Subgroups — #27136116

Order 12352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259954	C₂	1	1	yes	maximal, characteristic
2	#20259954	C₂	1	1	yes	maximal, characteristic
3	#20259890	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095400	—	2	1	no	—
5	C₂×#13095400	—	2	1	no	—
6	C₄⋊Dic₁₉₃	—	2	1	no	—
7	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic
8	#15690534	—	2	1	no	—
9	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic
11	#13095400	—	2	2	no	—
12	#13095400	—	2	2	no	—
13	C₂×Dic₁₉₃	—	2	1	no	—
14	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
15	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₃	—	2	1	no	—
17	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₃	—	2	1	no	—
20	C₄×C₁₉₃	—	4	1	no	—
21	Dic₁₉₃	—	2	1	no	—
22	Dic₁₉₃	—	2	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	—	193	1	no	—
32	C₂×C₈	—	386	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	#11715871	1	1	yes	characteristic
38	C₄²	—	193	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	386	1	no	—
41	C₂×C₄	—	193	1	no	—
42	C₂×C₄	—	193	1	no	—
43	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
44	C₈	—	386	2	no	—
45	C₈	—	386	2	no	—
46	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	—	772	1	no	—
50	C₄	—	386	1	no	—
51	C₄	—	386	1	no	—
52	C₂	#20259960	1	1	yes	characteristic
53	C₂	#20259963	1	1	yes	characteristic
54	C₂	#20259964	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715871
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₄⋊C₄×C₁₉₃	C₄
socle, prime radical	C₂²×C₁₉₃	C₂²⋊C₄
cosocle	C₄×Dic₁₉₃	C₂²
hypercenter	C₄⋊C₄	#11715871
hypocenter	C₁₉₃	#330
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

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
#20259954	2
#20259890	2
#330	193