Subgroups — #21438808

Order 7552.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838169	C₂	1	15	yes	maximal, complemented
2	C₂×#14273734	C₂	1	15	yes	maximal, complemented
3	C₅₉×#585	C₂	1	1	yes	maximal, characteristic
4	#14273734	C₂²	1	20	yes	complemented
5	#14273734	C₂²	1	60	yes	complemented
6	C₂×#11745664	C₂²	1	45	yes	complemented
7	C₂²×C₅₉⋊C₈	C₂²	1	15	yes	complemented
8	2₋¹⁺⁴×C₅₉	C₄	1	6	yes	—
9	H(5,𝔽₂)×C₅₉	C₄	1	10	yes	—
10	C₂×#41×C₅₉	C₂²	1	15	yes	—
11	#41×C₅₉	C₂×C₄	1	60	yes	—
12	#41×C₅₉	C₂³	1	20	yes	—
13	C₂×Q₈×C₅₉	C₂×C₄	1	15	yes	—
14	C₂×D₄×C₅₉	C₂×C₄	1	45	yes	—
15	C₂²×C₄×C₅₉	C₂³	1	15	yes	—
16	#11745664	C₂³	1	60	yes	—
17	C₂×C₅₉⋊C₈	C₂³	1	15	yes	—
18	C₂×C₅₉⋊C₈	C₂³	1	45	yes	—
19	C₂³×C₅₉	C₂²×C₄	1	15	yes	—
20	Q₈×C₅₉	C₂²×C₄	1	20	yes	—
21	D₄×C₅₉	C₂²×C₄	1	60	yes	—
22	C₂×C₄×C₅₉	C₂²×C₄	1	45	yes	—
23	C₂×C₄×C₅₉	C₂⁴	1	15	yes	—
24	C₅₉⋊C₈	C₂⁴	1	1	yes	characteristic
25	C₅₉⋊C₈	C₂⁴	1	15	yes	—
26	C₂²×C₅₉	—	4	15	no	—
27	C₂²×C₅₉	C₂³×C₄	1	15	yes	—
28	C₄×C₅₉	C₂³×C₄	1	15	yes	—
29	C₄×C₅₉	C₂⁵	1	1	yes	characteristic
30	#3573	—	59	1	no	maximal, Sylow
31	C₂×C₅₉	—	2	15	no	—
32	C₂×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
33	#585	D₅₉	1	1	yes	characteristic
34	#568	—	59	15	no	—
35	C₂×#131	—	59	15	no	—
36	C₅₉	#3573	1	1	yes	characteristic, complemented, Sylow
37	2₋¹⁺⁴	Dic₅₉	1	6	yes	—
38	H(5,𝔽₂)	Dic₅₉	1	10	yes	—
39	C₂×#41	C₂×D₅₉	1	15	yes	—
40	#131	—	59	20	no	—
41	#131	—	59	60	no	—
42	C₂×M₁₆	—	59	45	no	—
43	C₂²×C₈	—	59	15	no	—
44	#41	C₂×Dic₅₉	1	60	yes	—
45	#41	C₂²×D₅₉	1	20	yes	—
46	C₂×Q₈	C₂×Dic₅₉	1	15	yes	—
47	C₂×D₄	C₂×Dic₅₉	1	45	yes	—
48	C₂²×C₄	C₂²×D₅₉	1	15	yes	—
49	M₁₆	—	59	60	no	—
50	C₂×C₈	—	59	15	no	—
51	C₂×C₈	—	59	45	no	—
52	C₂³	C₂²×Dic₅₉	1	15	yes	—
53	Q₈	C₂²×Dic₅₉	1	20	yes	—
54	D₄	C₂²×Dic₅₉	1	60	yes	—
55	C₂×C₄	C₂²×Dic₅₉	1	45	yes	—
56	C₂×C₄	C₂³×D₅₉	1	15	yes	—
57	C₈	—	59	1	no	—
58	C₈	—	59	15	no	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×Dic₅₉	1	15	yes	—
61	C₄	C₂³×Dic₅₉	1	15	yes	—
62	C₄	C₂⁴×D₅₉	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₅₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₅₉
commutator, socle	C₂×C₅₉	C₂⁴×C₄
Fitting	C₅₉×#585	C₂
hypercenter	#585	D₅₉
hypocenter	C₅₉	#3573
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #16838169	C₂	semidirect
2. C₂×#14273734	C₂	semidirect
4. #14273734	C₂²	semidirect
5. #14273734	C₂²	semidirect
6. C₂×#11745664	C₂²	semidirect
7. C₂²×C₅₉⋊C₈	C₂²	semidirect
36. C₅₉	#3573	semidirect