Subgroups — #27136380

Order 12352.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259996	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259995	C₂	1	2	yes	maximal, complemented
3	#20259994	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15690567	C₂	1	2	yes	maximal, complemented
5	#20259986	C₂	1	1	yes	maximal, characteristic
6	#15690567	C₂²	1	1	yes	characteristic
7	#15690567	C₂²	1	2	yes	—
8	#15690567	C₂²	1	1	yes	characteristic
9	#15690565	C₄	1	1	yes	characteristic
10	#15690564	C₄	1	2	yes	—
11	#15690563	C₂²	1	1	yes	characteristic
12	C₂×#13095400	C₂²	1	2	yes	—
13	#15690554	C₄	1	1	yes	characteristic
14	#13095400	C₂×C₄	1	2	yes	—
15	#13095400	C₂×C₄	1	1	yes	characteristic
16	#13095400	C₂³	1	1	yes	characteristic
17	Dic₃₈₆	C₈	1	1	yes	characteristic
18	C₂²⋊D₁₉₃	C₈	1	2	yes	—
19	C₂×Dic₁₉₃	C₂×C₄	1	2	yes	—
20	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
21	D₄×C₁₉₃	C₈	1	1	yes	characteristic
22	C₂²×C₁₉₃	C₂×C₈	1	2	yes	—
23	C₂×D₁₉₃	C₂×C₈	1	1	yes	characteristic
24	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
25	Dic₁₉₃	C₂×C₈	1	2	yes	—
26	Dic₁₉₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₁₉₃	—	2	2	no	—
28	C₂×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
29	D₁₉₃	—	2	1	no	—
30	C₁₉₃	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	193	1	no	maximal, Sylow
32	#131	—	193	1	no	—
33	M₃₂	—	193	2	no	—
34	M₃₂	—	193	1	no	—
35	C₂×C₁₆	—	193	2	no	—
36	C₂×C₁₆	—	193	1	no	—
37	#41	—	193	1	no	—
38	M₁₆	—	193	2	no	—
39	M₁₆	—	193	1	no	—
40	C₂×C₈	—	193	2	no	—
41	C₂×C₈	—	193	1	no	—
42	C₁₆	—	193	1	no	—
43	C₁₆	—	193	2	no	—
44	C₁₆	—	193	1	no	—
45	Q₈	—	193	1	no	—
46	D₄	—	193	2	no	—
47	D₄	#13095402	1	1	yes	characteristic
48	C₂×C₄	—	193	2	no	—
49	C₂×C₄	—	193	1	no	—
50	C₈	—	193	1	no	—
51	C₈	—	193	2	no	—
52	C₈	—	193	1	no	—
53	C₂²	—	193	1	no	—
54	C₂²	C₂×#13095402	1	2	yes	—
55	C₄	—	193	1	no	—
56	C₄	—	193	2	no	—
57	C₄	C₂×#13095402	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	386	1	no	—
60	C₂	C₂²×#13095402	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#13095402
commutator, socle	C₂×C₁₉₃	C₂²×C₈
Fitting	D₄×C₁₉₃	C₈
prime radical	#15690554	C₄
cosocle	#13095400	C₂³
hypercenter	D₄	#13095402
hypocenter	C₁₉₃	#504
LCS	G ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #20259996	C₂	semidirect
2. #20259995	C₂	semidirect
3. #20259994	C₂	semidirect
4. C₂×#15690567	C₂	semidirect
30. C₁₉₃	#504	semidirect

Maximal subgroups

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

subgroup	index
#20259996	2
#20259995	2
#20259994	2
C₂×#15690567	2
#20259986	2
#504	193