Subgroups — #27135040

Order 12352.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690538	C₂	1	1	yes	maximal, characteristic
2	#20259839	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₁₉₃	C₄	1	1	yes	characteristic
5	#15690538	—	2	1	no	—
6	#15690538	—	2	1	no	—
7	C₂×#13095393	C₄	1	1	yes	characteristic
8	#15690530	—	2	1	no	—
9	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₉₃	—	2	1	no	—
11	C₂²×D₁₉₃	C₈	1	1	yes	characteristic
12	C₂×Dic₁₉₃	C₈	1	1	yes	characteristic
13	C₄×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	C₂×C₈	1	1	yes	characteristic
20	C₂×D₁₉₃	—	2	1	no	—
21	C₂×D₁₉₃	—	4	1	no	—
22	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
24	Dic₁₉₃	—	2	1	no	—
25	C₂×C₁₉₃	—	2	1	no	—
26	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
27	D₁₉₃	—	4	1	no	—
28	C₁₉₃	#349	1	1	yes	characteristic, complemented, Sylow
29	#349	—	193	1	no	maximal, Sylow
30	C₂×M₁₆	—	193	1	no	—
31	M₃₂	—	193	1	no	—
32	M₃₂	D₁₉₃	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	193	1	no	—
34	M₁₆	—	386	1	no	—
35	M₁₆	—	386	1	no	—
36	C₂×C₈	—	193	1	no	—
37	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	—	386	1	no	—
40	C₂³	—	193	1	no	—
41	C₂×C₄	—	386	1	no	—
42	C₂×C₄	—	386	1	no	—
43	C₂×C₄	—	193	1	no	—
44	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
45	C₈	—	386	1	no	—
46	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
47	C₈	D₇₇₂	1	1	yes	characteristic
48	C₂²	—	772	1	no	—
49	C₂²	—	386	1	no	—
50	C₂²	C₈×D₁₉₃	1	1	yes	characteristic
51	C₄	—	386	1	no	—
52	C₄	#15690532	1	1	yes	characteristic
53	C₄	#15690538	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	772	1	no	—
56	C₂	#20259811	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690532
commutator	C₂²×C₁₉₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₉₃
Fitting	M₃₂×C₁₉₃	C₂
socle	C₂×C₁₉₃	C₂²⋊C₈
prime radical	C₂²×D₁₉₃	C₈
cosocle	C₂×C₈×C₁₉₃	C₂²
hypercenter	M₃₂	D₁₉₃
hypocenter	C₁₉₃	#349
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20259839	C₂	semidirect
3. M₃₂×C₁₉₃	C₂	semidirect
28. C₁₉₃	#349	semidirect
32. M₃₂	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690538	2
#20259839	2
M₃₂×C₁₉₃	2
#349	193