Subgroups — #27134989

Order 12352.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₈₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095393	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₆	C₄	1	1	yes	characteristic
5	C₂×#13095393	—	2	1	no	—
6	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic
7	#15690534	—	2	1	no	—
8	C₄×Dic₁₉₃	—	2	1	no	—
9	C₂×C₈×C₁₉₃	—	2	1	no	—
10	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
11	Dic₃₈₆	—	2	1	no	—
12	Dic₃₈₆	C₈	1	2	yes	complemented
13	C₂×Dic₁₉₃	—	2	1	no	—
14	C₂×Dic₁₉₃	—	2	1	no	—
15	#13095393	—	2	2	no	—
16	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
19	C₈×C₁₉₃	—	4	1	no	—
20	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₃	—	2	1	no	—
22	C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
24	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
25	C₄×C₁₉₃	Q₁₆	1	1	yes	characteristic
26	Dic₁₉₃	—	2	2	no	—
27	Dic₁₉₃	—	4	1	no	—
28	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₉₃	#103	1	1	yes	characteristic
30	C₂×C₁₉₃	#104	1	1	yes	characteristic
31	C₁₉₃	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	193	1	no	maximal, Sylow
33	C₄×Q₈	—	193	1	no	—
34	C₄⋊C₈	D₁₉₃	1	1	yes	characteristic
35	C₄×C₈	—	193	1	no	—
36	C₂×Q₈	—	193	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	386	1	no	—
39	C₄⋊C₄	—	386	1	no	—
40	C₄⋊C₄	—	193	1	no	—
41	C₄²	—	386	1	no	—
42	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
43	Q₈	—	386	1	no	—
44	Q₈	—	193	2	no	—
45	C₂×C₄	—	386	1	no	—
46	C₂×C₄	—	386	1	no	—
47	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
49	C₂×C₄	D₇₇₂	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	386	2	no	—
52	C₂²	#15690532	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	772	1	no	—
55	C₄	—	386	2	no	—
56	C₄	C₈×D₁₉₃	1	1	yes	characteristic
57	C₄	#15690538	1	1	yes	characteristic
58	C₄	#15690543	1	1	yes	characteristic
59	C₄	#15690545	1	1	yes	characteristic
60	C₂	#20259811	1	1	yes	characteristic
61	C₂	#20259822	1	1	yes	characteristic
62	C₂	#20259825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₃
commutator	C₄×C₁₉₃	C₂×C₈
Frattini	C₄²	C₂×D₁₉₃
Fitting	C₄⋊C₈×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂²⋊C₄
cosocle	C₄²×C₁₉₃	C₂²
hypercenter	C₄⋊C₈	D₁₉₃
hypocenter	C₁₉₃	Q₈⋊C₈
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₃₈₆	C₈	semidirect
31. C₁₉₃	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₈₆	2
C₄×#13095393	2
C₄⋊C₈×C₁₉₃	2
Q₈⋊C₈	193