Subgroups — #27135000

Order 12352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259891	C₂	1	1	yes	maximal, characteristic
2	#20259810	C₂	1	1	yes	maximal, characteristic
3	#97×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095393	—	2	1	no	—
5	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic
6	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic
7	#15690534	—	2	1	no	—
8	#15690534	—	2	1	no	—
9	C₂×C₈×C₁₉₃	—	2	1	no	—
10	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₃	—	2	1	no	—
12	C₂×Dic₁₉₃	—	2	1	no	—
13	#13095393	—	2	2	no	—
14	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
17	C₈×C₁₉₃	—	2	2	no	—
18	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₃	—	2	1	no	—
20	C₄×C₁₉₃	—	2	1	no	—
21	C₄×C₁₉₃	—	2	1	no	—
22	Dic₁₉₃	—	4	1	no	—
23	Dic₁₉₃	—	4	1	no	—
24	C₂×C₁₉₃	#101	1	1	yes	characteristic
25	C₂×C₁₉₃	#104	1	1	yes	characteristic
26	C₂×C₁₉₃	#104	1	1	yes	characteristic
27	C₁₉₃	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	193	1	no	maximal, Sylow
29	#125	—	193	1	no	—
30	#97	—	193	1	no	—
31	#97	D₁₉₃	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	386	1	no	—
34	C₄⋊C₄	—	386	1	no	—
35	C₄⋊C₄	—	386	1	no	—
36	C₄⋊C₄	—	193	1	no	—
37	C₄⋊C₄	—	193	1	no	—
38	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
39	C₂×C₄	—	386	1	no	—
40	C₂×C₄	—	386	1	no	—
41	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
43	C₂×C₄	D₇₇₂	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	386	2	no	—
46	C₂²	#15690532	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	772	1	no	—
51	C₄	—	772	1	no	—
52	C₂	#20259817	1	1	yes	characteristic
53	C₂	#20259825	1	1	yes	characteristic
54	C₂	#20259827	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690532
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₄²	C₂×D₁₉₃
Fitting	#97×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂²⋊C₄
cosocle	C₄²×C₁₉₃	C₂²
hypercenter	#97	D₁₉₃
hypocenter	C₁₉₃	#330
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
27. C₁₉₃	#330	semidirect

Maximal subgroups

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

subgroup	index
#20259891	2
#20259810	2
#97×C₁₉₃	2
#330	193