Subgroups — #21427273

Order 7552.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837282	C₂	1	12	yes	maximal, complemented
2	#16837270	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₅₉	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₅₉	C₂²	1	12	yes	—
6	C₅₉⋊#95	C₂²	1	12	yes	—
7	C₂²×D₄×C₅₉	C₂²	1	6	yes	—
8	#127×C₅₉	C₄	1	8	yes	complemented
9	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
10	C₂⁴×C₅₉	C₂³	1	4	yes	—
11	C₂×D₄×C₅₉	—	4	6	no	—
12	C₂×D₄×C₅₉	—	4	6	no	—
13	C₂×D₄×C₅₉	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
15	C₂²×Dic₅₉	—	4	12	no	—
16	C₂²×Dic₅₉	C₂³	1	8	yes	—
17	C₄²×C₅₉	C₂×C₄	1	4	yes	complemented
18	C₂²⋊Dic₅₉	—	4	12	no	—
19	C₂³×C₅₉	—	2	12	no	—
20	C₂³×C₅₉	—	2	12	no	—
21	C₂³×C₅₉	C₂²×C₄	1	8	yes	—
22	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
23	D₄×C₅₉	—	4	24	no	—
24	C₂×C₄×C₅₉	—	2	6	no	—
25	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
26	C₂×Dic₅₉	—	2	24	no	—
27	C₂×Dic₅₉	—	4	16	no	—
28	C₂²×C₅₉	—	2	24	no	—
29	C₂²×C₅₉	—	4	4	no	—
30	C₂²×C₅₉	—	4	12	no	—
31	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₅₉	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₅₉	—	2	6	no	—
34	Dic₅₉	—	4	8	no	—
35	#2380	—	59	1	no	maximal, Sylow
36	C₂×C₅₉	—	4	8	no	—
37	C₂×C₅₉	#518	1	3	yes	—
38	C₂×C₅₉	#560	1	4	yes	—
39	C₂×#127	D₅₉	1	1	yes	characteristic
40	#386	—	59	12	no	—
41	#383	—	59	2	no	—
42	C₅₉	#2380	1	1	yes	characteristic, complemented, Sylow
43	C₂²×D₄	C₂×D₅₉	1	6	yes	—
44	C₂³×C₄	—	59	4	no	—
45	#127	Dic₅₉	1	8	yes	complemented
46	C₂×C₂²⋊C₄	—	59	12	no	—
47	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
48	#95	—	59	12	no	—
49	C₂⁴	C₂²×D₅₉	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₅₉	1	24	yes	complemented
53	C₂²×C₄	—	59	8	no	—
54	C₂²×C₄	—	236	12	no	—
55	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
56	C₂²⋊C₄	—	236	12	no	—
57	C₄²	C₂×Dic₅₉	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₅₉	1	8	yes	—
61	C₂³	C₂³×D₅₉	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	118	24	no	—
65	C₂×C₄	—	236	16	no	—
66	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
71	C₂²	#14273772	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	236	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#16837877	1	3	yes	—
76	C₂	#16838139	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#127×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₂×#127	D₅₉
hypocenter	C₅₉	#2380
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #16837282	C₂	semidirect
2. #16837270	C₂	semidirect
8. #127×C₅₉	C₄	semidirect
13. C₂×D₄×C₅₉	C₂×C₄	semidirect
17. C₄²×C₅₉	C₂×C₄	semidirect
42. C₅₉	#2380	semidirect
45. #127	Dic₅₉	semidirect
52. C₂×D₄	C₂×Dic₅₉	semidirect
57. C₄²	C₂×Dic₅₉	semidirect