	—	4	8	no	—
43	C₁₉₁	#2397	1	1	yes	characteristic, complemented, Sylow
44	#2397	—	191	1	no	maximal, Sylow
45	C₂×C₄×D₄	—	191	6	no	—
46	#399	—	191	4	no	—
47	#392	—	191	4	no	—
48	#376	D₁₉₁	1	1	yes	characteristic, complemented
49	C₂²×D₄	—	191	6	no	—
50	C₂³×C₄	—	191	4	no	—
51	C₄×D₄	—	382	24	no	—
52	C₂×C₄⋊C₄	—	191	6	no	—
53	C₂×C₂²⋊C₄	—	191	12	no	—
54	C₂×C₄²	C₂×D₁₉₁	1	3	yes	—
55	#95	C₂×D₁₉₁	1	4	yes	—
56	C₂⁴	—	191	4	no	—
57	C₂×D₄	—	382	24	no	—
58	C₂×D₄	—	382	12	no	—
59	C₂²×C₄	—	382	24	no	—
60	C₂²×C₄	—	191	4	no	—
61	C₂²×C₄	C₂²×D₁₉₁	1	6	yes	—
62	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
63	C₄⋊C₄	—	382	12	no	—
64	C₂²⋊C₄	—	382	24	no	—
65	C₄²	D₇₆₄	1	12	yes	complemented
66	C₂³	—	382	12	no	—
67	C₂³	—	382	12	no	—
68	C₂³	—	191	8	no	—
69	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
70	D₄	—	382	48	no	—
71	C₂×C₄	—	2	12	no	—
72	C₂×C₄	—	382	12	no	—
73	C₂×C₄	—	764	16	no	—
74	C₂×C₄	C₂×D₇₆₄	1	12	yes	—
75	C₂×C₄	C₂×D₇₆₄	1	6	yes	—
76	C₂²	—	382	24	no	—
77	C₂²	—	764	4	no	—
78	C₂²	—	764	12	no	—
79	C₂²	C₂²×D₇₆₄	1	3	yes	—
80	C₂²	#41×D₁₉₁	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	764	4	no	—
83	C₄	#20259319	1	4	yes	—
84	C₂	—	764	8	no	—
85	C₂	#27133175	1	6	yes	—
86	C₂	C₂×#20259319	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Frattini	C₂³	C₂³×D₁₉₁
Fitting	#376×C₁₉₁	C₂
hypercenter	#376	D₁₉₁
hypocenter	C₁₉₁	#2397
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₇₆₄	C₂	semidirect
2. #27132600	C₂	semidirect
3. #27132573	C₂	semidirect
4. #376×C₁₉₁	C₂	semidirect
21. C₄²×C₁₉₁	D₄	semidirect
43. C₁₉₁	#2397	semidirect
48. #376	D₁₉₁	semidirect
65. C₄²	D₇₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₆₄	2
#27132600	2
#27132573	2
#376×C₁₉₁	2
#2397	191