Subgroups — #27135530

Order 12352.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259927	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259905	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259815	C₂	1	2	yes	maximal, complemented
5	#108×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690555	C₂²	1	1	yes	characteristic
7	C₂×D₇₇₂	C₂²	1	2	yes	—
8	#15690544	—	2	2	no	—
9	D₁₅₄₄	—	2	2	no	—
10	D₁₅₄₄	—	2	1	no	—
11	D₁₅₄₄	—	2	1	no	—
12	#15690539	C₂²	1	1	yes	characteristic
13	#15690538	—	2	1	no	—
14	C₈×D₁₉₃	—	2	1	no	—
15	M₁₆×C₁₉₃	C₂²	1	2	yes	—
16	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₉₃	—	2	2	no	—
18	Dic₃₈₆	D₄	1	1	yes	characteristic
19	D₇₇₂	—	2	2	no	—
20	D₇₇₂	—	2	2	no	—
21	D₇₇₂	D₄	1	1	yes	characteristic
22	C₂²⋊D₁₉₃	—	2	1	no	—
23	C₄×D₁₉₃	—	2	1	no	—
24	#13095393	—	2	1	no	—
25	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
26	C₈×C₁₉₃	—	2	2	no	—
27	C₈×C₁₉₃	D₄	1	1	yes	characteristic
28	C₈×C₁₉₃	D₄	1	1	yes	characteristic
29	C₂²×C₁₉₃	#41	1	1	yes	characteristic
30	C₂×D₁₉₃	—	2	1	no	—
31	C₂×D₁₉₃	—	4	2	no	—
32	C₂×D₁₉₃	—	8	2	no	—
33	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
35	Dic₁₉₃	—	2	1	no	—
36	C₂×C₁₉₃	—	2	1	no	—
37	C₂×C₁₉₃	#121	1	1	yes	characteristic
38	D₁₉₃	—	4	1	no	—
39	D₁₉₃	—	8	2	no	—
40	C₁₉₃	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	193	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	193	2	no	—
43	C₂×D₈	—	193	1	no	—
44	#131	—	193	1	no	—
45	#108	D₁₉₃	1	1	yes	characteristic, complemented
46	#100	—	193	2	no	—
47	#41	—	193	1	no	—
48	C₂×D₄	—	193	2	no	—
49	SD₁₆	—	386	2	no	—
50	D₈	—	386	2	no	—
51	D₈	—	386	1	no	—
52	D₈	—	386	1	no	—
53	M₁₆	—	386	1	no	—
54	M₁₆	—	193	1	no	—
55	M₁₆	C₂×D₁₉₃	1	2	yes	—
56	C₂×C₈	—	386	1	no	—
57	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
58	C₂³	—	386	2	no	—
59	Q₈	—	193	1	no	—
60	D₄	—	386	1	no	—
61	D₄	—	386	2	no	—
62	D₄	—	386	2	no	—
63	D₄	—	193	1	no	—
64	C₂×C₄	—	386	1	no	—
65	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	386	1	no	—
68	C₈	D₇₇₂	1	1	yes	characteristic
69	C₈	D₇₇₂	1	1	yes	characteristic
70	C₂²	—	772	2	no	—
71	C₂²	—	1544	2	no	—
72	C₂²	—	386	1	no	—
73	C₂²	#15690556	1	1	yes	characteristic
74	C₄	—	386	1	no	—
75	C₄	D₄×D₁₉₃	1	1	yes	characteristic
76	C₄	C₂×D₇₇₂	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	772	1	no	—
79	C₂	—	1544	2	no	—
80	C₂	#20259874	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259874
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#108×C₁₉₃	C₂
socle	C₂×C₁₉₃	#121
hypercenter	#108	D₁₉₃
hypocenter	C₁₉₃	#472
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #20259927	C₂	semidirect
2. C₂×D₁₅₄₄	C₂	semidirect
3. #20259905	C₂	semidirect
4. #20259815	C₂	semidirect
5. #108×C₁₉₃	C₂	semidirect
40. C₁₉₃	#472	semidirect
45. #108	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259927	2
C₂×D₁₅₄₄	2
#20259905	2
#20259815	2
#108×C₁₉₃	2
#472	193