Group #31902440 — subgroups

Subgroups — #31902440

Order 18176.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22588515	C₂	1	1	yes	maximal, characteristic
2	#22587114	C₂	1	2	yes	maximal
3	#22586769	C₂	1	2	yes	maximal
4	#22586171	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1985	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556064	—	2	2	no	—
8	#17986023	C₂²	1	2	yes	—
9	#17985751	—	2	4	no	—
10	Q₈⋊(C₇₁⋊C₈)	—	2	4	no	—
11	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
14	#336×C₇₁	—	2	2	no	—
15	C₂×#11765849	—	2	2	no	—
16	#14556064	—	4	4	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄×C₇₁⋊C₈	D₄	1	4	yes	—
22	C₂²×Q₈×C₇₁	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	—	2	1	no	—
24	C₂²×C₈×C₇₁	D₄	1	2	yes	—
25	C₄×Q₈×C₇₁	—	2	2	no	—
26	C₄×Q₈×C₇₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	—	2	1	no	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	#97×C₇₁	—	2	2	no	—
33	#11765849	—	4	4	no	—
34	C₂×C₇₁⋊C₈	—	2	4	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×Q₈×C₇₁	—	2	2	no	—
37	C₂×Q₈×C₇₁	—	4	2	no	—
38	C₂²×C₄×C₇₁	—	2	1	no	—
39	C₂²×C₄×C₇₁	—	2	1	no	—
40	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	2	2	no	—
45	C₂×C₈×C₇₁	—	4	1	no	—
46	C₂×C₈×C₇₁	—	4	2	no	—
47	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
48	C₄⋊C₄×C₇₁	—	2	2	no	—
49	C₄⋊C₄×C₇₁	—	4	2	no	—
50	C₄²×C₇₁	—	4	1	no	—
51	C₄²×C₇₁	—	4	1	no	—
52	C₄²×C₇₁	C₂×D₄	1	2	yes	—
53	C₄²×C₇₁	#41	1	2	yes	—
54	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
55	Q₈×C₇₁	—	2	4	no	—
56	Q₈×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	—	4	1	no	—
60	C₂×C₄×C₇₁	—	4	1	no	—
61	C₂×C₄×C₇₁	—	4	1	no	—
62	C₂×C₄×C₇₁	—	4	2	no	—
63	C₂×C₄×C₇₁	—	4	1	no	—
64	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#123	1	2	yes	—
67	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
70	C₈×C₇₁	—	2	4	no	—
71	C₈×C₇₁	—	8	1	no	—
72	C₇₁⋊C₈	—	4	4	no	—
73	C₇₁⋊C₈	—	8	2	no	—
74	C₂²×C₇₁	—	2	1	no	—
75	C₂²×C₇₁	—	2	1	no	—
76	C₂²×C₇₁	#399	1	1	yes	characteristic
77	C₂²×C₇₁	#477	1	2	yes	—
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	—	2	2	no	—
80	C₄×C₇₁	—	4	2	no	—
81	C₄×C₇₁	—	8	1	no	—
82	C₄×C₇₁	#484	1	2	yes	—
83	C₄×C₇₁	#494	1	1	yes	characteristic
84	C₄×C₇₁	#496	1	1	yes	characteristic
85	#10662	—	71	1	no	maximal, Sylow
86	C₂×C₇₁	—	2	2	no	—
87	C₂×C₇₁	#2091	1	1	yes	characteristic
88	C₂×C₇₁	#2095	1	1	yes	characteristic
89	C₂×C₇₁	#2098	1	1	yes	characteristic
90	#1985	D₇₁	1	1	yes	characteristic
91	#1849	—	71	1	no	—
92	#1571	—	71	2	no	—
93	#1487	—	71	2	no	—
94	#1275	—	71	1	no	—
95	C₇₁	#10662	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
97	#431	—	71	2	no	—
98	C₂×#108	—	142	2	no	—
99	#423	—	71	2	no	—
100	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#335	—	142	4	no	—
104	Q₈⋊C₈	—	142	4	no	—
105	C₂²×Q₈	Dic₁₄₂	1	1	yes	characteristic
106	C₂×M₁₆	—	142	2	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
115	#108	—	284	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	142	2	no	—
118	C₄⋊C₈	—	142	2	no	—
119	C₄⋊C₈	—	142	2	no	—
120	C₂²⋊C₈	—	142	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	71	4	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
129	C₂²×C₄	#11765865	1	1	yes	characteristic
130	M₁₆	—	284	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	284	2	no	—
136	C₂×C₈	—	142	4	no	—
137	C₂×C₈	#11765855	1	4	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	D₄×D₇₁	1	2	yes	—
143	C₄²	#11765864	1	2	yes	—
144	C₂³	#14556107	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	#14556105	1	2	yes	—
155	C₂×C₄	#14556109	1	1	yes	characteristic
156	C₂×C₄	#14556110	1	2	yes	—
157	C₂×C₄	#14556124	1	1	yes	characteristic
158	C₂×C₄	C₂×#11765855	1	2	yes	—
159	C₂×C₄	#14556153	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	1	no	—
162	C₈	—	284	4	no	—
163	C₈	—	568	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#17985935	1	1	yes	characteristic
167	C₂²	#17986294	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	#17986332	1	2	yes	—
173	C₄	#17986378	1	1	yes	characteristic
174	C₄	#17986389	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#22589856	1	1	yes	characteristic
177	C₂	#22589878	1	1	yes	characteristic
178	C₂	#22589893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588515

C₂

yes

maximal, characteristic

#22587114

C₂

yes

maximal

#22586769

C₂

yes

maximal

#22586171

C₂

yes

maximal, characteristic

C₇₁×#1985

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556064

—

#17986023

C₂²

yes

—

#17985751

—

Q₈⋊(C₇₁⋊C₈)

—

C₂×C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×#97×C₇₁

C₂²

yes

characteristic

#336×C₇₁

—

C₂×#11765849

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂²×Q₈×C₇₁

Q₈

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

C₂²⋊Q₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

#123

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#399

yes

characteristic

C₂²×C₇₁

#477

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#484

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

—

—

—

—

C₇₁

#10662

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

C₂×D₇₁

yes

characteristic

#431

—

C₂×#108

—

142

—

#423

—

100

C₂×C₄⋊C₈

C₂×D₇₁

yes

characteristic

101

C₂×#97

C₂×D₇₁

yes

characteristic

102

#336

—

103

#335

—

142

—

104

Q₈⋊C₈

—

142

—

105

C₂²×Q₈

Dic₁₄₂

yes

characteristic

106

C₂×M₁₆

—

142

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²⋊D₇₁

yes

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

Dic₁₄₂

yes

characteristic

113

C₂×C₄²

—

114

C₂×C₄²

C₂²×D₇₁

yes

characteristic

115

#108

—

284

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

142

—

118

C₄⋊C₈

—

142

—

119

C₄⋊C₈

—

142

—

120

C₂²⋊C₈

—

142

—

121

#97

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

128

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

129

C₂²×C₄

#11765865

yes

characteristic

130

M₁₆

—

284

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

284

—

136

C₂×C₈

—

142

—

137

C₂×C₈

#11765855

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

D₄×D₇₁

yes

—

143

C₄²

#11765864

yes

—

144

C₂³

#14556107

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#14556105

yes

—

155

C₂×C₄

#14556109

yes

characteristic

156

C₂×C₄

#14556110

yes

—

157

C₂×C₄

#14556124

yes

characteristic

158

C₂×C₄

C₂×#11765855

yes

—

159

C₂×C₄

#14556153

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

284

—

163

C₈

—

568

—

164

C₂²

—

165

C₂²

—

166

C₂²

#17985935

yes

characteristic

167

C₂²

#17986294

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#17986332

yes

—

173

C₄

#17986378

yes

characteristic

174

C₄

#17986389

yes

characteristic

175

C₂

—

176

C₂

#22589856

yes

characteristic

177

C₂

#22589878

yes

characteristic

178

C₂

#22589893

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1985	C₂
socle	C₂²×C₇₁	#399
prime radical	C₂³×C₇₁	C₂²⋊Q₈
hypercenter	#1985	D₇₁
hypocenter	C₇₁	#10662
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

subgroup

G/H

center

C₂×C₄

#14556124

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1985

C₂

socle

C₂²×C₇₁

#399

prime radical

C₂³×C₇₁

C₂²⋊Q₈

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

N	H ≅ G/N	type
95. C₇₁	#10662	semidirect

H ≅ G/N

type

95. C₇₁

#10662

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588515	2
#22587114	2
#22586769	2
#22586171	2
C₇₁×#1985	2
#10662	71

subgroup

index

C₇₁×#1985

Subgroups — #31902440

Conjugacy classes of subgroups

Complements

Maximal subgroups