Subgroups — #21426338

Order 7552.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837804	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837779	C₂	1	1	yes	maximal, characteristic
3	#16837204	C₂	1	4	yes	maximal, complemented
4	C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₇₂	—	2	1	no	—
6	C₂×D₄₇₂	C₂²	1	2	yes	—
7	#14273724	—	2	1	no	—
8	#14273719	—	2	1	no	—
9	#14273661	C₂²	1	2	yes	—
10	#14273659	—	2	1	no	—
11	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×D₂₃₆	—	2	2	no	—
14	C₂×D₂₃₆	—	4	2	no	—
15	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₄²×C₅₉	D₄	1	1	yes	characteristic
18	C₄⋊Dic₅₉	—	2	2	no	—
19	Dic₅₉⋊C₄	—	4	2	no	—
20	D₄₇₂	—	2	4	no	—
21	D₄₇₂	—	4	2	no	—
22	C₁₆×C₅₉	—	2	2	no	—
23	C₂²×D₅₉	—	4	2	no	—
24	C₂×C₄×C₅₉	D₈	1	2	yes	—
25	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
26	C₂×Dic₅₉	—	4	2	no	—
27	D₂₃₆	—	4	2	no	—
28	D₂₃₆	—	4	2	no	—
29	D₂₃₆	—	8	4	no	—
30	C₈×C₅₉	SD₁₆	1	2	yes	—
31	C₈×C₅₉	#41	1	2	yes	—
32	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
33	C₂×D₅₉	—	8	2	no	—
34	C₂×D₅₉	—	8	4	no	—
35	C₄×C₅₉	—	2	2	no	—
36	C₄×C₅₉	#124	1	1	yes	characteristic
37	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
38	Dic₅₉	—	8	2	no	—
39	#2244	—	59	1	no	maximal, Sylow
40	C₂×C₅₉	#486	1	1	yes	characteristic
41	C₂×C₅₉	#509	1	2	yes	—
42	D₅₉	—	16	2	no	—
43	#499	—	59	1	no	—
44	#493	—	59	1	no	—
45	#357	—	59	4	no	—
46	#346	D₅₉	1	1	yes	characteristic, complemented
47	C₅₉	#2244	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₈	—	59	2	no	—
49	C₂×D₈	—	118	1	no	—
50	#127	—	118	1	no	—
51	#125	—	118	1	no	—
52	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
53	#107	—	59	2	no	—
54	#106	—	118	1	no	—
55	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
56	C₂×D₄	—	118	2	no	—
57	C₂×D₄	—	236	2	no	—
58	D₈	—	118	4	no	—
59	D₈	—	236	2	no	—
60	C₂×C₈	D₂₃₆	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
62	C₄⋊C₄	—	118	2	no	—
63	C₄⋊C₄	—	236	2	no	—
64	C₄²	D₂₃₆	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₂³	—	236	2	no	—
67	D₄	—	472	4	no	—
68	D₄	—	236	2	no	—
69	D₄	—	236	2	no	—
70	C₂×C₄	—	236	2	no	—
71	C₂×C₄	D₄₇₂	1	2	yes	—
72	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
73	C₈	#11745660	1	2	yes	—
74	C₈	#11745685	1	2	yes	—
75	C₂²	—	472	2	no	—
76	C₂²	—	472	4	no	—
77	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	472	2	no	—
80	C₄	#14273715	1	1	yes	characteristic
81	C₄	C₂×#11745660	1	1	yes	characteristic
82	C₂	—	944	2	no	—
83	C₂	#16837749	1	1	yes	characteristic
84	C₂	#16837840	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#346	D₅₉
hypocenter	C₅₉	#2244
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
1. #16837804	C₂	semidirect
3. #16837204	C₂	semidirect
4. C₅₉×#346	C₂	semidirect
46. #346	D₅₉	semidirect
47. C₅₉	#2244	semidirect