Subgroups — #27136156

Order 12352.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259973	C₂	1	2	yes	maximal, complemented
2	C₂×#15690538	C₂	1	1	yes	maximal, characteristic
3	#15690558	—	2	2	no	—
4	C₂×C₄×D₁₉₃	C₄	1	1	yes	characteristic
5	#15690538	—	2	2	no	—
6	C₂×#13095393	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₉₃	C₄	1	1	yes	characteristic
8	C₂²×D₁₉₃	C₈	1	1	yes	characteristic
9	C₂×Dic₁₉₃	C₈	1	1	yes	characteristic
10	C₄×D₁₉₃	—	2	1	no	—
11	C₄×D₁₉₃	—	2	1	no	—
12	#13095393	D₄	1	2	yes	—
13	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
14	C₈×C₁₉₃	—	2	1	no	—
15	C₂²×C₁₉₃	C₂×C₈	1	1	yes	characteristic
16	C₂×D₁₉₃	—	2	1	no	—
17	C₂×D₁₉₃	—	4	1	no	—
18	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
20	Dic₁₉₃	—	2	1	no	—
21	C₂×C₁₉₃	—	2	1	no	—
22	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
23	D₁₉₃	—	4	1	no	—
24	C₁₉₃	#349	1	1	yes	characteristic, complemented, Sylow
25	#349	—	193	1	no	maximal, Sylow
26	C₂×M₁₆	—	193	1	no	—
27	M₃₂	—	193	2	no	—
28	C₂²×C₄	—	193	1	no	—
29	M₁₆	—	386	2	no	—
30	C₂×C₈	—	193	1	no	—
31	C₂×C₈	#11715871	1	1	yes	characteristic
32	C₁₆	—	386	2	no	—
33	C₂³	—	193	1	no	—
34	C₂×C₄	—	386	1	no	—
35	C₂×C₄	—	386	1	no	—
36	C₂×C₄	—	193	1	no	—
37	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	—	193	2	no	—
40	C₂²	—	772	1	no	—
41	C₂²	—	386	1	no	—
42	C₂²	#15690563	1	1	yes	characteristic
43	C₄	—	386	1	no	—
44	C₄	C₂²⋊#11715871	1	1	yes	characteristic
45	C₄	#15690565	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	772	1	no	—
48	C₂	#20259956	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11715871
commutator	C₂²×C₁₉₃	C₂×C₈
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₂×C₈×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₂²⋊C₈
prime radical	C₂²×D₁₉₃	C₈
cosocle	C₂×#13095393	C₂²
hypercenter	C₂×C₈	#11715871
hypocenter	C₁₉₃	#349
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #20259973	C₂	semidirect
24. C₁₉₃	#349	semidirect

Maximal subgroups

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

subgroup	index
#20259973	2
C₂×#15690538	2
#349	193