Subgroups — #27135613

Order 12352.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259891	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259823	C₂	1	2	yes	maximal, complemented
3	#20259820	C₂	1	2	yes	maximal
4	C₄×#13095393	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095393	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
8	#15690534	—	2	2	no	—
9	C₂×Q₈×C₁₉₃	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×C₁₉₃	—	2	2	no	—
12	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₃	—	2	2	no	—
14	#13095393	—	2	2	no	—
15	C₂³×C₁₉₃	—	2	1	no	—
16	Q₈×C₁₉₃	—	2	2	no	—
17	D₄×C₁₉₃	—	2	2	no	—
18	C₂×C₄×C₁₉₃	—	2	1	no	—
19	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
20	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₃	—	4	1	no	—
22	C₂²×C₁₉₃	—	4	2	no	—
23	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₃	—	2	2	no	—
25	C₄×C₁₉₃	—	4	1	no	—
26	C₄×C₁₉₃	#41	1	2	yes	—
27	Dic₁₉₃	—	4	2	no	—
28	C₂×C₁₉₃	—	8	1	no	—
29	C₂×C₁₉₃	#124	1	1	yes	characteristic
30	C₂×C₁₉₃	#135	1	2	yes	—
31	C₁₉₃	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	193	1	no	maximal, Sylow
33	#125	—	193	1	no	—
34	#124	D₁₉₃	1	1	yes	characteristic
35	#103	—	193	2	no	—
36	#102	—	193	2	no	—
37	C₄×C₈	—	193	1	no	—
38	C₂×Q₈	C₂×D₁₉₃	1	1	yes	characteristic
39	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
40	C₂×C₈	—	193	2	no	—
41	C₄⋊C₄	—	386	2	no	—
42	C₄⋊C₄	—	193	2	no	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
45	C₂³	—	2	1	no	—
46	Q₈	—	2	2	no	—
47	D₄	—	2	2	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	386	2	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	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	1	no	—
58	C₄	—	772	2	no	—
59	C₄	#15690554	1	2	yes	—
60	C₂	—	8	1	no	—
61	C₂	#20259886	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	#124×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₄×C₁₉₃	C₂²
hypercenter	#124	D₁₉₃
hypocenter	C₁₉₃	#488
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20259891	C₂	semidirect
2. #20259823	C₂	semidirect
4. C₄×#13095393	C₂	semidirect
31. C₁₉₃	#488	semidirect

Maximal subgroups

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

subgroup	index
#20259891	2
#20259823	2
#20259820	2
C₄×#13095393	2
#124×C₁₉₃	2
#488	193