Subgroups — #14581171

Order 2624.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050211	C₂	1	6	yes	maximal, complemented
2	#13050209	C₂	1	9	yes	maximal, complemented
3	C₂×#10620308	C₂	1	9	yes	maximal, complemented
4	#41×D₄₁	C₂	1	6	yes	maximal, complemented
5	H(5,𝔽₂)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁	C₂²	1	6	yes	complemented
7	C₂×D₄×C₄₁	C₂²	1	9	yes	complemented
8	C₂×C₂²⋊D₄₁	C₂²	1	18	yes	complemented
9	C₂²×Dic₄₁	C₂²	1	6	yes	complemented
10	#10620310	C₂²	1	2	yes	complemented
11	Q₈×D₄₁	C₂²	1	6	yes	complemented
12	#10620308	C₂²	1	18	yes	complemented
13	#10620308	C₂²	1	36	yes	complemented
14	D₄×D₄₁	C₂²	1	18	yes	complemented
15	#10620306	C₂²	1	18	yes	complemented
16	C₂×C₄×D₄₁	C₂²	1	9	yes	complemented
17	C₂×Dic₈₂	C₂²	1	9	yes	complemented
18	C₂³×C₄₁	C₂³	1	6	yes	—
19	C₂²×D₄₁	C₂³	1	9	yes	—
20	Q₈×C₄₁	C₂³	1	2	yes	—
21	D₄×C₄₁	C₂³	1	18	yes	—
22	C₂×C₄×C₄₁	C₂³	1	9	yes	—
23	C₂²⋊D₄₁	C₂³	1	36	yes	—
24	C₂×Dic₄₁	C₂³	1	18	yes	—
25	C₂×Dic₄₁	C₂³	1	9	yes	—
26	D₁₆₄	C₂³	1	6	yes	—
27	C₄×D₄₁	C₂³	1	6	yes	—
28	C₄×D₄₁	C₂³	1	18	yes	—
29	Dic₈₂	C₂³	1	18	yes	—
30	C₂²×C₄₁	—	4	6	no	—
31	C₂²×C₄₁	C₂⁴	1	9	yes	—
32	C₂×D₄₁	—	4	9	no	—
33	C₂×D₄₁	C₂⁴	1	6	yes	—
34	C₄×C₄₁	C₂⁴	1	6	yes	—
35	Dic₄₁	C₂⁴	1	9	yes	—
36	Dic₄₁	C₂⁴	1	1	yes	characteristic
37	C₂×C₄₁	—	2	9	no	—
38	C₂×C₄₁	C₂⁵	1	1	yes	characteristic
39	D₄₁	—	2	6	no	—
40	#585	—	41	1	no	maximal, Sylow
41	C₄₁	#585	1	1	yes	characteristic, complemented, Sylow
42	2₋¹⁺⁴	—	41	6	no	—
43	H(5,𝔽₂)	—	41	9	no	—
44	H(5,𝔽₂)	D₄₁	1	1	yes	characteristic, complemented
45	C₂×#41	—	41	6	no	—
46	C₂×#41	—	41	9	no	—
47	#41	—	41	2	no	—
48	#41	—	41	18	no	—
49	#41	—	41	36	no	—
50	#41	—	41	18	no	—
51	#41	C₂×D₄₁	1	6	yes	complemented
52	C₂×Q₈	—	41	6	no	—
53	C₂×Q₈	—	41	9	no	—
54	C₂×D₄	—	41	18	no	—
55	C₂×D₄	—	41	18	no	—
56	C₂×D₄	C₂×D₄₁	1	9	yes	complemented
57	C₂²×C₄	—	41	6	no	—
58	C₂²×C₄	—	41	9	no	—
59	C₂³	—	41	9	no	—
60	C₂³	C₂²×D₄₁	1	6	yes	—
61	Q₈	—	41	18	no	—
62	Q₈	C₂²×D₄₁	1	2	yes	—
63	D₄	—	41	36	no	—
64	D₄	—	41	6	no	—
65	D₄	C₂²×D₄₁	1	18	yes	—
66	C₂×C₄	—	41	9	no	—
67	C₂×C₄	—	41	18	no	—
68	C₂×C₄	—	41	6	no	—
69	C₂×C₄	—	41	18	no	—
70	C₂×C₄	C₂²×D₄₁	1	9	yes	—
71	C₂²	—	4	6	no	—
72	C₂²	—	41	6	no	—
73	C₂²	—	164	9	no	—
74	C₂²	C₂³×D₄₁	1	9	yes	—
75	C₄	—	41	1	no	—
76	C₄	—	41	9	no	—
77	C₄	C₂³×D₄₁	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	82	6	no	—
80	C₂	C₂⁴×D₄₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₄₁
commutator, socle, cosocle	C₂×C₄₁	C₂⁵
Fitting	H(5,𝔽₂)×C₄₁	C₂
hypercenter	H(5,𝔽₂)	D₄₁
hypocenter	C₄₁	#585
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #13050211	C₂	semidirect
2. #13050209	C₂	semidirect
3. C₂×#10620308	C₂	semidirect
4. #41×D₄₁	C₂	semidirect
5. H(5,𝔽₂)×C₄₁	C₂	semidirect
6. #41×C₄₁	C₂²	semidirect
7. C₂×D₄×C₄₁	C₂²	semidirect
8. C₂×C₂²⋊D₄₁	C₂²	semidirect
9. C₂²×Dic₄₁	C₂²	semidirect
10. #10620310	C₂²	semidirect
11. Q₈×D₄₁	C₂²	semidirect
12. #10620308	C₂²	semidirect
13. #10620308	C₂²	semidirect
14. D₄×D₄₁	C₂²	semidirect
15. #10620306	C₂²	semidirect
16. C₂×C₄×D₄₁	C₂²	semidirect
17. C₂×Dic₈₂	C₂²	semidirect
41. C₄₁	#585	semidirect
44. H(5,𝔽₂)	D₄₁	semidirect
51. #41	C₂×D₄₁	semidirect
56. C₂×D₄	C₂×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050211	2
#13050209	2
C₂×#10620308	2
#41×D₄₁	2
H(5,𝔽₂)×C₄₁	2
#585	41