Subgroups — #27136127

Order 12352.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259978	C₂	1	2	yes	maximal
2	C₂×#15690538	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715871	—	2	2	no	—
4	C₂²⋊#11715871	—	2	2	no	—
5	C₄×#11715871	C₄	1	4	yes	complemented
6	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
7	#15690538	—	2	2	no	—
8	C₂×#13095393	C₄	1	1	yes	characteristic, complemented
9	C₂×C₈×C₁₉₃	C₄	1	1	yes	characteristic, complemented
10	C₂×#11715871	—	2	4	no	—
11	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
12	C₂×Dic₁₉₃	Q₈	1	1	yes	characteristic
13	C₄×D₁₉₃	C₂×C₄	1	2	yes	—
14	C₄×D₁₉₃	D₄	1	2	yes	—
15	#13095393	—	2	1	no	—
16	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₉₃	—	2	1	no	—
18	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₁₉₃	—	4	1	no	—
20	C₂×D₁₉₃	C₂²⋊C₄	1	2	yes	—
21	#11715871	—	4	4	no	—
22	C₄×C₁₉₃	C₄²	1	1	yes	characteristic
23	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₁₉₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₁₉₃	—	2	1	no	—
26	C₂×C₁₉₃	#95	1	1	yes	characteristic
27	D₁₉₃	—	2	2	no	—
28	C₁₉₃	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
29	H(3,ℤ/4)	—	193	1	no	maximal, Sylow
30	C₂×M₁₆	—	193	1	no	—
31	#117	—	193	2	no	—
32	C₂²×C₄	—	193	1	no	—
33	M₁₆	—	386	2	no	—
34	C₂×C₈	—	193	1	no	—
35	C₂×C₈	#11715871	1	1	yes	characteristic, complemented
36	C₄⋊C₄	—	386	2	no	—
37	C₂²⋊C₄	—	386	2	no	—
38	C₄²	—	193	4	no	—
39	C₂³	—	193	1	no	—
40	C₂×C₄	—	386	4	no	—
41	C₂×C₄	—	193	1	no	—
42	C₂×C₄	—	193	2	no	—
43	C₂×C₄	—	193	2	no	—
44	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	386	1	no	—
47	C₂²	—	772	1	no	—
48	C₂²	—	193	2	no	—
49	C₂²	C₄⋊#11715871	1	1	yes	characteristic
50	C₄	—	772	4	no	—
51	C₄	—	193	2	no	—
52	C₄	C₄×#11715871	1	1	yes	characteristic
53	C₄	C₂²⋊#11715871	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	386	2	no	—
56	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11715871
commutator	C₄×C₁₉₃	C₄²
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₂×C₈×C₁₉₃	C₄
socle	C₂×C₁₉₃	#95
prime radical	C₂²×D₁₉₃	D₄
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	C₂×C₈	#11715871
hypocenter	C₁₉₃	H(3,ℤ/4)
LCS	G ▷ C₄×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#11715871	C₄	semidirect
8. C₂×#13095393	C₄	semidirect
9. C₂×C₈×C₁₉₃	C₄	semidirect
28. C₁₉₃	H(3,ℤ/4)	semidirect
35. C₂×C₈	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259978	2
C₂×#15690538	2
H(3,ℤ/4)	193