Group #21443121 — subgroups

Subgroups — C₆₁×#3080 #21443121

Order 7808.

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#479	C₂	1	4	yes	maximal, complemented
4	C₆₁×#478	C₂	1	4	yes	maximal, complemented
5	C₆₁×#428	C₂	1	2	yes	maximal, complemented
6	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#417	C₂	1	2	yes	maximal
8	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
10	#125×C₆₁	—	2	1	no	—
11	#125×C₆₁	—	2	1	no	—
12	#125×C₆₁	C₂²	1	4	yes	—
13	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
14	C₄×D₄×C₆₁	—	2	2	no	—
15	C₄×D₄×C₆₁	C₂²	1	2	yes	—
16	#117×C₆₁	—	2	1	no	—
17	#117×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₁	—	2	1	no	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	#107×C₆₁	C₂²	1	4	yes	—
23	#106×C₆₁	C₂²	1	4	yes	—
24	C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
25	#103×C₆₁	C₂²	1	4	yes	—
26	#102×C₆₁	C₂²	1	4	yes	—
27	#41×C₆₁	—	2	2	no	—
28	#41×C₆₁	Q₈	1	4	yes	—
29	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₆₁	—	2	1	no	—
31	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	—	2	1	no	—
33	C₂²×C₄×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	—	2	2	no	—
35	C₂²×C₄×C₆₁	D₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₆₁	—	2	4	no	—
38	C₂×C₈×C₆₁	C₂³	1	4	yes	—
39	C₄⋊C₄×C₆₁	—	2	4	no	—
40	C₄⋊C₄×C₆₁	—	2	4	no	—
41	C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
44	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₆₁	—	2	1	no	—
46	C₂²⋊C₄×C₆₁	—	2	2	no	—
47	C₄²×C₆₁	—	2	1	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	D₄	1	1	yes	characteristic
50	C₄²×C₆₁	D₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂³	1	2	yes	—
52	C₂³×C₆₁	—	2	1	no	—
53	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₆₁	—	2	1	no	—
55	Q₈×C₆₁	C₂×Q₈	1	2	yes	—
56	D₄×C₆₁	—	2	4	no	—
57	D₄×C₆₁	—	2	1	no	—
58	D₄×C₆₁	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	—	2	4	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	4	no	—
63	C₂×C₄×C₆₁	—	4	2	no	—
64	C₂×C₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	4	4	no	—
66	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₆₁	#41	1	2	yes	—
71	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₈×C₆₁	—	2	4	no	—
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	—	2	2	no	—
75	C₂²×C₆₁	—	4	2	no	—
76	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
77	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₆₁	—	2	2	no	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	—	2	2	no	—
81	C₄×C₆₁	—	4	2	no	—
82	C₄×C₆₁	—	4	4	no	—
83	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
84	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
86	#3080	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₆₁	—	2	2	no	—
88	C₂×C₆₁	—	4	2	no	—
89	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₆₁	#575	1	2	yes	—
91	C₂×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
92	C₄×#41	C₂×C₆₁	1	1	yes	characteristic
93	#479	C₂×C₆₁	1	4	yes	complemented
94	#478	C₂×C₆₁	1	4	yes	complemented
95	#428	C₂×C₆₁	1	2	yes	complemented
96	#423	C₂×C₆₁	1	1	yes	characteristic, complemented
97	#417	C₂×C₆₁	1	2	yes	—
98	C₆₁	#3080	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
100	C₂×M₁₆	C₂²×C₆₁	1	2	yes	—
101	#125	—	2	1	no	—
102	#125	—	2	1	no	—
103	#125	C₂²×C₆₁	1	4	yes	—
104	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
105	C₄×D₄	—	2	2	no	—
106	C₄×D₄	C₂²×C₆₁	1	2	yes	—
107	#117	—	2	1	no	—
108	#117	C₂²×C₆₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
111	C₂×C₄²	—	2	1	no	—
112	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
113	#107	C₂²×C₆₁	1	4	yes	—
114	#106	C₂²×C₆₁	1	4	yes	—
115	C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
116	#103	C₂²×C₆₁	1	4	yes	—
117	#102	C₂²×C₆₁	1	4	yes	—
118	#41	—	2	2	no	—
119	#41	Q₈×C₆₁	1	4	yes	—
120	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
121	C₂×D₄	—	2	1	no	—
122	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	D₄×C₆₁	1	2	yes	—
127	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
128	M₁₆	—	2	4	no	—
129	C₂×C₈	C₂³×C₆₁	1	4	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
135	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	2	no	—
140	C₄²	D₄×C₆₁	1	1	yes	characteristic
141	C₄²	D₄×C₆₁	1	1	yes	characteristic
142	C₄²	C₂³×C₆₁	1	2	yes	—
143	C₂³	—	2	1	no	—
144	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
145	Q₈	—	2	1	no	—
146	Q₈	C₂×Q₈×C₆₁	1	2	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	1	no	—
149	D₄	C₂×Q₈×C₆₁	1	2	yes	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
159	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₈×C₆₁	1	2	yes	—
161	C₂×C₄	#41×C₆₁	1	2	yes	—
162	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	C₂²⋊Q₈×C₆₁	1	2	yes	—
168	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	4	no	—
174	C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
175	C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
176	C₄	C₂×#41×C₆₁	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	4	2	no	—
179	C₂	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
180	C₂	C₆₁×#575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₆₁

C₂

yes

maximal, characteristic, complemented

C₄×#41×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#479

C₂

yes

maximal, complemented

C₆₁×#478

C₂

yes

maximal, complemented

C₆₁×#428

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic, complemented

C₆₁×#417

C₂

yes

maximal

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂×M₁₆×C₆₁

C₂²

yes

—

#125×C₆₁

—

#125×C₆₁

—

#125×C₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

—

C₄×D₄×C₆₁

—

C₄×D₄×C₆₁

C₂²

yes

—

#117×C₆₁

—

#117×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#107×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₄⋊C₈×C₆₁

C₂²

yes

—

#103×C₆₁

C₂²

yes

—

#102×C₆₁

C₂²

yes

—

#41×C₆₁

—

#41×C₆₁

Q₈

yes

—

C₂×Q₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

C₂×Q₈

yes

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

#3080

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₁

#575

yes

—

C₂×#125

C₂×C₆₁

yes

characteristic, complemented

C₄×#41

C₂×C₆₁

yes

characteristic

#479

C₂×C₆₁

yes

complemented

#478

C₂×C₆₁

yes

complemented

#428

C₂×C₆₁

yes

complemented

#423

C₂×C₆₁

yes

characteristic, complemented

#417

C₂×C₆₁

yes

—

C₆₁

#3080

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₆₁

yes

characteristic

100

C₂×M₁₆

C₂²×C₆₁

yes

—

101

#125

—

102

#125

—

103

#125

C₂²×C₆₁

yes

—

104

C₄×Q₈

C₂²×C₆₁

yes

—

105

C₄×D₄

—

106

C₄×D₄

C₂²×C₆₁

yes

—

107

#117

—

108

#117

C₂²×C₆₁

yes

characteristic

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂²×C₆₁

yes

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂²×C₆₁

yes

characteristic

113

#107

C₂²×C₆₁

yes

—

114

#106

C₂²×C₆₁

yes

—

115

C₄⋊C₈

C₂²×C₆₁

yes

—

116

#103

C₂²×C₆₁

yes

—

117

#102

C₂²×C₆₁

yes

—

118

#41

—

119

#41

Q₈×C₆₁

yes

—

120

C₂×Q₈

C₂³×C₆₁

yes

characteristic

121

C₂×D₄

—

122

C₂×D₄

C₂³×C₆₁

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×C₆₁

yes

—

127

C₂²×C₄

C₂³×C₆₁

yes

characteristic

128

M₁₆

—

129

C₂×C₈

C₂³×C₆₁

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₂³×C₆₁

yes

—

135

C₄⋊C₄

C₂³×C₆₁

yes

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

D₄×C₆₁

yes

characteristic

141

C₄²

D₄×C₆₁

yes

characteristic

142

C₄²

C₂³×C₆₁

yes

—

143

C₂³

—

144

C₂³

C₂×D₄×C₆₁

yes

characteristic

145

Q₈

—

146

Q₈

C₂×Q₈×C₆₁

yes

—

147

D₄

—

148

D₄

—

149

D₄

C₂×Q₈×C₆₁

yes

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×D₄×C₆₁

yes

—

158

C₂×C₄

C₂×D₄×C₆₁

yes

—

159

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

160

C₂×C₄

C₂×Q₈×C₆₁

yes

—

161

C₂×C₄

#41×C₆₁

yes

—

162

C₂×C₄

C₂⁴×C₆₁

yes

characteristic

163

C₈

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

C₂²⋊Q₈×C₆₁

yes

—

168

C₂²

C₂²×D₄×C₆₁

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

C₂²⋊Q₈×C₆₁

yes

—

175

C₄

C₂²×Q₈×C₆₁

yes

characteristic

176

C₄

C₂×#41×C₆₁

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

C₂×C₂²⋊Q₈×C₆₁

yes

characteristic

180

C₂

C₆₁×#575

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₆₁

prime radical

C₂×#41×C₆₁

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
1. C₂×#125×C₆₁	C₂	semidirect
3. C₆₁×#479	C₂	semidirect
4. C₆₁×#478	C₂	semidirect
5. C₆₁×#428	C₂	semidirect
6. C₆₁×#423	C₂	semidirect
86. #3080	C₆₁	direct (N × H)
91. C₂×#125	C₂×C₆₁	semidirect
93. #479	C₂×C₆₁	semidirect
94. #478	C₂×C₆₁	semidirect
95. #428	C₂×C₆₁	semidirect
96. #423	C₂×C₆₁	semidirect
98. C₆₁	#3080	direct (N × H)

H ≅ G/N

type

1. C₂×#125×C₆₁

C₂

semidirect

3. C₆₁×#479

C₂

semidirect

4. C₆₁×#478

C₂

semidirect

5. C₆₁×#428

C₂

semidirect

6. C₆₁×#423

C₂

semidirect

86. #3080

C₆₁

direct (N × H)

91. C₂×#125

C₂×C₆₁

semidirect

93. #479

C₂×C₆₁

semidirect

94. #478

C₂×C₆₁

semidirect

95. #428

C₂×C₆₁

semidirect

96. #423

C₂×C₆₁

semidirect

98. C₆₁

#3080

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#125×C₆₁	2
C₄×#41×C₆₁	2
C₆₁×#479	2
C₆₁×#478	2
C₆₁×#428	2
C₆₁×#423	2
C₆₁×#417	2
#3080	61

subgroup

index

C₂×#125×C₆₁

C₄×#41×C₆₁

C₆₁×#479

C₆₁×#478

C₆₁×#428

C₆₁×#423

C₆₁×#417

#3080

Subgroups — C₆₁×#3080 #21443121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#3080 #21443121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3080 #21443121