Subgroups — #27135662

Order 12352.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259899	C₂	1	1	yes	maximal, characteristic
2	#20259832	C₂	1	4	yes	maximal
3	C₄×#13095393	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₆	—	2	1	no	—
6	C₂×#13095393	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₃	—	2	1	no	—
8	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₃	—	2	1	no	—
10	C₄⋊C₄×C₁₉₃	—	2	1	no	—
11	C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	—
12	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
13	Dic₃₈₆	—	4	2	no	—
14	C₂×Dic₁₉₃	—	2	2	no	—
15	#13095393	Q₈	1	4	yes	complemented
16	Q₈×C₁₉₃	—	4	2	no	—
17	C₂×C₄×C₁₉₃	—	2	2	no	—
18	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₉₃	—	4	2	no	—
23	C₄×C₁₉₃	D₈	1	2	yes	—
24	C₄×C₁₉₃	Q₁₆	1	2	yes	—
25	C₄×C₁₉₃	C₂×Q₈	1	2	yes	—
26	Dic₁₉₃	—	4	2	no	—
27	C₂×C₁₉₃	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₉₃	C₂×D₈	1	1	yes	characteristic
29	C₂×C₁₉₃	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₉₃	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	193	1	no	maximal, Sylow
32	C₄⋊Q₈	—	193	1	no	—
33	C₄⋊Q₈	D₁₉₃	1	1	yes	characteristic
34	#107	—	193	4	no	—
35	C₄×C₈	—	193	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	386	1	no	—
38	C₂×C₈	—	193	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	386	1	no	—
41	C₄⋊C₄	—	193	2	no	—
42	C₄⋊C₄	C₂×D₁₉₃	1	2	yes	—
43	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	772	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	386	2	no	—
48	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
51	C₈	—	193	4	no	—
52	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	772	2	no	—
55	C₄	#15690540	1	2	yes	—
56	C₄	#15690545	1	2	yes	—
57	C₄	Q₈×D₁₉₃	1	2	yes	—
58	C₂	#20259897	1	1	yes	characteristic
59	C₂	C₂×#15690540	1	1	yes	characteristic
60	C₂	C₂×#15690545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	C₄⋊Q₈×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₉₃
hypocenter	C₁₉₃	#500
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
15. #13095393	Q₈	semidirect
30. C₁₉₃	#500	semidirect

Maximal subgroups

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

subgroup	index
#20259899	2
#20259832	2
C₄×#13095393	2
C₄⋊Q₈×C₁₉₃	2
#500	193