Subgroups — #27135076

Order 12352.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₇₂	C₂	1	1	yes	maximal, characteristic
2	#20259835	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₆	—	2	1	no	—
5	Dic₇₇₂	—	2	1	no	—
6	Dic₇₇₂	C₄	1	2	yes	—
7	#15690539	—	2	1	no	—
8	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₉₃	—	2	1	no	—
10	Dic₃₈₆	—	2	2	no	—
11	Dic₃₈₆	—	4	1	no	—
12	C₂×Dic₁₉₃	—	4	1	no	—
13	#13095393	—	4	1	no	—
14	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
15	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₉₃	D₄	1	1	yes	characteristic
17	C₂²×C₁₉₃	SD₁₆	1	1	yes	characteristic
18	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₃	D₈	1	1	yes	characteristic
20	Dic₁₉₃	—	4	2	no	—
21	C₂×C₁₉₃	—	2	1	no	—
22	C₂×C₁₉₃	#102	1	1	yes	characteristic
23	C₁₉₃	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	193	1	no	maximal, Sylow
25	C₂×Q₁₆	—	193	1	no	—
26	M₃₂	D₁₉₃	1	1	yes	characteristic
27	#108	—	193	1	no	—
28	C₂×Q₈	—	386	1	no	—
29	Q₁₆	—	386	1	no	—
30	Q₁₆	—	193	2	no	—
31	M₁₆	—	386	1	no	—
32	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	Q₈	—	772	1	no	—
35	Q₈	—	386	2	no	—
36	C₂×C₄	—	772	1	no	—
37	C₂×C₄	D₇₇₂	1	1	yes	characteristic
38	C₈	—	772	1	no	—
39	C₈	C₄×D₁₉₃	1	1	yes	characteristic
40	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
41	C₂²	#15690544	1	1	yes	characteristic
42	C₄	—	772	2	no	—
43	C₄	#15690532	1	1	yes	characteristic
44	C₄	D₁₅₄₄	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20259821	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259821
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₃
Fitting	M₃₂×C₁₉₃	C₂
socle	C₂×C₁₉₃	#102
prime radical	C₂²×C₁₉₃	SD₁₆
cosocle	C₂×C₈×C₁₉₃	C₂²
hypercenter	M₃₂	D₁₉₃
hypocenter	C₁₉₃	#362
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
23. C₁₉₃	#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₇₂	2
#20259835	2
M₃₂×C₁₉₃	2
#362	193