Subgroups — #27135618

Order 12352.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259899	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259823	C₂	1	2	yes	maximal, complemented
3	#20259820	C₂	1	2	yes	maximal
4	#20259810	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₆	—	2	1	no	—
7	C₂×#13095393	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₃	—	2	1	no	—
9	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
10	C₂×Q₈×C₁₉₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
12	C₂²⋊C₄×C₁₉₃	—	2	1	no	—
13	C₂²⋊C₄×C₁₉₃	—	2	1	no	—
14	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
15	Dic₃₈₆	—	4	2	no	—
16	C₂×Dic₁₉₃	—	2	2	no	—
17	#13095393	—	2	2	no	—
18	C₂³×C₁₉₃	—	2	1	no	—
19	Q₈×C₁₉₃	—	2	2	no	—
20	D₄×C₁₉₃	—	2	2	no	—
21	C₂×C₄×C₁₉₃	—	2	1	no	—
22	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₃	—	4	1	no	—
26	C₂²×C₁₉₃	—	4	1	no	—
27	C₂²×C₁₉₃	—	4	1	no	—
28	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₃	—	2	1	no	—
30	C₄×C₁₉₃	—	2	1	no	—
31	C₄×C₁₉₃	—	4	1	no	—
32	C₄×C₁₉₃	#41	1	2	yes	—
33	Dic₁₉₃	—	4	2	no	—
34	C₂×C₁₉₃	—	8	1	no	—
35	C₂×C₁₉₃	#124	1	1	yes	characteristic
36	C₂×C₁₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₂×C₁₉₃	#137	1	1	yes	characteristic
38	C₁₉₃	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	193	1	no	maximal, Sylow
40	C₄⋊Q₈	—	193	1	no	—
41	#124	D₁₉₃	1	1	yes	characteristic
42	#103	—	193	2	no	—
43	#102	—	193	2	no	—
44	#97	—	193	1	no	—
45	C₂×Q₈	—	386	1	no	—
46	C₂×Q₈	C₂×D₁₉₃	1	1	yes	characteristic
47	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
48	C₂×C₈	—	193	2	no	—
49	C₄⋊C₄	—	386	1	no	—
50	C₄⋊C₄	—	193	2	no	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	772	2	no	—
57	D₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	386	2	no	—
60	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
63	C₈	—	386	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	772	2	no	—
72	C₄	#15690554	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	#20259886	1	1	yes	characteristic
75	C₂	#20259924	1	1	yes	characteristic
76	C₂	#20259929	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#124×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₄×C₁₉₃	C₂²
hypercenter	#124	D₁₉₃
hypocenter	C₁₉₃	#489
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20259899	C₂	semidirect
2. #20259823	C₂	semidirect
4. #20259810	C₂	semidirect
38. C₁₉₃	#489	semidirect

Maximal subgroups

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

subgroup	index
#20259899	2
#20259823	2
#20259820	2
#20259810	2
#124×C₁₉₃	2
#489	193