Group #31913839 — subgroups

Subgroups — #31913839

Order 18176.

318 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 155 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×#17985775	C₂	1	1	yes	maximal, characteristic
2	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	4	yes	maximal
4	C₇₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	C₂²	1	2	yes	—
7	C₂×#14556060	—	2	1	no	—
8	#17985865	—	2	1	no	—
9	#17985775	C₄	1	4	yes	complemented
10	C₂×#107×C₇₁	—	2	1	no	—
11	C₂×#107×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#384×C₇₁	—	2	1	no	—
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14556062	—	2	4	no	—
19	#14556060	—	4	2	no	—
20	C₇₁⋊#95	—	4	2	no	—
21	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
25	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
26	#107×C₇₁	—	2	4	no	—
27	#107×C₇₁	—	4	2	no	—
28	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₁	Q₈	1	2	yes	—
30	C₂²×Dic₇₁	—	4	2	no	—
31	C₄⋊Dic₇₁	—	4	2	no	—
32	C₄⋊Dic₇₁	—	4	2	no	—
33	C₇₁⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₇₁	—	4	2	no	—
35	C₂²×C₄×C₇₁	D₈	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
39	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₇₁	#41	1	2	yes	—
43	C₂×C₈×C₇₁	#41	1	2	yes	—
44	C₄⋊C₄×C₇₁	—	4	2	no	—
45	C₄⋊C₄×C₇₁	—	4	2	no	—
46	C₄⋊C₄×C₇₁	—	8	4	no	—
47	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₄²×C₇₁	Q₁₆	1	2	yes	—
49	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	—	8	4	no	—
54	C₂×C₄×C₇₁	#107	1	4	yes	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×Dic₇₁	—	8	2	no	—
62	C₂×Dic₇₁	—	8	2	no	—
63	C₂×Dic₇₁	—	8	2	no	—
64	C₈×C₇₁	#106	1	4	yes	—
65	C₈×C₇₁	#117	1	2	yes	—
66	C₈×C₇₁	#125	1	2	yes	—
67	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
68	C₂²×C₇₁	#486	1	1	yes	characteristic
69	C₂²×C₇₁	#486	1	1	yes	characteristic
70	C₂²×C₇₁	#509	1	2	yes	—
71	C₂²×C₇₁	#510	1	2	yes	—
72	C₄×C₇₁	—	2	4	no	—
73	C₄×C₇₁	—	16	2	no	—
74	C₄×C₇₁	#382	1	1	yes	characteristic
75	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
76	C₄×C₇₁	#487	1	1	yes	characteristic
77	C₄×C₇₁	#487	1	1	yes	characteristic
78	Dic₇₁	—	16	2	no	—
79	#12890	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	#1834	1	1	yes	characteristic
81	C₂×C₇₁	#2159	1	2	yes	—
82	C₂×C₇₁	#2243	1	2	yes	—
83	C₂×C₇₁	#2244	1	1	yes	characteristic
84	C₂×C₇₁	#2245	1	1	yes	characteristic
85	C₂×#346	—	71	1	no	—
86	#1846	—	71	1	no	—
87	#1845	D₇₁	1	1	yes	characteristic
88	#1380	—	71	4	no	—
89	C₇₁	#12890	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	71	2	no	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	—	71	2	no	—
93	C₂×#107	C₂×D₇₁	1	2	yes	—
94	C₂×#106	—	142	1	no	—
95	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	#382	—	142	1	no	—
98	#346	—	71	4	no	—
99	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₄⋊C₄	—	142	2	no	—
104	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
105	C₂×C₁₆	—	71	4	no	—
106	C₂×C₁₆	—	142	2	no	—
107	C₂×C₁₆	—	142	2	no	—
108	#107	—	2	4	no	—
109	#107	—	4	2	no	—
110	#107	—	142	4	no	—
111	#106	—	284	2	no	—
112	C₄×C₈	Dic₁₄₂	1	2	yes	—
113	C₄×C₈	C₄×D₇₁	1	2	yes	—
114	#95	—	284	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	284	2	no	—
117	C₂²×C₄	#11765850	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
120	C₂×C₈	#11765852	1	2	yes	—
121	C₂×C₈	#11765853	1	2	yes	—
122	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
124	C₂×C₈	#11765864	1	2	yes	—
125	C₂×C₈	#11765866	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₄⋊C₄	—	284	2	no	—
130	C₄⋊C₄	—	284	2	no	—
131	C₄²	Dic₇₁⋊C₄	1	2	yes	—
132	C₄²	#11765855	1	2	yes	—
133	C₁₆	—	142	4	no	—
134	C₂³	C₂×#11765850	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	568	2	no	—
140	C₂×C₄	—	568	2	no	—
141	C₂×C₄	—	568	2	no	—
142	C₂×C₄	#14556061	1	4	yes	—
143	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14556115	1	1	yes	characteristic
145	C₂×C₄	#14556117	1	1	yes	characteristic
146	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
147	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
148	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
149	C₈	#14556059	1	4	yes	—
150	C₈	#14556086	1	2	yes	—
151	C₈	#14556119	1	2	yes	—
152	C₂²	C₂×#14556061	1	1	yes	characteristic
153	C₂²	#17986345	1	1	yes	characteristic
154	C₂²	#17986347	1	1	yes	characteristic
155	C₂²	#17986435	1	2	yes	—
156	C₂²	#17986440	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	1136	2	no	—
160	C₄	#17985864	1	1	yes	characteristic
161	C₄	C₂×#14556059	1	1	yes	characteristic
162	C₄	#17986349	1	1	yes	characteristic
163	C₄	#17986351	1	1	yes	characteristic
164	C₂	#22588467	1	1	yes	characteristic
165	C₂	#22590126	1	2	yes	—
166	C₂	#22590579	1	2	yes	—
167	C₂	#22590581	1	1	yes	characteristic
168	C₂	#22590585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985775

C₂

yes

maximal, characteristic

#22588504

C₂

yes

maximal, characteristic

#22586431

C₂

yes

maximal

C₇₁×#1845

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

#17985865

—

#17985775

C₄

yes

complemented

C₂×#107×C₇₁

—

C₂×#107×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556062

—

#14556060

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

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

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

Q₁₆

yes

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

#125

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#510

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2243

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1380

—

C₇₁

#12890

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

C₂×#107

C₂×D₇₁

yes

—

C₂×#106

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#384

—

#382

—

142

—

#346

—

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

100

C₂²×C₈

C₂²×D₇₁

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

142

—

104

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

142

—

107

C₂×C₁₆

—

142

—

108

#107

—

109

#107

—

110

#107

—

142

—

111

#106

—

284

—

112

C₄×C₈

Dic₁₄₂

yes

—

113

C₄×C₈

C₄×D₇₁

yes

—

114

#95

—

284

—

115

C₂²×C₄

—

116

C₂²×C₄

—

284

—

117

C₂²×C₄

#11765850

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

119

C₂×C₈

Dic₇₁⋊C₄

yes

—

120

C₂×C₈

#11765852

yes

—

121

C₂×C₈

#11765853

yes

—

122

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

123

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

124

C₂×C₈

#11765864

yes

—

125

C₂×C₈

#11765866

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

284

—

130

C₄⋊C₄

—

284

—

131

C₄²

Dic₇₁⋊C₄

yes

—

132

C₄²

#11765855

yes

—

133

C₁₆

—

142

—

134

C₂³

C₂×#11765850

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

568

—

140

C₂×C₄

—

568

—

141

C₂×C₄

—

568

—

142

C₂×C₄

#14556061

yes

—

143

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

144

C₂×C₄

#14556115

yes

characteristic

145

C₂×C₄

#14556117

yes

characteristic

146

C₂×C₄

C₂×#11765852

yes

characteristic

147

C₂×C₄

C₂×#11765853

yes

characteristic

148

C₂×C₄

C₂×#11765855

yes

characteristic

149

C₈

#14556059

yes

—

150

C₈

#14556086

yes

—

151

C₈

#14556119

yes

—

152

C₂²

C₂×#14556061

yes

characteristic

153

C₂²

#17986345

yes

characteristic

154

C₂²

#17986347

yes

characteristic

155

C₂²

#17986435

yes

—

156

C₂²

#17986440

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

1136

—

160

C₄

#17985864

yes

characteristic

161

C₄

C₂×#14556059

yes

characteristic

162

C₄

#17986349

yes

characteristic

163

C₄

#17986351

yes

characteristic

164

C₂

#22588467

yes

characteristic

165

C₂

#22590126

yes

—

166

C₂

#22590579

yes

—

167

C₂

#22590581

yes

characteristic

168

C₂

#22590585

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1845	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1845	D₇₁
hypocenter	C₇₁	#12890
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1845

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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
9. #17985775	C₄	semidirect
89. C₇₁	#12890	semidirect

H ≅ G/N

type

9. #17985775

C₄

semidirect

89. C₇₁

#12890

semidirect

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

subgroup	index
C₂×#17985775	2
#22588504	2
#22586431	2
C₇₁×#1845	2
#12890	71

subgroup

index

C₇₁×#1845

Subgroups — #31913839

Conjugacy classes of subgroups

Complements

Maximal subgroups