Subgroups — #27136155

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×#15690539	C₂	1	1	yes	maximal, characteristic
3	#15690558	—	2	2	no	—
4	#15690539	—	2	2	no	—
5	C₂×#13095393	C₄	1	1	yes	characteristic
6	C₂×#13095393	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₁₉₃	C₄	1	1	yes	characteristic
8	#13095393	—	2	1	no	—
9	#13095393	D₄	1	2	yes	—
10	C₂³×C₁₉₃	C₈	1	1	yes	characteristic
11	C₂×C₄×C₁₉₃	—	2	1	no	—
12	C₂×C₄×C₁₉₃	—	2	1	no	—
13	C₂×C₄×C₁₉₃	C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₉₃	—	2	1	no	—
16	C₂²×C₁₉₃	—	4	1	no	—
17	C₂²×C₁₉₃	C₂×C₈	1	1	yes	characteristic
18	C₄×C₁₉₃	—	2	1	no	—
19	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
21	C₂×C₁₉₃	—	2	1	no	—
22	C₂×C₁₉₃	—	4	1	no	—
23	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
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	#11715871	1	1	yes	characteristic
29	M₁₆	—	386	2	no	—
30	C₂×C₈	—	193	1	no	—
31	C₂×C₈	—	193	1	no	—
32	C₁₆	—	386	2	no	—
33	C₂³	#13095400	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	#13095400	1	1	yes	characteristic
37	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
38	C₈	—	386	1	no	—
39	C₈	—	193	2	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×#13095400	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₂²⋊#11715871	1	1	yes	characteristic
45	C₄	#15690564	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂²⋊#13095400	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11715871
commutator	C₂²×C₁₉₃	C₂×C₈
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₂²×C₄×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₂²⋊C₈
prime radical	C₂³×C₁₉₃	C₈
cosocle	C₂×#13095393	C₂²
hypercenter	C₂²×C₄	#11715871
hypocenter	C₁₉₃	#349
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄

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×#15690539	2
#349	193