Group #31891156 — subgroups

Subgroups — #31891156

Order 18176.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#22588504	C₂	1	1	yes	maximal, characteristic
2	#22586632	C₂	1	1	yes	maximal, characteristic
3	#22586200	C₂	1	2	yes	maximal
4	#22586200	C₂	1	2	yes	maximal
5	C₇₁×#1837	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556062	—	2	1	no	—
8	C₂×#14556060	—	2	1	no	—
9	#17985865	C₂²	1	2	yes	—
10	#17985721	C₄	1	4	yes	complemented
11	#423×C₇₁	C₂²	1	1	yes	characteristic
12	#423×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#341×C₇₁	—	2	1	no	—
15	#341×C₇₁	—	2	1	no	—
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	#14556062	—	4	2	no	—
18	#14556060	—	4	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
22	C₇₁⋊#95	—	2	4	no	—
23	C₂×M₁₆×C₇₁	—	2	1	no	—
24	C₂×M₁₆×C₇₁	—	2	1	no	—
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₁	—	2	2	no	—
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₁	Q₈	1	2	yes	—
32	C₂²×Dic₇₁	—	4	2	no	—
33	C₂×C₇₁⋊C₈	—	2	4	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	#41	1	2	yes	—
38	M₁₆×C₇₁	—	4	1	no	—
39	M₁₆×C₇₁	—	4	1	no	—
40	M₁₆×C₇₁	—	4	1	no	—
41	M₁₆×C₇₁	—	4	1	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	4	1	no	—
47	C₂×C₈×C₇₁	—	4	1	no	—
48	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇₁	#41	1	2	yes	—
53	C₂³×C₇₁	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	4	2	no	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#117	1	2	yes	—
57	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#125	1	2	yes	—
59	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
63	C₂×Dic₇₁	—	8	4	no	—
64	C₂×Dic₇₁	—	8	2	no	—
65	C₈×C₇₁	—	2	2	no	—
66	C₈×C₇₁	—	2	2	no	—
67	C₈×C₇₁	—	8	1	no	—
68	C₈×C₇₁	—	8	1	no	—
69	C₇₁⋊C₈	—	4	4	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	#382	1	1	yes	characteristic
72	C₂²×C₇₁	#490	1	1	yes	characteristic
73	C₂²×C₇₁	#491	1	1	yes	characteristic
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	#428	1	1	yes	characteristic
77	C₄×C₇₁	#428	1	1	yes	characteristic
78	C₄×C₇₁	#489	1	1	yes	characteristic
79	C₄×C₇₁	#489	1	1	yes	characteristic
80	Dic₇₁	—	16	2	no	—
81	#8795	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	—	2	1	no	—
84	C₂×C₇₁	#1833	1	2	yes	—
85	C₂×C₇₁	#1836	1	1	yes	characteristic
86	#1846	—	71	1	no	—
87	#1837	D₇₁	1	1	yes	characteristic
88	#1452	—	71	1	no	—
89	#1286	—	71	2	no	—
90	#1286	—	71	2	no	—
91	C₇₁	#8795	1	1	yes	characteristic, complemented, Sylow
92	#431	—	71	2	no	—
93	C₂×#107	—	142	1	no	—
94	C₂×#106	—	142	1	no	—
95	#423	C₂×D₇₁	1	1	yes	characteristic
96	#423	C₂×D₇₁	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
98	#382	—	71	2	no	—
99	#341	—	2	1	no	—
100	#341	—	2	1	no	—
101	#322	—	71	4	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	142	2	no	—
107	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
108	#107	—	284	2	no	—
109	#106	—	284	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	142	2	no	—
113	C₂²⋊C₈	—	142	2	no	—
114	C₄×C₈	—	71	4	no	—
115	C₄×C₈	Dic₁₄₂	1	2	yes	—
116	C₄×C₈	C₄×D₇₁	1	2	yes	—
117	#95	—	142	4	no	—
118	C₂²×C₄	—	284	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
120	C₂²×C₄	#11765866	1	2	yes	—
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	1	no	—
124	M₁₆	—	4	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	142	4	no	—
132	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
133	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
134	C₄⋊C₄	—	284	2	no	—
135	C₄⋊C₄	—	284	2	no	—
136	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
138	C₄²	#11765864	1	2	yes	—
139	C₂³	#14556117	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	568	4	no	—
142	C₂×C₄	—	568	2	no	—
143	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14556086	1	2	yes	—
145	C₂×C₄	#14556115	1	1	yes	characteristic
146	C₂×C₄	#14556119	1	2	yes	—
147	C₂×C₄	#14556153	1	1	yes	characteristic
148	C₂×C₄	#14556153	1	1	yes	characteristic
149	C₂×C₄	#14556158	1	1	yes	characteristic
150	C₂×C₄	#14556158	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	—	8	1	no	—
155	C₈	—	284	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#17985864	1	1	yes	characteristic
158	C₂²	#17986365	1	1	yes	characteristic
159	C₂²	#17986369	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1136	2	no	—
163	C₄	#17986042	1	1	yes	characteristic
164	C₄	#17986042	1	1	yes	characteristic
165	C₄	#17986358	1	1	yes	characteristic
166	C₄	#17986358	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22588465	1	2	yes	—
170	C₂	#22588473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588504

C₂

yes

maximal, characteristic

#22586632

C₂

yes

maximal, characteristic

#22586200

C₂

yes

maximal

#22586200

C₂

yes

maximal

C₇₁×#1837

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556062

—

C₂×#14556060

—

#17985865

C₂²

yes

—

#17985721

C₄

yes

complemented

#423×C₇₁

C₂²

yes

characteristic

#423×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#341×C₇₁

—

#341×C₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556062

—

#14556060

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₇₁⋊#95

—

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#8795

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1833

yes

—

C₂×C₇₁

yes

—

D₇₁

yes

—

—

—

C₇₁

#8795

yes

characteristic, complemented, Sylow

#431

—

C₂×#107

—

142

—

C₂×#106

—

142

—

#423

C₂×D₇₁

yes

characteristic

#423

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

—

#341

—

100

#341

—

101

#322

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

105

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

106

C₂×C₄⋊C₄

—

142

—

107

C₂×C₄²

C₂²×D₇₁

yes

characteristic

108

#107

—

284

—

109

#106

—

284

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

142

—

113

C₂²⋊C₈

—

142

—

114

C₄×C₈

—

115

C₄×C₈

Dic₁₄₂

yes

—

116

C₄×C₈

C₄×D₇₁

yes

—

117

#95

—

142

—

118

C₂²×C₄

—

284

—

119

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

120

C₂²×C₄

#11765866

yes

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

142

—

132

C₂×C₈

Dic₇₁⋊C₄

yes

—

133

C₂×C₈

Dic₇₁⋊C₄

yes

—

134

C₄⋊C₄

—

284

—

135

C₄⋊C₄

—

284

—

136

C₄²

C₂×C₄×D₇₁

yes

characteristic

137

C₄²

C₂×Dic₁₄₂

yes

characteristic

138

C₄²

#11765864

yes

—

139

C₂³

#14556117

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

568

—

142

C₂×C₄

—

568

—

143

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

144

C₂×C₄

#14556086

yes

—

145

C₂×C₄

#14556115

yes

characteristic

146

C₂×C₄

#14556119

yes

—

147

C₂×C₄

#14556153

yes

characteristic

148

C₂×C₄

#14556153

yes

characteristic

149

C₂×C₄

#14556158

yes

characteristic

150

C₂×C₄

#14556158

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

284

—

156

C₂²

—

157

C₂²

#17985864

yes

characteristic

158

C₂²

#17986365

yes

characteristic

159

C₂²

#17986369

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

1136

—

163

C₄

#17986042

yes

characteristic

164

C₄

#17986042

yes

characteristic

165

C₄

#17986358

yes

characteristic

166

C₄

#17986358

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#22588465

yes

—

170

C₂

#22588473

yes

characteristic

	subgroup	G/H
center	C₂²	#17985864
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1837	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1837	D₇₁
hypocenter	C₇₁	#8795
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#17985864

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1837

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
10. #17985721	C₄	semidirect
91. C₇₁	#8795	semidirect

H ≅ G/N

type

10. #17985721

C₄

semidirect

91. C₇₁

#8795

semidirect

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

subgroup	index
#22588504	2
#22586632	2
#22586200	2
C₇₁×#1837	2
#8795	71

subgroup

index

C₇₁×#1837

Subgroups — #31891156

Conjugacy classes of subgroups

Complements

Maximal subgroups