Subgroups — #27135010

Order 12352.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095393	C₂	1	2	yes	maximal
2	C₄×C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095393	—	2	2	no	—
4	C₂×C₈×C₁₉₃	C₄	1	1	yes	characteristic
5	C₂×C₈×C₁₉₃	C₄	1	1	yes	characteristic
6	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
7	#13095393	—	4	2	no	—
8	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
10	C₂×C₄×C₁₉₃	Q₈	1	1	yes	characteristic
11	C₈×C₁₉₃	—	2	1	no	—
12	C₈×C₁₉₃	C₈	1	2	yes	complemented
13	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₁₉₃	—	2	1	no	—
15	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
16	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
17	C₄×C₁₉₃	SD₁₆	1	2	yes	—
18	C₂×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₁₉₃	#106	1	1	yes	characteristic
20	C₂×C₁₉₃	#108	1	1	yes	characteristic
21	C₁₉₃	#334	1	1	yes	characteristic, complemented, Sylow
22	#334	—	193	1	no	maximal, Sylow
23	C₄⋊C₈	—	193	2	no	—
24	C₄×C₈	D₁₉₃	1	1	yes	characteristic
25	C₂×C₈	—	386	2	no	—
26	C₂×C₈	Dic₁₉₃	1	1	yes	characteristic
27	C₂×C₈	Dic₁₉₃	1	1	yes	characteristic
28	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
29	C₂×C₄	C₂×Dic₁₉₃	1	1	yes	characteristic
30	C₂×C₄	D₇₇₂	1	1	yes	characteristic
31	C₂×C₄	Dic₃₈₆	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	772	2	no	—
34	C₈	#13095393	1	2	yes	complemented
35	C₂²	C₄⋊Dic₁₉₃	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	C₂×#13095393	1	1	yes	characteristic
38	C₄	#15690539	1	1	yes	characteristic
39	C₄	#15690544	1	2	yes	—
40	C₂	C₄⋊#13095393	1	1	yes	characteristic
41	C₂	#20259831	1	1	yes	characteristic
42	C₂	#20259835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₇₂
commutator	C₄×C₁₉₃	C₂×C₈
Frattini	C₄²	C₂×D₁₉₃
Fitting	C₄×C₈×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₄⋊C₄
cosocle	C₄²×C₁₉₃	C₂²
hypercenter	C₄×C₈	D₁₉₃
hypocenter	C₁₉₃	#334
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₁₉₃	C₈	semidirect
21. C₁₉₃	#334	semidirect
34. C₈	#13095393	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095393	2
C₄×C₈×C₁₉₃	2
#334	193