Group #21427282 — subgroups

Subgroups — #21427282

Order 7552.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837301	C₂	1	4	yes	maximal
2	#16837288	C₂	1	4	yes	maximal
3	#16837288	C₂	1	4	yes	maximal
4	#16837268	C₂	1	2	yes	maximal
5	C₂×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅₉	C₂²	1	4	yes	—
10	C₅₉⋊#95	C₂²	1	4	yes	—
11	C₅₉⋊#95	C₂²	1	4	yes	—
12	C₅₉⋊#95	C₂²	1	4	yes	—
13	C₅₉⋊#95	C₂²	1	4	yes	—
14	#125×C₅₉	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
21	C₂²×Dic₅₉	C₂³	1	4	yes	—
22	C₂²×Dic₅₉	C₂³	1	4	yes	—
23	C₄⋊C₄×C₅₉	C₂×C₄	1	8	yes	—
24	C₄⋊C₄×C₅₉	C₂×C₄	1	16	yes	—
25	C₄²×C₅₉	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₅₉	—	4	2	no	—
27	C₄⋊Dic₅₉	—	4	4	no	—
28	C₄⋊Dic₅₉	—	4	2	no	—
29	C₄×Dic₅₉	—	4	4	no	—
30	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₅₉	—	2	4	no	—
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	4	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	2	1	no	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₅₉	C₂²×C₄	1	8	yes	—
40	C₂×Dic₅₉	—	2	4	no	—
41	C₂×Dic₅₉	—	2	4	no	—
42	C₂×Dic₅₉	—	2	4	no	—
43	C₂×Dic₅₉	—	2	4	no	—
44	C₂×Dic₅₉	—	2	4	no	—
45	C₂×Dic₅₉	—	2	4	no	—
46	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
47	C₂²×C₅₉	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₅₉	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
51	C₄×C₅₉	—	2	2	no	—
52	C₄×C₅₉	—	2	8	no	—
53	C₄×C₅₉	—	2	4	no	—
54	Dic₅₉	—	4	4	no	—
55	Dic₅₉	—	4	4	no	—
56	#2382	—	59	1	no	maximal, Sylow
57	C₂×C₅₉	#519	1	1	yes	characteristic
58	C₂×C₅₉	#520	1	2	yes	—
59	C₂×C₅₉	#562	1	1	yes	characteristic
60	C₂×C₅₉	#563	1	2	yes	—
61	C₂×C₅₉	#564	1	1	yes	characteristic
62	C₂×#125	D₅₉	1	1	yes	characteristic
63	#389	—	59	4	no	—
64	#387	—	59	4	no	—
65	#387	—	59	4	no	—
66	#382	—	59	2	no	—
67	C₅₉	#2382	1	1	yes	characteristic, complemented, Sylow
68	#125	Dic₅₉	1	8	yes	complemented
69	C₂×C₄⋊C₄	—	59	2	no	—
70	C₂×C₄⋊C₄	—	59	2	no	—
71	C₂×C₄⋊C₄	—	59	4	no	—
72	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
73	C₂×C₄⋊C₄	C₂×D₅₉	1	4	yes	—
74	C₂×C₄²	—	59	4	no	—
75	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
76	#95	—	59	4	no	—
77	#95	—	59	4	no	—
78	#95	—	59	4	no	—
79	#95	—	59	4	no	—
80	C₂²×C₄	—	59	4	no	—
81	C₂²×C₄	—	59	4	no	—
82	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
84	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
85	C₄⋊C₄	—	236	4	no	—
86	C₄⋊C₄	—	236	2	no	—
87	C₄⋊C₄	—	236	2	no	—
88	C₄⋊C₄	C₂×Dic₅₉	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₅₉	1	16	yes	—
90	C₄²	—	236	4	no	—
91	C₄²	C₂×Dic₅₉	1	4	yes	—
92	C₂³	C₂³×D₅₉	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	118	4	no	—
100	C₂×C₄	—	118	4	no	—
101	C₂×C₄	—	118	4	no	—
102	C₂×C₄	—	118	4	no	—
103	C₂×C₄	—	118	4	no	—
104	C₂×C₄	—	118	4	no	—
105	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
106	C₂×C₄	C₂²×Dic₅₉	1	4	yes	—
107	C₂×C₄	C₂²×Dic₅₉	1	8	yes	—
108	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
109	C₂²	#14273773	1	2	yes	—
110	C₂²	#14273774	1	2	yes	—
111	C₂²	#14273775	1	1	yes	characteristic
112	C₂²	#14273775	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	236	4	no	—
117	C₄	—	236	4	no	—
118	C₂	#16837880	1	1	yes	characteristic
119	C₂	#16837885	1	2	yes	—
120	C₂	#16838150	1	1	yes	characteristic
121	C₂	#16838156	1	2	yes	—
122	C₂	C₅₉⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837301

C₂

yes

maximal

#16837288

C₂

yes

maximal

#16837288

C₂

yes

maximal

#16837268

C₂

yes

maximal

C₂×#125×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

#125×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂²×C₅₉

H(5,𝔽₂)

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₅₉

2₋¹⁺⁴

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#562

yes

characteristic

C₂×C₅₉

#563

yes

—

C₂×C₅₉

#564

yes

characteristic

C₂×#125

D₅₉

yes

—

—

—

—

C₅₉

#2382

yes

characteristic, complemented, Sylow

#125

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄²

—

236

—

C₄²

C₂×Dic₅₉

yes

—

C₂³

C₂³×D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

100

C₂×C₄

—

118

—

101

C₂×C₄

—

118

—

102

C₂×C₄

—

118

—

103

C₂×C₄

—

118

—

104

C₂×C₄

—

118

—

105

C₂×C₄

C₂²×Dic₅₉

yes

—

106

C₂×C₄

C₂²×Dic₅₉

yes

—

107

C₂×C₄

C₂²×Dic₅₉

yes

—

108

C₂²

C₂³×Dic₅₉

yes

characteristic

109

C₂²

#14273773

yes

—

110

C₂²

#14273774

yes

—

111

C₂²

#14273775

yes

characteristic

112

C₂²

#14273775

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

236

—

117

C₄

—

236

—

118

C₂

#16837880

yes

characteristic

119

C₂

#16837885

yes

—

120

C₂

#16838150

yes

characteristic

121

C₂

#16838156

yes

—

122

C₂

C₅₉⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#125×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₂×#125	D₅₉
hypocenter	C₅₉	#2382
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#125×C₅₉

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₅₉	C₄	semidirect
67. C₅₉	#2382	semidirect
68. #125	Dic₅₉	semidirect

H ≅ G/N

type

14. #125×C₅₉

C₄

semidirect

67. C₅₉

#2382

semidirect

68. #125

Dic₅₉

semidirect

Subgroups — #21427282

Conjugacy classes of subgroups

Complements