Group #22587587 — subgroups

Subgroups — #22587587

Order 9088.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic
2	#17986402	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986276	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985726	C₂	1	1	yes	maximal, characteristic
7	#462×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	#14556128	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556108	—	2	2	no	—
12	#14556108	—	2	1	no	—
13	C₄×D₂₈₄	—	2	1	no	—
14	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
15	#14556061	—	2	2	no	—
16	#14556061	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	#14556048	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₇₁	—	2	1	no	—
21	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#103×C₇₁	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	2	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	Q₈×D₇₁	—	4	2	no	—
28	Q₈×D₇₁	—	4	2	no	—
29	C₂×D₂₈₄	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₁	—	2	2	no	—
31	C₂×C₄×D₇₁	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₄⋊Dic₇₁	—	2	2	no	—
35	C₄⋊Dic₇₁	—	2	1	no	—
36	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₁	D₄	1	2	yes	—
38	(C₂×D₇₁)⋊C₄	—	2	2	no	—
39	(C₂×D₇₁)⋊C₄	—	2	1	no	—
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
42	Q₁₆×C₇₁	—	4	2	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
48	C₄²×C₇₁	—	2	1	no	—
49	C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂²×D₇₁	—	2	1	no	—
51	Q₈×C₇₁	—	2	1	no	—
52	Q₈×C₇₁	—	4	1	no	—
53	Q₈×C₇₁	—	4	1	no	—
54	Q₈×C₇₁	D₈	1	2	yes	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	—	2	1	no	—
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇₁	—	2	2	no	—
62	C₂×Dic₇₁	—	2	1	no	—
63	D₂₈₄	—	2	1	no	—
64	D₂₈₄	Q₁₆	1	2	yes	complemented
65	C₄×D₇₁	—	4	4	no	—
66	C₄×D₇₁	—	4	1	no	—
67	C₄×D₇₁	—	4	1	no	—
68	Dic₁₄₂	—	4	4	no	—
69	Dic₁₄₂	—	4	1	no	—
70	Dic₁₄₂	—	4	1	no	—
71	C₈×C₇₁	—	4	1	no	—
72	C₇₁⋊C₈	—	2	2	no	—
73	C₇₁⋊C₈	—	4	1	no	—
74	C₂²×C₇₁	#120	1	1	yes	characteristic
75	C₂×D₇₁	—	2	2	no	—
76	C₂×D₇₁	—	4	1	no	—
77	C₂×D₇₁	—	4	1	no	—
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	—	4	1	no	—
81	C₄×C₇₁	—	8	1	no	—
82	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
83	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
84	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₇₁	#137	1	1	yes	characteristic
86	Dic₇₁	—	4	1	no	—
87	Dic₇₁	—	4	2	no	—
88	C₂×C₇₁	#447	1	1	yes	characteristic
89	C₂×C₇₁	#451	1	1	yes	characteristic
90	C₂×C₇₁	#456	1	1	yes	characteristic
91	D₇₁	—	4	2	no	—
92	#1649	—	71	1	no	maximal, Sylow
93	C₇₁	#1649	1	1	yes	characteristic, complemented, Sylow
94	D₄×Q₈	—	71	1	no	—
95	#500	—	71	1	no	—
96	#474	—	71	1	no	—
97	#462	D₇₁	1	1	yes	characteristic, complemented
98	#332	—	71	1	no	—
99	Q₈⋊C₈	—	71	1	no	—
100	D₄⋊C₈	—	71	1	no	—
101	C₂²×Q₈	—	142	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₄⋊Q₈	—	71	1	no	—
104	C₄⋊Q₈	—	142	1	no	—
105	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂²⋊Q₈	—	142	2	no	—
107	C₂²⋊Q₈	—	142	1	no	—
108	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
109	C₄×D₄	—	71	1	no	—
110	C₄×D₄	—	142	1	no	—
111	#107	—	142	1	no	—
112	#107	—	142	2	no	—
113	C₄⋊C₈	—	71	1	no	—
114	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
115	#103	—	2	1	no	—
116	#102	—	142	1	no	—
117	C₄×C₈	—	71	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	284	2	no	—
120	C₂×Q₈	—	284	2	no	—
121	C₂×Q₈	—	142	2	no	—
122	C₂×Q₈	—	142	1	no	—
123	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
124	C₂×D₄	—	71	1	no	—
125	C₂²×C₄	—	142	2	no	—
126	C₂²×C₄	—	142	1	no	—
127	Q₁₆	—	4	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	142	1	no	—
130	C₂×C₈	—	142	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	71	1	no	—
135	C₄⋊C₄	—	71	2	no	—
136	C₄⋊C₄	—	142	2	no	—
137	C₄⋊C₄	—	142	1	no	—
138	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂²⋊C₄	—	142	2	no	—
140	C₂²⋊C₄	—	142	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂²×D₇₁	1	1	yes	characteristic
143	C₂³	—	142	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	284	4	no	—
148	Q₈	—	284	1	no	—
149	Q₈	—	284	1	no	—
150	Q₈	#11765850	1	2	yes	—
151	D₄	—	71	2	no	—
152	D₄	—	142	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	284	4	no	—
157	C₂×C₄	—	284	1	no	—
158	C₂×C₄	—	284	1	no	—
159	C₂×C₄	—	142	2	no	—
160	C₂×C₄	—	142	1	no	—
161	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
162	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	284	1	no	—
166	C₈	—	142	2	no	—
167	C₂²	—	284	1	no	—
168	C₂²	—	284	1	no	—
169	C₂²	—	142	2	no	—
170	C₂²	#14556096	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	284	2	no	—
176	C₄	—	284	1	no	—
177	C₄	C₂×#11765850	1	1	yes	characteristic
178	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
179	C₄	#14556153	1	1	yes	characteristic
180	C₄	#14556160	1	1	yes	characteristic
181	C₂	—	284	2	no	—
182	C₂	#17986132	1	1	yes	characteristic
183	C₂	#17986153	1	1	yes	characteristic
184	C₂	#17986180	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₈₄

C₂

yes

maximal, characteristic

#17986402

C₂

yes

maximal, characteristic, complemented

#17986276

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

#17985726

C₂

yes

maximal, characteristic

#462×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

#14556128

—

#14556128

C₂²

yes

characteristic

#14556108

—

#14556108

—

C₄×D₂₈₄

—

C₄×D₂₈₄

C₂²

yes

characteristic

#14556061

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556048

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Q₈×D₇₁

—

Q₈×D₇₁

—

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

D₈

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

Q₁₆

yes

complemented

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#120

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1649

—

maximal, Sylow

C₇₁

#1649

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#500

—

#474

—

#462

D₇₁

yes

characteristic, complemented

#332

—

Q₈⋊C₈

—

100

D₄⋊C₈

—

101

C₂²×Q₈

—

142

—

102

C₂×Q₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

142

—

105

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

106

C₂²⋊Q₈

—

142

—

107

C₂²⋊Q₈

—

142

—

108

C₄×Q₈

C₂×D₇₁

yes

characteristic

109

C₄×D₄

—

110

C₄×D₄

—

142

—

111

#107

—

142

—

112

#107

—

142

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₇₁

yes

characteristic

115

#103

—

116

#102

—

142

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

284

—

120

C₂×Q₈

—

284

—

121

C₂×Q₈

—

142

—

122

C₂×Q₈

—

142

—

123

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

124

C₂×D₄

—

125

C₂²×C₄

—

142

—

126

C₂²×C₄

—

142

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

142

—

130

C₂×C₈

—

142

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

142

—

137

C₄⋊C₄

—

142

—

138

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

139

C₂²⋊C₄

—

142

—

140

C₂²⋊C₄

—

142

—

141

C₄²

—

142

C₄²

C₂²×D₇₁

yes

characteristic

143

C₂³

—

142

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

284

—

148

Q₈

—

284

—

149

Q₈

—

284

—

150

Q₈

#11765850

yes

—

151

D₄

—

152

D₄

—

142

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

284

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

284

—

159

C₂×C₄

—

142

—

160

C₂×C₄

—

142

—

161

C₂×C₄

D₄×D₇₁

yes

characteristic

162

C₂×C₄

D₄×D₇₁

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₈

—

165

C₈

—

284

—

166

C₈

—

142

—

167

C₂²

—

284

—

168

C₂²

—

284

—

169

C₂²

—

142

—

170

C₂²

#14556096

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

284

—

176

C₄

—

284

—

177

C₄

C₂×#11765850

yes

characteristic

178

C₄

Q₁₆×D₇₁

yes

characteristic

179

C₄

#14556153

yes

characteristic

180

C₄

#14556160

yes

characteristic

181

C₂

—

284

—

182

C₂

#17986132

yes

characteristic

183

C₂

#17986153

yes

characteristic

184

C₂

#17986180

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#462×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#462	D₇₁
hypocenter	C₇₁	#1649
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#462×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
2. #17986402	C₂	semidirect
4. #17985746	C₂	semidirect
5. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
7. #462×C₇₁	C₂	semidirect
64. D₂₈₄	Q₁₆	semidirect
93. C₇₁	#1649	semidirect
97. #462	D₇₁	semidirect

H ≅ G/N

type

2. #17986402

C₂

semidirect

4. #17985746

C₂

semidirect

5. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

7. #462×C₇₁

C₂

semidirect

64. D₂₈₄

Q₁₆

semidirect

93. C₇₁

#1649

semidirect

97. #462

D₇₁

semidirect

Subgroups — #22587587

Conjugacy classes of subgroups

Complements