Group #21457539 — subgroups

Subgroups — #21457539

Order 7808.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#14519800	C₂	1	2	yes	maximal, complemented
2	#16840415	C₂	1	2	yes	maximal, complemented
3	#16840000	C₂	1	2	yes	maximal, complemented
4	#16839964	C₂	1	2	yes	maximal, complemented
5	#16839923	C₂	1	4	yes	maximal, complemented
6	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11758226	C₂²	1	4	yes	—
10	#14519800	—	2	8	no	—
11	C₂×#11758201	C₂²	1	4	yes	—
12	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₆₁	C₂²	1	2	yes	—
14	#14519762	—	2	4	no	—
15	#14519753	C₂²	1	4	yes	—
16	#14519752	—	2	2	no	—
17	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
18	C₄×D₂₄₄	C₂²	1	2	yes	—
19	C₄⋊C₄×D₆₁	—	2	2	no	—
20	#14519711	C₂²	1	1	yes	characteristic
21	C₄⋊#92418	C₂²	1	1	yes	characteristic
22	#14519701	C₂²	1	4	yes	—
23	#14519699	C₂²	1	2	yes	—
24	#14519689	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
28	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
30	#11758226	—	2	8	no	—
31	#11758226	—	4	4	no	—
32	C₂×D₂₄₄	—	2	4	no	—
33	C₂×D₂₄₄	D₄	1	1	yes	characteristic
34	C₂×D₂₄₄	C₂³	1	2	yes	—
35	C₂×C₄×D₆₁	—	2	2	no	—
36	C₂×C₄×D₆₁	—	2	4	no	—
37	C₂×C₄×D₆₁	C₂³	1	2	yes	—
38	Q₁₆×C₆₁	—	2	4	no	—
39	Q₁₆×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	#11758201	—	2	8	no	—
46	#11758198	—	2	4	no	—
47	#11758198	—	2	2	no	—
48	#11758198	D₄	1	2	yes	—
49	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
50	C₄⋊Dic₆₁	C₂³	1	2	yes	—
51	#11758196	—	2	4	no	—
52	C₂×#92418	C₂³	1	2	yes	—
53	D₄₈₈	D₄	1	4	yes	complemented
54	C₈×D₆₁	—	2	4	no	—
55	C₂²×D₆₁	—	2	2	no	—
56	C₂²×D₆₁	C₂×D₄	1	2	yes	—
57	Q₈×C₆₁	—	2	4	no	—
58	Q₈×C₆₁	—	2	4	no	—
59	C₂×C₄×C₆₁	—	2	4	no	—
60	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₆₁	—	2	2	no	—
64	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
65	D₂₄₄	—	2	2	no	—
66	D₂₄₄	—	2	2	no	—
67	D₂₄₄	—	4	8	no	—
68	D₂₄₄	C₂×D₄	1	4	yes	—
69	C₄×D₆₁	—	2	2	no	—
70	C₄×D₆₁	—	2	2	no	—
71	C₄×D₆₁	—	4	8	no	—
72	C₄×D₆₁	—	4	4	no	—
73	C₈×C₆₁	—	2	1	no	—
74	C₈×C₆₁	C₂×D₄	1	2	yes	—
75	#92418	—	2	2	no	—
76	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×D₆₁	—	2	2	no	—
78	C₂×D₆₁	—	2	4	no	—
79	C₂×D₆₁	—	2	2	no	—
80	C₂×D₆₁	—	2	2	no	—
81	C₂×D₆₁	—	4	2	no	—
82	C₂×D₆₁	—	4	2	no	—
83	C₄×C₆₁	—	2	1	no	—
84	C₄×C₆₁	—	4	4	no	—
85	C₄×C₆₁	#135	1	2	yes	complemented
86	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
88	Dic₆₁	—	2	2	no	—
89	Dic₆₁	—	4	2	no	—
90	#3283	—	61	1	no	maximal, Sylow
91	C₂×C₆₁	D₄²	1	1	yes	characteristic
92	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
93	C₂×C₆₁	#578	1	1	yes	characteristic
94	D₆₁	—	4	2	no	—
95	D₆₁	—	4	4	no	—
96	C₂×#135	—	61	2	no	—
97	#547	—	61	2	no	—
98	#494	D₆₁	1	1	yes	characteristic, complemented
99	#467	—	61	2	no	—
100	#461	—	61	2	no	—
101	#452	—	61	4	no	—
102	C₄×D₈	—	61	1	no	—
103	C₈×D₄	—	61	1	no	—
104	C₆₁	#3283	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	61	4	no	—
106	#135	—	122	8	no	—
107	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
108	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
109	C₂×SD₁₆	—	61	4	no	—
110	C₂×D₈	—	61	1	no	—
111	C₂²×C₈	—	61	2	no	—
112	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
113	#123	—	122	4	no	—
114	C₂²⋊Q₈	—	61	4	no	—
115	#121	—	122	2	no	—
116	C₄×D₄	—	61	1	no	—
117	C₄×D₄	—	61	2	no	—
118	C₂×C₄⋊C₄	—	122	2	no	—
119	#107	—	61	1	no	—
120	C₄⋊C₈	—	61	1	no	—
121	#103	—	61	4	no	—
122	#102	—	61	2	no	—
123	C₂²⋊C₈	—	61	2	no	—
124	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
125	#41	—	122	8	no	—
126	#41	—	244	4	no	—
127	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
128	C₂×D₄	—	61	1	no	—
129	C₂×D₄	—	61	2	no	—
130	C₂×D₄	—	122	4	no	—
131	C₂²×C₄	—	61	2	no	—
132	C₂²×C₄	—	122	4	no	—
133	C₂²×C₄	—	122	2	no	—
134	Q₁₆	—	2	4	no	—
135	Q₁₆	—	4	2	no	—
136	SD₁₆	—	122	8	no	—
137	D₈	—	61	4	no	—
138	C₂×C₈	—	61	2	no	—
139	C₂×C₈	—	122	4	no	—
140	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
141	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	61	1	no	—
145	C₄⋊C₄	—	61	2	no	—
146	C₄⋊C₄	—	122	4	no	—
147	C₂²⋊C₄	—	61	2	no	—
148	C₂²⋊C₄	—	122	4	no	—
149	C₂²⋊C₄	—	122	2	no	—
150	C₄²	C₂²×D₆₁	1	1	yes	characteristic
151	C₂³	—	61	2	no	—
152	C₂³	—	122	2	no	—
153	Q₈	—	2	4	no	—
154	Q₈	—	2	4	no	—
155	D₄	—	61	4	no	—
156	D₄	—	122	2	no	—
157	D₄	—	122	2	no	—
158	D₄	—	244	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	61	2	no	—
161	C₂×C₄	—	122	2	no	—
162	C₂×C₄	—	122	2	no	—
163	C₂×C₄	—	122	2	no	—
164	C₂×C₄	—	244	8	no	—
165	C₂×C₄	—	244	4	no	—
166	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
167	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
168	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	122	2	no	—
171	C₈	D₄×D₆₁	1	2	yes	—
172	C₂²	—	122	2	no	—
173	C₂²	—	122	2	no	—
174	C₂²	—	122	2	no	—
175	C₂²	—	122	4	no	—
176	C₂²	—	244	2	no	—
177	C₂²	—	244	2	no	—
178	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	122	2	no	—
182	C₄	—	244	2	no	—
183	C₄	#14519800	1	2	yes	complemented
184	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
185	C₄	#14519822	1	1	yes	characteristic
186	C₂	—	244	2	no	—
187	C₂	—	244	4	no	—
188	C₂	#16840397	1	1	yes	characteristic
189	C₂	C₂×#14519800	1	1	yes	characteristic
190	C₂	#16840575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519800

C₂

yes

maximal, complemented

#16840415

C₂

yes

maximal, complemented

#16840000

C₂

yes

maximal, complemented

#16839964

C₂

yes

maximal, complemented

#16839923

C₂

yes

maximal, complemented

C₄×D₄₈₈

C₂

yes

maximal, characteristic, complemented

C₈×D₂₄₄

C₂

yes

maximal, characteristic, complemented

C₆₁×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

C₂²

yes

—

#14519800

—

C₂×#11758201

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₂²

yes

—

#14519762

—

#14519753

C₂²

yes

—

#14519752

—

C₄×D₂₄₄

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

C₄⋊C₄×D₆₁

—

#14519711

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519701

C₂²

yes

—

#14519699

C₂²

yes

—

#14519689

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

#11758226

—

#11758226

—

C₂×D₂₄₄

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂³

yes

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

D₄

yes

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₂×#92418

C₂³

yes

—

D₄₈₈

D₄

yes

complemented

C₈×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂×D₄

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

C₂×D₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

—

#92418

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

complemented

C₄×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#3283

—

maximal, Sylow

C₂×C₆₁

D₄²

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

C₂×#135

—

#547

—

#494

D₆₁

yes

characteristic, complemented

#467

—

100

#461

—

101

#452

—

102

C₄×D₈

—

103

C₈×D₄

—

104

C₆₁

#3283

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

#135

—

122

—

107

C₂×Q₁₆

C₂×D₆₁

yes

—

108

C₂×Q₁₆

C₂×D₆₁

yes

—

109

C₂×SD₁₆

—

110

C₂×D₈

—

111

C₂²×C₈

—

112

C₄⋊Q₈

C₂×D₆₁

yes

—

113

#123

—

122

—

114

C₂²⋊Q₈

—

115

#121

—

122

—

116

C₄×D₄

—

117

C₄×D₄

—

118

C₂×C₄⋊C₄

—

122

—

119

#107

—

120

C₄⋊C₈

—

121

#103

—

122

#102

—

123

C₂²⋊C₈

—

124

C₄×C₈

C₂×D₆₁

yes

characteristic

125

#41

—

122

—

126

#41

—

244

—

127

C₂×Q₈

C₂²×D₆₁

yes

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

122

—

131

C₂²×C₄

—

132

C₂²×C₄

—

122

—

133

C₂²×C₄

—

122

—

134

Q₁₆

—

135

Q₁₆

—

136

SD₁₆

—

122

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

122

—

140

C₂×C₈

C₂²×D₆₁

yes

characteristic

141

C₂×C₈

C₂²×D₆₁

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

122

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

122

—

149

C₂²⋊C₄

—

122

—

150

C₄²

C₂²×D₆₁

yes

characteristic

151

C₂³

—

152

C₂³

—

122

—

153

Q₈

—

154

Q₈

—

155

D₄

—

156

D₄

—

122

—

157

D₄

—

122

—

158

D₄

—

244

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

122

—

162

C₂×C₄

—

122

—

163

C₂×C₄

—

122

—

164

C₂×C₄

—

244

—

165

C₂×C₄

—

244

—

166

C₂×C₄

D₄×D₆₁

yes

characteristic

167

C₂×C₄

D₄×D₆₁

yes

characteristic

168

C₂×C₄

C₂³×D₆₁

yes

characteristic

169

C₈

—

170

C₈

—

122

—

171

C₈

D₄×D₆₁

yes

—

172

C₂²

—

122

—

173

C₂²

—

122

—

174

C₂²

—

122

—

175

C₂²

—

122

—

176

C₂²

—

244

—

177

C₂²

—

244

—

178

C₂²

C₂×D₄×D₆₁

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

122

—

182

C₄

—

244

—

183

C₄

#14519800

yes

complemented

184

C₄

C₂×D₄×D₆₁

yes

characteristic

185

C₄

#14519822

yes

characteristic

186

C₂

—

244

—

187

C₂

—

244

—

188

C₂

#16840397

yes

characteristic

189

C₂

C₂×#14519800

yes

characteristic

190

C₂

#16840575

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#494	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#494	D₆₁
hypocenter	C₆₁	#3283
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#494

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14519800	C₂	semidirect
2. #16840415	C₂	semidirect
3. #16840000	C₂	semidirect
4. #16839964	C₂	semidirect
5. #16839923	C₂	semidirect
6. C₄×D₄₈₈	C₂	semidirect
7. C₈×D₂₄₄	C₂	semidirect
8. C₆₁×#494	C₂	semidirect
53. D₄₈₈	D₄	semidirect
85. C₄×C₆₁	#135	semidirect
98. #494	D₆₁	semidirect
104. C₆₁	#3283	semidirect
183. C₄	#14519800	semidirect

H ≅ G/N

type

1. C₂×#14519800

C₂

semidirect

2. #16840415

C₂

semidirect

3. #16840000

C₂

semidirect

4. #16839964

C₂

semidirect

5. #16839923

C₂

semidirect

6. C₄×D₄₈₈

C₂

semidirect

7. C₈×D₂₄₄

C₂

semidirect

8. C₆₁×#494

C₂

semidirect

53. D₄₈₈

D₄

semidirect

85. C₄×C₆₁

#135

semidirect

98. #494

D₆₁

semidirect

104. C₆₁

#3283

semidirect

183. C₄

#14519800

semidirect

Subgroups — #21457539

Conjugacy classes of subgroups

Complements