Group #22587055 — subgroups

Subgroups — #22587055

Order 9088.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#17986055	C₂	1	1	yes	maximal, characteristic
3	#17986055	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985744	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556127	—	2	1	no	—
11	#14556127	C₄	1	2	yes	—
12	#14556126	—	2	1	no	—
13	C₄⋊C₄×D₇₁	—	2	1	no	—
14	C₄⋊C₄×D₇₁	C₄	1	2	yes	—
15	C₄²×D₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₇₁)⋊C₈	—	2	1	no	—
19	(C₂×D₇₁)⋊C₈	—	2	1	no	—
20	#14556038	—	2	1	no	—
21	#14556038	—	2	1	no	—
22	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	2	no	—
27	Q₈×D₇₁	—	4	2	no	—
28	Q₈×D₇₁	—	4	2	no	—
29	C₂×C₄×D₇₁	—	2	2	no	—
30	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₄⋊Dic₇₁	—	2	1	no	—
36	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
37	Dic₇₁⋊C₄	—	2	2	no	—
38	Dic₇₁⋊C₄	—	2	2	no	—
39	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₁	D₄	1	1	yes	characteristic
41	C₄×Dic₇₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₇₁	—	4	1	no	—
50	Q₈×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	2	no	—
56	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₇₁	—	2	2	no	—
60	C₄×D₇₁	—	2	1	no	—
61	C₄×D₇₁	—	2	1	no	—
62	C₄×D₇₁	—	2	1	no	—
63	C₄×D₇₁	—	2	1	no	—
64	C₄×D₇₁	—	4	4	no	—
65	Dic₁₄₂	—	4	4	no	—
66	Dic₁₄₂	—	4	1	no	—
67	Dic₁₄₂	—	4	1	no	—
68	C₈×C₇₁	—	4	1	no	—
69	C₈×C₇₁	—	4	1	no	—
70	C₇₁⋊C₈	—	4	1	no	—
71	C₇₁⋊C₈	—	4	1	no	—
72	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₇₁	—	2	1	no	—
74	C₂×D₇₁	—	2	1	no	—
75	C₂×D₇₁	#101	1	2	yes	—
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	4	2	no	—
78	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₇₁	#137	1	1	yes	characteristic
81	C₄×C₇₁	#137	1	1	yes	characteristic
82	Dic₇₁	—	2	1	no	—
83	Dic₇₁	—	2	1	no	—
84	Dic₇₁	—	2	1	no	—
85	Dic₇₁	—	4	2	no	—
86	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
87	C₂×C₇₁	#418	1	1	yes	characteristic
88	C₂×C₇₁	#419	1	1	yes	characteristic
89	D₇₁	—	2	2	no	—
90	#1556	—	71	1	no	maximal, Sylow
91	C₇₁	#1556	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊Q₈	—	71	1	no	—
93	#432	—	71	1	no	—
94	#432	—	71	1	no	—
95	#332	—	71	1	no	—
96	#332	—	71	1	no	—
97	#332	—	71	1	no	—
98	#332	D₇₁	1	1	yes	characteristic, complemented
99	C₂²×Q₈	—	142	1	no	—
100	C₄⋊Q₈	—	71	2	no	—
101	C₄⋊Q₈	—	71	1	no	—
102	C₄⋊Q₈	—	142	1	no	—
103	C₄⋊Q₈	—	142	1	no	—
104	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	71	2	no	—
106	C₂×C₄⋊C₄	—	142	1	no	—
107	C₂×C₄²	—	71	1	no	—
108	C₄⋊C₈	—	71	1	no	—
109	C₄⋊C₈	—	71	1	no	—
110	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
112	C₂²⋊C₈	—	142	1	no	—
113	C₂²⋊C₈	—	142	1	no	—
114	#97	—	142	1	no	—
115	#97	—	142	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	284	2	no	—
118	C₂×Q₈	—	284	2	no	—
119	C₂×Q₈	—	142	2	no	—
120	C₂×Q₈	—	142	1	no	—
121	C₂²×C₄	—	71	1	no	—
122	C₂²×C₄	—	71	1	no	—
123	C₂²×C₄	—	71	1	no	—
124	C₂²×C₄	—	142	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	142	1	no	—
128	C₂×C₈	—	142	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	71	2	no	—
131	C₄⋊C₄	—	142	2	no	—
132	C₄⋊C₄	—	142	2	no	—
133	C₄⋊C₄	—	142	1	no	—
134	C₄⋊C₄	C₄×D₇₁	1	2	yes	—
135	C₄²	—	71	1	no	—
136	C₄²	—	71	1	no	—
137	C₄²	—	71	1	no	—
138	C₄²	C₂²×D₇₁	1	1	yes	characteristic
139	C₂³	—	71	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	284	4	no	—
143	Q₈	—	284	1	no	—
144	Q₈	—	284	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	284	4	no	—
147	C₂×C₄	—	71	1	no	—
148	C₂×C₄	—	71	1	no	—
149	C₂×C₄	—	71	1	no	—
150	C₂×C₄	—	142	2	no	—
151	C₂×C₄	—	142	2	no	—
152	C₂×C₄	—	142	1	no	—
153	C₂×C₄	—	142	1	no	—
154	C₂×C₄	—	142	1	no	—
155	C₂×C₄	—	142	1	no	—
156	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
157	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
158	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	284	1	no	—
162	C₈	—	284	1	no	—
163	C₂²	—	71	2	no	—
164	C₂²	—	142	1	no	—
165	C₂²	—	142	1	no	—
166	C₂²	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	284	2	no	—
170	C₄	—	142	1	no	—
171	C₄	—	142	1	no	—
172	C₄	—	142	1	no	—
173	C₄	#14556152	1	1	yes	characteristic
174	C₄	#14556155	1	1	yes	characteristic
175	C₄	#14556157	1	1	yes	characteristic
176	C₄	#14556160	1	1	yes	characteristic
177	C₂	—	142	2	no	—
178	C₂	#101×D₇₁	1	1	yes	characteristic
179	C₂	#17986002	1	1	yes	characteristic
180	C₂	#17986006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₇₁

C₂

yes

maximal, characteristic

#17986055

C₂

yes

maximal, characteristic

#17986055

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#17985744

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

#14556128

C₂²

yes

characteristic

#14556127

—

#14556127

C₄

yes

—

#14556126

—

C₄⋊C₄×D₇₁

—

C₄⋊C₄×D₇₁

C₄

yes

—

C₄²×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₈

—

(C₂×D₇₁)⋊C₈

—

#14556038

—

#14556038

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Q₈×D₇₁

—

Q₈×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#101

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1556

—

maximal, Sylow

C₇₁

#1556

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂²×Q₈

—

142

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

142

—

103

C₄⋊Q₈

—

142

—

104

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

142

—

107

C₂×C₄²

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₇₁

yes

characteristic

111

C₄⋊C₈

C₂×D₇₁

yes

characteristic

112

C₂²⋊C₈

—

142

—

113

C₂²⋊C₈

—

142

—

114

#97

—

142

—

115

#97

—

142

—

116

C₂×Q₈

—

117

C₂×Q₈

—

284

—

118

C₂×Q₈

—

284

—

119

C₂×Q₈

—

142

—

120

C₂×Q₈

—

142

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

142

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

142

—

128

C₂×C₈

—

142

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

142

—

132

C₄⋊C₄

—

142

—

133

C₄⋊C₄

—

142

—

134

C₄⋊C₄

C₄×D₇₁

yes

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂²×D₇₁

yes

characteristic

139

C₂³

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

284

—

143

Q₈

—

284

—

144

Q₈

—

284

—

145

C₂×C₄

—

146

C₂×C₄

—

284

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

142

—

151

C₂×C₄

—

142

—

152

C₂×C₄

—

142

—

153

C₂×C₄

—

142

—

154

C₂×C₄

—

142

—

155

C₂×C₄

—

142

—

156

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

157

C₂×C₄

D₄×D₇₁

yes

characteristic

158

C₂×C₄

D₄×D₇₁

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

284

—

162

C₈

—

284

—

163

C₂²

—

164

C₂²

—

142

—

165

C₂²

—

142

—

166

C₂²

C₂²⋊C₄×D₇₁

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

284

—

170

C₄

—

142

—

171

C₄

—

142

—

172

C₄

—

142

—

173

C₄

#14556152

yes

characteristic

174

C₄

#14556155

yes

characteristic

175

C₄

#14556157

yes

characteristic

176

C₄

#14556160

yes

characteristic

177

C₂

—

142

—

178

C₂

#101×D₇₁

yes

characteristic

179

C₂

#17986002

yes

characteristic

180

C₂

#17986006

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1556
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #17985746	C₂	semidirect
5. #17985745	C₂	semidirect
6. #17985744	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
91. C₇₁	#1556	semidirect
98. #332	D₇₁	semidirect

H ≅ G/N

type

4. #17985746

C₂

semidirect

5. #17985745

C₂

semidirect

6. #17985744

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

91. C₇₁

#1556

semidirect

98. #332

D₇₁

semidirect

Subgroups — #22587055

Conjugacy classes of subgroups

Complements