Group #31913833 — subgroups

Subgroups — #31913833

Order 18176.

310 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 131 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×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#22588503	C₂	1	2	yes	maximal
3	#22586432	C₂	1	1	yes	maximal, characteristic
4	#22586431	C₂	1	2	yes	maximal
5	C₇₁×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₂×#14556059	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985864	—	2	2	no	—
11	C₁₆×Dic₇₁	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₇₁	C₂²	1	2	yes	—
14	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₁	—	2	2	no	—
16	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
19	#14556062	—	2	2	no	—
20	#14556062	—	2	2	no	—
21	#14556059	—	4	4	no	—
22	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
23	C₈×Dic₇₁	Q₈	1	2	yes	—
24	C₇₁⋊#95	—	4	2	no	—
25	C₇₁⋊#95	—	4	2	no	—
26	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₂×C₁₆×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
30	#107×C₇₁	—	2	2	no	—
31	#107×C₇₁	—	2	2	no	—
32	C₂²×Dic₇₁	—	4	2	no	—
33	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
34	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
35	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₇₁⋊C₈	SD₁₆	1	4	yes	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
40	C₄×Dic₇₁	Q₁₆	1	2	yes	—
41	C₇₁⋊C₁₆	—	2	2	no	—
42	C₂²×C₄×C₇₁	—	4	2	no	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₇₁	#41	1	2	yes	—
47	C₂×C₈×C₇₁	#41	1	2	yes	—
48	C₄⋊C₄×C₇₁	—	4	2	no	—
49	C₄⋊C₄×C₇₁	—	4	2	no	—
50	C₁₆×C₇₁	—	2	2	no	—
51	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	—	8	2	no	—
54	C₂×C₄×C₇₁	—	8	2	no	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#124	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
58	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₇₁	—	2	1	no	—
60	C₂×Dic₇₁	—	2	2	no	—
61	C₂×Dic₇₁	—	2	1	no	—
62	C₂×Dic₇₁	—	8	2	no	—
63	C₂×Dic₇₁	—	8	2	no	—
64	C₂×Dic₇₁	—	8	2	no	—
65	C₂×Dic₇₁	#107	1	2	yes	—
66	C₂×Dic₇₁	#107	1	2	yes	—
67	C₈×C₇₁	#117	1	1	yes	characteristic
68	C₈×C₇₁	#117	1	1	yes	characteristic
69	C₈×C₇₁	#125	1	2	yes	—
70	C₇₁⋊C₈	#106	1	4	yes	—
71	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
72	C₂²×C₇₁	#486	1	2	yes	—
73	C₂²×C₇₁	#508	1	1	yes	characteristic
74	C₂²×C₇₁	#508	1	2	yes	—
75	C₂²×C₇₁	#508	1	1	yes	characteristic
76	C₄×C₇₁	—	16	2	no	—
77	C₄×C₇₁	#382	1	1	yes	characteristic
78	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
79	C₄×C₇₁	#487	1	2	yes	—
80	Dic₇₁	—	2	2	no	—
81	Dic₇₁	—	2	2	no	—
82	Dic₇₁	—	16	2	no	—
83	#12889	—	71	1	no	maximal, Sylow
84	C₂×C₇₁	#1834	1	1	yes	characteristic
85	C₂×C₇₁	#2158	1	1	yes	characteristic
86	C₂×C₇₁	#2158	1	1	yes	characteristic
87	C₂×C₇₁	#2242	1	2	yes	—
88	C₂×C₇₁	#2242	1	2	yes	—
89	C₂×C₄×C₁₆	—	71	1	no	—
90	#1846	—	71	2	no	—
91	#1380	—	71	1	no	—
92	#1380	—	71	2	no	—
93	#1380	D₇₁	1	1	yes	characteristic
94	C₇₁	#12889	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	71	1	no	—
96	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
97	C₂×#107	—	71	2	no	—
98	C₂×#107	C₂×D₇₁	1	2	yes	—
99	C₂×#106	—	142	2	no	—
100	C₂×C₄×C₈	—	71	1	no	—
101	#382	—	142	2	no	—
102	C₄×C₁₆	—	71	4	no	—
103	C₂²×C₈	—	71	1	no	—
104	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	142	2	no	—
107	C₂×C₄²	—	71	1	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	71	2	no	—
110	C₂×C₁₆	—	142	2	no	—
111	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
112	#107	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	—	142	2	no	—
115	#107	—	142	2	no	—
116	#106	—	284	4	no	—
117	C₄×C₈	—	71	2	no	—
118	C₄×C₈	—	71	2	no	—
119	#95	—	284	2	no	—
120	#95	—	284	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	284	2	no	—
123	C₂²×C₄	—	71	1	no	—
124	C₂²×C₄	—	71	1	no	—
125	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
126	C₂×C₈	—	71	2	no	—
127	C₂×C₈	—	71	4	no	—
128	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
129	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
130	C₂×C₈	#11765864	1	2	yes	—
131	C₂×C₈	#11765865	1	2	yes	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	284	2	no	—
135	C₄⋊C₄	—	284	2	no	—
136	C₄²	—	71	2	no	—
137	C₄²	—	71	2	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	142	2	no	—
140	C₂³	D₈×D₇₁	1	1	yes	characteristic
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	568	2	no	—
145	C₂×C₄	—	568	2	no	—
146	C₂×C₄	—	568	2	no	—
147	C₂×C₄	—	71	2	no	—
148	C₂×C₄	—	71	2	no	—
149	C₂×C₄	—	142	1	no	—
150	C₂×C₄	—	142	2	no	—
151	C₂×C₄	—	142	1	no	—
152	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
153	C₂×C₄	#14556114	1	2	yes	—
154	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
155	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
156	C₈	—	71	4	no	—
157	C₈	#14556084	1	1	yes	characteristic
158	C₈	#14556085	1	1	yes	characteristic
159	C₈	#14556118	1	2	yes	—
160	C₂²	#107×D₇₁	1	1	yes	characteristic
161	C₂²	#17986344	1	2	yes	—
162	C₂²	#17986428	1	1	yes	characteristic
163	C₂²	#17986430	1	2	yes	—
164	C₂²	#17986432	1	1	yes	characteristic
165	C₄	—	16	2	no	—
166	C₄	—	1136	2	no	—
167	C₄	—	142	2	no	—
168	C₄	—	142	2	no	—
169	C₄	#17985863	1	1	yes	characteristic
170	C₄	#106×D₇₁	1	1	yes	characteristic
171	C₄	#17986348	1	2	yes	—
172	C₂	#22588466	1	1	yes	characteristic
173	C₂	#22590121	1	1	yes	characteristic
174	C₂	#22590123	1	1	yes	characteristic
175	C₂	#22590570	1	2	yes	—
176	C₂	#22590573	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#22588503

C₂

yes

maximal

#22586432

C₂

yes

maximal, characteristic

#22586431

C₂

yes

maximal

C₇₁×#1380

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556062

C₂²

yes

—

C₂×#14556059

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985864

—

C₁₆×Dic₇₁

C₄

yes

complemented

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#107×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556062

—

#14556062

—

#14556059

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₈

yes

characteristic

C₂²×Dic₇₁

D₈

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₄×Dic₇₁

Q₁₆

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

C₂×Dic₇₁

#107

yes

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

—

C₇₁⋊C₈

#106

yes

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#486

yes

—

C₂²×C₇₁

#508

yes

characteristic

C₂²×C₇₁

#508

yes

—

C₂²×C₇₁

#508

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2242

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₇₁

yes

—

C₂×#106

—

142

—

100

C₂×C₄×C₈

—

101

#382

—

142

—

102

C₄×C₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂²×D₇₁

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

142

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

142

—

111

C₂×C₁₆

C₄×D₇₁

yes

—

112

#107

—

113

#107

—

114

#107

—

142

—

115

#107

—

142

—

116

#106

—

284

—

117

C₄×C₈

—

118

C₄×C₈

—

119

#95

—

284

—

120

#95

—

284

—

121

C₂²×C₄

—

122

C₂²×C₄

—

284

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×D₇₁

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

129

C₂×C₈

Q₈×D₇₁

yes

characteristic

130

C₂×C₈

#11765864

yes

—

131

C₂×C₈

#11765865

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

284

—

135

C₄⋊C₄

—

284

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₁₆

—

142

—

140

C₂³

D₈×D₇₁

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

568

—

145

C₂×C₄

—

568

—

146

C₂×C₄

—

568

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

142

—

150

C₂×C₄

—

142

—

151

C₂×C₄

—

142

—

152

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

153

C₂×C₄

#14556114

yes

—

154

C₂×C₄

SD₁₆×D₇₁

yes

—

155

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

156

C₈

—

157

C₈

#14556084

yes

characteristic

158

C₈

#14556085

yes

characteristic

159

C₈

#14556118

yes

—

160

C₂²

#107×D₇₁

yes

characteristic

161

C₂²

#17986344

yes

—

162

C₂²

#17986428

yes

characteristic

163

C₂²

#17986430

yes

—

164

C₂²

#17986432

yes

characteristic

165

C₄

—

166

C₄

—

1136

—

167

C₄

—

142

—

168

C₄

—

142

—

169

C₄

#17985863

yes

characteristic

170

C₄

#106×D₇₁

yes

characteristic

171

C₄

#17986348

yes

—

172

C₂

#22588466

yes

characteristic

173

C₂

#22590121

yes

characteristic

174

C₂

#22590123

yes

characteristic

175

C₂

#22590570

yes

—

176

C₂

#22590573

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1380	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1380	D₇₁
hypocenter	C₇₁	#12889
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1380

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
11. C₁₆×Dic₇₁	C₄	semidirect
94. C₇₁	#12889	semidirect

H ≅ G/N

type

11. C₁₆×Dic₇₁

C₄

semidirect

94. C₇₁

#12889

semidirect

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

subgroup	index
C₂×C₁₆×Dic₇₁	2
#22588503	2
#22586432	2
#22586431	2
C₇₁×#1380	2
#12889	71

subgroup

index

C₂×C₁₆×Dic₇₁

C₇₁×#1380

Subgroups — #31913833

Conjugacy classes of subgroups

Complements

Maximal subgroups