Subgroups — #27136300

Order 12352.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11715871	C₂	1	2	yes	maximal, complemented
2	#20259967	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259961	C₂	1	2	yes	maximal, complemented
4	#20259953	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715871	—	2	2	no	—
7	#15690563	C₂²	1	1	yes	characteristic
8	C₄⋊#11715871	C₂²	1	2	yes	—
9	C₂²⋊#11715871	—	2	2	no	—
10	C₄×#11715871	C₂²	1	1	yes	characteristic
11	D₄×D₁₉₃	C₂²	1	2	yes	—
12	D₁₅₄₄	C₄	1	1	yes	characteristic, complemented
13	#15690540	C₄	1	2	yes	complemented
14	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
15	D₈×C₁₉₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#11715871	—	2	2	no	—
17	C₂×#11715871	—	4	4	no	—
18	C₂×#11715871	D₄	1	2	yes	—
19	#13095400	—	2	1	no	—
20	C₂²×D₁₉₃	—	2	2	no	—
21	D₇₇₂	C₂×C₄	1	2	yes	complemented
22	C₂²⋊D₁₉₃	—	2	2	no	—
23	C₄×D₁₉₃	C₂³	1	1	yes	characteristic
24	#13095393	C₂×C₄	1	1	yes	characteristic, complemented
25	D₄×C₁₉₃	C₂×C₄	1	2	yes	complemented
26	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₁₉₃	—	2	2	no	—
28	C₂×D₁₉₃	—	2	2	no	—
29	C₂×D₁₉₃	—	4	2	no	—
30	C₂×D₁₉₃	—	4	2	no	—
31	C₂×D₁₉₃	C₂×D₄	1	1	yes	characteristic
32	#11715871	—	2	2	no	—
33	#11715871	—	4	2	no	—
34	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
35	Dic₁₉₃	#41	1	1	yes	characteristic
36	C₂×C₁₉₃	—	4	2	no	—
37	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
38	D₁₉₃	—	4	2	no	—
39	D₁₉₃	AGL(1,ℤ/8)	1	2	yes	—
40	C₁₉₃	#442	1	1	yes	characteristic, complemented, Sylow
41	#442	—	193	1	no	maximal, Sylow
42	C₂×D₈	—	193	1	no	—
43	C₄×D₄	—	193	2	no	—
44	#106	—	193	1	no	—
45	#102	—	193	2	no	—
46	#97	—	193	1	no	—
47	C₂×D₄	—	193	2	no	—
48	C₂²×C₄	—	386	2	no	—
49	D₈	—	193	2	no	—
50	D₈	—	193	1	no	—
51	D₈	#11715871	1	1	yes	characteristic, complemented
52	C₂×C₈	—	193	1	no	—
53	C₂×C₈	—	193	1	no	—
54	C₄⋊C₄	—	193	2	no	—
55	C₂²⋊C₄	—	386	2	no	—
56	C₄²	—	193	1	no	—
57	C₂³	—	386	2	no	—
58	D₄	—	386	2	no	—
59	D₄	—	193	2	no	—
60	D₄	C₂×#11715871	1	2	yes	complemented
61	C₂×C₄	—	772	4	no	—
62	C₂×C₄	—	386	2	no	—
63	C₂×C₄	—	193	2	no	—
64	C₂×C₄	—	193	1	no	—
65	C₈	—	386	1	no	—
66	C₈	—	193	1	no	—
67	C₈	C₂×#11715871	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	772	2	no	—
70	C₂²	—	772	2	no	—
71	C₂²	—	386	2	no	—
72	C₂²	—	193	1	no	—
73	C₄	—	772	2	no	—
74	C₄	—	386	2	no	—
75	C₄	—	193	1	no	—
76	C₄	C₂²×#11715871	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	772	2	no	—
79	C₂	—	193	2	no	—
80	C₂	D₄×#11715871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11715871
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₄	C₂²×#11715871
Fitting	D₈×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₄×D₄
prime radical	D₈×D₁₉₃	C₂
cosocle	C₄×D₁₉₃	C₂³
hypercenter	D₈	#11715871
hypocenter	C₁₉₃	#442
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#11715871	C₂	semidirect
2. #20259967	C₂	semidirect
3. #20259961	C₂	semidirect
4. #20259953	C₂	semidirect
12. D₁₅₄₄	C₄	semidirect
13. #15690540	C₄	semidirect
15. D₈×C₁₉₃	C₄	semidirect
21. D₇₇₂	C₂×C₄	semidirect
24. #13095393	C₂×C₄	semidirect
25. D₄×C₁₉₃	C₂×C₄	semidirect
26. C₈×C₁₉₃	C₂×C₄	semidirect
40. C₁₉₃	#442	semidirect
51. D₈	#11715871	semidirect
60. D₄	C₂×#11715871	semidirect
67. C₈	C₂×#11715871	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11715871	2
#20259967	2
#20259961	2
#20259953	2
D₈×D₁₉₃	2
#442	193