Subgroups — #21423693

Order 7552.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837807	C₂	1	1	yes	maximal, characteristic
2	#16837800	C₂	1	2	yes	maximal
3	#16837782	C₂	1	1	yes	maximal, characteristic
4	#16837753	C₂	1	2	yes	maximal
5	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	—	2	2	no	—
7	#14273727	C₂²	1	2	yes	—
8	#14273727	C₂²	1	2	yes	—
9	#14273661	—	2	2	no	—
10	#14273659	—	2	4	no	—
11	#14273652	—	2	4	no	—
12	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₁₈	—	2	2	no	—
16	C₂×Dic₁₁₈	—	2	2	no	—
17	C₂×C₈×C₅₉	D₄	1	2	yes	—
18	C₂×C₈×C₅₉	D₄	1	2	yes	—
19	C₂×C₈×C₅₉	D₄	1	2	yes	—
20	C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₅₉	—	2	2	no	—
22	C₄⋊Dic₅₉	—	2	2	no	—
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	Dic₂₃₆	—	4	4	no	—
26	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₅₉	—	4	2	no	—
30	C₂×Dic₅₉	—	4	2	no	—
31	Dic₁₁₈	—	4	4	no	—
32	Dic₁₁₈	—	4	4	no	—
33	C₈×C₅₉	—	2	2	no	—
34	C₈×C₅₉	SD₁₆	1	4	yes	—
35	C₈×C₅₉	Q₁₆	1	4	yes	complemented
36	C₂²×C₅₉	#127	1	1	yes	characteristic
37	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
38	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
39	C₄×C₅₉	#135	1	2	yes	—
40	Dic₅₉	—	8	2	no	—
41	Dic₅₉	—	8	2	no	—
42	#1707	—	59	1	no	maximal, Sylow
43	C₂×C₅₉	#492	1	1	yes	characteristic
44	C₂×C₅₉	#494	1	1	yes	characteristic
45	C₂×C₅₉	#495	1	1	yes	characteristic
46	#500	—	59	1	no	—
47	#498	—	59	2	no	—
48	#494	—	59	1	no	—
49	#487	—	59	2	no	—
50	C₈²	D₅₉	1	1	yes	characteristic
51	C₅₉	#1707	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	118	2	no	—
53	C₄⋊Q₈	—	59	2	no	—
54	C₄⋊Q₈	—	59	2	no	—
55	#107	—	118	2	no	—
56	#106	—	118	4	no	—
57	#103	—	118	4	no	—
58	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
59	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
60	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
61	C₂×Q₈	—	118	2	no	—
62	C₂×Q₈	—	118	2	no	—
63	Q₁₆	—	236	4	no	—
64	C₂×C₈	D₂₃₆	1	2	yes	—
65	C₂×C₈	D₂₃₆	1	2	yes	—
66	C₂×C₈	D₂₃₆	1	2	yes	—
67	C₄⋊C₄	—	118	2	no	—
68	C₄⋊C₄	—	118	2	no	—
69	C₄⋊C₄	—	118	2	no	—
70	C₄⋊C₄	—	118	2	no	—
71	C₄²	C₂²×D₅₉	1	1	yes	characteristic
72	Q₈	—	236	4	no	—
73	Q₈	—	236	4	no	—
74	C₂×C₄	—	236	2	no	—
75	C₂×C₄	—	236	2	no	—
76	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
77	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
78	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#11745660	1	4	yes	—
81	C₈	Dic₂₃₆	1	4	yes	complemented
82	C₂²	#14273724	1	1	yes	characteristic
83	C₄	—	472	2	no	—
84	C₄	—	472	2	no	—
85	C₄	C₂×#11745660	1	2	yes	—
86	C₄	C₂×Dic₂₃₆	1	2	yes	—
87	C₄	#14273749	1	2	yes	—
88	C₂	#16837775	1	1	yes	characteristic
89	C₂	#16837782	1	1	yes	characteristic
90	C₂	#16837786	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14273724
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈²×C₅₉	C₂
socle	C₂²×C₅₉	#127
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1707
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
35. C₈×C₅₉	Q₁₆	semidirect
51. C₅₉	#1707	semidirect
81. C₈	Dic₂₃₆	semidirect