Subgroups — #27136112

Order 12352.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259956	C₂	1	1	yes	maximal, characteristic
2	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#20259879	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715871	—	2	1	no	—
5	C₂×#13095400	—	2	1	no	—
6	C₂×Dic₃₈₆	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
16	Dic₃₈₆	—	4	1	no	—
17	C₂×Dic₁₉₃	—	2	1	no	—
18	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
19	C₄×D₁₉₃	—	4	1	no	—
20	C₂×C₄×C₁₉₃	—	2	1	no	—
21	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₁₉₃	—	2	1	no	—
24	C₂×D₁₉₃	—	2	1	no	—
25	C₂×D₁₉₃	SD₁₆	1	1	yes	characteristic
26	C₂×D₁₉₃	Q₁₆	1	1	yes	characteristic
27	#11715871	—	4	2	no	—
28	C₄×C₁₉₃	—	2	1	no	—
29	C₄×C₁₉₃	—	4	1	no	—
30	Dic₁₉₃	—	2	1	no	—
31	Dic₁₉₃	—	4	1	no	—
32	C₂×C₁₉₃	#99	1	1	yes	characteristic
33	C₂×C₁₉₃	#103	1	1	yes	characteristic
34	C₂×C₁₉₃	#104	1	1	yes	characteristic
35	D₁₉₃	—	2	1	no	—
36	D₁₉₃	—	2	1	no	—
37	C₁₉₃	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	193	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	193	1	no	—
40	C₂²⋊C₈	—	193	1	no	—
41	#95	—	193	1	no	—
42	C₂×Q₈	—	193	1	no	—
43	C₂²×C₄	—	386	1	no	—
44	C₂²×C₄	—	193	1	no	—
45	C₂×C₈	—	386	1	no	—
46	C₄⋊C₄	—	386	1	no	—
47	C₄⋊C₄	#11715871	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	386	1	no	—
49	C₂³	—	193	1	no	—
50	Q₈	—	772	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	772	1	no	—
53	C₂×C₄	—	772	1	no	—
54	C₂×C₄	—	772	1	no	—
55	C₂×C₄	—	386	2	no	—
56	C₂×C₄	—	386	1	no	—
57	C₂×C₄	—	193	1	no	—
58	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
59	C₈	—	772	1	no	—
60	C₂²	—	386	1	no	—
61	C₂²	—	386	1	no	—
62	C₂²	—	193	1	no	—
63	C₂²	—	193	1	no	—
64	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	772	2	no	—
68	C₄	—	772	1	no	—
69	C₄	—	386	1	no	—
70	C₂	—	386	1	no	—
71	C₂	—	386	1	no	—
72	C₂	#20259958	1	1	yes	characteristic
73	C₂	#20259962	1	1	yes	characteristic
74	C₂	#20259963	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	C₂²×D₁₉₃	D₄
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	C₄⋊C₄	#11715871
hypocenter	C₁₉₃	#328
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
6. C₂×Dic₃₈₆	C₄	semidirect
10. C₄⋊C₄×C₁₉₃	C₄	semidirect
37. C₁₉₃	#328	semidirect
47. 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
#20259879	2
#328	193