Subgroups — #27135044

Order 12352.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259905	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690530	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690555	C₄	1	1	yes	characteristic
5	#15690539	C₄	1	1	yes	characteristic
6	#15690538	—	2	1	no	—
7	C₈×D₁₉₃	—	2	1	no	—
8	#15690530	—	2	1	no	—
9	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₉₃	—	2	1	no	—
11	Dic₃₈₆	C₈	1	1	yes	characteristic
12	D₇₇₂	C₈	1	1	yes	characteristic
13	C₂²⋊D₁₉₃	—	2	1	no	—
14	C₄×D₁₉₃	—	2	1	no	—
15	#13095393	—	2	1	no	—
16	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₉₃	D₄	1	1	yes	characteristic
18	C₈×C₁₉₃	D₄	1	1	yes	characteristic
19	C₂²×C₁₉₃	M₁₆	1	1	yes	characteristic
20	C₂×D₁₉₃	—	2	1	no	—
21	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
23	Dic₁₉₃	—	2	1	no	—
24	C₂×C₁₉₃	—	2	1	no	—
25	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
26	D₁₉₃	—	4	1	no	—
27	C₁₉₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	193	1	no	maximal, Sylow
29	#131	—	193	1	no	—
30	M₃₂	D₁₉₃	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	193	1	no	—
32	#41	—	193	1	no	—
33	M₁₆	—	386	1	no	—
34	M₁₆	—	193	1	no	—
35	C₂×C₈	—	386	1	no	—
36	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	386	1	no	—
39	Q₈	—	193	1	no	—
40	D₄	—	386	1	no	—
41	D₄	—	193	1	no	—
42	C₂×C₄	—	386	1	no	—
43	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
44	C₈	—	386	1	no	—
45	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
46	C₈	D₇₇₂	1	1	yes	characteristic
47	C₂²	—	386	1	no	—
48	C₂²	#15690538	1	1	yes	characteristic
49	C₄	—	386	1	no	—
50	C₄	#15690532	1	1	yes	characteristic
51	C₄	C₈×D₁₉₃	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	772	1	no	—
54	C₂	#20259811	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₁₉₃
commutator	C₄×C₁₉₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₉₃
Fitting	M₃₂×C₁₉₃	C₂
socle	C₂×C₁₉₃	C₂²⋊C₈
prime radical	#15690555	C₄
cosocle	C₂×C₈×C₁₉₃	C₂²
hypercenter	M₃₂	D₁₉₃
hypocenter	C₁₉₃	#350
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#15690530	C₂	semidirect
3. M₃₂×C₁₉₃	C₂	semidirect
27. C₁₉₃	#350	semidirect
30. M₃₂	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259905	2
C₂×#15690530	2
M₃₂×C₁₉₃	2
#350	193