Subgroups — #27133060

Order 12224.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259321	C₂	1	2	yes	maximal
2	#20259254	C₂	1	1	yes	maximal, characteristic
3	#20259253	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	—	2	2	no	—
7	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	Dic₁₉₁⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
13	Dic₃₈₂	—	4	4	no	—
14	C₂×Dic₁₉₁	—	2	2	no	—
15	C₂×Dic₁₉₁	D₄	1	2	yes	—
16	C₁₉₁⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₁₉₁	—	2	2	no	—
18	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
19	C₈×C₁₉₁	Q₈	1	2	yes	complemented
20	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₉₁	—	4	2	no	—
22	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₉₁	—	4	2	no	—
25	Dic₁₉₁	SD₁₆	1	4	yes	—
26	C₂×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₁₉₁	C₂×SD₁₆	1	2	yes	—
28	C₁₉₁	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	191	1	no	maximal, Sylow
30	C₄⋊Q₈	—	191	2	no	—
31	#106	—	191	1	no	—
32	#106	—	191	2	no	—
33	#106	D₁₉₁	1	1	yes	characteristic
34	C₄×C₈	—	191	1	no	—
35	C₂×Q₈	—	382	2	no	—
36	C₂×C₈	—	191	1	no	—
37	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
38	C₄⋊C₄	—	382	2	no	—
39	C₄⋊C₄	—	191	2	no	—
40	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
41	C₄²	—	191	1	no	—
42	Q₈	—	764	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	382	2	no	—
45	C₂×C₄	—	191	2	no	—
46	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
47	C₈	—	191	2	no	—
48	C₈	Dic₃₈₂	1	2	yes	complemented
49	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	191	4	no	—
52	C₄	—	764	2	no	—
53	C₄	Q₈×D₁₉₁	1	1	yes	characteristic
54	C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
55	C₂	#20259321	1	1	yes	characteristic
56	C₂	SD₁₆×D₁₉₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#106×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
hypercenter	#106	D₁₉₁
hypocenter	C₁₉₁	#498
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. C₁₉₁⋊C₈	Q₈	semidirect
19. C₈×C₁₉₁	Q₈	semidirect
28. C₁₉₁	#498	semidirect
48. C₈	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
#20259321	2
#20259254	2
#20259253	2
C₈×Dic₁₉₁	2
#106×C₁₉₁	2
#498	191