Subgroups — #14580401

Order 2624.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₁	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₆₄	C₂	1	1	yes	maximal, characteristic
3	#13050076	C₂	1	2	yes	maximal, complemented
4	C₄×#66491	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₁	C₂²	1	2	yes	—
8	C₂×C₈×C₄₁	C₂²	1	2	yes	—
9	C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	#10620281	C₄	1	2	yes	—
11	C₄⋊Dic₄₁	C₄	1	1	yes	characteristic
12	C₂×#66491	C₂²	1	1	yes	characteristic
13	C₂×#66491	C₂²	1	1	yes	characteristic
14	C₈×D₄₁	—	2	4	no	—
15	C₂²×D₄₁	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₁	C₂×C₄	1	2	yes	—
20	D₁₆₄	C₈	1	4	yes	complemented
21	C₄×D₄₁	—	2	2	no	—
22	C₄×D₄₁	—	2	2	no	—
23	C₈×C₄₁	—	2	2	no	—
24	#66491	—	2	1	no	—
25	#66491	D₄	1	2	yes	complemented
26	C₂²×C₄₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₄₁	—	2	2	no	—
28	C₂×D₄₁	—	2	2	no	—
29	C₂×D₄₁	C₂×C₈	1	4	yes	complemented
30	C₄×C₄₁	—	2	1	no	—
31	C₄×C₄₁	C₂×C₈	1	2	yes	complemented
32	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₁	#41	1	1	yes	characteristic
34	Dic₄₁	—	2	2	no	—
35	C₂×C₄₁	C₄×D₄	1	1	yes	characteristic
36	C₂×C₄₁	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₄₁	#131	1	1	yes	characteristic
38	D₄₁	—	2	4	no	—
39	C₈×D₄	—	41	1	no	maximal, Sylow
40	C₄₁	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	—	41	2	no	—
42	C₄×D₄	—	41	1	no	—
43	C₄⋊C₈	D₄₁	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	41	2	no	—
45	C₄×C₈	—	41	1	no	—
46	C₂×D₄	—	41	1	no	—
47	C₂²×C₄	—	41	2	no	—
48	C₂×C₈	—	41	1	no	—
49	C₂×C₈	—	41	1	no	—
50	C₂×C₈	—	82	4	no	—
51	C₂×C₈	C₂×D₄₁	1	2	yes	—
52	C₄⋊C₄	—	41	1	no	—
53	C₂²⋊C₄	—	41	2	no	—
54	C₄²	C₂×D₄₁	1	1	yes	characteristic
55	C₂³	—	41	2	no	—
56	D₄	—	41	4	no	—
57	C₂×C₄	—	41	2	no	—
58	C₂×C₄	—	82	2	no	—
59	C₂×C₄	—	82	2	no	—
60	C₂×C₄	C₄×D₄₁	1	1	yes	characteristic
61	C₂×C₄	C₄×D₄₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	41	2	no	—
65	C₈	—	82	1	no	—
66	C₂²	—	41	4	no	—
67	C₂²	—	82	2	no	—
68	C₂²	—	82	2	no	—
69	C₂²	C₂×C₄×D₄₁	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	82	2	no	—
72	C₄	C₈×D₄₁	1	2	yes	complemented
73	C₄	D₄×D₄₁	1	1	yes	characteristic
74	C₄	#10620310	1	1	yes	characteristic
75	C₂	—	82	4	no	—
76	C₂	#13050126	1	1	yes	characteristic
77	C₂	C₂×C₈×D₄₁	1	1	yes	characteristic
78	C₂	#13050172	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₁
commutator	C₂×C₄₁	C₂²×C₈
Fitting	C₄⋊C₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂²×C₄
prime radical	C₂×D₁₆₄	C₄
cosocle	C₂×C₄×C₄₁	C₂³
hypercenter	C₄⋊C₈	D₄₁
hypocenter	C₄₁	C₈×D₄
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄₁	C₂	semidirect
3. #13050076	C₂	semidirect
4. C₄×#66491	C₂	semidirect
5. C₄⋊C₈×C₄₁	C₂	semidirect
20. D₁₆₄	C₈	semidirect
25. #66491	D₄	semidirect
29. C₂×D₄₁	C₂×C₈	semidirect
31. C₄×C₄₁	C₂×C₈	semidirect
40. C₄₁	C₈×D₄	semidirect
43. C₄⋊C₈	D₄₁	semidirect
72. C₄	C₈×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄₁	2
C₄×D₁₆₄	2
#13050076	2
C₄×#66491	2
C₄⋊C₈×C₄₁	2
C₈×D₄	41