Group #22589703 — subgroups

Subgroups — #22589703

Order 9088.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#14556063	C₂	1	1	yes	maximal, characteristic
2	#17986041	C₂	1	1	yes	maximal, characteristic
3	#17986027	C₂	1	1	yes	maximal, characteristic
4	#17986008	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556052	C₂	1	1	yes	maximal, characteristic
6	#17985767	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#14556087	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
12	#14556063	—	2	2	no	—
13	#14556062	—	2	1	no	—
14	#14556060	—	2	1	no	—
15	Dic₇₁⋊C₈	—	2	2	no	—
16	#14556052	—	2	2	no	—
17	#14556052	—	2	1	no	—
18	#14556052	—	2	1	no	—
19	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#101×C₇₁	—	2	2	no	—
23	C₂²×Dic₇₁	—	2	1	no	—
24	#11765849	—	2	1	no	—
25	#11765849	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
33	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
34	C₂²⋊Dic₇₁	—	2	1	no	—
35	C₄×Dic₇₁	—	2	1	no	—
36	C₂×Q₈×C₇₁	—	2	1	no	—
37	C₂×Q₈×C₇₁	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	C₂×Q₈×C₇₁	—	2	1	no	—
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×Q₈×C₇₁	Q₈	1	2	yes	—
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
44	M₁₆×C₇₁	—	2	1	no	—
45	M₁₆×C₇₁	—	2	1	no	—
46	M₁₆×C₇₁	—	2	1	no	—
47	M₁₆×C₇₁	D₄	1	2	yes	—
48	C₂×C₈×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	—	2	1	no	—
50	C₂³×C₇₁	#41	1	1	yes	characteristic
51	Q₈×C₇₁	—	2	2	no	—
52	Q₈×C₇₁	—	2	2	no	—
53	Q₈×C₇₁	—	4	1	no	—
54	Q₈×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	4	1	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×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
64	C₂×Dic₇₁	—	2	1	no	—
65	C₂×Dic₇₁	—	2	1	no	—
66	C₂×Dic₇₁	—	4	1	no	—
67	C₂×Dic₇₁	—	8	1	no	—
68	C₈×C₇₁	—	2	2	no	—
69	C₈×C₇₁	—	4	1	no	—
70	C₇₁⋊C₈	—	2	2	no	—
71	C₇₁⋊C₈	—	4	1	no	—
72	C₂²×C₇₁	—	2	1	no	—
73	C₂²×C₇₁	—	2	1	no	—
74	C₂²×C₇₁	#121	1	1	yes	characteristic
75	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇₁	#126	1	1	yes	characteristic
77	C₄×C₇₁	—	4	2	no	—
78	C₄×C₇₁	#121	1	1	yes	characteristic
79	C₄×C₇₁	#121	1	1	yes	characteristic
80	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
82	Dic₇₁	—	4	1	no	—
83	Dic₇₁	—	4	1	no	—
84	Dic₇₁	—	8	1	no	—
85	C₂×C₇₁	—	2	1	no	—
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	#396	1	1	yes	characteristic
88	C₂×C₇₁	#471	1	1	yes	characteristic
89	C₂×C₇₁	#473	1	1	yes	characteristic
90	#2064	—	71	1	no	maximal, Sylow
91	C₇₁	#2064	1	1	yes	characteristic, complemented, Sylow
92	C₂×#108	—	71	1	no	—
93	#428	—	71	1	no	—
94	#424	—	71	1	no	—
95	#419	—	71	1	no	—
96	C₂×#103	—	71	1	no	—
97	C₂×#101	D₇₁	1	1	yes	characteristic
98	#343	—	71	1	no	—
99	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
100	C₂×M₁₆	—	71	1	no	—
101	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
102	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
103	C₂²×C₈	—	71	1	no	—
104	#117	—	71	1	no	—
105	C₂×C₄⋊C₄	—	71	1	no	—
106	#108	—	142	2	no	—
107	#107	—	142	1	no	—
108	#106	—	142	1	no	—
109	C₄⋊C₈	—	142	2	no	—
110	#103	—	142	1	no	—
111	#103	—	142	1	no	—
112	#103	—	142	2	no	—
113	#101	—	2	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₁₄₂	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	142	1	no	—
122	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	142	1	no	—
127	M₁₆	—	142	1	no	—
128	M₁₆	C₂²⋊D₇₁	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	71	2	no	—
132	C₂×C₈	—	142	1	no	—
133	C₂×C₈	—	142	1	no	—
134	C₂×C₈	—	142	1	no	—
135	C₄⋊C₄	—	71	1	no	—
136	C₄⋊C₄	—	71	1	no	—
137	C₄⋊C₄	—	142	1	no	—
138	C₄⋊C₄	—	142	1	no	—
139	C₂²⋊C₄	—	142	1	no	—
140	C₄²	—	142	1	no	—
141	C₂³	#11765865	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	284	1	no	—
150	C₂×C₄	—	568	1	no	—
151	C₂×C₄	—	142	1	no	—
152	C₂×C₄	—	142	1	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	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
157	C₂×C₄	#11765864	1	1	yes	characteristic
158	C₂×C₄	#11765864	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	4	1	no	—
161	C₈	—	284	1	no	—
162	C₈	—	142	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14556101	1	1	yes	characteristic
166	C₂²	#14556107	1	1	yes	characteristic
167	C₂²	#14556121	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	284	1	no	—
170	C₄	—	284	1	no	—
171	C₄	—	568	1	no	—
172	C₄	#14556099	1	1	yes	characteristic
173	C₄	#14556100	1	1	yes	characteristic
174	C₄	#14556105	1	1	yes	characteristic
175	C₄	#14556105	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#17985922	1	1	yes	characteristic
179	C₂	#17986265	1	1	yes	characteristic
180	C₂	#17986273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556063

C₂

yes

maximal, characteristic

#17986041

C₂

yes

maximal, characteristic

#17986027

C₂

yes

maximal, characteristic

#17986008

C₂

yes

maximal, characteristic

C₂×#14556052

C₂

yes

maximal, characteristic

#17985767

C₂

yes

maximal, characteristic

C₂×#101×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556087

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

—

—

Dic₇₁⋊C₈

—

#14556052

—

#14556052

—

#14556052

—

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#101×C₇₁

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

characteristic

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×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×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

#126

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic

#343

—

C₂²×Q₈

C₂×D₇₁

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×D₇₁

yes

characteristic

102

C₂×M₁₆

C₂×D₇₁

yes

characteristic

103

C₂²×C₈

—

104

#117

—

105

C₂×C₄⋊C₄

—

106

#108

—

142

—

107

#107

—

142

—

108

#106

—

142

—

109

C₄⋊C₈

—

142

—

110

#103

—

142

—

111

#103

—

142

—

112

#103

—

142

—

113

#101

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₁₄₂

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

142

—

122

C₂²×C₄

C₂²×D₇₁

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

142

—

127

M₁₆

—

142

—

128

M₁₆

C₂²⋊D₇₁

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

142

—

133

C₂×C₈

—

142

—

134

C₂×C₈

—

142

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

142

—

138

C₄⋊C₄

—

142

—

139

C₂²⋊C₄

—

142

—

140

C₄²

—

142

—

141

C₂³

#11765865

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

284

—

150

C₂×C₄

—

568

—

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

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

157

C₂×C₄

#11765864

yes

characteristic

158

C₂×C₄

#11765864

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

284

—

162

C₈

—

142

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14556101

yes

characteristic

166

C₂²

#14556107

yes

characteristic

167

C₂²

#14556121

yes

characteristic

168

C₄

—

169

C₄

—

284

—

170

C₄

—

284

—

171

C₄

—

568

—

172

C₄

#14556099

yes

characteristic

173

C₄

#14556100

yes

characteristic

174

C₄

#14556105

yes

characteristic

175

C₄

#14556105

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#17985922

yes

characteristic

179

C₂

#17986265

yes

characteristic

180

C₂

#17986273

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#101×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	C₂×#101	D₇₁
hypocenter	C₇₁	#2064
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#14556101

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#101×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
91. C₇₁	#2064	semidirect

H ≅ G/N

type

91. C₇₁

#2064

semidirect

Subgroups — #22589703

Conjugacy classes of subgroups

Complements