Group #22587549 — subgroups

Subgroups — #22587549

Order 9088.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986706	C₂	1	1	yes	maximal, characteristic
2	#17986350	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986285	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986204	C₂	1	1	yes	maximal, characteristic
5	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985731	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556119	—	2	1	no	—
12	#14556107	—	2	1	no	—
13	#14556105	—	2	2	no	—
14	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
15	D₄×Dic₇₁	—	2	1	no	—
16	C₂×C₄⋊Dic₇₁	—	2	1	no	—
17	#14556060	—	2	1	no	—
18	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
19	#14556053	—	2	2	no	—
20	#14556053	—	2	1	no	—
21	#14556051	—	2	1	no	—
22	C₄×D₄×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	D₄	1	1	yes	characteristic
27	C₂×Dic₁₄₂	D₄	1	2	yes	—
28	C₂²×Dic₇₁	—	2	2	no	—
29	#11765853	—	4	2	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	2	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	—	2	1	no	—
35	C₄⋊Dic₇₁	—	2	1	no	—
36	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
37	Dic₇₁⋊C₄	—	2	2	no	—
38	Dic₇₁⋊C₄	—	2	1	no	—
39	C₂²⋊Dic₇₁	—	2	2	no	—
40	C₄×Dic₇₁	—	2	1	no	—
41	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₇₁	—	2	1	no	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂³×C₇₁	—	2	1	no	—
49	D₄×C₇₁	—	2	1	no	—
50	D₄×C₇₁	SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	4	1	no	—
53	C₂×C₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₇₁	—	2	1	no	—
58	C₂×Dic₇₁	—	2	1	no	—
59	C₂×Dic₇₁	—	2	2	no	—
60	C₂×Dic₇₁	—	4	1	no	—
61	C₂×Dic₇₁	—	8	2	no	—
62	Dic₁₄₂	—	2	1	no	—
63	Dic₁₄₂	—	2	1	no	—
64	Dic₁₄₂	—	4	2	no	—
65	Dic₁₄₂	—	4	1	no	—
66	Dic₁₄₂	—	4	1	no	—
67	C₈×C₇₁	—	4	1	no	—
68	C₈×C₇₁	—	4	1	no	—
69	C₇₁⋊C₈	—	4	1	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	—	4	1	no	—
72	C₂²×C₇₁	—	4	1	no	—
73	C₂²×C₇₁	#120	1	1	yes	characteristic
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	4	1	no	—
76	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₇₁	#135	1	1	yes	characteristic
78	C₄×C₇₁	#137	1	1	yes	characteristic
79	C₄×C₇₁	#137	1	1	yes	characteristic
80	Dic₇₁	—	4	1	no	—
81	Dic₇₁	—	4	2	no	—
82	Dic₇₁	—	4	1	no	—
83	Dic₇₁	—	8	1	no	—
84	C₂×C₇₁	—	4	2	no	—
85	C₂×C₇₁	#448	1	1	yes	characteristic
86	C₂×C₇₁	#452	1	1	yes	characteristic
87	C₂×C₇₁	#455	1	1	yes	characteristic
88	#1643	—	71	1	no	maximal, Sylow
89	C₇₁	#1643	1	1	yes	characteristic, complemented, Sylow
90	#554	—	71	1	no	—
91	#487	—	71	1	no	—
92	#475	—	71	1	no	—
93	#461	—	71	1	no	—
94	#333	—	71	1	no	—
95	Q₈⋊C₈	—	71	1	no	—
96	D₄⋊C₈	D₇₁	1	1	yes	characteristic
97	C₂×SD₁₆	—	142	1	no	—
98	C₄⋊Q₈	—	71	1	no	—
99	C₄⋊Q₈	—	71	1	no	—
100	#125	—	142	1	no	—
101	C₂²⋊Q₈	—	142	2	no	—
102	C₂²⋊Q₈	—	142	1	no	—
103	C₄×Q₈	—	71	1	no	—
104	C₄×D₄	—	142	1	no	—
105	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	142	1	no	—
107	#106	—	142	1	no	—
108	C₄⋊C₈	—	71	1	no	—
109	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
110	#103	—	142	1	no	—
111	#103	—	142	2	no	—
112	#103	—	142	1	no	—
113	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₂×Q₈	—	71	2	no	—
115	C₂×Q₈	—	71	1	no	—
116	C₂×Q₈	—	142	1	no	—
117	C₂×D₄	D₂₈₄	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	142	2	no	—
120	SD₁₆	—	284	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	142	1	no	—
124	C₄⋊C₄	—	71	1	no	—
125	C₄⋊C₄	—	142	2	no	—
126	C₄⋊C₄	—	142	2	no	—
127	C₄⋊C₄	—	142	1	no	—
128	C₄⋊C₄	—	142	1	no	—
129	C₄⋊C₄	—	142	1	no	—
130	C₄⋊C₄	—	142	1	no	—
131	C₄⋊C₄	—	142	1	no	—
132	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	142	2	no	—
135	C₄²	—	142	1	no	—
136	C₄²	C₂²×D₇₁	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	Q₈	—	284	2	no	—
139	Q₈	—	284	1	no	—
140	Q₈	—	284	1	no	—
141	Q₈	—	142	1	no	—
142	Q₈	—	142	1	no	—
143	D₄	—	2	1	no	—
144	D₄	#11765854	1	2	yes	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	284	1	no	—
149	C₂×C₄	—	568	2	no	—
150	C₂×C₄	—	142	1	no	—
151	C₂×C₄	—	142	1	no	—
152	C₂×C₄	—	142	2	no	—
153	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
154	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
155	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	284	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#14556098	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	284	1	no	—
166	C₄	—	284	1	no	—
167	C₄	—	284	2	no	—
168	C₄	—	568	1	no	—
169	C₄	C₂×#11765854	1	1	yes	characteristic
170	C₄	#14556147	1	1	yes	characteristic
171	C₄	#14556157	1	1	yes	characteristic
172	C₄	#14556161	1	1	yes	characteristic
173	C₂	—	4	2	no	—
174	C₂	#17986140	1	1	yes	characteristic
175	C₂	#17986162	1	1	yes	characteristic
176	C₂	#17986178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986706

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic, complemented

#17986285

C₂

yes

maximal, characteristic, complemented

#17986204

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, characteristic, complemented

#17985731

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

—

—

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

D₄×Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556060

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

—

—

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

—

C₂²×Dic₇₁

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

SD₁₆

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

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

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

—

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

D₇₁

yes

characteristic

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#125

—

142

—

101

C₂²⋊Q₈

—

142

—

102

C₂²⋊Q₈

—

142

—

103

C₄×Q₈

—

104

C₄×D₄

—

142

—

105

C₄×D₄

C₂×D₇₁

yes

characteristic

106

C₂×C₄⋊C₄

—

142

—

107

#106

—

142

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×D₇₁

yes

characteristic

110

#103

—

142

—

111

#103

—

142

—

112

#103

—

142

—

113

C₄×C₈

C₂×D₇₁

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

142

—

117

C₂×D₄

D₂₈₄

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

142

—

120

SD₁₆

—

284

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

142

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

142

—

126

C₄⋊C₄

—

142

—

127

C₄⋊C₄

—

142

—

128

C₄⋊C₄

—

142

—

129

C₄⋊C₄

—

142

—

130

C₄⋊C₄

—

142

—

131

C₄⋊C₄

—

142

—

132

C₄⋊C₄

D₂₈₄

yes

characteristic

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

142

—

135

C₄²

—

142

—

136

C₄²

C₂²×D₇₁

yes

characteristic

137

C₂³

—

138

Q₈

—

284

—

139

Q₈

—

284

—

140

Q₈

—

284

—

141

Q₈

—

142

—

142

Q₈

—

142

—

143

D₄

—

144

D₄

#11765854

yes

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

284

—

149

C₂×C₄

—

568

—

150

C₂×C₄

—

142

—

151

C₂×C₄

—

142

—

152

C₂×C₄

—

142

—

153

C₂×C₄

D₄×D₇₁

yes

characteristic

154

C₂×C₄

D₄×D₇₁

yes

characteristic

155

C₂×C₄

C₂×D₂₈₄

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

284

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#14556098

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

284

—

166

C₄

—

284

—

167

C₄

—

284

—

168

C₄

—

568

—

169

C₄

C₂×#11765854

yes

characteristic

170

C₄

#14556147

yes

characteristic

171

C₄

#14556157

yes

characteristic

172

C₄

#14556161

yes

characteristic

173

C₂

—

174

C₂

#17986140

yes

characteristic

175

C₂

#17986162

yes

characteristic

176

C₂

#17986178

yes

characteristic

	subgroup	G/H
center	C₂²	#14556098
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	D₄⋊C₈×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	D₄⋊C₈	D₇₁
hypocenter	C₇₁	#1643
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

subgroup

G/H

center

C₂²

#14556098

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

D₄⋊C₈×C₇₁

C₂

socle

C₂²×C₇₁

#120

hypercenter

D₄⋊C₈

D₇₁

hypocenter

C₇₁

#1643

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

N	H ≅ G/N	type
2. #17986350	C₂	semidirect
3. #17986285	C₂	semidirect
5. #17985747	C₂	semidirect
6. #17985731	C₂	semidirect
89. C₇₁	#1643	semidirect

H ≅ G/N

type

2. #17986350

C₂

semidirect

3. #17986285

C₂

semidirect

5. #17985747

C₂

semidirect

6. #17985731

C₂

semidirect

89. C₇₁

#1643

semidirect

Subgroups — #22587549

Conjugacy classes of subgroups

Complements