Group #22587433 — subgroups

Subgroups — #22587433

Order 9088.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986724	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986196	C₂	1	1	yes	maximal, characteristic
3	#17986196	C₂	1	1	yes	maximal, characteristic
4	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985731	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985731	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	—	2	1	no	—
9	C₂×#11765853	—	2	1	no	—
10	#14556126	—	2	1	no	—
11	#14556119	—	2	1	no	—
12	#14556118	—	2	2	no	—
13	Q₈×Dic₇₁	—	2	1	no	—
14	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
16	#14556093	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
19	#14556049	—	2	1	no	—
20	#14556049	—	2	1	no	—
21	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	#127×C₇₁	C₂²	1	1	yes	characteristic
23	#102×C₇₁	—	2	2	no	—
24	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
26	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
27	#11765853	—	4	2	no	—
28	#11765853	—	4	2	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	2	no	—
32	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
33	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
34	Dic₇₁⋊C₄	—	2	2	no	—
35	Dic₇₁⋊C₄	—	2	2	no	—
36	Dic₇₁⋊C₄	—	2	1	no	—
37	Dic₇₁⋊C₄	—	2	1	no	—
38	C₄×Dic₇₁	—	2	2	no	—
39	C₄×Dic₇₁	—	2	1	no	—
40	C₄×Dic₇₁	—	2	1	no	—
41	C₂×Q₈×C₇₁	—	2	1	no	—
42	C₂×D₄×C₇₁	—	2	1	no	—
43	C₂×D₄×C₇₁	—	2	1	no	—
44	C₂×D₄×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
48	C₄²×C₇₁	C₂³	1	1	yes	characteristic
49	C₂³×C₇₁	—	4	1	no	—
50	Q₈×C₇₁	—	4	1	no	—
51	Q₈×C₇₁	—	4	1	no	—
52	D₄×C₇₁	—	4	2	no	—
53	D₄×C₇₁	—	4	1	no	—
54	D₄×C₇₁	—	4	1	no	—
55	D₄×C₇₁	—	4	1	no	—
56	D₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	2	2	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	1	no	—
62	C₂×Dic₇₁	—	2	1	no	—
63	C₂×Dic₇₁	—	2	1	no	—
64	C₂×Dic₇₁	—	2	1	no	—
65	Dic₁₄₂	—	2	1	no	—
66	Dic₁₄₂	—	2	1	no	—
67	Dic₁₄₂	—	2	1	no	—
68	Dic₁₄₂	SD₁₆	1	2	yes	complemented
69	C₈×C₇₁	—	4	1	no	—
70	C₇₁⋊C₈	—	4	1	no	—
71	C₇₁⋊C₈	—	4	1	no	—
72	C₂²×C₇₁	—	8	1	no	—
73	C₂²×C₇₁	—	8	1	no	—
74	C₂²×C₇₁	—	8	1	no	—
75	C₂²×C₇₁	#120	1	1	yes	characteristic
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	4	2	no	—
78	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₄×C₇₁	#135	1	1	yes	characteristic
80	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₄×C₇₁	#137	1	1	yes	characteristic
82	Dic₇₁	—	2	2	no	—
83	Dic₇₁	—	4	1	no	—
84	Dic₇₁	—	4	1	no	—
85	Dic₇₁	—	4	1	no	—
86	C₂×C₇₁	—	16	1	no	—
87	C₂×C₇₁	#448	1	1	yes	characteristic
88	C₂×C₇₁	#449	1	1	yes	characteristic
89	C₂×C₇₁	#454	1	1	yes	characteristic
90	#1626	—	71	1	no	maximal, Sylow
91	C₇₁	#1626	1	1	yes	characteristic, complemented, Sylow
92	#557	—	71	1	no	—
93	#486	D₇₁	1	1	yes	characteristic
94	#460	—	71	1	no	—
95	#460	—	71	1	no	—
96	#332	—	71	1	no	—
97	Q₈⋊C₈	—	71	1	no	—
98	Q₈⋊C₈	—	71	1	no	—
99	C₂×SD₁₆	—	142	1	no	—
100	C₂×SD₁₆	—	142	1	no	—
101	C₄⋊Q₈	—	142	1	no	—
102	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
103	#127	C₂×D₇₁	1	1	yes	characteristic
104	#125	—	142	1	no	—
105	#125	—	142	2	no	—
106	C₄×Q₈	—	71	1	no	—
107	C₄×Q₈	—	71	1	no	—
108	C₄×Q₈	—	142	1	no	—
109	C₄×Q₈	—	142	1	no	—
110	C₄⋊C₈	—	71	1	no	—
111	C₄⋊C₈	—	71	1	no	—
112	#102	—	2	2	no	—
113	#102	—	142	1	no	—
114	#102	—	142	1	no	—
115	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	71	1	no	—
118	C₂×Q₈	—	71	1	no	—
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	2	1	no	—
122	SD₁₆	—	284	2	no	—
123	SD₁₆	—	284	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	142	1	no	—
126	C₂×C₈	—	142	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	71	1	no	—
129	C₄⋊C₄	—	71	1	no	—
130	C₄⋊C₄	—	142	2	no	—
131	C₄⋊C₄	—	142	2	no	—
132	C₄⋊C₄	—	142	2	no	—
133	C₄⋊C₄	—	142	1	no	—
134	C₄⋊C₄	—	142	1	no	—
135	C₄⋊C₄	C₂²⋊D₇₁	1	2	yes	—
136	C₄²	—	142	2	no	—
137	C₄²	—	142	1	no	—
138	C₄²	—	142	1	no	—
139	C₄²	C₂²×D₇₁	1	1	yes	characteristic
140	C₂³	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	71	2	no	—
144	Q₈	—	142	1	no	—
145	Q₈	—	142	1	no	—
146	Q₈	—	142	1	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	142	1	no	—
154	C₂×C₄	—	142	1	no	—
155	C₂×C₄	—	142	1	no	—
156	C₂×C₄	—	142	1	no	—
157	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
158	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	284	1	no	—
162	C₈	—	284	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	#14556096	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	284	1	no	—
170	C₄	—	284	1	no	—
171	C₄	—	284	1	no	—
172	C₄	—	142	2	no	—
173	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
174	C₄	#14556147	1	1	yes	characteristic
175	C₄	#14556152	1	1	yes	characteristic
176	C₄	#14556157	1	1	yes	characteristic
177	C₂	—	16	1	no	—
178	C₂	#17986136	1	1	yes	characteristic
179	C₂	#17986141	1	1	yes	characteristic
180	C₂	#17986171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986724

C₂

yes

maximal, characteristic, complemented

#17986196

C₂

yes

maximal, characteristic

#17986196

C₂

yes

maximal, characteristic

#17985745

C₂

yes

maximal, characteristic, complemented

#17985731

C₂

yes

maximal, characteristic, complemented

#17985731

C₂

yes

maximal, characteristic, complemented

#486×C₇₁

C₂

yes

maximal, characteristic

—

—

—

—

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556093

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556049

—

#14556049

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

#102×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

#11765853

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×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

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

SD₁₆

yes

complemented

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#557

—

#486

D₇₁

yes

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×SD₁₆

—

142

—

100

C₂×SD₁₆

—

142

—

101

C₄⋊Q₈

—

142

—

102

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

103

#127

C₂×D₇₁

yes

characteristic

104

#125

—

142

—

105

#125

—

142

—

106

C₄×Q₈

—

107

C₄×Q₈

—

108

C₄×Q₈

—

142

—

109

C₄×Q₈

—

142

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

142

—

114

#102

—

142

—

115

C₄×C₈

C₂×D₇₁

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

SD₁₆

—

284

—

123

SD₁₆

—

284

—

124

C₂×C₈

—

125

C₂×C₈

—

142

—

126

C₂×C₈

—

142

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

142

—

131

C₄⋊C₄

—

142

—

132

C₄⋊C₄

—

142

—

133

C₄⋊C₄

—

142

—

134

C₄⋊C₄

—

142

—

135

C₄⋊C₄

C₂²⋊D₇₁

yes

—

136

C₄²

—

142

—

137

C₄²

—

142

—

138

C₄²

—

142

—

139

C₄²

C₂²×D₇₁

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

142

—

145

Q₈

—

142

—

146

Q₈

—

142

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

142

—

154

C₂×C₄

—

142

—

155

C₂×C₄

—

142

—

156

C₂×C₄

—

142

—

157

C₂×C₄

D₄×D₇₁

yes

characteristic

158

C₂×C₄

D₄×D₇₁

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

160

C₈

—

161

C₈

—

284

—

162

C₈

—

284

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#14556096

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

284

—

170

C₄

—

284

—

171

C₄

—

284

—

172

C₄

—

142

—

173

C₄

SD₁₆×D₇₁

yes

characteristic

174

C₄

#14556147

yes

characteristic

175

C₄

#14556152

yes

characteristic

176

C₄

#14556157

yes

characteristic

177

C₂

—

178

C₂

#17986136

yes

characteristic

179

C₂

#17986141

yes

characteristic

180

C₂

#17986171

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#486×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#486	D₇₁
hypocenter	C₇₁	#1626
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#486×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #17986724	C₂	semidirect
4. #17985745	C₂	semidirect
5. #17985731	C₂	semidirect
6. #17985731	C₂	semidirect
68. Dic₁₄₂	SD₁₆	semidirect
91. C₇₁	#1626	semidirect

H ≅ G/N

type

1. #17986724

C₂

semidirect

4. #17985745

C₂

semidirect

5. #17985731

C₂

semidirect

6. #17985731

C₂

semidirect

68. Dic₁₄₂

SD₁₆

semidirect

91. C₇₁

#1626

semidirect

Subgroups — #22587433

Conjugacy classes of subgroups

Complements