Subgroups — #27136131

Order 12352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259978	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11715871	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715871	—	2	1	no	—
5	C₄⋊#11715871	C₄	1	2	yes	complemented
6	C₂²⋊#11715871	—	2	1	no	—
7	C₄×#11715871	—	2	1	no	—
8	C₄×#11715871	—	2	1	no	—
9	C₄×#11715871	C₄	1	2	yes	complemented
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	4	no	—
16	C₂×#11715871	—	2	2	no	—
17	C₂×#11715871	—	2	2	no	—
18	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
19	C₂×Dic₁₉₃	Q₈	1	1	yes	characteristic
20	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
21	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₉₃	D₄	1	1	yes	characteristic
23	C₄×D₁₉₃	D₄	1	1	yes	characteristic
24	#13095393	—	2	1	no	—
25	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
26	C₈×C₁₉₃	—	2	1	no	—
27	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₁₉₃	—	2	2	no	—
29	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
31	#11715871	—	2	4	no	—
32	#11715871	—	4	2	no	—
33	C₄×C₁₉₃	C₄²	1	1	yes	characteristic
34	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
35	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
36	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₁₉₃	—	2	1	no	—
38	C₂×C₁₉₃	#95	1	1	yes	characteristic
39	D₁₉₃	—	2	1	no	—
40	D₁₉₃	#104	1	2	yes	—
41	C₁₉₃	#339	1	1	yes	characteristic, complemented, Sylow
42	#339	—	193	1	no	maximal, Sylow
43	C₂×M₁₆	—	193	1	no	—
44	#117	—	193	1	no	—
45	C₂×C₄²	—	193	1	no	—
46	C₂²×C₄	—	386	1	no	—
47	C₂²×C₄	—	193	1	no	—
48	M₁₆	—	386	1	no	—
49	M₁₆	—	193	1	no	—
50	M₁₆	#11715871	1	1	yes	characteristic, complemented
51	C₂×C₈	—	386	1	no	—
52	C₄⋊C₄	—	193	2	no	—
53	C₂²⋊C₄	—	386	1	no	—
54	C₄²	—	386	1	no	—
55	C₄²	—	386	1	no	—
56	C₄²	—	193	2	no	—
57	C₂³	—	193	1	no	—
58	C₂×C₄	—	386	2	no	—
59	C₂×C₄	—	386	4	no	—
60	C₂×C₄	—	386	2	no	—
61	C₂×C₄	—	193	1	no	—
62	C₂×C₄	—	193	1	no	—
63	C₂×C₄	—	193	1	no	—
64	C₂×C₄	—	193	1	no	—
65	C₂×C₄	—	193	1	no	—
66	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	386	1	no	—
69	C₂²	—	386	2	no	—
70	C₂²	—	193	1	no	—
71	C₂²	—	193	1	no	—
72	C₂²	C₄⋊#11715871	1	1	yes	characteristic
73	C₄	—	772	2	no	—
74	C₄	—	386	4	no	—
75	C₄	—	193	1	no	—
76	C₄	—	193	1	no	—
77	C₄	C₄×#11715871	1	1	yes	characteristic
78	C₄	C₂²⋊#11715871	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	386	1	no	—
81	C₂	—	193	2	no	—
82	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11715871
commutator	C₄×C₁₉₃	C₄²
Frattini	C₂×C₄	C₂×#11715871
Fitting	M₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	#95
prime radical	C₂²×D₁₉₃	D₄
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	M₁₆	#11715871
hypocenter	C₁₉₃	#339
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊#11715871	C₄	semidirect
9. C₄×#11715871	C₄	semidirect
11. #15690539	C₄	semidirect
14. M₁₆×C₁₉₃	C₄	semidirect
41. C₁₉₃	#339	semidirect
50. M₁₆	#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×C₄×#11715871	2
M₁₆×D₁₉₃	2
#339	193