Subgroups — #21422270

Order 7552.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837756	C₂	1	1	yes	maximal, characteristic
2	#16837199	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#354	C₂	1	1	yes	maximal, characteristic
4	#14273719	C₄	1	1	yes	characteristic, complemented
5	#14273651	—	2	1	no	—
6	#14273648	—	2	1	no	—
7	#14273646	—	2	1	no	—
8	C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#124×C₅₉	C₂²	1	1	yes	characteristic
10	#99×C₅₉	—	2	1	no	—
11	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
12	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
13	C₂²⋊C₄×C₅₉	—	4	1	no	—
14	C₂²⋊C₄×C₅₉	—	4	1	no	—
15	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₅₉	—	2	1	no	—
17	Dic₅₉⋊C₄	—	4	1	no	—
18	#11745664	—	4	1	no	—
19	C₂×C₅₉⋊C₈	—	2	1	no	—
20	C₂³×C₅₉	—	2	1	no	—
21	Q₈×C₅₉	—	4	1	no	—
22	D₄×C₅₉	—	4	1	no	—
23	C₂×C₄×C₅₉	—	2	1	no	—
24	C₂×C₄×C₅₉	—	2	1	no	—
25	C₂×C₄×C₅₉	—	8	1	no	—
26	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₅₉	—	4	1	no	—
28	C₅₉⋊C₈	—	4	1	no	—
29	C₅₉⋊C₈	—	8	1	no	—
30	C₂²×C₅₉	—	4	1	no	—
31	C₂²×C₅₉	—	8	1	no	—
32	C₂²×C₅₉	#99	1	1	yes	characteristic
33	C₄×C₅₉	—	2	1	no	—
34	C₄×C₅₉	—	4	1	no	—
35	C₄×C₅₉	—	4	1	no	—
36	C₄×C₅₉	—	16	1	no	—
37	Dic₅₉	—	8	1	no	—
38	#1414	—	59	1	no	maximal, Sylow
39	C₂×C₅₉	—	2	1	no	—
40	C₂×C₅₉	—	8	1	no	—
41	C₂×C₅₉	#354	1	1	yes	characteristic
42	#488	—	59	1	no	—
43	#355	—	59	1	no	—
44	#354	D₅₉	1	1	yes	characteristic
45	C₅₉	#1414	1	1	yes	characteristic, complemented, Sylow
46	#125	—	59	1	no	—
47	#124	C₂×D₅₉	1	1	yes	characteristic
48	#103	—	118	1	no	—
49	#102	—	118	1	no	—
50	#101	—	118	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	59	1	no	—
53	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
55	M₁₆	—	236	1	no	—
56	C₂×C₈	—	118	1	no	—
57	C₄⋊C₄	—	118	1	no	—
58	C₄⋊C₄	—	236	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₅₉	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	236	1	no	—
69	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
70	C₈	—	472	1	no	—
71	C₈	—	236	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#14273641	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	472	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#16837196	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837196
commutator	C₄²×C₅₉	C₂×C₄
Frattini	#124	C₂×D₅₉
Fitting	C₅₉×#354	C₂
socle	C₂×C₅₉	#354
hypercenter	#354	D₅₉
hypocenter	C₅₉	#1414
LCS	G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #14273719	C₄	semidirect
8. C₄×C₅₉⋊C₈	C₄	semidirect
45. C₅₉	#1414	semidirect