Subgroups — #27135129

Order 12352.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₉₃	C₂	1	6	yes	maximal
2	C₂×C₄²×C₁₉₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₁₉₃	C₂²	1	4	yes	—
4	C₄⋊Dic₁₉₃	—	2	12	no	—
5	C₂²×C₄×C₁₉₃	C₂²	1	3	yes	—
6	C₄²×C₁₉₃	C₄	1	4	yes	complemented
7	C₂×Dic₁₉₃	—	2	12	no	—
8	C₂³×C₁₉₃	C₂³	1	1	yes	characteristic
9	C₂×C₄×C₁₉₃	C₂×C₄	1	6	yes	—
10	C₂×C₄×C₁₉₃	D₄	1	6	yes	—
11	C₂×C₄×C₁₉₃	Q₈	1	6	yes	—
12	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
13	C₂²×C₁₉₃	C₂×D₄	1	3	yes	—
14	C₂²×C₁₉₃	C₂×Q₈	1	3	yes	—
15	C₄×C₁₉₃	C₄⋊C₄	1	12	yes	complemented
16	Dic₁₉₃	—	4	4	no	—
17	C₂×C₁₉₃	C₂×C₄⋊C₄	1	3	yes	—
18	C₂×C₁₉₃	#127	1	1	yes	characteristic
19	C₂×C₁₉₃	C₄⋊Q₈	1	3	yes	—
20	C₁₉₃	#384	1	1	yes	characteristic, complemented, Sylow
21	#384	—	193	1	no	maximal, Sylow
22	C₂×C₄⋊C₄	—	193	6	no	—
23	C₂×C₄²	D₁₉₃	1	1	yes	characteristic
24	C₂²×C₄	—	193	4	no	—
25	C₂²×C₄	C₂×D₁₉₃	1	3	yes	—
26	C₄⋊C₄	—	386	12	no	—
27	C₄²	Dic₁₉₃	1	4	yes	complemented
28	C₂³	C₂²×D₁₉₃	1	1	yes	characteristic
29	C₂×C₄	—	386	12	no	—
30	C₂×C₄	C₂×Dic₁₉₃	1	6	yes	—
31	C₂×C₄	D₇₇₂	1	6	yes	—
32	C₂×C₄	Dic₃₈₆	1	6	yes	—
33	C₂²	C₂²×Dic₁₉₃	1	1	yes	characteristic
34	C₂²	C₂×D₇₇₂	1	3	yes	—
35	C₂²	C₂×Dic₃₈₆	1	3	yes	—
36	C₄	—	772	4	no	—
37	C₄	C₄⋊Dic₁₉₃	1	12	yes	complemented
38	C₂	C₂×C₄⋊Dic₁₉₃	1	3	yes	—
39	C₂	#20259896	1	1	yes	characteristic
40	C₂	#20259899	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₃
commutator	C₂²×C₁₉₃	C₂²×C₄
Fitting	C₂×C₄²×C₁₉₃	C₂
socle, prime radical, cosocle	C₂³×C₁₉₃	C₂³
hypercenter	C₂×C₄²	D₁₉₃
hypocenter	C₁₉₃	#384
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×C₁₉₃	C₄	semidirect
15. C₄×C₁₉₃	C₄⋊C₄	semidirect
20. C₁₉₃	#384	semidirect
27. C₄²	Dic₁₉₃	semidirect
37. C₄	C₄⋊Dic₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₉₃	2
C₂×C₄²×C₁₉₃	2
#384	193