Group #22589959 — subgroups

Subgroups — #22589959

Order 9088.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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×#14556136	C₂	1	1	yes	maximal, characteristic
2	M₃₂×D₇₁	C₂	1	2	yes	maximal, complemented
3	#17985781	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985780	C₂	1	2	yes	maximal, complemented
5	#349×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765865	C₄	1	1	yes	characteristic
7	#14556136	—	2	2	no	—
8	#14556136	—	2	2	no	—
9	C₂×#11765848	C₄	1	1	yes	characteristic
10	C₂×#11765848	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
13	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
14	#14556068	C₂²	1	2	yes	—
15	#14556067	—	2	2	no	—
16	C₁₆×D₇₁	—	2	2	no	—
17	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	M₃₂×C₇₁	C₂²	1	2	yes	—
19	#11765865	—	2	2	no	—
20	#11765865	—	2	2	no	—
21	C₂×Dic₁₄₂	C₈	1	1	yes	characteristic
22	C₂×D₂₈₄	C₈	1	1	yes	characteristic
23	C₂×C₂²⋊D₇₁	C₈	1	1	yes	characteristic
24	C₂×C₂²⋊D₇₁	C₈	1	1	yes	characteristic
25	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
27	#11765848	—	2	2	no	—
28	#11765848	—	2	2	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₈×D₇₁	—	2	1	no	—
34	C₈×D₇₁	—	2	1	no	—
35	C₈×D₇₁	D₄	1	2	yes	—
36	C₈×D₇₁	D₄	1	2	yes	—
37	C₇₁⋊C₁₆	—	2	2	no	—
38	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
39	M₁₆×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	2	no	—
43	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₇₁	—	2	2	no	—
51	C₂²⋊D₇₁	—	2	2	no	—
52	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
54	D₂₈₄	—	2	2	no	—
55	C₄×D₇₁	—	2	1	no	—
56	C₄×D₇₁	—	2	1	no	—
57	C₄×D₇₁	C₂²⋊C₄	1	2	yes	—
58	C₄×D₇₁	M₁₆	1	2	yes	—
59	Dic₁₄₂	—	2	2	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₇₁⋊C₈	—	2	1	no	—
63	C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	4	1	no	—
67	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₇₁	—	2	1	no	—
69	C₂×D₇₁	—	4	1	no	—
70	C₂×D₇₁	—	4	1	no	—
71	C₂×D₇₁	C₂²⋊C₈	1	2	yes	—
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	1	no	—
78	C₂×C₇₁	—	4	1	no	—
79	C₂×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	D₇₁	—	2	2	no	—
81	D₇₁	—	4	1	no	—
82	#2115	—	71	1	no	maximal, Sylow
83	C₇₁	#2115	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	71	1	no	—
85	C₂×M₃₂	—	71	2	no	—
86	#349	—	71	1	no	—
87	#349	—	71	2	no	—
88	#349	D₇₁	1	1	yes	characteristic, complemented
89	C₂×#41	—	71	1	no	—
90	#131	—	142	2	no	—
91	#131	—	142	2	no	—
92	C₂×M₁₆	—	71	1	no	—
93	C₂×M₁₆	—	71	1	no	—
94	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₂²×C₈	—	71	1	no	—
96	C₂²×C₈	—	71	1	no	—
97	C₂²×C₈	—	71	1	no	—
98	M₃₂	—	71	2	no	—
99	M₃₂	—	142	2	no	—
100	M₃₂	C₂×D₇₁	1	2	yes	—
101	C₂×C₁₆	—	142	2	no	—
102	#41	—	142	2	no	—
103	#41	—	142	2	no	—
104	C₂×Q₈	—	71	1	no	—
105	C₂×D₄	—	71	1	no	—
106	C₂×D₄	—	71	1	no	—
107	C₂×D₄	—	71	1	no	—
108	C₂²×C₄	—	71	1	no	—
109	C₂²×C₄	—	71	1	no	—
110	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	142	2	no	—
113	M₁₆	—	142	2	no	—
114	C₂×C₈	—	71	1	no	—
115	C₂×C₈	—	71	2	no	—
116	C₂×C₈	—	71	2	no	—
117	C₂×C₈	—	71	1	no	—
118	C₂×C₈	—	142	1	no	—
119	C₂×C₈	—	142	1	no	—
120	C₂×C₈	—	142	1	no	—
121	C₂×C₈	—	142	1	no	—
122	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	—	142	2	no	—
126	C₂³	—	71	1	no	—
127	C₂³	—	71	1	no	—
128	C₂³	C₈×D₇₁	1	1	yes	characteristic
129	Q₈	—	142	2	no	—
130	D₄	—	142	2	no	—
131	D₄	—	142	2	no	—
132	D₄	—	142	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	71	1	no	—
136	C₂×C₄	—	71	2	no	—
137	C₂×C₄	—	71	2	no	—
138	C₂×C₄	—	71	1	no	—
139	C₂×C₄	—	142	1	no	—
140	C₂×C₄	—	142	1	no	—
141	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	71	1	no	—
145	C₈	—	71	1	no	—
146	C₈	—	142	1	no	—
147	C₈	D₄×D₇₁	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	284	1	no	—
151	C₂²	—	284	1	no	—
152	C₂²	—	71	2	no	—
153	C₂²	—	142	1	no	—
154	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	71	2	no	—
157	C₄	—	142	1	no	—
158	C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
159	C₄	M₁₆×D₇₁	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	4	1	no	—
162	C₂	—	284	1	no	—
163	C₂	—	142	2	no	—
164	C₂	C₂²⋊C₈×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556136

C₂

yes

maximal, characteristic

M₃₂×D₇₁

C₂

yes

maximal, complemented

#17985781

C₂

yes

maximal, characteristic, complemented

#17985780

C₂

yes

maximal, complemented

#349×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765865

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11765848

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

#14556068

C₂²

yes

—

#14556067

—

C₁₆×D₇₁

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

—

#11765865

—

#11765865

—

C₂×Dic₁₄₂

C₈

yes

characteristic

C₂×D₂₈₄

C₈

yes

characteristic

C₂×C₂²⋊D₇₁

C₈

yes

characteristic

C₂×C₂²⋊D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

#11765848

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

D₄

yes

—

C₈×D₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

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

C₂²⋊D₇₁

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

C₂²⋊C₄

yes

—

C₄×D₇₁

M₁₆

yes

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

C₂²⋊C₈

yes

—

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×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

D₇₁

—

D₇₁

—

#2115

—

maximal, Sylow

C₇₁

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

#349

—

#349

—

#349

D₇₁

yes

characteristic, complemented

C₂×#41

—

#131

—

142

—

#131

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

142

—

100

M₃₂

C₂×D₇₁

yes

—

101

C₂×C₁₆

—

142

—

102

#41

—

142

—

103

#41

—

142

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₇₁

yes

characteristic

111

M₁₆

—

112

M₁₆

—

142

—

113

M₁₆

—

142

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

—

142

—

121

C₂×C₈

—

142

—

122

C₂×C₈

C₄×D₇₁

yes

characteristic

123

C₂×C₈

C₂²×D₇₁

yes

characteristic

124

C₁₆

—

125

C₁₆

—

142

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₈×D₇₁

yes

characteristic

129

Q₈

—

142

—

130

D₄

—

142

—

131

D₄

—

142

—

132

D₄

—

142

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

142

—

140

C₂×C₄

—

142

—

141

C₂×C₄

C₈×D₇₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

142

—

147

C₈

D₄×D₇₁

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

284

—

151

C₂²

—

284

—

152

C₂²

—

153

C₂²

—

142

—

154

C₂²

C₂×C₈×D₇₁

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

142

—

158

C₄

C₂²⋊C₄×D₇₁

yes

characteristic

159

C₄

M₁₆×D₇₁

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

—

284

—

163

C₂

—

142

—

164

C₂

C₂²⋊C₈×D₇₁

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₇₁
commutator	C₂²×C₇₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#349×C₇₁	C₂
socle	C₂×C₇₁	C₂×C₂²⋊C₈
hypercenter	#349	D₇₁
hypocenter	C₇₁	#2115
LCS	G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

subgroup

G/H

center

C₄

C₂²⋊C₄×D₇₁

commutator

C₂²×C₇₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#349×C₇₁

C₂

socle

C₂×C₇₁

C₂×C₂²⋊C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

N	H ≅ G/N	type
2. M₃₂×D₇₁	C₂	semidirect
3. #17985781	C₂	semidirect
4. #17985780	C₂	semidirect
5. #349×C₇₁	C₂	semidirect
83. C₇₁	#2115	semidirect
88. #349	D₇₁	semidirect

H ≅ G/N

type

2. M₃₂×D₇₁

C₂

semidirect

3. #17985781

C₂

semidirect

4. #17985780

C₂

semidirect

5. #349×C₇₁

C₂

semidirect

83. C₇₁

#2115

semidirect

88. #349

D₇₁

semidirect

Subgroups — #22589959

Conjugacy classes of subgroups

Complements