Group #21427288 — subgroups

Subgroups — #21427288

Order 7552.

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

normal subgroups only

This group has 456 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	3	yes	maximal, complemented
2	#16837295	C₂	1	3	yes	maximal
3	#16837288	C₂	1	3	yes	maximal, complemented
4	#16837277	C₂	1	3	yes	maximal
5	#16837272	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837270	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	3	yes	—
12	C₂×C₂²⋊Dic₅₉	C₂²	1	3	yes	—
13	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
14	C₅₉⋊#95	C₂²	1	3	yes	—
15	C₅₉⋊#95	C₂²	1	3	yes	—
16	C₅₉⋊#95	C₂²	1	3	yes	—
17	C₅₉⋊#95	C₂²	1	3	yes	—
18	#126×C₅₉	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
25	C₂²×Dic₅₉	—	4	3	no	—
26	C₂²×Dic₅₉	C₂³	1	3	yes	—
27	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₉	C₂³	1	3	yes	—
30	C₄⋊C₄×C₅₉	C₂×C₄	1	12	yes	complemented
31	C₂²⋊C₄×C₅₉	C₂×C₄	1	12	yes	—
32	C₄²×C₅₉	C₂×C₄	1	4	yes	complemented
33	C₂²⋊Dic₅₉	—	4	3	no	—
34	C₄⋊Dic₅₉	—	4	3	no	—
35	C₄⋊Dic₅₉	—	4	3	no	—
36	C₄⋊Dic₅₉	—	4	3	no	—
37	C₄×Dic₅₉	—	4	3	no	—
38	C₂³×C₅₉	—	2	3	no	—
39	C₂³×C₅₉	—	2	3	no	—
40	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
41	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	3	no	—
43	C₂×C₄×C₅₉	—	2	3	no	—
44	C₂×C₄×C₅₉	—	2	3	no	—
45	C₂×C₄×C₅₉	—	2	3	no	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
47	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
48	C₂×Dic₅₉	—	2	3	no	—
49	C₂×Dic₅₉	—	2	3	no	—
50	C₂×Dic₅₉	—	2	3	no	—
51	C₂×Dic₅₉	—	2	3	no	—
52	C₂×Dic₅₉	—	2	3	no	—
53	C₂×Dic₅₉	—	2	3	no	—
54	C₂×Dic₅₉	—	2	6	no	—
55	C₂×Dic₅₉	—	4	4	no	—
56	C₂²×C₅₉	—	2	6	no	—
57	C₂²×C₅₉	—	4	1	no	—
58	C₂²×C₅₉	—	4	3	no	—
59	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₅₉	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₅₉	2₋¹⁺⁴	1	3	yes	—
62	C₄×C₅₉	—	2	6	no	—
63	C₄×C₅₉	—	2	6	no	—
64	Dic₅₉	—	4	3	no	—
65	Dic₅₉	—	4	1	no	—
66	Dic₅₉	—	4	1	no	—
67	Dic₅₉	—	4	3	no	—
68	#2383	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	4	2	no	—
70	C₂×C₅₉	#520	1	3	yes	—
71	C₂×C₅₉	#560	1	1	yes	characteristic
72	C₂×C₅₉	#563	1	3	yes	—
73	C₂×#126	D₅₉	1	1	yes	characteristic
74	#389	—	59	3	no	—
75	#388	—	59	3	no	—
76	#387	—	59	3	no	—
77	#385	—	59	3	no	—
78	#384	—	59	1	no	—
79	#383	—	59	1	no	—
80	C₅₉	#2383	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	—	59	1	no	—
82	#126	Dic₅₉	1	8	yes	complemented
83	C₂×C₄⋊C₄	—	59	3	no	—
84	C₂×C₄⋊C₄	—	59	3	no	—
85	C₂×C₄⋊C₄	—	59	3	no	—
86	C₂×C₄⋊C₄	C₂×D₅₉	1	3	yes	—
87	C₂×C₂²⋊C₄	—	59	3	no	—
88	C₂×C₂²⋊C₄	C₂×D₅₉	1	3	yes	—
89	C₂×C₄²	—	59	3	no	—
90	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
91	#95	—	59	3	no	—
92	#95	—	59	3	no	—
93	#95	—	59	3	no	—
94	#95	—	59	3	no	—
95	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
96	C₂²×C₄	—	59	3	no	—
97	C₂²×C₄	—	59	1	no	—
98	C₂²×C₄	—	59	1	no	—
99	C₂²×C₄	—	59	3	no	—
100	C₂²×C₄	—	236	3	no	—
101	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
102	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
103	C₄⋊C₄	—	236	3	no	—
104	C₄⋊C₄	—	236	3	no	—
105	C₄⋊C₄	—	236	3	no	—
106	C₄⋊C₄	C₂×Dic₅₉	1	12	yes	complemented
107	C₂²⋊C₄	—	236	3	no	—
108	C₂²⋊C₄	C₂×Dic₅₉	1	12	yes	—
109	C₄²	—	236	3	no	—
110	C₄²	C₂×Dic₅₉	1	4	yes	complemented
111	C₂³	—	2	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	C₂²×Dic₅₉	1	2	yes	—
114	C₂³	C₂³×D₅₉	1	1	yes	characteristic
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	118	3	no	—
120	C₂×C₄	—	118	3	no	—
121	C₂×C₄	—	118	3	no	—
122	C₂×C₄	—	118	3	no	—
123	C₂×C₄	—	118	3	no	—
124	C₂×C₄	—	118	3	no	—
125	C₂×C₄	—	118	6	no	—
126	C₂×C₄	—	236	4	no	—
127	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
128	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	3	no	—
131	C₂²	—	4	1	no	—
132	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
133	C₂²	#14273773	1	3	yes	—
134	C₂²	#14273774	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	2	6	no	—
137	C₄	—	236	3	no	—
138	C₄	—	236	1	no	—
139	C₄	—	236	1	no	—
140	C₄	—	236	3	no	—
141	C₂	—	4	2	no	—
142	C₂	#16837885	1	3	yes	—
143	C₂	#16838143	1	1	yes	characteristic
144	C₂	#16838154	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837301

C₂

yes

maximal, complemented

#16837295

C₂

yes

maximal

#16837288

C₂

yes

maximal, complemented

#16837277

C₂

yes

maximal

#16837272

C₂

yes

maximal, characteristic, complemented

#16837270

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₅₉

C₂

yes

maximal, characteristic

C₂³×Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₂²⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

#126×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

complemented

C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

complemented

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²×C₄

yes

—

C₂³×C₅₉

C₂⁴

yes

characteristic

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×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂²×C₅₉

H(5,𝔽₂)

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2383

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#520

yes

—

C₂×C₅₉

#560

yes

characteristic

C₂×C₅₉

#563

yes

—

C₂×#126

D₅₉

yes

—

—

—

—

—

—

C₅₉

#2383

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#126

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

—

—

—

—

C₂⁴

C₂²×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

236

—

101

C₂²×C₄

C₂²×D₅₉

yes

—

102

C₂²×C₄

C₂²×D₅₉

yes

—

103

C₄⋊C₄

—

236

—

104

C₄⋊C₄

—

236

—

105

C₄⋊C₄

—

236

—

106

C₄⋊C₄

C₂×Dic₅₉

yes

complemented

107

C₂²⋊C₄

—

236

—

108

C₂²⋊C₄

C₂×Dic₅₉

yes

—

109

C₄²

—

236

—

110

C₄²

C₂×Dic₅₉

yes

complemented

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²×Dic₅₉

yes

—

114

C₂³

C₂³×D₅₉

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

236

—

127

C₂×C₄

C₂²×Dic₅₉

yes

—

128

C₂×C₄

C₂²×Dic₅₉

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₅₉

yes

characteristic

133

C₂²

#14273773

yes

—

134

C₂²

#14273774

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

236

—

138

C₄

—

236

—

139

C₄

—

236

—

140

C₄

—

236

—

141

C₂

—

142

C₂

#16837885

yes

—

143

C₂

#16838143

yes

characteristic

144

C₂

#16838154

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#126×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₂×#126	D₅₉
hypocenter	C₅₉	#2383
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#126×C₅₉

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. #16837301	C₂	semidirect
3. #16837288	C₂	semidirect
5. #16837272	C₂	semidirect
6. #16837270	C₂	semidirect
18. #126×C₅₉	C₄	semidirect
30. C₄⋊C₄×C₅₉	C₂×C₄	semidirect
32. C₄²×C₅₉	C₂×C₄	semidirect
80. C₅₉	#2383	semidirect
82. #126	Dic₅₉	semidirect
106. C₄⋊C₄	C₂×Dic₅₉	semidirect
110. C₄²	C₂×Dic₅₉	semidirect

H ≅ G/N

type

1. #16837301

C₂

semidirect

3. #16837288

C₂

semidirect

5. #16837272

C₂

semidirect

6. #16837270

C₂

semidirect

18. #126×C₅₉

C₄

semidirect

30. C₄⋊C₄×C₅₉

C₂×C₄

semidirect

32. C₄²×C₅₉

C₂×C₄

semidirect

80. C₅₉

#2383

semidirect

82. #126

Dic₅₉

semidirect

106. C₄⋊C₄

C₂×Dic₅₉

semidirect

110. C₄²

C₂×Dic₅₉

semidirect

Subgroups — #21427288

Conjugacy classes of subgroups

Complements