Group #31917596 — subgroups

Subgroups — #31917596

Order 18176.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22589088	C₂	1	1	yes	maximal, characteristic
2	#22586447	C₂	1	2	yes	maximal
3	#22586438	C₂	1	2	yes	maximal
4	#22586422	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556064	C₂²	1	1	yes	characteristic
8	#17986037	C₂²	1	1	yes	characteristic
9	#17986025	—	2	1	no	—
10	#17985759	—	2	1	no	—
11	C₂×#107×C₇₁	—	2	1	no	—
12	C₂×#106×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#382×C₇₁	—	2	1	no	—
15	C₂×#11765849	—	2	1	no	—
16	#14556087	—	2	1	no	—
17	#14556068	—	2	2	no	—
18	#14556068	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	#14556064	—	2	2	no	—
21	#14556062	Q₈	1	2	yes	—
22	#14556060	—	2	1	no	—
23	Dic₇₁⋊C₈	—	4	2	no	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
28	#107×C₇₁	—	4	2	no	—
29	#106×C₇₁	—	2	4	no	—
30	C₄×C₈×C₇₁	—	2	2	no	—
31	#95×C₇₁	—	4	2	no	—
32	#11765849	—	4	1	no	—
33	#11765849	—	4	1	no	—
34	C₂×C₇₁⋊C₈	—	4	1	no	—
35	C₄⋊Dic₇₁	—	2	2	no	—
36	C₂²⋊Dic₇₁	—	4	1	no	—
37	C₄×Dic₇₁	—	4	1	no	—
38	C₇₁⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₁	—	2	1	no	—
40	C₂²×C₄×C₇₁	—	4	2	no	—
41	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
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	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	4	2	no	—
50	C₄⋊C₄×C₇₁	—	4	2	no	—
51	C₄²×C₇₁	—	2	1	no	—
52	C₄²×C₇₁	Q₁₆	1	2	yes	—
53	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	4	2	no	—
56	C₂×C₄×C₇₁	—	8	4	no	—
57	C₂×C₄×C₇₁	—	8	2	no	—
58	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₇₁	—	4	2	no	—
65	C₈×C₇₁	—	2	2	no	—
66	C₈×C₇₁	#125	1	2	yes	—
67	C₈×C₇₁	#126	1	2	yes	—
68	C₇₁⋊C₈	—	8	1	no	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	#476	1	1	yes	characteristic
71	C₂²×C₇₁	#484	1	1	yes	characteristic
72	C₂²×C₇₁	#490	1	1	yes	characteristic
73	C₄×C₇₁	—	4	2	no	—
74	C₄×C₇₁	—	16	2	no	—
75	C₄×C₇₁	#400	1	1	yes	characteristic
76	C₄×C₇₁	#477	1	1	yes	characteristic
77	C₄×C₇₁	#481	1	1	yes	characteristic
78	C₄×C₇₁	#487	1	1	yes	characteristic
79	Dic₇₁	—	8	2	no	—
80	#13542	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	#2100	1	1	yes	characteristic
84	C₂×C₇₁	#2231	1	2	yes	—
85	#1945	—	71	1	no	—
86	#1846	D₇₁	1	1	yes	characteristic
87	#1386	—	71	2	no	—
88	#1383	—	71	2	no	—
89	#1376	—	71	1	no	—
90	C₇₁	#13542	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	71	2	no	—
92	C₂×#108	—	71	1	no	—
93	#427	—	71	1	no	—
94	C₂×#107	—	2	1	no	—
95	C₂×#106	C₂×D₇₁	1	2	yes	—
96	#424	—	142	1	no	—
97	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
98	#382	—	2	1	no	—
99	#339	—	142	1	no	—
100	C₂×M₁₆	—	142	1	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	#117	—	142	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
106	M₃₂	—	142	2	no	—
107	M₃₂	—	142	2	no	—
108	C₂×C₁₆	—	142	2	no	—
109	#108	—	142	2	no	—
110	#107	—	4	2	no	—
111	#107	—	71	2	no	—
112	#106	—	2	4	no	—
113	#106	—	142	1	no	—
114	C₄⋊C₈	—	284	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	4	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
120	M₁₆	—	284	1	no	—
121	M₁₆	—	284	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	284	1	no	—
125	C₂×C₈	#11765852	1	2	yes	—
126	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
127	C₂×C₈	#11765864	1	2	yes	—
128	C₂×C₈	#11765865	1	2	yes	—
129	C₂×C₈	#11765866	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	142	2	no	—
133	C₂²⋊C₄	—	284	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	284	1	no	—
136	C₄²	#11765855	1	2	yes	—
137	C₁₆	—	284	2	no	—
138	C₂³	#14556154	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	284	2	no	—
144	C₂×C₄	#14556104	1	1	yes	characteristic
145	C₂×C₄	#14556111	1	1	yes	characteristic
146	C₂×C₄	#14556115	1	1	yes	characteristic
147	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
148	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
149	C₂×C₄	#14556154	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	568	1	no	—
152	C₈	#14556118	1	2	yes	—
153	C₈	#14556123	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#17986286	1	1	yes	characteristic
156	C₂²	#17986334	1	1	yes	characteristic
157	C₂²	#17986363	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	16	2	no	—
160	C₄	—	568	2	no	—
161	C₄	#17985937	1	1	yes	characteristic
162	C₄	#17986291	1	1	yes	characteristic
163	C₄	#17986317	1	1	yes	characteristic
164	C₄	#17986349	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#22589902	1	1	yes	characteristic
168	C₂	#22590513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589088

C₂

yes

maximal, characteristic

#22586447

C₂

yes

maximal

#22586438

C₂

yes

maximal

#22586422

C₂

yes

maximal, characteristic

C₇₁×#1846

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

characteristic

#17986025

—

#17985759

—

C₂×#107×C₇₁

—

C₂×#106×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556062

Q₈

yes

—

#14556060

—

Dic₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

—

#95×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

Q₁₆

yes

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#125

yes

—

C₈×C₇₁

#126

yes

—

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

—

#13542

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2100

yes

characteristic

C₂×C₇₁

#2231

yes

—

#1945

—

#1846

D₇₁

yes

—

—

—

C₇₁

#13542

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

C₂×D₇₁

yes

—

#424

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

—

#339

—

142

—

100

C₂×M₁₆

—

142

—

101

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

102

C₂²×C₈

C₂²×D₇₁

yes

characteristic

103

#117

—

142

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

106

M₃₂

—

142

—

107

M₃₂

—

142

—

108

C₂×C₁₆

—

142

—

109

#108

—

142

—

110

#107

—

111

#107

—

112

#106

—

113

#106

—

142

—

114

C₄⋊C₈

—

284

—

115

C₄×C₈

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

120

M₁₆

—

284

—

121

M₁₆

—

284

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

284

—

125

C₂×C₈

#11765852

yes

—

126

C₂×C₈

Q₈×D₇₁

yes

characteristic

127

C₂×C₈

#11765864

yes

—

128

C₂×C₈

#11765865

yes

—

129

C₂×C₈

#11765866

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

142

—

133

C₂²⋊C₄

—

284

—

134

C₄²

—

135

C₄²

—

284

—

136

C₄²

#11765855

yes

—

137

C₁₆

—

284

—

138

C₂³

#14556154

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

284

—

144

C₂×C₄

#14556104

yes

characteristic

145

C₂×C₄

#14556111

yes

characteristic

146

C₂×C₄

#14556115

yes

characteristic

147

C₂×C₄

C₂×#11765852

yes

characteristic

148

C₂×C₄

C₂×#11765855

yes

characteristic

149

C₂×C₄

#14556154

yes

characteristic

150

C₈

—

151

C₈

—

568

—

152

C₈

#14556118

yes

—

153

C₈

#14556123

yes

—

154

C₂²

—

155

C₂²

#17986286

yes

characteristic

156

C₂²

#17986334

yes

characteristic

157

C₂²

#17986363

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

568

—

161

C₄

#17985937

yes

characteristic

162

C₄

#17986291

yes

characteristic

163

C₄

#17986317

yes

characteristic

164

C₄

#17986349

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#22589902

yes

characteristic

168

C₂

#22590513

yes

—

	subgroup	G/H
center	C₂²	#17986286
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1846	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
hypercenter	#1846	D₇₁
hypocenter	C₇₁	#13542
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#17986286

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1846

C₂

C₂²×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
90. C₇₁	#13542	semidirect

H ≅ G/N

type

90. C₇₁

#13542

semidirect

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

subgroup	index
#22589088	2
#22586447	2
#22586438	2
#22586422	2
C₇₁×#1846	2
#13542	71

subgroup

index

C₇₁×#1846

Subgroups — #31917596

Conjugacy classes of subgroups

Complements

Maximal subgroups