Subgroups — #27132948

Order 12224.

96 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	#20259253	C₂	1	2	yes	maximal
3	C₄⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#20259246	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	—	2	1	no	—
7	C₂×C₁₉₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₁	—	2	1	no	—
9	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
10	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₁₉₁	—	2	1	no	—
12	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₁	—	2	1	no	—
14	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	Dic₃₈₂	—	4	2	no	—
16	C₂×Dic₁₉₁	—	2	2	no	—
17	C₁₉₁⋊C₈	—	2	2	no	—
18	Q₈×C₁₉₁	—	2	1	no	—
19	Q₈×C₁₉₁	Q₈	1	2	yes	complemented
20	C₂×C₄×C₁₉₁	—	2	1	no	—
21	C₂×C₄×C₁₉₁	—	2	1	no	—
22	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₉₁	—	2	2	no	—
27	C₄×C₁₉₁	—	2	1	no	—
28	C₄×C₁₉₁	—	4	1	no	—
29	C₄×C₁₉₁	SD₁₆	1	2	yes	—
30	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₁₉₁	#41	1	1	yes	characteristic
32	Dic₁₉₁	—	4	2	no	—
33	C₂×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₉₁	#137	1	1	yes	characteristic
36	C₁₉₁	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	191	1	no	maximal, Sylow
38	C₄⋊Q₈	—	191	1	no	—
39	C₄×Q₈	D₁₉₁	1	1	yes	characteristic
40	#106	—	191	2	no	—
41	C₄⋊C₈	—	191	1	no	—
42	#103	—	191	2	no	—
43	C₂×Q₈	—	382	1	no	—
44	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
45	C₂×C₈	—	191	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	382	1	no	—
48	C₄⋊C₄	—	191	2	no	—
49	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	764	2	no	—
54	Q₈	Dic₃₈₂	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	382	2	no	—
58	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
61	C₈	—	382	2	no	—
62	C₂²	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	764	2	no	—
67	C₄	#15690274	1	2	yes	—
68	C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
69	C₄	#15690287	1	1	yes	characteristic
70	C₂	#20259301	1	1	yes	characteristic
71	C₂	C₂×#15690274	1	1	yes	characteristic
72	C₂	#20259352	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	#475
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
19. Q₈×C₁₉₁	Q₈	semidirect
36. C₁₉₁	#475	semidirect
54. Q₈	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
#20259322	2
#20259253	2
C₄⋊(C₁₉₁⋊C₈)	2
#20259246	2
C₄×Q₈×C₁₉₁	2
#475	191