Subgroups — #42119339

Order 27584.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28352189	C₂	1	1	yes	maximal, characteristic, complemented
2	#28352122	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₄₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#28352110	C₂	1	2	yes	maximal
5	C₄×D₄×C₄₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₂	—	2	1	no	—
7	C₂×C₄₃₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₄₃₁	—	2	1	no	—
9	C₄⋊Dic₄₃₁	C₂²	1	2	yes	—
10	C₂×D₄×C₄₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₃₁	—	2	1	no	—
12	C₄⋊C₄×C₄₃₁	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₄₃₁	—	2	1	no	—
14	C₄²×C₄₃₁	C₂²	1	1	yes	characteristic
15	Dic₈₆₂	—	4	2	no	—
16	C₂×Dic₄₃₁	—	2	2	no	—
17	C₄₃₁⋊C₈	—	2	2	no	—
18	C₂³×C₄₃₁	—	2	1	no	—
19	D₄×C₄₃₁	—	2	1	no	—
20	D₄×C₄₃₁	Q₈	1	2	yes	complemented
21	C₂×C₄×C₄₃₁	—	2	1	no	—
22	C₂×C₄×C₄₃₁	—	4	2	no	—
23	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄₃₁	—	2	2	no	—
27	C₂²×C₄₃₁	—	4	1	no	—
28	C₂²×C₄₃₁	—	4	1	no	—
29	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₃₁	—	2	1	no	—
31	C₄×C₄₃₁	—	4	1	no	—
32	C₄×C₄₃₁	D₈	1	2	yes	—
33	C₄×C₄₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₄₃₁	#41	1	1	yes	characteristic
35	Dic₄₃₁	—	4	2	no	—
36	C₂×C₄₃₁	—	4	2	no	—
37	C₂×C₄₃₁	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₄₃₁	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄₃₁	#137	1	1	yes	characteristic
40	C₄₃₁	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	431	1	no	maximal, Sylow
42	C₄⋊Q₈	—	431	1	no	—
43	C₄×D₄	D₄₃₁	1	1	yes	characteristic
44	#107	—	431	2	no	—
45	C₄⋊C₈	—	431	1	no	—
46	#102	—	431	2	no	—
47	C₂×Q₈	—	862	1	no	—
48	C₂×D₄	C₂×D₄₃₁	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	—	431	2	no	—
51	C₄⋊C₄	—	862	1	no	—
52	C₄⋊C₄	—	431	2	no	—
53	C₄⋊C₄	C₂×D₄₃₁	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	1724	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Dic₈₆₂	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	862	2	no	—
63	C₂×C₄	C₂²⋊D₄₃₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₄₃₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
66	C₈	—	862	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₄₃₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	1724	2	no	—
74	C₄	#20293785	1	2	yes	—
75	C₄	C₂×Dic₈₆₂	1	1	yes	characteristic
76	C₄	#20293800	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#28352168	1	1	yes	characteristic
79	C₂	C₂×#20293785	1	1	yes	characteristic
80	C₂	#28352219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃₁
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	C₄×D₄×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
prime radical	C₂×D₄×C₄₃₁	C₂²
hypercenter	C₄×D₄	D₄₃₁
hypocenter	C₄₃₁	#474
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #28352189	C₂	semidirect
2. #28352122	C₂	semidirect
3. C₄⋊(C₄₃₁⋊C₈)	C₂	semidirect
20. D₄×C₄₃₁	Q₈	semidirect
40. C₄₃₁	#474	semidirect
59. D₄	Dic₈₆₂	semidirect

Maximal subgroups

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

subgroup	index
#28352189	2
#28352122	2
C₄⋊(C₄₃₁⋊C₈)	2
#28352110	2
C₄×D₄×C₄₃₁	2
#474	431