Subgroups — #27136345

Order 12352.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13095402	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11715871	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095402	—	2	2	no	—
4	C₂²×#11715871	C₄	1	1	yes	characteristic
5	C₂²×#11715871	C₂²	1	1	yes	characteristic
6	C₄⋊#11715871	—	2	2	no	—
7	C₂×C₄×D₁₉₃	C₄	1	1	yes	characteristic
8	#13095402	—	4	2	no	—
9	C₂×#11715871	—	2	1	no	—
10	C₂×#11715871	—	4	1	no	—
11	C₂×#11715871	—	4	1	no	—
12	C₂×#11715871	D₄	1	2	yes	—
13	C₂²×D₁₉₃	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₁₉₃	C₈	1	1	yes	characteristic, complemented
15	C₄×D₁₉₃	—	2	1	no	—
16	C₄×D₁₉₃	—	2	1	no	—
17	C₂×C₄×C₁₉₃	C₈	1	1	yes	characteristic, complemented
18	C₂²×C₁₉₃	C₂×C₈	1	1	yes	characteristic
19	C₂×D₁₉₃	—	2	1	no	—
20	C₂×D₁₉₃	—	2	1	no	—
21	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₉₃	M₁₆	1	1	yes	characteristic
23	#11715871	—	2	2	no	—
24	#11715871	—	4	1	no	—
25	C₄×C₁₉₃	—	2	1	no	—
26	Dic₁₉₃	—	2	1	no	—
27	C₂×C₁₉₃	—	2	1	no	—
28	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
29	D₁₉₃	—	2	1	no	—
30	D₁₉₃	#99	1	1	yes	characteristic
31	D₁₉₃	#101	1	1	yes	characteristic
32	C₁₉₃	#324	1	1	yes	characteristic, complemented, Sylow
33	#324	—	193	1	no	maximal, Sylow
34	C₂×C₄⋊C₄	—	193	1	no	—
35	C₂²⋊C₈	—	193	2	no	—
36	C₂²×C₄	—	193	1	no	—
37	C₂²×C₄	—	193	1	no	—
38	C₂²×C₄	—	193	1	no	—
39	C₂×C₈	—	386	2	no	—
40	C₄⋊C₄	—	386	2	no	—
41	C₂³	—	193	1	no	—
42	C₂×C₄	—	772	1	no	—
43	C₂×C₄	—	772	1	no	—
44	C₂×C₄	—	386	1	no	—
45	C₂×C₄	—	386	1	no	—
46	C₂×C₄	—	386	1	no	—
47	C₂×C₄	—	193	2	no	—
48	C₂×C₄	—	193	1	no	—
49	C₂×C₄	#13095402	1	1	yes	characteristic, complemented
50	C₈	—	772	2	no	—
51	C₂²	—	386	1	no	—
52	C₂²	—	386	1	no	—
53	C₂²	—	193	1	no	—
54	C₂²	—	193	1	no	—
55	C₂²	C₂×#13095402	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	772	1	no	—
58	C₄	—	386	2	no	—
59	C₄	—	386	1	no	—
60	C₂	—	2	1	no	—
61	C₂	—	386	1	no	—
62	C₂	—	193	1	no	—
63	C₂	—	193	1	no	—
64	C₂	C₂²⋊#13095402	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#13095402
commutator	C₂²×C₁₉₃	C₂×C₈
Frattini	C₂²	C₂×#13095402
Fitting	C₂×C₄×C₁₉₃	C₈
socle	C₂×C₁₉₃	C₂²⋊C₈
prime radical	C₂²×D₁₉₃	C₂×C₄
cosocle	C₂²×#11715871	C₂²
hypercenter	C₂×C₄	#13095402
hypocenter	C₁₉₃	#324
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
14. C₂×Dic₁₉₃	C₈	semidirect
17. C₂×C₄×C₁₉₃	C₈	semidirect
32. C₁₉₃	#324	semidirect
49. C₂×C₄	#13095402	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13095402	2
C₂×C₄⋊#11715871	2
#324	193