Subgroups — #14581173

Order 2624.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050212	C₂	1	5	yes	maximal, complemented
2	#13050210	C₂	1	10	yes	maximal, complemented
3	C₂×#10620310	C₂	1	5	yes	maximal, complemented
4	#41×D₄₁	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁	C₂²	1	10	yes	complemented
7	C₂×Q₈×C₄₁	C₂²	1	5	yes	complemented
8	#10620310	C₂²	1	10	yes	complemented
9	#10620310	C₂²	1	20	yes	complemented
10	Q₈×D₄₁	C₂²	1	10	yes	complemented
11	#10620308	C₂²	1	10	yes	complemented
12	D₄×D₄₁	C₂²	1	30	yes	complemented
13	#10620306	C₂²	1	30	yes	complemented
14	C₂×D₁₆₄	C₂²	1	15	yes	complemented
15	C₂×C₄×D₄₁	C₂²	1	15	yes	complemented
16	C₂²×D₄₁	C₂³	1	15	yes	—
17	Q₈×C₄₁	C₂³	1	10	yes	—
18	D₄×C₄₁	C₂³	1	10	yes	—
19	C₂×C₄×C₄₁	C₂³	1	15	yes	—
20	C₂²⋊D₄₁	C₂³	1	20	yes	—
21	C₂×Dic₄₁	C₂³	1	5	yes	—
22	D₁₆₄	C₂³	1	30	yes	—
23	C₄×D₄₁	C₂³	1	10	yes	—
24	C₄×D₄₁	C₂³	1	30	yes	—
25	Dic₈₂	C₂³	1	10	yes	—
26	C₂²×C₄₁	C₂⁴	1	5	yes	—
27	C₂×D₄₁	—	4	15	no	—
28	C₂×D₄₁	C₂⁴	1	10	yes	—
29	C₄×C₄₁	C₂⁴	1	10	yes	—
30	Dic₄₁	C₂⁴	1	5	yes	—
31	Dic₄₁	C₂⁴	1	1	yes	characteristic
32	C₂×C₄₁	—	2	5	no	—
33	C₂×C₄₁	C₂⁵	1	1	yes	characteristic
34	D₄₁	—	2	10	no	—
35	#585	—	41	1	no	maximal, Sylow
36	C₄₁	#585	1	1	yes	characteristic, complemented, Sylow
37	2₋¹⁺⁴	—	41	5	no	—
38	2₋¹⁺⁴	D₄₁	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	41	10	no	—
40	C₂×#41	—	41	10	no	—
41	C₂×#41	—	41	5	no	—
42	#41	—	41	10	no	—
43	#41	—	41	20	no	—
44	#41	—	41	10	no	—
45	#41	—	41	30	no	—
46	#41	C₂×D₄₁	1	10	yes	complemented
47	C₂×Q₈	—	41	10	no	—
48	C₂×Q₈	C₂×D₄₁	1	5	yes	complemented
49	C₂×D₄	—	41	30	no	—
50	C₂×D₄	—	41	15	no	—
51	C₂²×C₄	—	41	15	no	—
52	C₂³	—	41	15	no	—
53	Q₈	—	41	10	no	—
54	Q₈	C₂²×D₄₁	1	10	yes	—
55	D₄	—	41	20	no	—
56	D₄	—	41	30	no	—
57	D₄	C₂²×D₄₁	1	10	yes	—
58	C₂×C₄	—	41	5	no	—
59	C₂×C₄	—	41	10	no	—
60	C₂×C₄	—	41	30	no	—
61	C₂×C₄	C₂²×D₄₁	1	15	yes	—
62	C₂²	—	41	10	no	—
63	C₂²	—	164	15	no	—
64	C₂²	C₂³×D₄₁	1	5	yes	—
65	C₄	—	41	1	no	—
66	C₄	—	41	5	no	—
67	C₄	C₂³×D₄₁	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	82	10	no	—
70	C₂	C₂⁴×D₄₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₄₁
commutator, socle, cosocle	C₂×C₄₁	C₂⁵
Fitting	2₋¹⁺⁴×C₄₁	C₂
hypercenter	2₋¹⁺⁴	D₄₁
hypocenter	C₄₁	#585
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #13050212	C₂	semidirect
2. #13050210	C₂	semidirect
3. C₂×#10620310	C₂	semidirect
4. #41×D₄₁	C₂	semidirect
5. 2₋¹⁺⁴×C₄₁	C₂	semidirect
6. #41×C₄₁	C₂²	semidirect
7. C₂×Q₈×C₄₁	C₂²	semidirect
8. #10620310	C₂²	semidirect
9. #10620310	C₂²	semidirect
10. Q₈×D₄₁	C₂²	semidirect
11. #10620308	C₂²	semidirect
12. D₄×D₄₁	C₂²	semidirect
13. #10620306	C₂²	semidirect
14. C₂×D₁₆₄	C₂²	semidirect
15. C₂×C₄×D₄₁	C₂²	semidirect
36. C₄₁	#585	semidirect
38. 2₋¹⁺⁴	D₄₁	semidirect
46. #41	C₂×D₄₁	semidirect
48. C₂×Q₈	C₂×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050212	2
#13050210	2
C₂×#10620310	2
#41×D₄₁	2
2₋¹⁺⁴×C₄₁	2
#585	41