Subgroups — #27136142

Order 12352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690565	C₂	1	1	yes	maximal, characteristic
2	#20259978	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₉₃	C₂	1	1	yes	maximal, characteristic
4	#15690565	—	2	1	no	—
5	#15690565	—	2	1	no	—
6	C₂×#13095400	C₄	1	2	yes	complemented
7	C₄⋊#11715871	—	2	1	no	—
8	C₂²⋊#11715871	C₄	1	2	yes	—
9	C₄×#11715871	—	2	1	no	—
10	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
11	#15690539	C₄	1	1	yes	characteristic, complemented
12	#15690538	—	2	1	no	—
13	C₈×D₁₉₃	—	2	1	no	—
14	M₁₆×C₁₉₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#11715871	—	2	2	no	—
16	#13095400	—	2	2	no	—
17	C₂²×D₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₉₃	D₄	1	1	yes	characteristic
20	C₄×D₁₉₃	D₄	1	1	yes	characteristic
21	C₄×D₁₉₃	D₄	1	1	yes	characteristic
22	C₄×D₁₉₃	Q₈	1	1	yes	characteristic
23	#13095393	—	2	1	no	—
24	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₉₃	—	2	1	no	—
26	C₂²×C₁₉₃	C₄²	1	1	yes	characteristic
27	C₂×D₁₉₃	—	4	1	no	—
28	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₉₃	C₄⋊C₄	1	1	yes	characteristic
30	#11715871	—	4	2	no	—
31	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₉₃	—	2	1	no	—
36	C₂×C₁₉₃	#95	1	1	yes	characteristic
37	D₁₉₃	—	2	1	no	—
38	D₁₉₃	—	2	1	no	—
39	C₁₉₃	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	193	1	no	maximal, Sylow
41	C₂×M₁₆	—	193	1	no	—
42	C₂×M₁₆	—	193	1	no	—
43	#117	—	193	1	no	—
44	C₂²×C₄	—	193	1	no	—
45	M₁₆	—	386	1	no	—
46	M₁₆	—	386	1	no	—
47	M₁₆	—	386	1	no	—
48	M₁₆	—	193	1	no	—
49	M₁₆	#11715871	1	1	yes	characteristic, complemented
50	C₂×C₈	—	386	1	no	—
51	C₂×C₈	—	193	2	no	—
52	C₄⋊C₄	—	386	1	no	—
53	C₂²⋊C₄	—	193	2	no	—
54	C₄²	—	386	1	no	—
55	C₂³	—	193	1	no	—
56	C₂×C₄	—	386	2	no	—
57	C₂×C₄	—	193	1	no	—
58	C₂×C₄	—	193	1	no	—
59	C₂×C₄	—	193	1	no	—
60	C₂×C₄	—	193	1	no	—
61	C₂×C₄	—	193	1	no	—
62	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	386	2	no	—
65	C₈	—	386	1	no	—
66	C₂²	—	772	1	no	—
67	C₂²	—	193	1	no	—
68	C₂²	—	193	1	no	—
69	C₂²	C₄×#11715871	1	1	yes	characteristic
70	C₄	—	772	2	no	—
71	C₄	—	193	1	no	—
72	C₄	—	193	1	no	—
73	C₄	C₂²⋊#11715871	1	1	yes	characteristic
74	C₄	C₄⋊#11715871	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	386	1	no	—
77	C₂	—	386	1	no	—
78	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#11715871)⋊C₄
commutator	C₂²×C₁₉₃	C₄²
Frattini	C₂×C₄	C₂×#11715871
Fitting	M₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	#95
prime radical	C₂²×D₁₉₃	C₂×C₄
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	M₁₆	#11715871
hypocenter	C₁₉₃	#344
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. C₂×#13095400	C₄	semidirect
11. #15690539	C₄	semidirect
14. M₁₆×C₁₉₃	C₄	semidirect
39. C₁₉₃	#344	semidirect
49. M₁₆	#11715871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690565	2
#20259978	2
M₁₆×D₁₉₃	2
#344	193