Subgroups — #27133100

Order 12224.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259345	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259265	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₀₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259263	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259261	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690288	—	2	1	no	—
9	D₄×D₁₉₁	—	2	1	no	—
10	#15690274	—	2	1	no	—
11	D₁₅₂₈	C₂²	1	1	yes	characteristic
12	D₁₅₂₈	C₂²	1	1	yes	characteristic, complemented
13	#15690272	—	2	1	no	—
14	C₈×D₁₉₁	C₂²	1	1	yes	characteristic
15	C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
16	Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
17	D₈×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₁₉₁	—	4	1	no	—
20	D₇₆₄	—	2	1	no	—
21	D₇₆₄	—	2	1	no	—
22	D₇₆₄	—	4	1	no	—
23	C₂²⋊D₁₉₁	—	4	1	no	—
24	C₄×D₁₉₁	—	4	1	no	—
25	C₄×D₁₉₁	D₄	1	1	yes	characteristic
26	C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₁₉₁	—	2	1	no	—
28	D₄×C₁₉₁	—	2	1	no	—
29	C₈×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉₁	—	4	1	no	—
31	C₂×D₁₉₁	—	4	1	no	—
32	C₂×D₁₉₁	—	4	1	no	—
33	C₂×D₁₉₁	—	8	2	no	—
34	C₂×D₁₉₁	D₈	1	1	yes	characteristic
35	C₄×C₁₉₁	—	4	1	no	—
36	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	Dic₁₉₁	D₈	1	1	yes	characteristic
38	C₂×C₁₉₁	—	8	1	no	—
39	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
40	D₁₉₁	—	2	1	no	—
41	D₁₉₁	—	8	1	no	—
42	D₁₉₁	—	8	1	no	—
43	C₁₉₁	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	191	1	no	maximal, Sylow
45	#135	—	191	1	no	—
46	C₂×D₈	—	191	1	no	—
47	SD₃₂	—	191	1	no	—
48	SD₃₂	D₁₉₁	1	1	yes	characteristic, complemented
49	D₁₆	—	191	1	no	—
50	D₁₆	—	191	1	no	—
51	M₃₂	—	191	1	no	—
52	#41	—	382	1	no	—
53	C₂×D₄	—	382	1	no	—
54	Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic, complemented
55	SD₁₆	—	382	1	no	—
56	D₈	—	382	1	no	—
57	D₈	—	191	1	no	—
58	D₈	—	191	1	no	—
59	D₈	C₂×D₁₉₁	1	1	yes	characteristic
60	C₂×C₈	—	191	1	no	—
61	C₁₆	—	191	1	no	—
62	C₁₆	C₂×D₁₉₁	1	1	yes	characteristic, complemented
63	C₂³	—	764	1	no	—
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	382	1	no	—
67	D₄	—	382	1	no	—
68	D₄	—	764	1	no	—
69	D₄	—	764	1	no	—
70	C₂×C₄	—	191	1	no	—
71	C₂×C₄	—	764	1	no	—
72	C₈	—	191	1	no	—
73	C₈	C₂²×D₁₉₁	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	191	1	no	—
76	C₂²	—	1528	2	no	—
77	C₂²	—	764	1	no	—
78	C₂²	—	764	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	191	1	no	—
81	C₄	D₄×D₁₉₁	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	382	1	no	—
84	C₂	—	1528	1	no	—
85	C₂	—	1528	1	no	—
86	C₂	D₈×D₁₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₉₁
commutator, cosocle	C₈×C₁₉₁	C₂³
Frattini	C₈	C₂²×D₁₉₁
Fitting	SD₃₂×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂×D₈
hypercenter	SD₃₂	D₁₉₁
hypocenter	C₁₉₁	#509
LCS	G ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20259345	C₂	semidirect
2. D₈×D₁₉₁	C₂	semidirect
3. #20259265	C₂	semidirect
4. D₃₀₅₆	C₂	semidirect
5. #20259263	C₂	semidirect
6. #20259261	C₂	semidirect
7. SD₃₂×C₁₉₁	C₂	semidirect
12. D₁₅₂₈	C₂²	semidirect
15. C₁₉₁⋊C₁₆	C₂²	semidirect
16. Q₁₆×C₁₉₁	C₂²	semidirect
18. C₁₆×C₁₉₁	C₂²	semidirect
43. C₁₉₁	#509	semidirect
48. SD₃₂	D₁₉₁	semidirect
54. Q₁₆	C₂×D₁₉₁	semidirect
62. C₁₆	C₂×D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#20259345	2
D₈×D₁₉₁	2
#20259265	2
D₃₀₅₆	2
#20259263	2
#20259261	2
SD₃₂×C₁₉₁	2
#509	191