Subgroups — #28475670

Order 15104.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	2	yes	maximal
2	C₅₉×#1452	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	—
5	C₄×C₅₉⋊C₁₆	—	2	2	no	—
6	C₅₉×#431	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#322	—	2	2	no	—
9	C₂×C₅₉⋊C₁₆	—	2	4	no	—
10	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	C₈	1	4	yes	—
13	C₂²⋊C₈×C₅₉	—	2	2	no	—
14	C₄×C₈×C₅₉	—	2	2	no	—
15	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₉	D₄	1	2	yes	—
17	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
20	C₂×C₈×C₅₉	—	2	4	no	—
21	C₂×C₈×C₅₉	—	4	2	no	—
22	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
24	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₅₉	C₂×C₈	1	2	yes	—
27	C₅₉⋊C₁₆	—	4	4	no	—
28	C₂³×C₅₉	#97	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	2	4	no	—
31	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
34	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
35	C₈×C₅₉	—	2	2	no	—
36	C₈×C₅₉	—	2	2	no	—
37	C₈×C₅₉	—	4	4	no	—
38	#7198	—	59	1	no	maximal, Sylow
39	C₂²×C₅₉	—	2	1	no	—
40	C₂²×C₅₉	—	2	2	no	—
41	C₂²×C₅₉	#336	1	1	yes	characteristic
42	C₄×C₅₉	—	2	1	no	—
43	C₄×C₅₉	—	2	2	no	—
44	C₄×C₅₉	#344	1	2	yes	—
45	C₄×C₅₉	#350	1	4	yes	—
46	#2162	—	59	2	no	—
47	#1452	D₅₉	1	1	yes	characteristic
48	C₂×C₅₉	—	4	1	no	—
49	C₂×C₅₉	#1281	1	1	yes	characteristic
50	C₂×C₅₉	#1379	1	2	yes	—
51	#431	Dic₅₉	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
53	C₄⋊C₁₆	—	118	2	no	—
54	C₂²⋊C₁₆	—	59	4	no	—
55	C₄×C₁₆	—	118	2	no	—
56	#322	—	2	2	no	—
57	C₅₉	#7198	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	C₄×D₅₉	1	2	yes	—
59	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
60	C₂×C₁₆	—	118	4	no	—
61	C₄⋊C₈	C₅₉⋊C₈	1	4	yes	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
65	C₄×C₈	D₂₃₆	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
67	C₂²×C₄	#11745664	1	2	yes	—
68	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
72	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
73	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
74	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
75	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
76	C₁₆	—	236	4	no	—
77	C₂³	#14273638	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
81	C₂×C₄	C₄×C₅₉⋊C₈	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
83	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#16837157	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	#16837173	1	2	yes	—
93	C₄	#16837186	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	#21421941	1	1	yes	characteristic
96	C₂	#21422185	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#1452	C₂
socle	C₂²×C₅₉	#336
prime radical	C₂³×C₅₉	#97
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#1452	D₅₉
hypocenter	C₅₉	#7198
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
57. C₅₉	#7198	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21425895	2
C₅₉×#1452	2
#7198	59