Group #21458792 — subgroups

Subgroups — #21458792

Order 7808.

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

normal subgroups only

This group has 460 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	#16840432	C₂	1	2	yes	maximal, complemented
2	#16840137	C₂	1	2	yes	maximal, complemented
3	#16839955	C₂	1	4	yes	maximal, complemented
4	#16839954	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₆₁	C₂	1	2	yes	maximal, complemented
8	C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11758224	—	2	2	no	—
10	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
11	C₂×#11758199	C₂²	1	4	yes	—
12	#14519773	C₂²	1	4	yes	—
13	#14519749	—	2	4	no	—
14	#14519748	—	2	2	no	—
15	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	C₄×D₂₄₄	C₂²	1	2	yes	—
17	D₄×Dic₆₁	C₂²	1	4	yes	—
18	#14519711	C₂²	1	1	yes	characteristic
19	C₄⋊#92418	C₂²	1	1	yes	characteristic
20	#14519706	C₂²	1	2	yes	—
21	#14519699	C₂²	1	2	yes	—
22	#14519698	C₂²	1	4	yes	—
23	C₈×Dic₆₁	C₂²	1	2	yes	—
24	C₂×D₈×C₆₁	C₂²	1	2	yes	—
25	C₂×D₈×C₆₁	C₂²	1	2	yes	—
26	#127×C₆₁	C₂²	1	2	yes	—
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×D₄×C₆₁	—	2	2	no	—
29	C₂×D₄×C₆₁	—	2	2	no	—
30	C₂×D₄×C₆₁	C₂³	1	4	yes	—
31	C₂×C₂²⋊D₆₁	—	2	4	no	—
32	C₂×C₂²⋊D₆₁	—	2	4	no	—
33	C₂²×Dic₆₁	—	2	4	no	—
34	#11758224	—	4	8	no	—
35	C₂×D₂₄₄	C₂³	1	2	yes	—
36	C₂×C₄×D₆₁	—	2	2	no	—
37	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
38	D₈×C₆₁	—	2	4	no	—
39	D₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₆₁	—	2	4	no	—
44	#11758199	—	4	4	no	—
45	#11758198	—	2	2	no	—
46	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
47	C₄⋊Dic₆₁	C₂³	1	2	yes	—
48	#11758196	D₄	1	2	yes	—
49	C₄×Dic₆₁	C₂³	1	2	yes	—
50	C₂×#92418	C₂³	1	2	yes	—
51	D₄₈₈	—	2	2	no	—
52	C₂³×C₆₁	—	2	4	no	—
53	C₂²×D₆₁	—	2	2	no	—
54	D₄×C₆₁	—	2	4	no	—
55	D₄×C₆₁	—	2	4	no	—
56	D₄×C₆₁	—	4	4	no	—
57	D₄×C₆₁	—	8	2	no	—
58	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₆₁	—	4	8	no	—
62	C₂²⋊D₆₁	—	8	4	no	—
63	C₂×Dic₆₁	—	2	2	no	—
64	C₂×Dic₆₁	—	4	8	no	—
65	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
66	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
67	D₂₄₄	—	2	2	no	—
68	D₂₄₄	—	2	2	no	—
69	C₄×D₆₁	—	4	4	no	—
70	Dic₁₂₂	D₈	1	4	yes	complemented
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	#41	1	2	yes	complemented
73	#92418	—	2	2	no	—
74	C₂²×C₆₁	—	4	4	no	—
75	C₂²×C₆₁	—	4	4	no	—
76	C₂²×C₆₁	—	4	4	no	—
77	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×D₆₁	—	4	2	no	—
79	C₂×D₆₁	—	4	2	no	—
80	C₂×D₆₁	—	4	2	no	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	C₂×D₈	1	2	yes	—
83	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
84	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
85	Dic₆₁	—	2	2	no	—
86	Dic₆₁	—	4	2	no	—
87	Dic₆₁	C₂×D₈	1	4	yes	—
88	#3391	—	61	1	no	maximal, Sylow
89	C₂×C₆₁	—	8	4	no	—
90	C₂×C₆₁	#550	1	1	yes	characteristic
91	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
92	C₂×C₆₁	#576	1	1	yes	characteristic
93	D₆₁	—	8	2	no	—
94	#550	—	61	2	no	—
95	#493	—	61	2	no	—
96	#493	D₆₁	1	1	yes	characteristic, complemented
97	#459	—	61	4	no	—
98	#459	—	61	2	no	—
99	C₈×Q₈	—	61	1	no	—
100	C₄×D₈	—	61	2	no	—
101	C₄×D₈	—	61	1	no	—
102	C₆₁	#3391	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	122	2	no	—
104	C₂×D₈	—	61	4	no	—
105	C₂×D₈	—	61	1	no	—
106	C₂×D₈	C₂×D₆₁	1	2	yes	—
107	C₂×D₈	C₂×D₆₁	1	2	yes	—
108	#127	—	61	4	no	—
109	#127	C₂×D₆₁	1	2	yes	—
110	#121	—	122	2	no	—
111	#121	—	122	4	no	—
112	C₄×Q₈	—	61	1	no	—
113	C₄×D₄	—	61	4	no	—
114	C₄×D₄	—	61	2	no	—
115	#107	—	61	1	no	—
116	C₄⋊C₈	—	61	2	no	—
117	C₄⋊C₈	—	61	1	no	—
118	#102	—	61	4	no	—
119	#102	—	61	2	no	—
120	C₄×C₈	—	61	2	no	—
121	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
122	#41	—	244	8	no	—
123	C₂×Q₈	—	61	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	61	2	no	—
127	C₂×D₄	—	122	4	no	—
128	C₂×D₄	—	122	4	no	—
129	C₂×D₄	C₂²×D₆₁	1	4	yes	—
130	C₂²×C₄	—	122	4	no	—
131	C₂²×C₄	—	122	2	no	—
132	D₈	—	2	4	no	—
133	D₈	—	4	2	no	—
134	D₈	—	122	2	no	—
135	D₈	—	244	4	no	—
136	C₂×C₈	—	61	2	no	—
137	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
138	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
139	C₄⋊C₄	—	61	2	no	—
140	C₄⋊C₄	—	61	1	no	—
141	C₄⋊C₄	—	61	2	no	—
142	C₂²⋊C₄	—	122	4	no	—
143	C₂²⋊C₄	—	122	2	no	—
144	C₄²	—	61	2	no	—
145	C₄²	C₂²×D₆₁	1	1	yes	characteristic
146	C₂³	—	2	4	no	—
147	C₂³	—	122	2	no	—
148	Q₈	—	61	4	no	—
149	D₄	—	2	4	no	—
150	D₄	—	2	4	no	—
151	D₄	—	4	4	no	—
152	D₄	—	8	2	no	—
153	D₄	—	122	2	no	—
154	D₄	—	122	2	no	—
155	D₄	—	488	4	no	—
156	D₄	—	244	8	no	—
157	C₂×C₄	—	61	2	no	—
158	C₂×C₄	—	61	2	no	—
159	C₂×C₄	—	122	2	no	—
160	C₂×C₄	—	244	8	no	—
161	C₂×C₄	—	244	4	no	—
162	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
163	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	122	2	no	—
167	C₈	#11758224	1	2	yes	complemented
168	C₂²	—	4	4	no	—
169	C₂²	—	4	4	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	244	2	no	—
172	C₂²	—	244	2	no	—
173	C₂²	—	244	2	no	—
174	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	61	4	no	—
177	C₄	—	122	2	no	—
178	C₄	—	244	2	no	—
179	C₄	D₈×D₆₁	1	2	yes	—
180	C₄	C₂×#11758224	1	1	yes	characteristic
181	C₄	#14519822	1	1	yes	characteristic
182	C₂	—	8	4	no	—
183	C₂	—	488	2	no	—
184	C₂	#16840432	1	1	yes	characteristic
185	C₂	C₂×D₈×D₆₁	1	1	yes	characteristic
186	C₂	#16840563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840432

C₂

yes

maximal, complemented

#16840137

C₂

yes

maximal, complemented

#16839955

C₂

yes

maximal, complemented

#16839954

C₂

yes

maximal, complemented

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

C₄×D₄₈₈

C₂

yes

maximal, characteristic, complemented

D₈×Dic₆₁

C₂

yes

maximal, complemented

C₆₁×#493

C₂

yes

maximal, characteristic, complemented

C₂×#11758224

—

C₂×D₄₈₈

C₂²

yes

characteristic

C₂×#11758199

C₂²

yes

—

#14519773

C₂²

yes

—

#14519749

—

#14519748

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

D₄×Dic₆₁

C₂²

yes

—

#14519711

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519706

C₂²

yes

—

#14519699

C₂²

yes

—

#14519698

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×D₈×C₆₁

C₂²

yes

—

C₂×D₈×C₆₁

C₂²

yes

—

#127×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

#11758224

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

D₄

yes

characteristic

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

#11758199

—

#11758198

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

D₄₈₈

—

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

D₈

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

#41

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×D₈

yes

—

#3391

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#550

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

#576

yes

characteristic

D₆₁

—

#550

—

#493

—

#493

D₆₁

yes

characteristic, complemented

#459

—

#459

—

C₈×Q₈

—

100

C₄×D₈

—

101

C₄×D₈

—

102

C₆₁

#3391

yes

characteristic, complemented, Sylow

103

C₂×#41

—

122

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

C₂×D₆₁

yes

—

107

C₂×D₈

C₂×D₆₁

yes

—

108

#127

—

109

#127

C₂×D₆₁

yes

—

110

#121

—

122

—

111

#121

—

122

—

112

C₄×Q₈

—

113

C₄×D₄

—

114

C₄×D₄

—

115

#107

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#102

—

119

#102

—

120

C₄×C₈

—

121

C₄×C₈

C₂×D₆₁

yes

characteristic

122

#41

—

244

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

122

—

128

C₂×D₄

—

122

—

129

C₂×D₄

C₂²×D₆₁

yes

—

130

C₂²×C₄

—

122

—

131

C₂²×C₄

—

122

—

132

D₈

—

133

D₈

—

134

D₈

—

122

—

135

D₈

—

244

—

136

C₂×C₈

—

137

C₂×C₈

C₂²×D₆₁

yes

characteristic

138

C₂×C₈

C₂²×D₆₁

yes

characteristic

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

122

—

143

C₂²⋊C₄

—

122

—

144

C₄²

—

145

C₄²

C₂²×D₆₁

yes

characteristic

146

C₂³

—

147

C₂³

—

122

—

148

Q₈

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

122

—

154

D₄

—

122

—

155

D₄

—

488

—

156

D₄

—

244

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

122

—

160

C₂×C₄

—

244

—

161

C₂×C₄

—

244

—

162

C₂×C₄

D₄×D₆₁

yes

characteristic

163

C₂×C₄

D₄×D₆₁

yes

characteristic

164

C₂×C₄

C₂³×D₆₁

yes

characteristic

165

C₈

—

166

C₈

—

122

—

167

C₈

#11758224

yes

complemented

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

244

—

172

C₂²

—

244

—

173

C₂²

—

244

—

174

C₂²

C₂×D₄×D₆₁

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

122

—

178

C₄

—

244

—

179

C₄

D₈×D₆₁

yes

—

180

C₄

C₂×#11758224

yes

characteristic

181

C₄

#14519822

yes

characteristic

182

C₂

—

183

C₂

—

488

—

184

C₂

#16840432

yes

characteristic

185

C₂

C₂×D₈×D₆₁

yes

characteristic

186

C₂

#16840563

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#493	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#493	D₆₁
hypocenter	C₆₁	#3391
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#493

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16840432	C₂	semidirect
2. #16840137	C₂	semidirect
3. #16839955	C₂	semidirect
4. #16839954	C₂	semidirect
5. C₈×Dic₁₂₂	C₂	semidirect
6. C₄×D₄₈₈	C₂	semidirect
7. D₈×Dic₆₁	C₂	semidirect
8. C₆₁×#493	C₂	semidirect
70. Dic₁₂₂	D₈	semidirect
72. C₈×C₆₁	#41	semidirect
96. #493	D₆₁	semidirect
102. C₆₁	#3391	semidirect
167. C₈	#11758224	semidirect

H ≅ G/N

type

1. #16840432

C₂

semidirect

2. #16840137

C₂

semidirect

3. #16839955

C₂

semidirect

4. #16839954

C₂

semidirect

5. C₈×Dic₁₂₂

C₂

semidirect

6. C₄×D₄₈₈

C₂

semidirect

7. D₈×Dic₆₁

C₂

semidirect

8. C₆₁×#493

C₂

semidirect

70. Dic₁₂₂

D₈

semidirect

72. C₈×C₆₁

#41

semidirect

96. #493

D₆₁

semidirect

102. C₆₁

#3391

semidirect

167. C₈

#11758224

semidirect

Subgroups — #21458792

Conjugacy classes of subgroups

Complements