Group #21426705 — subgroups

Subgroups — #21426705

Order 7552.

484 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 229 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	#16837295	C₂	1	4	yes	maximal
2	#16837288	C₂	1	4	yes	maximal, complemented
3	#16837272	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837268	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837262	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₉	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₅₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₅₉	C₂²	1	4	yes	—
13	C₅₉⋊#95	C₂²	1	4	yes	—
14	C₅₉⋊#95	C₂²	1	4	yes	—
15	C₅₉⋊#95	C₂²	1	4	yes	—
16	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
17	#117×C₅₉	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	—	2	2	no	—
24	C₂²×C₄×C₅₉	—	2	2	no	—
25	C₂²×C₄×C₅₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₉	C₂³	1	4	yes	—
31	C₂²×Dic₅₉	C₂³	1	4	yes	—
32	C₄⋊C₄×C₅₉	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₅₉	C₂×C₄	1	8	yes	—
34	C₄²×C₅₉	C₂×C₄	1	4	yes	—
35	C₄²×C₅₉	C₂×C₄	1	4	yes	—
36	C₂²⋊Dic₅₉	—	4	4	no	—
37	C₄⋊Dic₅₉	—	2	4	no	—
38	C₄⋊Dic₅₉	—	2	4	no	—
39	C₄⋊Dic₅₉	—	4	4	no	—
40	C₄×Dic₅₉	—	4	4	no	—
41	C₂³×C₅₉	—	2	1	no	—
42	C₂³×C₅₉	—	2	1	no	—
43	C₂³×C₅₉	—	2	2	no	—
44	C₂³×C₅₉	—	2	2	no	—
45	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
46	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	2	8	no	—
54	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₅₉	#41	1	2	yes	—
59	C₂×C₄×C₅₉	#41	1	2	yes	—
60	C₂×Dic₅₉	—	2	4	no	—
61	C₂×Dic₅₉	—	2	4	no	—
62	C₂×Dic₅₉	—	2	4	no	—
63	C₂×Dic₅₉	—	2	4	no	—
64	C₂×Dic₅₉	—	2	4	no	—
65	C₂×Dic₅₉	—	2	4	no	—
66	C₂²×C₅₉	—	2	2	no	—
67	C₂²×C₅₉	—	2	4	no	—
68	C₂²×C₅₉	—	4	1	no	—
69	C₂²×C₅₉	—	4	1	no	—
70	C₂²×C₅₉	—	4	2	no	—
71	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
73	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
74	C₂²×C₅₉	H(5,𝔽₂)	1	2	yes	—
75	C₂²×C₅₉	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	2	4	no	—
78	C₄×C₅₉	—	2	4	no	—
79	C₄×C₅₉	#117	1	4	yes	—
80	Dic₅₉	—	4	4	no	—
81	Dic₅₉	—	4	4	no	—
82	#2314	—	59	1	no	maximal, Sylow
83	C₂×C₅₉	—	4	2	no	—
84	C₂×C₅₉	C₂×#117	1	1	yes	characteristic
85	C₂×C₅₉	#520	1	2	yes	—
86	C₂×C₅₉	#540	1	1	yes	characteristic
87	C₂×C₅₉	#541	1	1	yes	characteristic
88	C₂×C₅₉	#542	1	2	yes	—
89	C₂×#117	D₅₉	1	1	yes	characteristic
90	#388	—	59	4	no	—
91	#387	—	59	4	no	—
92	#384	—	59	1	no	—
93	#382	—	59	1	no	—
94	#380	—	59	2	no	—
95	C₄×C₄⋊C₄	—	59	2	no	—
96	C₅₉	#2314	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
98	#117	Dic₅₉	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	59	2	no	—
100	C₂×C₄⋊C₄	—	59	2	no	—
101	C₂×C₄⋊C₄	—	59	4	no	—
102	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
103	C₂×C₂²⋊C₄	—	59	4	no	—
104	C₂×C₂²⋊C₄	C₂×D₅₉	1	2	yes	—
105	C₂×C₄²	—	59	4	no	—
106	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
108	#95	—	59	4	no	—
109	#95	—	59	4	no	—
110	#95	—	59	4	no	—
111	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	59	4	no	—
115	C₂²×C₄	—	59	4	no	—
116	C₂²×C₄	C₂×Dic₅₉	1	4	yes	—
117	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
119	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
121	C₄⋊C₄	—	118	4	no	—
122	C₄⋊C₄	—	118	4	no	—
123	C₄⋊C₄	—	236	4	no	—
124	C₄⋊C₄	C₂×Dic₅₉	1	8	yes	—
125	C₂²⋊C₄	—	236	4	no	—
126	C₂²⋊C₄	C₂×Dic₅₉	1	8	yes	—
127	C₄²	—	236	4	no	—
128	C₄²	C₂×Dic₅₉	1	4	yes	—
129	C₄²	C₂×Dic₅₉	1	4	yes	—
130	C₂³	—	2	1	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	2	2	no	—
134	C₂³	C₂²×Dic₅₉	1	2	yes	—
135	C₂³	C₂³×D₅₉	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	118	4	no	—
144	C₂×C₄	—	118	4	no	—
145	C₂×C₄	—	118	4	no	—
146	C₂×C₄	—	118	4	no	—
147	C₂×C₄	—	118	4	no	—
148	C₂×C₄	—	118	4	no	—
149	C₂×C₄	#11745685	1	2	yes	—
150	C₂×C₄	#11745685	1	2	yes	—
151	C₂×C₄	C₂²×Dic₅₉	1	4	yes	—
152	C₂×C₄	C₂²×Dic₅₉	1	4	yes	—
153	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
154	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
161	C₂²	C₂×#11745685	1	1	yes	characteristic
162	C₂²	C₂×#11745685	1	1	yes	characteristic
163	C₂²	#14273773	1	2	yes	—
164	C₂²	#14273774	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	236	4	no	—
169	C₄	—	236	4	no	—
170	C₄	#14273686	1	4	yes	—
171	C₂	—	4	2	no	—
172	C₂	C₂×#14273686	1	1	yes	characteristic
173	C₂	#16837885	1	2	yes	—
174	C₂	#16838007	1	1	yes	characteristic
175	C₂	#16838014	1	1	yes	characteristic
176	C₂	#16838023	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837295

C₂

yes

maximal

#16837288

C₂

yes

maximal, complemented

#16837272

C₂

yes

maximal, characteristic, complemented

#16837268

C₂

yes

maximal, characteristic, complemented

#16837262

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, complemented

C₂×#117×C₅₉

C₂

yes

maximal, 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

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

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

—

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×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

characteristic

C₂²×C₅₉

H(5,𝔽₂)

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

—

Dic₅₉

—

Dic₅₉

—

#2314

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#117

yes

characteristic

C₂×C₅₉

#520

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#117

D₅₉

yes

—

—

—

—

—

C₄×C₄⋊C₄

—

C₅₉

#2314

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

#117

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₂²⋊C₄

C₂×D₅₉

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₅₉

yes

characteristic

107

C₂×C₄²

C₂×D₅₉

yes

characteristic

108

#95

—

109

#95

—

110

#95

—

111

C₂⁴

C₂²×D₅₉

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×Dic₅₉

yes

—

117

C₂²×C₄

C₂²×D₅₉

yes

characteristic

118

C₂²×C₄

C₂²×D₅₉

yes

—

119

C₂²×C₄

C₂²×D₅₉

yes

—

120

C₂²×C₄

C₂²×D₅₉

yes

characteristic

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

—

236

—

124

C₄⋊C₄

C₂×Dic₅₉

yes

—

125

C₂²⋊C₄

—

236

—

126

C₂²⋊C₄

C₂×Dic₅₉

yes

—

127

C₄²

—

236

—

128

C₄²

C₂×Dic₅₉

yes

—

129

C₄²

C₂×Dic₅₉

yes

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²×Dic₅₉

yes

—

135

C₂³

C₂³×D₅₉

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

118

—

146

C₂×C₄

—

118

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

118

—

149

C₂×C₄

#11745685

yes

—

150

C₂×C₄

#11745685

yes

—

151

C₂×C₄

C₂²×Dic₅₉

yes

—

152

C₂×C₄

C₂²×Dic₅₉

yes

—

153

C₂×C₄

C₂²×Dic₅₉

yes

—

154

C₂×C₄

C₂²×Dic₅₉

yes

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂³×Dic₅₉

yes

characteristic

161

C₂²

C₂×#11745685

yes

characteristic

162

C₂²

C₂×#11745685

yes

characteristic

163

C₂²

#14273773

yes

—

164

C₂²

#14273774

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

236

—

169

C₄

—

236

—

170

C₄

#14273686

yes

—

171

C₂

—

172

C₂

C₂×#14273686

yes

characteristic

173

C₂

#16837885

yes

—

174

C₂

#16838007

yes

characteristic

175

C₂

#16838014

yes

characteristic

176

C₂

#16838023

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#117×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₂×#117	D₅₉
hypocenter	C₅₉	#2314
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#117×C₅₉

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×#117

N	H ≅ G/N	type
2. #16837288	C₂	semidirect
3. #16837272	C₂	semidirect
4. #16837268	C₂	semidirect
6. C₄×C₄⋊Dic₅₉	C₂	semidirect
17. #117×C₅₉	C₄	semidirect
96. C₅₉	#2314	semidirect
98. #117	Dic₅₉	semidirect

H ≅ G/N

type

2. #16837288

C₂

semidirect

3. #16837272

C₂

semidirect

4. #16837268

C₂

semidirect

6. C₄×C₄⋊Dic₅₉

C₂

semidirect

17. #117×C₅₉

C₄

semidirect

96. C₅₉

#2314

semidirect

98. #117

Dic₅₉

semidirect

Subgroups — #21426705

Conjugacy classes of subgroups

Complements