Group #22589949 — subgroups

Subgroups — #22589949

Order 9088.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556067	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇₁)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×D₇₁	—	2	4	no	—
9	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
13	#14556067	—	2	4	no	—
14	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂²	1	2	yes	complemented
17	C₂³×D₇₁	C₈	1	1	yes	characteristic
18	C₂²×Dic₇₁	C₈	1	1	yes	characteristic
19	C₂×C₄×D₇₁	—	2	2	no	—
20	C₂×C₄×D₇₁	—	2	2	no	—
21	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
22	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
23	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	—	2	2	no	—
26	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₈×D₇₁	—	2	2	no	—
29	C₈×D₇₁	—	2	2	no	—
30	C₈×D₇₁	—	2	2	no	—
31	C₈×D₇₁	D₄	1	4	yes	—
32	C₇₁⋊C₁₆	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₁₆×C₇₁	—	2	2	no	—
38	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₇₁	—	2	2	no	—
40	C₂²×D₇₁	—	2	2	no	—
41	C₂²×D₇₁	—	2	2	no	—
42	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₇₁	—	2	2	no	—
49	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
51	C₄×D₇₁	—	2	2	no	—
52	C₄×D₇₁	—	2	1	no	—
53	C₄×D₇₁	—	2	1	no	—
54	C₄×D₇₁	—	2	2	no	—
55	C₄×D₇₁	C₂²⋊C₄	1	2	yes	—
56	C₄×D₇₁	M₁₆	1	2	yes	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	C₂×D₄	1	2	yes	—
59	C₇₁⋊C₈	—	2	1	no	—
60	C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	M₃₂	1	2	yes	complemented
64	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₇₁	—	2	8	no	—
66	C₂×D₇₁	—	2	1	no	—
67	C₂×D₇₁	—	2	1	no	—
68	C₂×D₇₁	—	2	1	no	—
69	C₂×D₇₁	—	2	1	no	—
70	C₂×D₇₁	C₂²⋊C₈	1	2	yes	—
71	C₂×D₇₁	M₃₂	1	2	yes	complemented
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
75	Dic₇₁	—	2	1	no	—
76	Dic₇₁	C₂²⋊C₈	1	2	yes	—
77	C₂×C₇₁	—	2	2	no	—
78	C₂×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
80	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
81	D₇₁	—	2	2	no	—
82	D₇₁	—	2	2	no	—
83	#2112	—	71	1	no	maximal, Sylow
84	C₇₁	#2112	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₈	—	71	1	no	—
86	C₂×M₃₂	—	71	2	no	—
87	C₂²⋊C₁₆	—	71	1	no	—
88	C₂²⋊C₁₆	—	71	2	no	—
89	C₂²⋊C₁₆	D₇₁	1	1	yes	characteristic, complemented
90	C₂³×C₄	—	71	1	no	—
91	C₂²×C₈	—	71	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	C₂²×C₈	—	71	1	no	—
94	C₂²×C₈	—	71	1	no	—
95	C₂²×C₈	—	71	1	no	—
96	C₂²×C₈	—	142	4	no	—
97	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
98	M₃₂	—	142	4	no	—
99	C₂×C₁₆	—	71	2	no	—
100	C₂×C₁₆	C₂×D₇₁	1	2	yes	complemented
101	C₂⁴	—	71	1	no	—
102	C₂²×C₄	—	71	1	no	—
103	C₂²×C₄	—	71	1	no	—
104	C₂²×C₄	—	71	1	no	—
105	C₂²×C₄	—	71	1	no	—
106	C₂²×C₄	—	71	1	no	—
107	C₂²×C₄	—	142	2	no	—
108	C₂²×C₄	—	142	2	no	—
109	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	—	71	4	no	—
113	C₂×C₈	—	71	1	no	—
114	C₂×C₈	—	142	2	no	—
115	C₂×C₈	—	142	2	no	—
116	C₂×C₈	—	142	2	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	142	2	no	—
122	C₂³	—	71	1	no	—
123	C₂³	—	71	1	no	—
124	C₂³	—	142	2	no	—
125	C₂³	—	142	2	no	—
126	C₂³	—	142	2	no	—
127	C₂³	C₈×D₇₁	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	71	1	no	—
131	C₂×C₄	—	71	2	no	—
132	C₂×C₄	—	71	2	no	—
133	C₂×C₄	—	71	1	no	—
134	C₂×C₄	—	142	2	no	—
135	C₂×C₄	—	142	2	no	—
136	C₂×C₄	—	142	2	no	—
137	C₂×C₄	—	142	1	no	—
138	C₂×C₄	—	142	1	no	—
139	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	71	2	no	—
143	C₈	—	142	1	no	—
144	C₈	D₄×D₇₁	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	71	2	no	—
148	C₂²	—	71	2	no	—
149	C₂²	—	142	8	no	—
150	C₂²	—	142	1	no	—
151	C₂²	—	142	1	no	—
152	C₂²	—	142	1	no	—
153	C₂²	—	142	1	no	—
154	C₂²	#14556067	1	2	yes	complemented
155	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	71	2	no	—
158	C₄	—	142	1	no	—
159	C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
160	C₄	M₁₆×D₇₁	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	—	142	2	no	—
163	C₂	—	142	2	no	—
164	C₂	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
165	C₂	M₃₂×D₇₁	1	1	yes	characteristic
166	C₂	C₂×#14556067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₇₁

C₂

yes

maximal, characteristic

C₂×#14556067

C₂

yes

maximal, complemented

C₂²⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₁)⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₁

C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

complemented

C₂³×D₇₁

C₈

yes

characteristic

C₂²×Dic₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

C₂×C₈

yes

characteristic

C₂²×D₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

C₂²⋊C₄

yes

—

C₄×D₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

M₃₂

yes

complemented

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

C₂²⋊C₈

yes

—

C₂×D₇₁

M₃₂

yes

complemented

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

C₂²⋊C₈

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

D₇₁

—

D₇₁

—

#2112

—

maximal, Sylow

C₇₁

#2112

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₇₁

yes

characteristic, complemented

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

M₃₂

—

142

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂×D₇₁

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

142

—

108

C₂²×C₄

—

142

—

109

C₂²×C₄

C₄×D₇₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

142

—

115

C₂×C₈

—

142

—

116

C₂×C₈

—

142

—

117

C₂×C₈

—

142

—

118

C₂×C₈

C₄×D₇₁

yes

characteristic

119

C₂×C₈

C₂²×D₇₁

yes

characteristic

120

C₁₆

—

121

C₁₆

—

142

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

142

—

125

C₂³

—

142

—

126

C₂³

—

142

—

127

C₂³

C₈×D₇₁

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

—

142

—

137

C₂×C₄

—

142

—

138

C₂×C₄

—

142

—

139

C₂×C₄

C₈×D₇₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

142

—

144

C₈

D₄×D₇₁

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

142

—

150

C₂²

—

142

—

151

C₂²

—

142

—

152

C₂²

—

142

—

153

C₂²

—

142

—

154

C₂²

#14556067

yes

complemented

155

C₂²

C₂×C₈×D₇₁

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

142

—

159

C₄

C₂²⋊C₄×D₇₁

yes

characteristic

160

C₄

M₁₆×D₇₁

yes

characteristic

161

C₂

—

162

C₂

—

142

—

163

C₂

—

142

—

164

C₂

C₂²⋊C₈×D₇₁

yes

characteristic

165

C₂

M₃₂×D₇₁

yes

characteristic

166

C₂

C₂×#14556067

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₂²⋊C₁₆×C₇₁	C₂
hypercenter	C₂²⋊C₁₆	D₇₁
hypocenter	C₇₁	#2112
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₈

Fitting

C₂²⋊C₁₆×C₇₁

C₂

hypercenter

C₂²⋊C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#14556067	C₂	semidirect
3. C₂²⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. (C₂×D₇₁)⋊C₁₆	C₂	semidirect
5. C₂²⋊C₁₆×C₇₁	C₂	semidirect
14. C₂×C₇₁⋊C₁₆	C₂²	semidirect
16. C₂×C₁₆×C₇₁	C₂²	semidirect
63. C₂²×C₇₁	M₃₂	semidirect
71. C₂×D₇₁	M₃₂	semidirect
84. C₇₁	#2112	semidirect
89. C₂²⋊C₁₆	D₇₁	semidirect
100. C₂×C₁₆	C₂×D₇₁	semidirect
154. C₂²	#14556067	semidirect

H ≅ G/N

type

2. C₂×#14556067

C₂

semidirect

3. C₂²⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. (C₂×D₇₁)⋊C₁₆

C₂

semidirect

5. C₂²⋊C₁₆×C₇₁

C₂

semidirect

14. C₂×C₇₁⋊C₁₆

C₂²

semidirect

16. C₂×C₁₆×C₇₁

C₂²

semidirect

63. C₂²×C₇₁

M₃₂

semidirect

71. C₂×D₇₁

M₃₂

semidirect

84. C₇₁

#2112

semidirect

89. C₂²⋊C₁₆

D₇₁

semidirect

100. C₂×C₁₆

C₂×D₇₁

semidirect

154. C₂²

#14556067

semidirect

Subgroups — #22589949

Conjugacy classes of subgroups

Complements