Subgroups — #27133038

Order 12224.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690272	C₂	1	2	yes	maximal, complemented
3	#20259318	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₆₄	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₁₉₁	—	2	4	no	—
8	D₁₅₂₈	—	2	2	no	—
9	#15690272	—	2	4	no	—
10	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₁₉₁	C₂²	1	2	yes	—
13	D₈×C₁₉₁	—	2	2	no	—
14	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₁	—	2	2	no	—
16	D₇₆₄	—	2	2	no	—
17	D₇₆₄	—	2	2	no	—
18	C₂²⋊D₁₉₁	—	4	8	no	—
19	C₂×Dic₁₉₁	D₄	1	2	yes	—
20	C₁₉₁⋊C₈	D₄	1	2	yes	complemented
21	C₂³×C₁₉₁	—	2	2	no	—
22	D₄×C₁₉₁	—	2	2	no	—
23	D₄×C₁₉₁	—	2	2	no	—
24	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₈×C₁₉₁	D₄	1	2	yes	complemented
26	C₂²×C₁₉₁	—	4	2	no	—
27	C₂²×C₁₉₁	—	4	4	no	—
28	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₁₉₁	—	4	2	no	—
30	C₂×D₁₉₁	—	4	4	no	—
31	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
33	Dic₁₉₁	D₈	1	4	yes	complemented
34	C₂×C₁₉₁	—	8	2	no	—
35	C₂×C₁₉₁	#127	1	1	yes	characteristic
36	C₂×C₁₉₁	C₂×D₈	1	2	yes	—
37	D₁₉₁	—	8	2	no	—
38	C₁₉₁	#493	1	1	yes	characteristic, complemented, Sylow
39	#493	—	191	1	no	maximal, Sylow
40	C₂×D₈	—	191	2	no	—
41	C₂×D₈	—	191	1	no	—
42	C₂×D₈	D₁₉₁	1	1	yes	characteristic, complemented
43	#127	—	191	2	no	—
44	C₄×C₈	—	191	1	no	—
45	C₂×D₄	—	382	4	no	—
46	C₂×D₄	—	191	2	no	—
47	C₂×D₄	C₂×D₁₉₁	1	2	yes	—
48	D₈	—	2	2	no	—
49	D₈	—	382	4	no	—
50	D₈	—	382	2	no	—
51	C₂×C₈	—	191	1	no	—
52	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
53	C₄²	—	191	1	no	—
54	C₂³	—	2	2	no	—
55	C₂³	—	382	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	382	2	no	—
59	D₄	—	382	2	no	—
60	D₄	—	764	8	no	—
61	C₂×C₄	—	191	2	no	—
62	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
63	C₈	—	191	2	no	—
64	C₈	C₂²⋊D₁₉₁	1	2	yes	complemented
65	C₂²	—	4	4	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	764	4	no	—
68	C₂²	—	764	2	no	—
69	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
70	C₄	—	191	4	no	—
71	C₄	D₄×D₁₉₁	1	1	yes	characteristic
72	C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
73	C₂	—	8	2	no	—
74	C₂	—	1528	2	no	—
75	C₂	#20259318	1	1	yes	characteristic
76	C₂	D₈×D₁₉₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₂×D₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
hypercenter	C₂×D₈	D₁₉₁
hypocenter	C₁₉₁	#493
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₂₈	C₂	semidirect
2. C₂×#15690272	C₂	semidirect
3. #20259318	C₂	semidirect
4. C₈×Dic₁₉₁	C₂	semidirect
5. C₂×D₈×C₁₉₁	C₂	semidirect
20. C₁₉₁⋊C₈	D₄	semidirect
25. C₈×C₁₉₁	D₄	semidirect
33. Dic₁₉₁	D₈	semidirect
38. C₁₉₁	#493	semidirect
42. C₂×D₈	D₁₉₁	semidirect
64. C₈	C₂²⋊D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₂₈	2
C₂×#15690272	2
#20259318	2
C₈×Dic₁₉₁	2
C₂×D₈×C₁₉₁	2
#493	191