Group #22585415 — subgroups

Subgroups — C₇₁×#2853 #22585415

Order 9088.

582 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 216 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×#123×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#388×C₇₁	C₂	1	4	yes	maximal, complemented
4	#387×C₇₁	C₂	1	4	yes	maximal, complemented
5	#386×C₇₁	C₂	1	2	yes	maximal
6	#385×C₇₁	C₂	1	2	yes	maximal, complemented
7	#381×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₇₁	C₂²	1	2	yes	—
10	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
11	#123×C₇₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₇₁	C₂²	1	4	yes	—
18	C₂×C₄²×C₇₁	—	2	4	no	—
19	C₂×C₄²×C₇₁	C₂²	1	4	yes	—
20	#95×C₇₁	C₂²	1	4	yes	—
21	#95×C₇₁	C₂²	1	4	yes	—
22	#95×C₇₁	C₂²	1	4	yes	—
23	C₂⁴×C₇₁	C₂³	1	1	yes	characteristic
24	C₂⁴×C₇₁	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₇₁	—	2	2	no	—
26	C₂×D₄×C₇₁	—	2	4	no	—
27	C₂²×C₄×C₇₁	—	2	4	no	—
28	C₂²×C₄×C₇₁	—	2	4	no	—
29	C₂²×C₄×C₇₁	—	2	8	no	—
30	C₂²×C₄×C₇₁	D₄	1	4	yes	—
31	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
35	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
36	C₄⋊C₄×C₇₁	—	2	4	no	—
37	C₄⋊C₄×C₇₁	—	2	8	no	—
38	C₂²⋊C₄×C₇₁	—	2	2	no	—
39	C₂²⋊C₄×C₇₁	—	2	4	no	—
40	C₂²⋊C₄×C₇₁	—	2	4	no	—
41	C₂²⋊C₄×C₇₁	—	2	8	no	—
42	C₄²×C₇₁	—	2	8	no	—
43	C₂³×C₇₁	—	2	4	no	—
44	C₂³×C₇₁	—	2	1	no	—
45	C₂³×C₇₁	—	2	4	no	—
46	C₂³×C₇₁	—	2	2	no	—
47	C₂³×C₇₁	C₂×D₄	1	2	yes	—
48	C₂³×C₇₁	#41	1	4	yes	—
49	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
50	D₄×C₇₁	—	4	4	no	—
51	C₂×C₄×C₇₁	—	2	16	no	—
52	C₂×C₄×C₇₁	—	2	8	no	—
53	C₂×C₄×C₇₁	—	2	4	no	—
54	C₂×C₄×C₇₁	—	2	4	no	—
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	2	1	no	—
58	C₂×C₄×C₇₁	—	2	4	no	—
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	—	2	4	no	—
61	C₂×C₄×C₇₁	—	2	4	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₇₁	#41	1	8	yes	—
65	C₂²×C₇₁	—	2	4	no	—
66	C₂²×C₇₁	—	2	8	no	—
67	C₂²×C₇₁	—	4	1	no	—
68	C₂²×C₇₁	—	4	2	no	—
69	C₂²×C₇₁	—	4	4	no	—
70	C₂²×C₇₁	#123	1	4	yes	—
71	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₇₁	C₂×#41	1	2	yes	—
73	C₂²×C₇₁	C₂×#41	1	4	yes	—
74	C₄×C₇₁	—	2	4	no	—
75	C₄×C₇₁	—	2	8	no	—
76	C₄×C₇₁	—	4	1	no	—
77	C₄×C₇₁	—	4	2	no	—
78	C₄×C₇₁	—	4	4	no	—
79	C₂×C₇₁	—	2	4	no	—
80	C₂×C₇₁	—	8	1	no	—
81	C₂×C₇₁	C₂×#123	1	1	yes	characteristic
82	C₂×C₇₁	#525	1	2	yes	—
83	C₂×C₇₁	#529	1	4	yes	—
84	#2853	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₇₁	#2853	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#123	C₂×C₇₁	1	1	yes	characteristic
87	C₂²×C₄²	C₂×C₇₁	1	1	yes	characteristic, complemented
88	#388	C₂×C₇₁	1	4	yes	complemented
89	#387	C₂×C₇₁	1	4	yes	complemented
90	#386	C₂×C₇₁	1	2	yes	—
91	#385	C₂×C₇₁	1	2	yes	complemented
92	#381	C₂×C₇₁	1	1	yes	characteristic, complemented
93	C₂²×D₄	C₂²×C₇₁	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₇₁	1	2	yes	—
95	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
96	#123	—	2	4	no	—
97	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
100	C₂×C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₇₁	1	4	yes	—
103	C₂×C₄²	—	2	4	no	—
104	C₂×C₄²	C₂²×C₇₁	1	4	yes	—
105	#95	C₂²×C₇₁	1	4	yes	—
106	#95	C₂²×C₇₁	1	4	yes	—
107	#95	C₂²×C₇₁	1	4	yes	—
108	C₂⁴	C₂³×C₇₁	1	1	yes	characteristic
109	C₂⁴	C₂³×C₇₁	1	1	yes	characteristic
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	D₄×C₇₁	1	4	yes	—
116	C₂²×C₄	C₂³×C₇₁	1	4	yes	—
117	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
119	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
120	C₂²×C₄	C₂³×C₇₁	1	4	yes	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	8	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	2	4	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₄²	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	1	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	2	no	—
132	C₂³	C₂×D₄×C₇₁	1	2	yes	—
133	C₂³	#41×C₇₁	1	4	yes	—
134	C₂³	C₂⁴×C₇₁	1	1	yes	characteristic
135	D₄	—	4	4	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₂×D₄×C₇₁	1	4	yes	—
149	C₂×C₄	#41×C₇₁	1	8	yes	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	4	no	—
155	C₂²	#123×C₇₁	1	4	yes	—
156	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
157	C₂²	C₂×#41×C₇₁	1	2	yes	—
158	C₂²	C₂×#41×C₇₁	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	4	no	—
164	C₂	—	2	4	no	—
165	C₂	—	8	1	no	—
166	C₂	C₂×#123×C₇₁	1	1	yes	characteristic
167	C₂	#525×C₇₁	1	2	yes	—
168	C₂	#529×C₇₁	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#123×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₄²×C₇₁

C₂

yes

maximal, characteristic, complemented

#388×C₇₁

C₂

yes

maximal, complemented

#387×C₇₁

C₂

yes

maximal, complemented

#386×C₇₁

C₂

yes

maximal

#385×C₇₁

C₂

yes

maximal, complemented

#381×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₇₁

C₂²

yes

characteristic

C₂³×C₄×C₇₁

C₂²

yes

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

#123×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

—

#95×C₇₁

C₂²

yes

—

#95×C₇₁

C₂²

yes

—

#95×C₇₁

C₂²

yes

—

C₂⁴×C₇₁

C₂³

yes

characteristic

C₂⁴×C₇₁

C₂³

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

#41

yes

—

C₂³×C₇₁

C₂⁴

yes

characteristic

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#123

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂²×C₇₁

C₂×#41

yes

—

C₂²×C₇₁

C₂×#41

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×#123

yes

characteristic

C₂×C₇₁

#525

yes

—

C₂×C₇₁

#529

yes

—

#2853

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₇₁

#2853

yes

characteristic, direct product factor, Sylow

C₂×#123

C₂×C₇₁

yes

characteristic

C₂²×C₄²

C₂×C₇₁

yes

characteristic, complemented

#388

C₂×C₇₁

yes

complemented

#387

C₂×C₇₁

yes

complemented

#386

C₂×C₇₁

yes

—

#385

C₂×C₇₁

yes

complemented

#381

C₂×C₇₁

yes

characteristic, complemented

C₂²×D₄

C₂²×C₇₁

yes

characteristic

C₂³×C₄

C₂²×C₇₁

yes

—

C₂³×C₄

C₂²×C₇₁

yes

characteristic

#123

—

C₂×C₄⋊C₄

C₂²×C₇₁

yes

—

C₂×C₄⋊C₄

C₂²×C₇₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₇₁

yes

characteristic

100

C₂×C₂²⋊C₄

C₂²×C₇₁

yes

—

101

C₂×C₂²⋊C₄

C₂²×C₇₁

yes

—

102

C₂×C₂²⋊C₄

C₂²×C₇₁

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

C₂²×C₇₁

yes

—

105

#95

C₂²×C₇₁

yes

—

106

#95

C₂²×C₇₁

yes

—

107

#95

C₂²×C₇₁

yes

—

108

C₂⁴

C₂³×C₇₁

yes

characteristic

109

C₂⁴

C₂³×C₇₁

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₄×C₇₁

yes

—

116

C₂²×C₄

C₂³×C₇₁

yes

—

117

C₂²×C₄

C₂³×C₇₁

yes

characteristic

118

C₂²×C₄

C₂³×C₇₁

yes

—

119

C₂²×C₄

C₂³×C₇₁

yes

—

120

C₂²×C₄

C₂³×C₇₁

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂×D₄×C₇₁

yes

—

133

C₂³

#41×C₇₁

yes

—

134

C₂³

C₂⁴×C₇₁

yes

characteristic

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×D₄×C₇₁

yes

—

149

C₂×C₄

#41×C₇₁

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#123×C₇₁

yes

—

156

C₂²

C₂²×D₄×C₇₁

yes

characteristic

157

C₂²

C₂×#41×C₇₁

yes

—

158

C₂²

C₂×#41×C₇₁

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₂

—

165

C₂

—

166

C₂

C₂×#123×C₇₁

yes

characteristic

167

C₂

#525×C₇₁

yes

—

168

C₂

#529×C₇₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇₁
prime radical	C₂²×D₄×C₇₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₇₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₇₁

prime radical

C₂²×D₄×C₇₁

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₂²×C₄²×C₇₁	C₂	semidirect
3. #388×C₇₁	C₂	semidirect
4. #387×C₇₁	C₂	semidirect
6. #385×C₇₁	C₂	semidirect
7. #381×C₇₁	C₂	semidirect
84. #2853	C₇₁	direct (N × H)
85. C₇₁	#2853	direct (N × H)
87. C₂²×C₄²	C₂×C₇₁	semidirect
88. #388	C₂×C₇₁	semidirect
89. #387	C₂×C₇₁	semidirect
91. #385	C₂×C₇₁	semidirect
92. #381	C₂×C₇₁	semidirect

H ≅ G/N

type

2. C₂²×C₄²×C₇₁

C₂

semidirect

3. #388×C₇₁

C₂

semidirect

4. #387×C₇₁

C₂

semidirect

6. #385×C₇₁

C₂

semidirect

7. #381×C₇₁

C₂

semidirect

84. #2853

C₇₁

direct (N × H)

85. C₇₁

#2853

direct (N × H)

87. C₂²×C₄²

C₂×C₇₁

semidirect

88. #388

C₂×C₇₁

semidirect

89. #387

C₂×C₇₁

semidirect

91. #385

C₂×C₇₁

semidirect

92. #381

C₂×C₇₁

semidirect

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

subgroup	index
C₂×#123×C₇₁	2
C₂²×C₄²×C₇₁	2
#388×C₇₁	2
#387×C₇₁	2
#386×C₇₁	2
#385×C₇₁	2
#381×C₇₁	2
#2853	71

subgroup

index

C₂×#123×C₇₁

C₂²×C₄²×C₇₁

#388×C₇₁

#387×C₇₁

#386×C₇₁

#385×C₇₁

#381×C₇₁

#2853

Subgroups — C₇₁×#2853 #22585415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#2853 #22585415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2853 #22585415