Subgroups — #27133066

Order 12224.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259322	C₂	1	1	yes	maximal, characteristic
2	#20259255	C₂	1	4	yes	maximal
3	C₄×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₂	—	2	1	no	—
6	C₂×C₁₉₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₁	—	2	1	no	—
8	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₁	—	2	1	no	—
10	C₄⋊C₄×C₁₉₁	—	2	1	no	—
11	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
12	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	Dic₃₈₂	—	4	2	no	—
14	C₂×Dic₁₉₁	—	2	2	no	—
15	C₁₉₁⋊C₈	Q₈	1	4	yes	complemented
16	Q₈×C₁₉₁	—	4	2	no	—
17	C₂×C₄×C₁₉₁	—	2	2	no	—
18	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₉₁	—	4	2	no	—
23	C₄×C₁₉₁	D₈	1	2	yes	—
24	C₄×C₁₉₁	Q₁₆	1	2	yes	—
25	C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
26	Dic₁₉₁	—	4	2	no	—
27	C₂×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
29	C₂×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₉₁	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	191	1	no	maximal, Sylow
32	C₄⋊Q₈	—	191	1	no	—
33	C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic
34	#107	—	191	4	no	—
35	C₄×C₈	—	191	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	382	1	no	—
38	C₂×C₈	—	191	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	382	1	no	—
41	C₄⋊C₄	—	191	2	no	—
42	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
43	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	764	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	382	2	no	—
48	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
51	C₈	—	191	4	no	—
52	C₂²	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	764	2	no	—
55	C₄	#15690272	1	2	yes	—
56	C₄	#15690277	1	2	yes	—
57	C₄	Q₈×D₁₉₁	1	2	yes	—
58	C₂	#20259320	1	1	yes	characteristic
59	C₂	C₂×#15690272	1	1	yes	characteristic
60	C₂	C₂×#15690277	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₄⋊Q₈×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#500
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
15. C₁₉₁⋊C₈	Q₈	semidirect
30. C₁₉₁	#500	semidirect

Maximal subgroups

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

subgroup	index
#20259322	2
#20259255	2
C₄×C₁₉₁⋊C₈	2
C₄⋊Q₈×C₁₉₁	2
#500	191