Group #22587545 — subgroups

Subgroups — #22587545

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	#17986351	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986281	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985729	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#461×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556119	—	2	1	no	—
10	#14556107	—	2	1	no	—
11	#14556105	—	2	2	no	—
12	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
13	D₄×Dic₇₁	—	2	1	no	—
14	C₂×C₄⋊Dic₇₁	—	2	1	no	—
15	#14556059	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#14556051	—	2	2	no	—
18	#14556051	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₇₁	—	2	1	no	—
21	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#103×C₇₁	—	2	1	no	—
25	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
26	C₂×Dic₁₄₂	D₄	1	2	yes	—
27	C₂²×Dic₇₁	—	2	2	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	2	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
34	Dic₇₁⋊C₄	—	2	2	no	—
35	Dic₇₁⋊C₄	—	2	1	no	—
36	C₂²⋊Dic₇₁	—	2	2	no	—
37	C₄×Dic₇₁	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	—	2	1	no	—
41	SD₁₆×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₄⋊C₄×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	Q₈×C₇₁	—	4	1	no	—
50	Q₈×C₇₁	—	4	1	no	—
51	D₄×C₇₁	—	2	1	no	—
52	D₄×C₇₁	SD₁₆	1	2	yes	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	—	4	1	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₇₁	—	2	1	no	—
61	C₂×Dic₇₁	—	2	1	no	—
62	C₂×Dic₇₁	—	2	2	no	—
63	C₂×Dic₇₁	—	8	2	no	—
64	Dic₁₄₂	—	2	1	no	—
65	Dic₁₄₂	—	2	1	no	—
66	Dic₁₄₂	—	4	2	no	—
67	C₈×C₇₁	—	4	1	no	—
68	C₇₁⋊C₈	—	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	—	8	1	no	—
77	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₄×C₇₁	#135	1	1	yes	characteristic
79	C₄×C₇₁	#137	1	1	yes	characteristic
80	C₄×C₇₁	#137	1	1	yes	characteristic
81	Dic₇₁	—	4	1	no	—
82	Dic₇₁	—	4	2	no	—
83	Dic₇₁	—	4	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	D₇₁	1	1	yes	characteristic
94	#333	—	71	1	no	—
95	Q₈⋊C₈	—	71	1	no	—
96	D₄⋊C₈	—	71	1	no	—
97	C₂×SD₁₆	—	2	1	no	—
98	C₄⋊Q₈	—	71	1	no	—
99	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
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	—	2	1	no	—
111	#103	—	142	1	no	—
112	#103	—	142	2	no	—
113	C₄×C₈	—	71	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	71	2	no	—
116	C₂×Q₈	—	71	1	no	—
117	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	142	2	no	—
120	SD₁₆	—	4	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	142	1	no	—
123	C₂×C₈	—	142	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	71	1	no	—
127	C₄⋊C₄	—	142	2	no	—
128	C₄⋊C₄	—	142	2	no	—
129	C₄⋊C₄	—	142	1	no	—
130	C₄⋊C₄	—	142	1	no	—
131	C₄⋊C₄	—	142	1	no	—
132	C₄⋊C₄	C₂²⋊D₇₁	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	—	4	1	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	284	2	no	—
141	Q₈	—	142	1	no	—
142	Q₈	—	142	1	no	—
143	D₄	—	2	1	no	—
144	D₄	#11765853	1	2	yes	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	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×C₂²⋊D₇₁	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	284	1	no	—
158	C₈	—	284	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#14556096	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	8	1	no	—
166	C₄	—	284	1	no	—
167	C₄	—	284	1	no	—
168	C₄	—	284	2	no	—
169	C₄	C₂×#11765853	1	1	yes	characteristic
170	C₄	#14556149	1	1	yes	characteristic
171	C₄	#14556157	1	1	yes	characteristic
172	C₄	#14556158	1	1	yes	characteristic
173	C₂	—	4	2	no	—
174	C₂	#17986137	1	1	yes	characteristic
175	C₂	#17986158	1	1	yes	characteristic
176	C₂	#17986175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986706

C₂

yes

maximal, characteristic

#17986351

C₂

yes

maximal, characteristic, complemented

#17986281

C₂

yes

maximal, characteristic, complemented

#17985747

C₂

yes

maximal, characteristic, complemented

#17985729

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#461×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

—

—

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

D₄×Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556051

—

#14556051

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

—

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

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic

#333

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

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

—

111

#103

—

142

—

112

#103

—

142

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

142

—

120

SD₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

142

—

123

C₂×C₈

—

142

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

142

—

128

C₄⋊C₄

—

142

—

129

C₄⋊C₄

—

142

—

130

C₄⋊C₄

—

142

—

131

C₄⋊C₄

—

142

—

132

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

142

—

135

C₄²

—

142

—

136

C₄²

C₂²×D₇₁

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

284

—

141

Q₈

—

142

—

142

Q₈

—

142

—

143

D₄

—

144

D₄

#11765853

yes

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

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×C₂²⋊D₇₁

yes

characteristic

156

C₈

—

157

C₈

—

284

—

158

C₈

—

284

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#14556096

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

284

—

167

C₄

—

284

—

168

C₄

—

284

—

169

C₄

C₂×#11765853

yes

characteristic

170

C₄

#14556149

yes

characteristic

171

C₄

#14556157

yes

characteristic

172

C₄

#14556158

yes

characteristic

173

C₂

—

174

C₂

#17986137

yes

characteristic

175

C₂

#17986158

yes

characteristic

176

C₂

#17986175

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#461×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#461	D₇₁
hypocenter	C₇₁	#1643
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#461×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
2. #17986351	C₂	semidirect
3. #17986281	C₂	semidirect
4. #17985747	C₂	semidirect
5. #17985729	C₂	semidirect
89. C₇₁	#1643	semidirect

H ≅ G/N

type

2. #17986351

C₂

semidirect

3. #17986281

C₂

semidirect

4. #17985747

C₂

semidirect

5. #17985729

C₂

semidirect

89. C₇₁

#1643

semidirect

Subgroups — #22587545

Conjugacy classes of subgroups

Complements