Subgroups — #14580819

Order 2624.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₄₁	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₄₁	C₂	1	6	yes	maximal, complemented
3	#13050123	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊Dic₄₁	C₂	1	3	yes	maximal, complemented
5	C₂×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
6	#41×C₄₁	C₄	1	8	yes	complemented
7	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₄₁	C₂²	1	3	yes	—
9	C₂²×C₄×C₄₁	C₂²	1	3	yes	—
10	C₂²×Dic₄₁	C₂²	1	6	yes	—
11	C₂²⋊Dic₄₁	C₂²	1	6	yes	—
12	C₄⋊Dic₄₁	C₂²	1	3	yes	—
13	C₄⋊Dic₄₁	C₂²	1	6	yes	—
14	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
15	C₄×Dic₄₁	C₂²	1	6	yes	—
16	C₂³×C₄₁	C₂³	1	3	yes	—
17	Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
18	D₄×C₄₁	C₂×C₄	1	12	yes	complemented
19	C₂×C₄×C₄₁	C₂×C₄	1	12	yes	complemented
20	C₂×C₄×C₄₁	C₂³	1	3	yes	—
21	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₄₁	—	4	6	no	—
23	C₂×Dic₄₁	C₂³	1	2	yes	—
24	C₂×Dic₄₁	C₂³	1	6	yes	—
25	C₂²×C₄₁	—	2	3	no	—
26	C₂²×C₄₁	—	2	3	no	—
27	C₂²×C₄₁	C₂²×C₄	1	6	yes	—
28	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
29	C₄×C₄₁	C₂²×C₄	1	2	yes	—
30	C₄×C₄₁	C₂²×C₄	1	6	yes	—
31	Dic₄₁	—	2	6	no	—
32	Dic₄₁	—	2	2	no	—
33	C₂×C₄₁	—	2	6	no	—
34	C₂×C₄₁	C₂³×C₄	1	1	yes	characteristic
35	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
36	C₂×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
37	#520	—	41	1	no	maximal, Sylow
38	C₄₁	#520	1	1	yes	characteristic, complemented, Sylow
39	C₂×#41	D₄₁	1	1	yes	characteristic
40	C₄×Q₈	—	41	2	no	—
41	C₄×D₄	—	41	6	no	—
42	#117	—	41	3	no	—
43	C₂×C₄⋊C₄	—	41	3	no	—
44	#41	Dic₄₁	1	8	yes	complemented
45	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
46	C₂×D₄	C₂×D₄₁	1	3	yes	—
47	C₂²×C₄	—	41	6	no	—
48	C₂²×C₄	C₂×D₄₁	1	3	yes	—
49	C₄⋊C₄	—	41	3	no	—
50	C₄⋊C₄	—	41	6	no	—
51	C₄⋊C₄	—	41	1	no	—
52	C₂²⋊C₄	—	41	6	no	—
53	C₄²	—	41	6	no	—
54	C₂³	C₂²×D₄₁	1	3	yes	—
55	Q₈	C₂×Dic₄₁	1	4	yes	complemented
56	D₄	C₂×Dic₄₁	1	12	yes	complemented
57	C₂×C₄	—	41	2	no	—
58	C₂×C₄	—	41	6	no	—
59	C₂×C₄	—	164	6	no	—
60	C₂×C₄	C₂×Dic₄₁	1	12	yes	complemented
61	C₂×C₄	C₂²×D₄₁	1	3	yes	—
62	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×Dic₄₁	1	6	yes	—
66	C₂²	C₂³×D₄₁	1	1	yes	characteristic
67	C₄	—	82	6	no	—
68	C₄	—	82	2	no	—
69	C₄	C₂²×Dic₄₁	1	2	yes	—
70	C₄	C₂²×Dic₄₁	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	C₂³×Dic₄₁	1	1	yes	characteristic
73	C₂	#13050210	1	1	yes	characteristic
74	C₂	#13050211	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator	C₂×C₄₁	C₂³×C₄
Fitting, prime radical	C₂×#41×C₄₁	C₂
socle, cosocle	C₂²×C₄₁	C₂⁴
hypercenter	C₂×#41	D₄₁
hypocenter	C₄₁	#520
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₄₁	C₂	semidirect
2. D₄×Dic₄₁	C₂	semidirect
3. #13050123	C₂	semidirect
4. C₂×C₄⋊Dic₄₁	C₂	semidirect
6. #41×C₄₁	C₄	semidirect
17. Q₈×C₄₁	C₂×C₄	semidirect
18. D₄×C₄₁	C₂×C₄	semidirect
19. C₂×C₄×C₄₁	C₂×C₄	semidirect
38. C₄₁	#520	semidirect
44. #41	Dic₄₁	semidirect
55. Q₈	C₂×Dic₄₁	semidirect
56. D₄	C₂×Dic₄₁	semidirect
60. C₂×C₄	C₂×Dic₄₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₄₁	2
D₄×Dic₄₁	2
#13050123	2
C₂×C₄⋊Dic₄₁	2
C₂×#41×C₄₁	2
#520	41