Subgroups — #27135677

Order 12352.

84 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	#20259838	C₂	1	2	yes	maximal, complemented
3	C₂×#15690530	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₁₉₃	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690555	C₄	1	1	yes	characteristic
7	#15690539	C₄	1	1	yes	characteristic
8	#15690538	C₄	1	2	yes	—
9	C₈×D₁₉₃	C₂²	1	2	yes	—
10	#15690530	C₂²	1	1	yes	characteristic
11	#15690530	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₁₆×C₁₉₃	C₂²	1	2	yes	—
14	Dic₃₈₆	C₈	1	1	yes	characteristic
15	D₇₇₂	C₈	1	1	yes	characteristic
16	C₂²⋊D₁₉₃	C₈	1	2	yes	—
17	C₄×D₁₉₃	C₂×C₄	1	2	yes	—
18	#13095393	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
20	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
21	C₈×C₁₉₃	C₂³	1	1	yes	characteristic
22	C₂²×C₁₉₃	C₂×C₈	1	1	yes	characteristic
23	C₂×D₁₉₃	C₂×C₈	1	2	yes	—
24	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
25	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
26	Dic₁₉₃	C₂×C₈	1	2	yes	—
27	C₂×C₁₉₃	—	2	1	no	—
28	C₂×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
29	D₁₉₃	—	2	2	no	—
30	C₁₉₃	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	193	1	no	maximal, Sylow
32	#131	—	193	1	no	—
33	M₃₂	—	193	2	no	—
34	M₃₂	D₁₉₃	1	1	yes	characteristic, complemented
35	C₂×C₁₆	—	193	1	no	—
36	C₂×C₁₆	—	193	2	no	—
37	#41	—	193	1	no	—
38	M₁₆	—	193	1	no	—
39	M₁₆	—	193	2	no	—
40	C₂×C₈	—	193	2	no	—
41	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
42	C₁₆	—	193	1	no	—
43	C₁₆	—	193	1	no	—
44	C₁₆	C₂×D₁₉₃	1	2	yes	—
45	Q₈	—	193	1	no	—
46	D₄	—	193	2	no	—
47	D₄	—	193	1	no	—
48	C₂×C₄	—	193	2	no	—
49	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
50	C₈	—	193	2	no	—
51	C₈	C₄×D₁₉₃	1	1	yes	characteristic
52	C₈	C₂²×D₁₉₃	1	1	yes	characteristic
53	C₂²	—	193	2	no	—
54	C₂²	C₈×D₁₉₃	1	1	yes	characteristic
55	C₄	—	193	2	no	—
56	C₄	C₈×D₁₉₃	1	1	yes	characteristic
57	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	386	2	no	—
60	C₂	C₂×C₈×D₁₉₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₁₉₃
commutator, socle	C₂×C₁₉₃	C₂²×C₈
Fitting	M₃₂×C₁₉₃	C₂
prime radical	#15690555	C₄
cosocle	C₈×C₁₉₃	C₂³
hypercenter	M₃₂	D₁₉₃
hypocenter	C₁₉₃	#504
LCS	G ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20259838	C₂	semidirect
3. C₂×#15690530	C₂	semidirect
4. C₁₆×D₁₉₃	C₂	semidirect
5. M₃₂×C₁₉₃	C₂	semidirect
30. C₁₉₃	#504	semidirect
34. 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
#20259838	2
C₂×#15690530	2
C₁₆×D₁₉₃	2
M₃₂×C₁₉₃	2
#504	193