Subgroups — #27135009

Order 12352.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095393	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095393	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095393	—	2	1	no	—
5	C₂×#13095393	C₄	1	1	yes	characteristic
6	C₂×#13095393	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₉₃	—	2	1	no	—
8	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
9	#13095393	—	2	1	no	—
10	#13095393	—	4	1	no	—
11	#13095393	C₈	1	2	yes	complemented
12	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉₃	Q₈	1	1	yes	characteristic
15	C₈×C₁₉₃	—	4	1	no	—
16	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₁₉₃	—	2	1	no	—
18	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
20	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
22	C₂×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₁₉₃	#106	1	1	yes	characteristic
24	C₂×C₁₉₃	#108	1	1	yes	characteristic
25	C₁₉₃	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	193	1	no	maximal, Sylow
27	C₄⋊C₈	—	193	1	no	—
28	C₄⋊C₈	D₁₉₃	1	1	yes	characteristic
29	C₄×C₈	—	193	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	386	1	no	—
32	C₂×C₈	—	193	1	no	—
33	C₂×C₈	—	193	1	no	—
34	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
35	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
37	C₂×C₄	Dic₃₈₆	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	772	1	no	—
40	C₈	—	386	1	no	—
41	C₈	—	193	2	no	—
42	C₂²	#15690534	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₁₉₃	1	1	yes	characteristic
45	C₄	#15690538	1	1	yes	characteristic
46	C₄	#15690542	1	1	yes	characteristic
47	C₄	#15690543	1	1	yes	characteristic
48	C₂	#20259828	1	1	yes	characteristic
49	C₂	#20259830	1	1	yes	characteristic
50	C₂	#20259834	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₃
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
11. #13095393	C₈	semidirect
25. C₁₉₃	#334	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×#13095393	2
C₄⋊C₈×C₁₉₃	2
#334	193