Group #22588520 — subgroups

Subgroups — #22588520

Order 9088.

348 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂³×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#17985754	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	—	2	4	no	—
7	C₂²×C₇₁⋊C₈	—	2	2	no	—
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₇₁	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
17	#95×C₇₁	C₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	—	2	8	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	1	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×C₇₁⋊C₈	—	2	2	no	—
25	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
26	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	—	2	2	no	—
28	C₂²×C₄×C₇₁	—	2	2	no	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₇₁	D₄	1	2	yes	—
33	C₂²×C₄×C₇₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₇₁	—	2	2	no	—
36	C₂²⋊C₄×C₇₁	C₈	1	4	yes	complemented
37	C₄²×C₇₁	—	2	2	no	—
38	C₄²×C₇₁	—	2	2	no	—
39	C₂³×C₇₁	—	2	2	no	—
40	C₂³×C₇₁	—	2	2	no	—
41	C₂³×C₇₁	C₄⋊C₄	1	2	yes	—
42	C₂³×C₇₁	C₂×C₈	1	2	yes	—
43	C₂³×C₇₁	M₁₆	1	2	yes	—
44	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	4	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×C₈	1	4	yes	complemented
56	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#41	1	2	yes	—
60	C₇₁⋊C₈	—	2	4	no	—
61	C₇₁⋊C₈	—	4	1	no	—
62	C₇₁⋊C₈	—	4	1	no	—
63	C₂²×C₇₁	—	2	2	no	—
64	C₂²×C₇₁	—	2	2	no	—
65	C₂²×C₇₁	—	2	4	no	—
66	C₂²×C₇₁	—	2	4	no	—
67	C₂²×C₇₁	C₄⋊C₈	1	4	yes	complemented
68	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₇₁	C₄×D₄	1	2	yes	—
70	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂²×C₇₁	#131	1	2	yes	—
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	—	2	4	no	—
75	C₄×C₇₁	—	4	2	no	—
76	C₄×C₇₁	#120	1	1	yes	characteristic
77	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
78	C₄×C₇₁	#123	1	1	yes	characteristic
79	C₂×C₇₁	—	2	4	no	—
80	C₂×C₇₁	#385	1	1	yes	characteristic
81	C₂×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₇₁	#424	1	1	yes	characteristic
83	C₂×C₇₁	C₈×D₄	1	2	yes	—
84	C₂×C₇₁	#435	1	2	yes	—
85	#1850	—	71	1	no	maximal, Sylow
86	C₇₁	#1850	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₈	—	71	1	no	—
88	C₂×C₄⋊C₈	—	71	2	no	—
89	C₂×C₂²⋊C₈	—	71	1	no	—
90	C₄×C₂²⋊C₄	D₇₁	1	1	yes	characteristic
91	#336	—	71	2	no	—
92	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
93	C₂²×C₈	—	71	2	no	—
94	C₂²×C₈	—	71	1	no	—
95	C₂²×C₈	—	71	1	no	—
96	C₂²×C₈	—	142	4	no	—
97	C₂²×C₈	—	142	2	no	—
98	C₂×C₂²⋊C₄	Dic₇₁	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	Dic₇₁	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₇₁	1	2	yes	—
101	C₄⋊C₈	—	142	4	no	—
102	C₂²⋊C₈	—	142	2	no	—
103	#95	Dic₇₁	1	2	yes	—
104	C₂⁴	C₂×Dic₇₁	1	1	yes	characteristic
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	Dic₁₄₂	1	2	yes	—
108	C₂²×C₄	D₂₈₄	1	2	yes	—
109	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
110	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
113	C₂×C₈	—	71	4	no	—
114	C₂×C₈	—	142	2	no	—
115	C₂×C₈	—	142	1	no	—
116	C₂×C₈	—	142	1	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	—	142	8	no	—
119	C₂×C₈	—	142	2	no	—
120	C₂×C₈	—	142	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	C₇₁⋊C₈	1	4	yes	complemented
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₄⋊Dic₇₁	1	2	yes	—
128	C₂³	C₂×C₇₁⋊C₈	1	2	yes	—
129	C₂³	#11765849	1	2	yes	—
130	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	C₄⋊Dic₇₁	1	2	yes	—
141	C₂×C₄	C₂×C₇₁⋊C₈	1	4	yes	complemented
142	C₂×C₄	D₄×D₇₁	1	2	yes	—
143	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
144	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
145	C₂×C₄	#11765864	1	2	yes	—
146	C₈	—	284	1	no	—
147	C₈	—	284	1	no	—
148	C₈	—	142	4	no	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₄⋊(C₇₁⋊C₈)	1	4	yes	complemented
154	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
155	C₂²	D₄×Dic₇₁	1	2	yes	—
156	C₂²	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
157	C₂²	C₂×#11765849	1	1	yes	characteristic
158	C₂²	#14556135	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	#14556098	1	1	yes	characteristic
163	C₄	#14556105	1	2	yes	—
164	C₄	#14556112	1	1	yes	characteristic
165	C₂	—	2	4	no	—
166	C₂	#17985874	1	1	yes	characteristic
167	C₂	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
168	C₂	#17986026	1	1	yes	characteristic
169	C₂	D₄×C₇₁⋊C₈	1	2	yes	—
170	C₂	#17986071	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

—

#95×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

C₈

yes

complemented

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

M₁₆

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

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

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

complemented

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊C₈

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#120

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

#123

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

#385

yes

characteristic

C₂×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₇₁

#424

yes

characteristic

C₂×C₇₁

C₈×D₄

yes

—

C₂×C₇₁

#435

yes

—

#1850

—

maximal, Sylow

C₇₁

#1850

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

D₇₁

yes

characteristic

#336

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

—

142

—

C₂×C₂²⋊C₄

Dic₇₁

yes

characteristic

C₂×C₂²⋊C₄

Dic₇₁

yes

characteristic

100

C₂×C₄²

C₂×D₇₁

yes

—

101

C₄⋊C₈

—

142

—

102

C₂²⋊C₈

—

142

—

103

#95

Dic₇₁

yes

—

104

C₂⁴

C₂×Dic₇₁

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

Dic₁₄₂

yes

—

108

C₂²×C₄

D₂₈₄

yes

—

109

C₂²×C₄

C₂×Dic₇₁

yes

—

110

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

111

C₂²×C₄

C₂×Dic₇₁

yes

—

112

C₂²×C₄

C₂²×D₇₁

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

142

—

115

C₂×C₈

—

142

—

116

C₂×C₈

—

142

—

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

—

142

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

C₇₁⋊C₈

yes

complemented

123

C₄²

—

124

C₄²

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₇₁

yes

—

128

C₂³

C₂×C₇₁⋊C₈

yes

—

129

C₂³

#11765849

yes

—

130

C₂³

C₂²×Dic₇₁

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₄⋊Dic₇₁

yes

—

141

C₂×C₄

C₂×C₇₁⋊C₈

yes

complemented

142

C₂×C₄

D₄×D₇₁

yes

—

143

C₂×C₄

C₂×D₂₈₄

yes

characteristic

144

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

145

C₂×C₄

#11765864

yes

—

146

C₈

—

284

—

147

C₈

—

284

—

148

C₈

—

142

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₄⋊(C₇₁⋊C₈)

yes

complemented

154

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

155

C₂²

D₄×Dic₇₁

yes

—

156

C₂²

C₂²×C₇₁⋊C₈

yes

characteristic

157

C₂²

C₂×#11765849

yes

characteristic

158

C₂²

#14556135

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#14556098

yes

characteristic

163

C₄

#14556105

yes

—

164

C₄

#14556112

yes

characteristic

165

C₂

—

166

C₂

#17985874

yes

characteristic

167

C₂

C₂×C₄⋊(C₇₁⋊C₈)

yes

characteristic

168

C₂

#17986026

yes

characteristic

169

C₂

D₄×C₇₁⋊C₈

yes

—

170

C₂

#17986071

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₈
Fitting	C₄×C₂²⋊C₄×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₄×C₂²⋊C₄	D₇₁
hypocenter	C₇₁	#1850
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂²×C₈

Fitting

C₄×C₂²⋊C₄×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₄×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₁⋊C₈)	C₂	semidirect
4. #17985754	C₂	semidirect
36. C₂²⋊C₄×C₇₁	C₈	semidirect
55. C₂×C₄×C₇₁	C₂×C₈	semidirect
67. C₂²×C₇₁	C₄⋊C₈	semidirect
86. C₇₁	#1850	semidirect
122. C₂²⋊C₄	C₇₁⋊C₈	semidirect
141. C₂×C₄	C₂×C₇₁⋊C₈	semidirect
153. C₂²	C₄⋊(C₇₁⋊C₈)	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₇₁⋊C₈)

C₂

semidirect

4. #17985754

C₂

semidirect

36. C₂²⋊C₄×C₇₁

C₈

semidirect

55. C₂×C₄×C₇₁

C₂×C₈

semidirect

67. C₂²×C₇₁

C₄⋊C₈

semidirect

86. C₇₁

#1850

semidirect

122. C₂²⋊C₄

C₇₁⋊C₈

semidirect

141. C₂×C₄

C₂×C₇₁⋊C₈

semidirect

153. C₂²

C₄⋊(C₇₁⋊C₈)

semidirect

Subgroups — #22588520

Conjugacy classes of subgroups

Complements