Group #22587823 — subgroups

Subgroups — #22587823

Order 9088.

184 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17986375	C₂	1	1	yes	maximal, characteristic
2	#17986200	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986196	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986126	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985749	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985742	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765854	—	2	1	no	—
9	C₂×#11765853	—	2	1	no	—
10	C₂×#11765850	—	2	1	no	—
11	C₂×#11765850	—	2	1	no	—
12	#14556125	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₇₁⋊C₈	—	2	1	no	—
16	#14556050	—	2	1	no	—
17	#14556049	—	2	1	no	—
18	#14556038	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
24	C₂×D₂₈₄	—	2	1	no	—
25	C₂×D₂₈₄	—	2	1	no	—
26	C₂×D₂₈₄	—	2	1	no	—
27	#11765854	—	4	2	no	—
28	#11765853	—	4	2	no	—
29	#11765850	—	4	2	no	—
30	#11765850	—	4	2	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	Dic₇₁⋊C₄	—	2	1	no	—
36	C₄×Dic₇₁	—	2	1	no	—
37	C₂×D₄×C₇₁	—	2	1	no	—
38	C₂×D₄×C₇₁	—	2	1	no	—
39	C₂×D₄×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	C₂³×C₇₁	—	4	1	no	—
44	C₂²×D₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	D₄×C₇₁	—	4	1	no	—
47	D₄×C₇₁	—	4	1	no	—
48	D₄×C₇₁	—	4	1	no	—
49	D₄×C₇₁	—	8	1	no	—
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	—	2	1	no	—
55	D₂₈₄	—	4	1	no	—
56	D₂₈₄	—	4	1	no	—
57	D₂₈₄	—	4	1	no	—
58	D₂₈₄	—	4	1	no	—
59	D₂₈₄	—	8	1	no	—
60	Dic₁₄₂	—	2	1	no	—
61	Dic₁₄₂	—	2	1	no	—
62	C₈×C₇₁	—	4	1	no	—
63	C₈×C₇₁	—	4	1	no	—
64	C₇₁⋊C₈	—	2	1	no	—
65	C₇₁⋊C₈	—	4	1	no	—
66	C₇₁⋊C₈	SD₁₆	1	2	yes	complemented
67	C₂²×C₇₁	—	8	1	no	—
68	C₂²×C₇₁	—	8	1	no	—
69	C₂²×C₇₁	—	8	1	no	—
70	C₂²×C₇₁	#121	1	1	yes	characteristic
71	C₂×D₇₁	—	8	1	no	—
72	C₂×D₇₁	—	8	1	no	—
73	C₂×D₇₁	—	8	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₇₁	—	4	1	no	—
80	Dic₇₁	—	4	1	no	—
81	C₂×C₇₁	—	16	1	no	—
82	C₂×C₇₁	#460	1	1	yes	characteristic
83	C₂×C₇₁	#469	1	1	yes	characteristic
84	C₂×C₇₁	#472	1	1	yes	characteristic
85	D₇₁	—	16	1	no	—
86	#1688	—	71	1	no	maximal, Sylow
87	C₇₁	#1688	1	1	yes	characteristic, complemented, Sylow
88	#493	—	71	1	no	—
89	#460	—	71	1	no	—
90	#460	—	71	1	no	—
91	#446	—	71	1	no	—
92	#334	—	71	1	no	—
93	#331	—	71	1	no	—
94	#331	D₇₁	1	1	yes	characteristic, complemented
95	C₂×SD₁₆	—	142	1	no	—
96	C₂×SD₁₆	—	142	1	no	—
97	C₂×D₈	—	142	1	no	—
98	C₂×D₈	—	142	1	no	—
99	#127	—	71	1	no	—
100	#127	C₂×D₇₁	1	1	yes	characteristic
101	C₄×Q₈	—	71	1	no	—
102	C₄⋊C₈	—	71	1	no	—
103	C₄⋊C₈	—	142	1	no	—
104	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
106	#102	—	142	1	no	—
107	#102	—	142	1	no	—
108	#97	—	142	1	no	—
109	C₄×C₈	—	71	1	no	—
110	C₂×Q₈	—	71	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	142	1	no	—
115	C₂×D₄	—	142	1	no	—
116	C₂×D₄	—	142	1	no	—
117	SD₁₆	—	284	2	no	—
118	SD₁₆	—	284	2	no	—
119	D₈	—	284	2	no	—
120	D₈	—	284	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	71	1	no	—
124	C₂×C₈	—	71	1	no	—
125	C₂×C₈	—	142	1	no	—
126	C₄⋊C₄	—	71	1	no	—
127	C₄⋊C₄	—	142	1	no	—
128	C₄²	—	142	1	no	—
129	C₄²	C₂²×D₇₁	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	284	1	no	—
132	Q₈	—	142	1	no	—
133	Q₈	—	142	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	284	1	no	—
140	D₄	—	284	1	no	—
141	D₄	—	284	1	no	—
142	D₄	—	284	1	no	—
143	D₄	—	568	1	no	—
144	C₂×C₄	—	142	1	no	—
145	C₂×C₄	—	142	1	no	—
146	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
148	C₂×C₄	#11765865	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	284	1	no	—
152	C₈	—	71	2	no	—
153	C₈	—	142	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	568	1	no	—
158	C₂²	—	568	1	no	—
159	C₂²	—	568	1	no	—
160	C₂²	#14556101	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	284	1	no	—
163	C₄	—	284	1	no	—
164	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
165	C₄	#14556152	1	1	yes	characteristic
166	C₄	#14556152	1	1	yes	characteristic
167	C₄	#14556155	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	1136	1	no	—
170	C₂	#17986198	1	1	yes	characteristic
171	C₂	#17986251	1	1	yes	characteristic
172	C₂	#17986268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986375

C₂

yes

maximal, characteristic

#17986200

C₂

yes

maximal, characteristic, complemented

#17986196

C₂

yes

maximal, characteristic, complemented

#17986126

C₂

yes

maximal, characteristic, complemented

#17985749

C₂

yes

maximal, characteristic, complemented

#17985742

C₂

yes

maximal, characteristic, complemented

#331×C₇₁

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556050

—

#14556049

—

#14556038

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂²×D₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1688

—

maximal, Sylow

C₇₁

#1688

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

#127

—

100

#127

C₂×D₇₁

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

142

—

104

C₄⋊C₈

C₂×D₇₁

yes

characteristic

105

C₄⋊C₈

C₂×D₇₁

yes

characteristic

106

#102

—

142

—

107

#102

—

142

—

108

#97

—

142

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

142

—

115

C₂×D₄

—

142

—

116

C₂×D₄

—

142

—

117

SD₁₆

—

284

—

118

SD₁₆

—

284

—

119

D₈

—

284

—

120

D₈

—

284

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

142

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

142

—

128

C₄²

—

142

—

129

C₄²

C₂²×D₇₁

yes

characteristic

130

C₂³

—

131

C₂³

—

284

—

132

Q₈

—

142

—

133

Q₈

—

142

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

284

—

140

D₄

—

284

—

141

D₄

—

284

—

142

D₄

—

284

—

143

D₄

—

568

—

144

C₂×C₄

—

142

—

145

C₂×C₄

—

142

—

146

C₂×C₄

D₄×D₇₁

yes

characteristic

147

C₂×C₄

D₄×D₇₁

yes

characteristic

148

C₂×C₄

#11765865

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

284

—

152

C₈

—

153

C₈

—

142

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

568

—

158

C₂²

—

568

—

159

C₂²

—

568

—

160

C₂²

#14556101

yes

characteristic

161

C₄

—

162

C₄

—

284

—

163

C₄

—

284

—

164

C₄

SD₁₆×D₇₁

yes

characteristic

165

C₄

#14556152

yes

characteristic

166

C₄

#14556152

yes

characteristic

167

C₄

#14556155

yes

characteristic

168

C₂

—

169

C₂

—

1136

—

170

C₂

#17986198

yes

characteristic

171

C₂

#17986251

yes

characteristic

172

C₂

#17986268

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#331×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#331	D₇₁
hypocenter	C₇₁	#1688
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#331×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #17986200	C₂	semidirect
3. #17986196	C₂	semidirect
4. #17986126	C₂	semidirect
5. #17985749	C₂	semidirect
6. #17985742	C₂	semidirect
7. #331×C₇₁	C₂	semidirect
66. C₇₁⋊C₈	SD₁₆	semidirect
87. C₇₁	#1688	semidirect
94. #331	D₇₁	semidirect

H ≅ G/N

type

2. #17986200

C₂

semidirect

3. #17986196

C₂

semidirect

4. #17986126

C₂

semidirect

5. #17985749

C₂

semidirect

6. #17985742

C₂

semidirect

7. #331×C₇₁

C₂

semidirect

66. C₇₁⋊C₈

SD₁₆

semidirect

87. C₇₁

#1688

semidirect

94. #331

D₇₁

semidirect

Subgroups — #22587823

Conjugacy classes of subgroups

Complements