Subgroups — #42119186

Order 27584.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₈₆₂	C₂	1	1	yes	maximal, characteristic
2	Dic₄₃₁⋊C₈	C₂	1	2	yes	maximal
3	#28352099	C₂	1	2	yes	maximal
4	C₄×C₄₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₂	C₄	1	1	yes	characteristic
7	C₂×C₄₃₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄₃₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₃₁	C₄	1	1	yes	characteristic
10	Dic₄₃₁⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₄₃₁	C₂²	1	2	yes	—
12	C₂×C₈×C₄₃₁	C₂²	1	2	yes	—
13	C₄²×C₄₃₁	C₂²	1	1	yes	characteristic
14	Dic₈₆₂	—	2	2	no	—
15	C₂×Dic₄₃₁	C₂×C₄	1	2	yes	—
16	C₂×Dic₄₃₁	C₂×C₄	1	2	yes	—
17	C₄₃₁⋊C₈	—	2	1	no	—
18	C₄₃₁⋊C₈	Q₈	1	2	yes	complemented
19	C₂×C₄×C₄₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
22	C₈×C₄₃₁	—	2	2	no	—
23	C₂²×C₄₃₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₃₁	—	2	1	no	—
25	C₄×C₄₃₁	M₁₆	1	2	yes	—
26	C₄×C₄₃₁	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₄₃₁	#41	1	1	yes	characteristic
28	Dic₄₃₁	—	2	2	no	—
29	Dic₄₃₁	—	2	2	no	—
30	C₂×C₄₃₁	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₄₃₁	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₄₃₁	#131	1	1	yes	characteristic
33	C₄₃₁	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	431	1	no	maximal, Sylow
35	C₄×Q₈	—	431	1	no	—
36	C₄⋊C₈	—	431	2	no	—
37	C₄⋊C₈	D₄₃₁	1	1	yes	characteristic
38	#97	—	431	2	no	—
39	C₄×C₈	—	431	1	no	—
40	C₂×Q₈	—	431	1	no	—
41	C₂×C₈	—	431	1	no	—
42	C₂×C₈	—	431	1	no	—
43	C₂×C₈	C₂×D₄₃₁	1	2	yes	—
44	C₄⋊C₄	—	431	2	no	—
45	C₄⋊C₄	—	431	1	no	—
46	C₄²	—	431	2	no	—
47	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
48	Q₈	—	862	2	no	—
49	C₂×C₄	—	431	2	no	—
50	C₂×C₄	—	431	2	no	—
51	C₂×C₄	C₄×D₄₃₁	1	1	yes	characteristic
52	C₂×C₄	C₄×D₄₃₁	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	862	1	no	—
56	C₈	—	431	2	no	—
57	C₂²	C₂×C₄×D₄₃₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	862	2	no	—
60	C₄	—	862	2	no	—
61	C₄	#20293783	1	2	yes	—
62	C₄	Q₈×D₄₃₁	1	1	yes	characteristic
63	C₄	#20293799	1	1	yes	characteristic
64	C₂	#28352154	1	1	yes	characteristic
65	C₂	C₂×#20293783	1	1	yes	characteristic
66	C₂	#28352197	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₃₁
commutator, socle, prime radical	C₂²×C₄₃₁	C₂²×C₄
Fitting	C₄⋊C₈×C₄₃₁	C₂
cosocle	C₂×C₄×C₄₃₁	C₂³
hypercenter	C₄⋊C₈	D₄₃₁
hypocenter	C₄₃₁	#446
LCS	G ▷ C₂²×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. C₄₃₁⋊C₈	Q₈	semidirect
33. C₄₃₁	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₈₆₂	2
Dic₄₃₁⋊C₈	2
#28352099	2
C₄×C₄₃₁⋊C₈	2
C₄⋊C₈×C₄₃₁	2
#446	431