Subgroups — #27135658

Order 12352.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259891	C₂	1	1	yes	maximal, characteristic
2	#20259832	C₂	1	2	yes	maximal
3	#20259830	C₂	1	2	yes	maximal
4	C₄×#13095393	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095393	C₂²	1	1	yes	characteristic
7	C₂×#13095393	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
9	#15690534	—	2	2	no	—
10	C₄⋊C₄×C₁₉₃	—	2	2	no	—
11	C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	—
12	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₃	—	2	2	no	—
14	#13095393	Q₈	1	2	yes	—
15	#13095393	Q₈	1	2	yes	—
16	C₂×C₄×C₁₉₃	—	2	2	no	—
17	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
19	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₉₃	—	2	2	no	—
21	C₄×C₁₉₃	—	4	2	no	—
22	C₄×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₉₃	—	4	2	no	—
25	C₂×C₁₉₃	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₉₃	#135	1	2	yes	—
27	C₁₉₃	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	193	1	no	maximal, Sylow
29	#125	—	193	1	no	—
30	#125	D₁₉₃	1	1	yes	characteristic
31	#107	—	193	2	no	—
32	#106	—	193	2	no	—
33	C₄×C₈	—	193	1	no	—
34	C₂×C₈	—	193	1	no	—
35	C₂×C₈	—	193	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	386	2	no	—
38	C₄⋊C₄	—	193	2	no	—
39	C₄⋊C₄	C₂×D₁₉₃	1	2	yes	—
40	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	386	2	no	—
43	C₂×C₄	C₂²⋊D₁₉₃	1	2	yes	—
44	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
45	C₈	—	193	2	no	—
46	C₈	—	193	2	no	—
47	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	772	2	no	—
51	C₄	Q₈×D₁₉₃	1	1	yes	characteristic
52	C₄	Q₈×D₁₉₃	1	1	yes	characteristic
53	C₂	#20259897	1	1	yes	characteristic
54	C₂	#20259919	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#125×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂×D₄
hypercenter	#125	D₁₉₃
hypocenter	C₁₉₃	#499
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. C₁₉₃	#499	semidirect

Maximal subgroups

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

subgroup	index
#20259891	2
#20259832	2
#20259830	2
C₄×#13095393	2
#125×C₁₉₃	2
#499	193