Subgroups — #27136174

Order 12352.

56 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	#20259970	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259922	C₂	1	1	yes	maximal, characteristic
4	#15690565	—	2	1	no	—
5	#15690558	—	2	1	no	—
6	#15690556	—	2	1	no	—
7	#15690542	—	2	1	no	—
8	D₁₅₄₄	C₄	1	1	yes	characteristic
9	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
10	Q₁₆×C₁₉₃	C₄	1	1	yes	characteristic
11	#13095400	—	4	1	no	—
12	D₇₇₂	—	2	1	no	—
13	D₇₇₂	—	4	1	no	—
14	C₄×D₁₉₃	—	4	1	no	—
15	C₄×D₁₉₃	D₄	1	1	yes	characteristic
16	#13095393	D₄	1	1	yes	characteristic
17	Q₈×C₁₉₃	—	2	1	no	—
18	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₁₉₃	—	4	1	no	—
20	C₂×D₁₉₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₉₃	—	4	1	no	—
22	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₁₉₃	D₈	1	1	yes	characteristic
24	C₂×C₁₉₃	#102	1	1	yes	characteristic
25	D₁₉₃	—	2	1	no	—
26	D₁₉₃	—	8	1	no	—
27	C₁₉₃	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	193	1	no	maximal, Sylow
29	#135	—	193	1	no	—
30	C₂×C₁₆	—	193	1	no	—
31	#108	—	193	1	no	—
32	#41	—	386	1	no	—
33	Q₁₆	#11715871	1	1	yes	characteristic
34	SD₁₆	—	386	1	no	—
35	D₈	—	193	1	no	—
36	M₁₆	—	386	1	no	—
37	C₂×C₈	—	193	1	no	—
38	C₁₆	—	386	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	772	1	no	—
41	D₄	—	386	1	no	—
42	C₂×C₄	—	772	1	no	—
43	C₂×C₄	—	193	1	no	—
44	C₈	—	772	1	no	—
45	C₈	—	193	1	no	—
46	C₈	C₂×#11715871	1	1	yes	characteristic
47	C₂²	—	772	1	no	—
48	C₂²	—	193	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	193	1	no	—
51	C₄	C₂²⋊#11715871	1	1	yes	characteristic
52	C₂	—	1544	1	no	—
53	C₂	—	386	1	no	—
54	C₂	#20259961	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259961
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₈	C₂×#11715871
Fitting	Q₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	#102
prime radical	#20259922	C₂
cosocle	C₈×D₁₉₃	C₂²
hypercenter	Q₁₆	#11715871
hypocenter	C₁₉₃	#359
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #20259972	C₂	semidirect
2. #20259970	C₂	semidirect
27. C₁₉₃	#359	semidirect

Maximal subgroups

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

subgroup	index
#20259972	2
#20259970	2
#20259922	2
#359	193