Subgroups — #27135616

Order 12352.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259887	C₂	1	1	yes	maximal, characteristic
2	#20259822	C₂	1	2	yes	maximal, complemented
3	#20259819	C₂	1	2	yes	maximal
4	C₄×#13095393	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₆	C₂²	1	1	yes	characteristic
7	C₂×D₇₇₂	C₂²	1	1	yes	characteristic
8	C₂×#13095393	C₂²	1	2	yes	—
9	#15690532	—	2	2	no	—
10	C₄⋊C₄×C₁₉₃	—	2	2	no	—
11	C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	—
12	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₂²×D₁₉₃	—	2	1	no	—
14	Dic₃₈₆	—	2	2	no	—
15	D₇₇₂	—	2	2	no	—
16	C₂×Dic₁₉₃	—	2	1	no	—
17	#13095393	—	2	2	no	—
18	C₂×C₄×C₁₉₃	—	2	2	no	—
19	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
20	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
22	C₂×D₁₉₃	—	4	1	no	—
23	C₂×D₁₉₃	—	4	2	no	—
24	C₄×C₁₉₃	—	2	2	no	—
25	C₄×C₁₉₃	—	4	2	no	—
26	C₄×C₁₉₃	#41	1	2	yes	—
27	Dic₁₉₃	—	4	1	no	—
28	C₂×C₁₉₃	#124	1	1	yes	characteristic
29	C₂×C₁₉₃	#135	1	2	yes	—
30	D₁₉₃	—	8	1	no	—
31	C₁₉₃	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	193	1	no	maximal, Sylow
33	#125	D₁₉₃	1	1	yes	characteristic, complemented
34	#124	—	193	1	no	—
35	#103	—	193	2	no	—
36	#102	—	193	2	no	—
37	C₄×C₈	—	193	1	no	—
38	C₂×Q₈	—	193	1	no	—
39	C₂×D₄	—	193	1	no	—
40	C₂×C₈	—	193	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₁₉₃	1	2	yes	—
43	C₂²⋊C₄	—	386	2	no	—
44	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
45	C₂³	—	386	1	no	—
46	Q₈	—	386	2	no	—
47	D₄	—	386	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	386	1	no	—
50	C₂×C₄	C₂²⋊D₁₉₃	1	2	yes	—
51	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
52	C₈	—	386	2	no	—
53	C₂²	—	772	1	no	—
54	C₂²	—	772	2	no	—
55	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	772	1	no	—
59	C₄	#15690556	1	2	yes	—
60	C₂	—	1544	1	no	—
61	C₂	#20259888	1	1	yes	characteristic
62	C₂	#20259919	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#125×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₇₇₂	C₂²
hypercenter	#125	D₁₉₃
hypocenter	C₁₉₃	#488
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #20259822	C₂	semidirect
4. C₄×#13095393	C₂	semidirect
5. #125×C₁₉₃	C₂	semidirect
31. C₁₉₃	#488	semidirect
33. #125	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259887	2
#20259822	2
#20259819	2
C₄×#13095393	2
#125×C₁₉₃	2
#488	193