Subgroups — #27136299

Order 12352.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11715871	C₂	1	2	yes	maximal
2	#20259967	C₂	1	1	yes	maximal, characteristic
3	#20259962	C₂	1	2	yes	maximal
4	#20259953	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₁₉₃	C₂	1	1	yes	maximal, characteristic
6	#15690563	C₂²	1	1	yes	characteristic
7	C₄⋊#11715871	—	2	2	no	—
8	C₄⋊#11715871	C₂²	1	2	yes	—
9	C₄×#11715871	—	2	2	no	—
10	C₄×#11715871	C₂²	1	1	yes	characteristic
11	Q₈×D₁₉₃	C₂²	1	2	yes	—
12	Dic₇₇₂	C₄	1	1	yes	characteristic, complemented
13	#15690545	C₄	1	2	yes	complemented
14	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₉₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#11715871	—	2	2	no	—
17	C₂×#11715871	D₄	1	2	yes	—
18	#13095400	—	2	1	no	—
19	Dic₃₈₆	—	2	2	no	—
20	Dic₃₈₆	C₂×C₄	1	2	yes	—
21	C₄×D₁₉₃	—	2	2	no	—
22	C₄×D₁₉₃	C₂³	1	1	yes	characteristic
23	#13095393	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₁₉₃	C₂×C₄	1	2	yes	—
25	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₉₃	C₂×D₄	1	1	yes	characteristic
27	#11715871	—	2	2	no	—
28	#11715871	—	4	2	no	—
29	C₄×C₁₉₃	—	2	2	no	—
30	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₉₃	—	2	2	no	—
32	Dic₁₉₃	#41	1	1	yes	characteristic
33	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
34	D₁₉₃	#137	1	2	yes	—
35	C₁₉₃	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	193	1	no	maximal, Sylow
37	C₂×Q₁₆	—	193	1	no	—
38	C₄×Q₈	—	193	2	no	—
39	#106	—	193	1	no	—
40	#103	—	193	2	no	—
41	#97	—	193	1	no	—
42	C₂×Q₈	—	193	2	no	—
43	Q₁₆	—	193	2	no	—
44	Q₁₆	—	193	1	no	—
45	Q₁₆	#11715871	1	1	yes	characteristic, complemented
46	C₂×C₈	—	193	1	no	—
47	C₂×C₈	—	193	1	no	—
48	C₄⋊C₄	—	386	2	no	—
49	C₄⋊C₄	—	193	2	no	—
50	C₄²	—	386	2	no	—
51	C₄²	—	193	1	no	—
52	Q₈	—	386	2	no	—
53	Q₈	—	193	2	no	—
54	Q₈	C₂×#11715871	1	2	yes	—
55	C₂×C₄	—	386	2	no	—
56	C₂×C₄	—	386	2	no	—
57	C₂×C₄	—	193	2	no	—
58	C₂×C₄	—	193	1	no	—
59	C₈	—	386	1	no	—
60	C₈	—	193	1	no	—
61	C₈	C₂×#11715871	1	1	yes	characteristic
62	C₂²	—	193	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	772	2	no	—
65	C₄	—	386	2	no	—
66	C₄	—	386	2	no	—
67	C₄	—	193	1	no	—
68	C₄	C₂²×#11715871	1	1	yes	characteristic
69	C₂	—	193	2	no	—
70	C₂	D₄×#11715871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11715871
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₄	C₂²×#11715871
Fitting	Q₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₄×D₄
prime radical	C₂×D₁₉₃	C₂×D₄
cosocle	C₄×D₁₉₃	C₂³
hypercenter	Q₁₆	#11715871
hypocenter	C₁₉₃	#441
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₇₇₂	C₄	semidirect
13. #15690545	C₄	semidirect
15. Q₁₆×C₁₉₃	C₄	semidirect
35. C₁₉₃	#441	semidirect
45. Q₁₆	#11715871	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11715871	2
#20259967	2
#20259962	2
#20259953	2
Q₁₆×D₁₉₃	2
#441	193