Subgroups — #27136111

Order 12352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259956	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259874	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715871	—	2	1	no	—
5	C₂×#13095400	—	2	1	no	—
6	C₂×D₇₇₂	—	2	1	no	—
7	C₂×D₇₇₂	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
9	#15690532	—	2	1	no	—
10	C₄⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11715871	—	2	2	no	—
12	C₂×#11715871	—	4	1	no	—
13	C₂×#11715871	—	4	1	no	—
14	#13095400	—	4	1	no	—
15	C₂²×D₁₉₃	—	2	1	no	—
16	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
17	D₇₇₂	—	4	1	no	—
18	D₇₇₂	—	4	1	no	—
19	D₇₇₂	—	4	1	no	—
20	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
21	C₄×D₁₉₃	—	4	1	no	—
22	C₂×C₄×C₁₉₃	—	2	1	no	—
23	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₁₉₃	—	2	1	no	—
26	C₂×D₁₉₃	—	2	1	no	—
27	C₂×D₁₉₃	—	4	1	no	—
28	C₂×D₁₉₃	—	4	1	no	—
29	C₂×D₁₉₃	—	4	1	no	—
30	C₂×D₁₉₃	D₈	1	1	yes	characteristic
31	C₂×D₁₉₃	SD₁₆	1	1	yes	characteristic
32	#11715871	—	4	2	no	—
33	C₄×C₁₉₃	—	2	1	no	—
34	C₄×C₁₉₃	—	4	1	no	—
35	Dic₁₉₃	—	2	1	no	—
36	C₂×C₁₉₃	#99	1	1	yes	characteristic
37	C₂×C₁₉₃	#102	1	1	yes	characteristic
38	C₂×C₁₉₃	#104	1	1	yes	characteristic
39	D₁₉₃	—	2	1	no	—
40	D₁₉₃	—	2	1	no	—
41	D₁₉₃	—	8	1	no	—
42	C₁₉₃	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	193	1	no	maximal, Sylow
44	#121	—	193	1	no	—
45	C₂²⋊C₈	—	193	1	no	—
46	#95	—	193	1	no	—
47	C₂×D₄	—	386	1	no	—
48	C₂×D₄	—	193	1	no	—
49	C₂²×C₄	—	386	1	no	—
50	C₂²×C₄	—	193	1	no	—
51	C₂×C₈	—	386	1	no	—
52	C₄⋊C₄	#11715871	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	386	1	no	—
54	C₂³	—	386	1	no	—
55	C₂³	—	193	1	no	—
56	D₄	—	772	1	no	—
57	D₄	—	772	1	no	—
58	D₄	—	772	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	772	1	no	—
61	C₂×C₄	—	772	1	no	—
62	C₂×C₄	—	772	1	no	—
63	C₂×C₄	—	386	2	no	—
64	C₂×C₄	—	193	1	no	—
65	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
66	C₈	—	772	1	no	—
67	C₂²	—	772	1	no	—
68	C₂²	—	772	1	no	—
69	C₂²	—	772	1	no	—
70	C₂²	—	386	1	no	—
71	C₂²	—	386	1	no	—
72	C₂²	—	193	1	no	—
73	C₂²	—	193	1	no	—
74	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	772	2	no	—
78	C₄	—	386	1	no	—
79	C₂	—	1544	1	no	—
80	C₂	—	386	1	no	—
81	C₂	—	386	1	no	—
82	C₂	#20259958	1	1	yes	characteristic
83	C₂	#20259961	1	1	yes	characteristic
84	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	C₂²×C₁₉₃	C₂²⋊C₄
prime radical	#20259874	C₂
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	C₄⋊C₄	#11715871
hypocenter	C₁₉₃	#327
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
1. #20259956	C₂	semidirect
2. (C₂×#11715871)⋊C₄	C₂	semidirect
7. C₂×D₇₇₂	C₄	semidirect
10. C₄⋊C₄×C₁₉₃	C₄	semidirect
42. C₁₉₃	#327	semidirect
52. C₄⋊C₄	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259956	2
(C₂×#11715871)⋊C₄	2
#20259874	2
#327	193