Subgroups — #21423727

Order 7552.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837800	C₂	1	1	yes	maximal, characteristic
2	#16837798	C₂	1	2	yes	maximal
3	#16837754	C₂	1	2	yes	maximal
4	#16837123	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#494	C₂	1	1	yes	maximal, characteristic
6	#14273727	C₂²	1	2	yes	—
7	#14273659	—	2	2	no	—
8	#14273658	—	2	4	no	—
9	#14273650	—	2	4	no	—
10	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₅₉	—	2	2	no	—
12	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	2	1	no	—
16	C₂×Q₈×C₅₉	—	2	1	no	—
17	C₂×Dic₁₁₈	—	2	2	no	—
18	Q₁₆×C₅₉	—	4	4	no	—
19	C₂×C₈×C₅₉	D₄	1	2	yes	—
20	C₄⋊C₄×C₅₉	—	2	2	no	—
21	C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
26	Q₈×C₅₉	—	4	2	no	—
27	Q₈×C₅₉	—	4	2	no	—
28	C₂×C₄×C₅₉	—	4	1	no	—
29	C₂×C₄×C₅₉	—	4	1	no	—
30	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₅₉	—	4	2	no	—
33	Dic₁₁₈	—	4	4	no	—
34	C₈×C₅₉	SD₁₆	1	4	yes	—
35	C₅₉⋊C₈	—	2	4	no	—
36	C₂²×C₅₉	#127	1	1	yes	characteristic
37	C₄×C₅₉	—	8	1	no	—
38	C₄×C₅₉	—	8	1	no	—
39	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
40	C₄×C₅₉	#137	1	4	yes	—
41	Dic₅₉	—	8	2	no	—
42	#1716	—	59	1	no	maximal, Sylow
43	C₂×C₅₉	#492	1	1	yes	characteristic
44	C₂×C₅₉	#497	1	2	yes	—
45	#498	—	59	2	no	—
46	#498	—	59	1	no	—
47	#494	D₅₉	1	1	yes	characteristic
48	#487	—	59	2	no	—
49	#322	—	59	1	no	—
50	C₅₉	#1716	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	2	2	no	—
52	C₄⋊Q₈	—	59	2	no	—
53	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
54	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
55	#106	—	118	4	no	—
56	#106	—	118	2	no	—
57	#103	—	118	4	no	—
58	C₄×C₈	—	59	2	no	—
59	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	118	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	59	4	no	—
65	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	118	2	no	—
69	C₄⋊C₄	—	118	2	no	—
70	C₄²	C₂²×D₅₉	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	236	4	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	236	2	no	—
77	C₂×C₄	D₄×D₅₉	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
79	C₈	—	118	4	no	—
80	C₈	#11745670	1	4	yes	—
81	C₂²	#14273723	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	472	2	no	—
85	C₄	C₂×#11745670	1	2	yes	—
86	C₄	#14273759	1	4	yes	—
87	C₂	#16837774	1	1	yes	characteristic
88	C₂	#16837795	1	2	yes	—

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#494	C₂
socle	C₂²×C₅₉	#127
hypercenter	#494	D₅₉
hypocenter	C₅₉	#1716
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
50. C₅₉	#1716	semidirect