Subgroups — #27136171

Order 12352.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259972	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259968	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11715871	—	2	1	no	—
5	#15690558	—	2	1	no	—
6	D₄×D₁₉₃	—	2	1	no	—
7	D₁₅₄₄	C₄	1	1	yes	characteristic, complemented
8	#15690540	—	2	1	no	—
9	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
10	D₈×C₁₉₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11715871	—	4	1	no	—
12	C₂²×D₁₉₃	—	4	1	no	—
13	D₇₇₂	—	2	1	no	—
14	C₂²⋊D₁₉₃	—	4	1	no	—
15	C₄×D₁₉₃	D₄	1	1	yes	characteristic
16	#13095393	D₄	1	1	yes	characteristic
17	D₄×C₁₉₃	—	2	1	no	—
18	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₉₃	—	4	1	no	—
20	C₂×D₁₉₃	—	4	1	no	—
21	C₂×D₁₉₃	—	8	1	no	—
22	C₂×D₁₉₃	—	8	1	no	—
23	C₂×D₁₉₃	D₈	1	1	yes	characteristic
24	#11715871	—	8	1	no	—
25	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₁₉₃	SD₁₆	1	1	yes	characteristic
27	C₂×C₁₉₃	—	8	1	no	—
28	C₂×C₁₉₃	#102	1	1	yes	characteristic
29	D₁₉₃	—	8	1	no	—
30	D₁₉₃	D₁₆	1	1	yes	characteristic
31	D₁₉₃	SD₃₂	1	1	yes	characteristic
32	C₁₉₃	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	193	1	no	maximal, Sylow
34	C₂×D₈	—	193	1	no	—
35	C₂×C₁₆	—	193	1	no	—
36	#107	—	193	1	no	—
37	C₂×D₄	—	386	1	no	—
38	D₈	—	386	1	no	—
39	D₈	—	193	1	no	—
40	D₈	#11715871	1	1	yes	characteristic, complemented
41	C₂×C₈	—	193	1	no	—
42	C₄⋊C₄	—	386	1	no	—
43	C₁₆	—	386	1	no	—
44	C₂³	—	772	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	772	1	no	—
47	D₄	—	386	1	no	—
48	C₂×C₄	—	772	1	no	—
49	C₂×C₄	—	193	1	no	—
50	C₈	—	193	1	no	—
51	C₈	C₂×#11715871	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	772	1	no	—
54	C₂²	—	1544	1	no	—
55	C₂²	—	1544	1	no	—
56	C₂²	—	193	1	no	—
57	C₄	—	1544	1	no	—
58	C₄	—	193	1	no	—
59	C₄	C₂²⋊#11715871	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	1544	1	no	—
62	C₂	—	193	1	no	—
63	C₂	—	193	1	no	—
64	C₂	#20259961	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259961
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₈	C₂×#11715871
Fitting	D₈×C₁₉₃	C₄
socle	C₂×C₁₉₃	#102
prime radical	D₈×D₁₉₃	C₂
cosocle	C₈×D₁₉₃	C₂²
hypercenter	D₈	#11715871
hypocenter	C₁₉₃	#357
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #20259972	C₂	semidirect
2. #20259968	C₂	semidirect
7. D₁₅₄₄	C₄	semidirect
10. D₈×C₁₉₃	C₄	semidirect
32. C₁₉₃	#357	semidirect
40. D₈	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259972	2
#20259968	2
D₈×D₁₉₃	2
#357	193