Subgroups — #29495160

Order 15104.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437199	C₂	1	15	yes	maximal, complemented
2	C₂×#16837819	C₂	1	15	yes	maximal, complemented
3	C₅₉×#3573	C₂	1	1	yes	maximal, characteristic
4	#16837819	C₂²	1	20	yes	complemented
5	#16837819	C₂²	1	60	yes	complemented
6	C₂×#14273667	C₂²	1	45	yes	complemented
7	C₂²×C₅₉⋊C₁₆	C₂²	1	15	yes	complemented
8	C₅₉×#585	C₄	1	1	yes	characteristic
9	C₅₉×#568	C₄	1	15	yes	—
10	C₂×#131×C₅₉	C₂²	1	15	yes	—
11	#14273667	C₂³	1	60	yes	—
12	C₂×C₅₉⋊C₁₆	C₂³	1	15	yes	—
13	C₂×C₅₉⋊C₁₆	C₂³	1	45	yes	—
14	2₋¹⁺⁴×C₅₉	C₈	1	6	yes	—
15	H(5,𝔽₂)×C₅₉	C₈	1	10	yes	—
16	C₂×#41×C₅₉	C₂×C₄	1	15	yes	—
17	#131×C₅₉	C₂×C₄	1	60	yes	—
18	#131×C₅₉	C₂³	1	20	yes	—
19	C₂×M₁₆×C₅₉	C₂×C₄	1	45	yes	—
20	C₂²×C₈×C₅₉	C₂³	1	15	yes	—
21	#41×C₅₉	C₂×C₈	1	60	yes	—
22	#41×C₅₉	C₂²×C₄	1	20	yes	—
23	C₂×Q₈×C₅₉	C₂×C₈	1	15	yes	—
24	C₂×D₄×C₅₉	C₂×C₈	1	45	yes	—
25	C₂²×C₄×C₅₉	C₂²×C₄	1	15	yes	—
26	M₁₆×C₅₉	C₂²×C₄	1	60	yes	—
27	C₂×C₈×C₅₉	C₂²×C₄	1	45	yes	—
28	C₂×C₈×C₅₉	C₂⁴	1	15	yes	—
29	C₅₉⋊C₁₆	C₂⁴	1	1	yes	characteristic
30	C₅₉⋊C₁₆	C₂⁴	1	15	yes	—
31	C₂³×C₅₉	C₂²×C₈	1	15	yes	—
32	Q₈×C₅₉	C₂²×C₈	1	20	yes	—
33	D₄×C₅₉	C₂²×C₈	1	60	yes	—
34	C₂×C₄×C₅₉	C₂²×C₈	1	45	yes	—
35	C₂×C₄×C₅₉	C₂³×C₄	1	15	yes	—
36	C₈×C₅₉	C₂³×C₄	1	15	yes	—
37	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
38	#62624	—	59	1	no	maximal, Sylow
39	C₂²×C₅₉	—	4	15	no	—
40	C₂²×C₅₉	C₂³×C₈	1	15	yes	—
41	C₄×C₅₉	C₂³×C₈	1	15	yes	—
42	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
43	#3573	D₅₉	1	1	yes	characteristic
44	#3407	—	59	15	no	—
45	C₂×#504	—	59	15	no	—
46	C₂×C₅₉	—	2	15	no	—
47	C₂×C₅₉	C₂⁴×C₈	1	1	yes	characteristic
48	#585	Dic₅₉	1	1	yes	characteristic
49	#568	Dic₅₉	1	15	yes	—
50	C₂×#131	C₂×D₅₉	1	15	yes	—
51	#504	—	59	20	no	—
52	#504	—	59	60	no	—
53	C₂×M₃₂	—	59	45	no	—
54	C₂²×C₁₆	—	59	15	no	—
55	C₅₉	#62624	1	1	yes	characteristic, complemented, Sylow
56	2₋¹⁺⁴	C₅₉⋊C₈	1	6	yes	—
57	H(5,𝔽₂)	C₅₉⋊C₈	1	10	yes	—
58	C₂×#41	C₂×Dic₅₉	1	15	yes	—
59	#131	C₂×Dic₅₉	1	60	yes	—
60	#131	C₂²×D₅₉	1	20	yes	—
61	C₂×M₁₆	C₂×Dic₅₉	1	45	yes	—
62	C₂²×C₈	C₂²×D₅₉	1	15	yes	—
63	M₃₂	—	59	60	no	—
64	C₂×C₁₆	—	59	15	no	—
65	C₂×C₁₆	—	59	45	no	—
66	#41	C₂×C₅₉⋊C₈	1	60	yes	—
67	#41	C₂²×Dic₅₉	1	20	yes	—
68	C₂×Q₈	C₂×C₅₉⋊C₈	1	15	yes	—
69	C₂×D₄	C₂×C₅₉⋊C₈	1	45	yes	—
70	C₂²×C₄	C₂²×Dic₅₉	1	15	yes	—
71	M₁₆	C₂²×Dic₅₉	1	60	yes	—
72	C₂×C₈	C₂²×Dic₅₉	1	45	yes	—
73	C₂×C₈	C₂³×D₅₉	1	15	yes	—
74	C₁₆	—	59	1	no	—
75	C₁₆	—	59	15	no	—
76	C₂³	C₂²×C₅₉⋊C₈	1	15	yes	—
77	Q₈	C₂²×C₅₉⋊C₈	1	20	yes	—
78	D₄	C₂²×C₅₉⋊C₈	1	60	yes	—
79	C₂×C₄	C₂²×C₅₉⋊C₈	1	45	yes	—
80	C₂×C₄	C₂³×Dic₅₉	1	15	yes	—
81	C₈	C₂³×Dic₅₉	1	15	yes	—
82	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
83	C₂²	—	4	15	no	—
84	C₂²	C₂³×C₅₉⋊C₈	1	15	yes	—
85	C₄	C₂³×C₅₉⋊C₈	1	15	yes	—
86	C₄	C₂⁴×Dic₅₉	1	1	yes	characteristic
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×C₅₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂⁴×D₅₉
commutator, socle	C₂×C₅₉	C₂⁴×C₈
Fitting	C₅₉×#3573	C₂
prime radical	C₅₉×#585	C₄
cosocle	C₈×C₅₉	C₂⁵
hypercenter	#3573	D₅₉
hypocenter	C₅₉	#62624
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ #3573

Complements

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

N	H ≅ G/N	type
1. #21437199	C₂	semidirect
2. C₂×#16837819	C₂	semidirect
4. #16837819	C₂²	semidirect
5. #16837819	C₂²	semidirect
6. C₂×#14273667	C₂²	semidirect
7. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
55. C₅₉	#62624	semidirect

Maximal subgroups

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

subgroup	index
#21437199	2
C₂×#16837819	2
C₅₉×#3573	2
#62624	59