Group #21433171 — subgroups

Subgroups — #21433171

Order 7552.

402 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	#16837481	C₂	1	2	yes	maximal, complemented
3	#16837472	C₂	1	4	yes	maximal, complemented
4	C₈×D₂₃₆	C₂	1	2	yes	maximal, complemented
5	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837367	C₂	1	4	yes	maximal, complemented
7	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745685	C₂²	1	2	yes	—
9	C₂×#11745659	C₂²	1	4	yes	—
10	C₂×C₈×D₅₉	C₂²	1	4	yes	—
11	#14273727	C₄	1	1	yes	characteristic
12	#14273724	C₄	1	1	yes	characteristic
13	#14273715	C₄	1	2	yes	—
14	#14273701	C₄	1	2	yes	—
15	#14273701	C₄	1	2	yes	—
16	C₄×D₂₃₆	C₂²	1	2	yes	—
17	C₄×D₂₃₆	C₂²	1	2	yes	—
18	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
19	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	(C₂×D₅₉)⋊C₈	C₂²	1	4	yes	—
21	(C₂×D₅₉)⋊C₈	C₂²	1	4	yes	—
22	#14273638	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
26	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₅₉	C₂×C₄	1	4	yes	—
32	#11745685	—	4	4	no	—
33	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
34	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
35	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₅₉	C₂³	1	4	yes	—
37	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂³	1	2	yes	—
39	C₂×C₈×C₅₉	C₂³	1	2	yes	—
40	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄²×C₅₉	C₂³	1	1	yes	characteristic
44	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
45	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
46	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
48	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
49	#11745659	—	4	4	no	—
50	C₈×D₅₉	—	4	4	no	—
51	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₅₉	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₂²⋊D₅₉	—	4	4	no	—
63	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
64	D₂₃₆	—	2	4	no	—
65	D₂₃₆	—	4	2	no	—
66	D₂₃₆	—	4	2	no	—
67	C₄×D₅₉	—	2	4	no	—
68	C₄×D₅₉	—	2	4	no	—
69	Dic₁₁₈	—	4	2	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	2	2	no	—
72	C₅₉⋊C₈	—	2	4	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	—	2	1	no	—
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₅₉	—	2	4	no	—
78	C₂×D₅₉	—	2	4	no	—
79	C₂×D₅₉	—	2	4	no	—
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	#131	1	2	yes	—
85	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
86	Dic₅₉	—	2	4	no	—
87	#2981	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	—	4	1	no	—
89	C₂×C₅₉	#518	1	1	yes	characteristic
90	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
91	C₂×C₅₉	#568	1	1	yes	characteristic
92	D₅₉	—	4	4	no	—
93	#532	—	59	1	no	—
94	#436	—	59	2	no	—
95	#435	—	59	4	no	—
96	C₈×D₄	—	59	2	no	—
97	#432	—	59	1	no	—
98	#431	D₅₉	1	1	yes	characteristic, complemented
99	#408	—	59	4	no	—
100	C₅₉	#2981	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	59	2	no	—
102	C₂×M₁₆	—	59	4	no	—
103	C₂²×C₈	—	59	4	no	—
104	C₄⋊Q₈	—	59	1	no	—
105	#127	—	59	1	no	—
106	#124	—	59	2	no	—
107	#121	—	59	2	no	—
108	#121	—	59	2	no	—
109	C₄×D₄	—	59	2	no	—
110	C₄×D₄	—	59	2	no	—
111	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
112	C₄⋊C₈	—	59	2	no	—
113	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
114	C₂²⋊C₈	—	59	2	no	—
115	C₂²⋊C₈	—	59	4	no	—
116	C₂²⋊C₈	—	59	4	no	—
117	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
118	#97	—	59	2	no	—
119	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
120	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
121	#41	—	236	4	no	—
122	C₂×Q₈	—	59	2	no	—
123	C₂×D₄	—	59	4	no	—
124	C₂×D₄	—	59	2	no	—
125	C₂×D₄	—	59	2	no	—
126	C₂×D₄	—	59	2	no	—
127	C₂²×C₄	—	59	4	no	—
128	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
131	M₁₆	—	236	4	no	—
132	C₂×C₈	—	59	4	no	—
133	C₂×C₈	—	236	4	no	—
134	C₂×C₈	C₂²×D₅₉	1	2	yes	—
135	C₂×C₈	C₂²×D₅₉	1	2	yes	—
136	C₄⋊C₄	—	59	2	no	—
137	C₄⋊C₄	—	59	2	no	—
138	C₂²⋊C₄	—	59	4	no	—
139	C₂²⋊C₄	—	59	4	no	—
140	C₄²	C₄×D₅₉	1	1	yes	characteristic
141	C₄²	C₄×D₅₉	1	1	yes	characteristic
142	C₄²	C₂²×D₅₉	1	1	yes	characteristic
143	C₄²	C₂²×D₅₉	1	1	yes	characteristic
144	C₂³	—	59	4	no	—
145	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
146	Q₈	—	236	2	no	—
147	D₄	—	118	4	no	—
148	D₄	—	236	2	no	—
149	D₄	—	236	2	no	—
150	D₄	—	236	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	59	4	no	—
155	C₂×C₄	—	118	4	no	—
156	C₂×C₄	—	118	4	no	—
157	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	118	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	118	4	no	—
170	C₂²	—	118	4	no	—
171	C₂²	—	118	4	no	—
172	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	118	4	no	—
178	C₄	#14273733	1	2	yes	—
179	C₄	#14273773	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	236	4	no	—
182	C₂	#16837876	1	1	yes	characteristic
183	C₂	C₂×#14273733	1	1	yes	characteristic
184	C₂	#16838168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

#16837481

C₂

yes

maximal, complemented

#16837472

C₂

yes

maximal, complemented

C₈×D₂₃₆

C₂

yes

maximal, complemented

#16837459

C₂

yes

maximal, characteristic, complemented

#16837367

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

C₂×#11745659

C₂²

yes

—

C₂×C₈×D₅₉

C₂²

yes

—

#14273727

C₄

yes

characteristic

#14273724

C₄

yes

characteristic

#14273715

C₄

yes

—

#14273701

C₄

yes

—

#14273701

C₄

yes

—

C₄×D₂₃₆

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

(C₂×D₅₉)⋊C₈

C₂²

yes

—

(C₂×D₅₉)⋊C₈

C₂²

yes

—

#14273638

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

C₂×C₄

yes

—

#11745685

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

C₂³

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

—

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

#11745659

—

C₈×D₅₉

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂²×D₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

H(5,𝔽₂)

yes

—

Dic₅₉

—

#2981

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#532

—

#436

—

#435

—

C₈×D₄

—

#432

—

#431

D₅₉

yes

characteristic, complemented

#408

—

100

C₅₉

#2981

yes

characteristic, complemented, Sylow

101

C₂×#41

—

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₄⋊Q₈

—

105

#127

—

106

#124

—

107

#121

—

108

#121

—

109

C₄×D₄

—

110

C₄×D₄

—

111

C₂×C₄²

C₂×D₅₉

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

C₂×D₅₉

yes

—

114

C₂²⋊C₈

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

117

C₂²⋊C₈

C₂×D₅₉

yes

—

118

#97

—

119

C₄×C₈

C₂×D₅₉

yes

characteristic

120

C₄×C₈

C₂×D₅₉

yes

characteristic

121

#41

—

236

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₄×D₅₉

yes

characteristic

129

C₂²×C₄

C₄×D₅₉

yes

characteristic

130

C₂²×C₄

C₂²×D₅₉

yes

characteristic

131

M₁₆

—

236

—

132

C₂×C₈

—

133

C₂×C₈

—

236

—

134

C₂×C₈

C₂²×D₅₉

yes

—

135

C₂×C₈

C₂²×D₅₉

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

C₄×D₅₉

yes

characteristic

141

C₄²

C₄×D₅₉

yes

characteristic

142

C₄²

C₂²×D₅₉

yes

characteristic

143

C₄²

C₂²×D₅₉

yes

characteristic

144

C₂³

—

145

C₂³

C₂×C₄×D₅₉

yes

characteristic

146

Q₈

—

236

—

147

D₄

—

118

—

148

D₄

—

236

—

149

D₄

—

236

—

150

D₄

—

236

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

118

—

156

C₂×C₄

—

118

—

157

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

158

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

159

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

160

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

161

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

162

C₂×C₄

C₂³×D₅₉

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

118

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

118

—

170

C₂²

—

118

—

171

C₂²

—

118

—

172

C₂²

C₂²×C₄×D₅₉

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

118

—

178

C₄

#14273733

yes

—

179

C₄

#14273773

yes

—

180

C₂

—

181

C₂

—

236

—

182

C₂

#16837876

yes

characteristic

183

C₂

C₂×#14273733

yes

characteristic

184

C₂

#16838168

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#431	C₂
hypercenter	#431	D₅₉
hypocenter	C₅₉	#2981
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₅₉×#431

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #16837481	C₂	semidirect
3. #16837472	C₂	semidirect
4. C₈×D₂₃₆	C₂	semidirect
5. #16837459	C₂	semidirect
6. #16837367	C₂	semidirect
7. C₅₉×#431	C₂	semidirect
98. #431	D₅₉	semidirect
100. C₅₉	#2981	semidirect

H ≅ G/N

type

2. #16837481

C₂

semidirect

3. #16837472

C₂

semidirect

4. C₈×D₂₃₆

C₂

semidirect

5. #16837459

C₂

semidirect

6. #16837367

C₂

semidirect

7. C₅₉×#431

C₂

semidirect

98. #431

D₅₉

semidirect

100. C₅₉

#2981

semidirect

Subgroups — #21433171

Conjugacy classes of subgroups

Complements