Subgroups — #21423701

Order 7552.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837748	C₂	1	4	yes	maximal, complemented
3	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₃₆	—	2	2	no	—
6	#14273727	C₂²	1	2	yes	—
7	#14273648	—	2	8	no	—
8	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	C₂×D₈×C₅₉	—	2	2	no	—
10	#127×C₅₉	C₂²	1	2	yes	—
11	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₅₉	—	2	2	no	—
13	C₂×D₄×C₅₉	—	2	4	no	—
14	C₂×Dic₁₁₈	—	2	2	no	—
15	D₈×C₅₉	—	4	4	no	—
16	C₂×C₈×C₅₉	D₄	1	2	yes	—
17	C₄²×C₅₉	C₂³	1	1	yes	characteristic
18	C₄⋊Dic₅₉	—	2	4	no	—
19	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
20	Dic₂₃₆	—	4	4	no	—
21	C₂³×C₅₉	—	4	2	no	—
22	D₄×C₅₉	—	4	4	no	—
23	D₄×C₅₉	—	4	8	no	—
24	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
26	C₂×Dic₅₉	—	4	2	no	—
27	Dic₁₁₈	—	4	4	no	—
28	C₈×C₅₉	D₈	1	4	yes	—
29	C₅₉⋊C₈	—	2	4	no	—
30	C₂²×C₅₉	—	8	2	no	—
31	C₂²×C₅₉	—	8	4	no	—
32	C₂²×C₅₉	#127	1	1	yes	characteristic
33	C₄×C₅₉	C₂×D₈	1	2	yes	—
34	C₄×C₅₉	#135	1	4	yes	—
35	Dic₅₉	—	8	2	no	—
36	#1709	—	59	1	no	maximal, Sylow
37	C₂×C₅₉	—	16	2	no	—
38	C₂×C₅₉	#493	1	1	yes	characteristic
39	C₂×C₅₉	#495	1	2	yes	—
40	#494	—	59	1	no	—
41	#493	D₅₉	1	1	yes	characteristic
42	#486	—	59	4	no	—
43	C₈²	—	59	1	no	—
44	C₅₉	#1709	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	118	2	no	—
46	C₂×D₈	—	2	2	no	—
47	C₄⋊Q₈	—	59	2	no	—
48	#127	C₂×D₅₉	1	2	yes	—
49	#102	—	118	8	no	—
50	C₄×C₈	—	59	2	no	—
51	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
52	C₂×Q₈	—	118	2	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	2	2	no	—
55	Q₁₆	—	236	4	no	—
56	D₈	—	4	4	no	—
57	C₂×C₈	—	59	4	no	—
58	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
59	C₄⋊C₄	—	118	4	no	—
60	C₄²	C₂²×D₅₉	1	1	yes	characteristic
61	C₂³	—	4	2	no	—
62	Q₈	—	236	4	no	—
63	D₄	—	4	4	no	—
64	D₄	—	4	8	no	—
65	C₂×C₄	—	236	2	no	—
66	C₂×C₄	D₄×D₅₉	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
68	C₈	—	118	4	no	—
69	C₈	#11745669	1	4	yes	—
70	C₂²	—	8	2	no	—
71	C₂²	—	8	4	no	—
72	C₂²	#14273723	1	1	yes	characteristic
73	C₄	—	472	2	no	—
74	C₄	C₂×#11745669	1	2	yes	—
75	C₄	#14273746	1	4	yes	—
76	C₂	—	16	2	no	—
77	C₂	#16837778	1	1	yes	characteristic
78	C₂	#16837785	1	2	yes	—

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#493	C₂
socle	C₂²×C₅₉	#127
hypercenter	#493	D₅₉
hypocenter	C₅₉	#1709
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16837782	C₂	semidirect
2. #16837748	C₂	semidirect
3. C₈×C₅₉⋊C₈	C₂	semidirect
44. C₅₉	#1709	semidirect