Group #31884494 — subgroups

Subgroups — #31884494

Order 18176.

196 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590139	C₂	1	1	yes	maximal, characteristic
2	#22587096	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1837	C₂	1	1	yes	maximal, characteristic
4	#17986023	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
6	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
7	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
8	#17985750	—	2	1	no	—
9	#17985750	C₄	1	2	yes	—
10	#423×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#341×C₇₁	—	2	1	no	—
13	#341×C₇₁	C₄	1	2	yes	—
14	C₂×#11765849	—	4	1	no	—
15	C₂×C₇₁⋊C₁₆	—	2	2	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
17	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
18	C₂×M₁₆×C₇₁	—	2	2	no	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	—	4	1	no	—
23	C₄⋊C₈×C₇₁	—	4	1	no	—
24	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
28	#11765849	—	8	2	no	—
29	C₂×C₇₁⋊C₈	—	4	2	no	—
30	C₇₁⋊C₁₆	—	4	2	no	—
31	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₇₁	—	4	2	no	—
35	M₁₆×C₇₁	—	4	2	no	—
36	C₂×C₈×C₇₁	—	4	1	no	—
37	C₂×C₈×C₇₁	—	4	1	no	—
38	C₂×C₈×C₇₁	—	4	2	no	—
39	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₁	D₈	1	1	yes	characteristic
46	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
47	C₂³×C₇₁	#103	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	—	2	1	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
58	C₈×C₇₁	—	2	1	no	—
59	C₈×C₇₁	—	2	1	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	8	2	no	—
63	C₇₁⋊C₈	—	8	2	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	#340	1	1	yes	characteristic
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₇₁	#344	1	1	yes	characteristic
73	C₄×C₇₁	#365	1	2	yes	—
74	C₄×C₇₁	#368	1	2	yes	—
75	#7219	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	4	1	no	—
77	C₂×C₇₁	#1286	1	1	yes	characteristic
78	C₂×C₇₁	#1381	1	1	yes	characteristic
79	C₂×C₇₁	#1383	1	1	yes	characteristic
80	#2162	—	71	1	no	—
81	#1837	D₇₁	1	1	yes	characteristic
82	#1566	—	71	1	no	—
83	C₇₁	#7219	1	1	yes	characteristic, complemented, Sylow
84	#423	—	2	1	no	—
85	#423	—	142	1	no	—
86	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	142	1	no	—
88	C₂²⋊C₁₆	—	142	1	no	—
89	C₄×C₁₆	—	71	2	no	—
90	#341	—	2	1	no	—
91	#341	Dic₇₁	1	2	yes	—
92	#334	—	71	2	no	—
93	#334	—	142	1	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	284	1	no	—
96	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
99	C₂×C₁₆	—	142	2	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	284	1	no	—
103	C₄⋊C₈	—	142	2	no	—
104	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
105	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
106	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
107	C₄×C₈	D₂₈₄	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
109	C₂²×C₄	#11765852	1	1	yes	characteristic
110	C₂²×C₄	#11765855	1	1	yes	characteristic
111	M₁₆	—	4	2	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	568	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	284	2	no	—
118	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
119	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
120	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
122	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
123	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
124	C₄²	#11765850	1	1	yes	characteristic
125	C₄²	#11765853	1	1	yes	characteristic
126	C₁₆	—	284	2	no	—
127	C₂³	#14556052	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
133	C₂×C₄	#14556048	1	1	yes	characteristic
134	C₂×C₄	#14556049	1	1	yes	characteristic
135	C₂×C₄	#14556051	1	1	yes	characteristic
136	C₂×C₄	#14556059	1	1	yes	characteristic
137	C₂×C₄	#14556061	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	568	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#17985761	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#17985755	1	1	yes	characteristic
152	C₄	#17985769	1	1	yes	characteristic
153	C₄	#17985823	1	2	yes	—
154	C₄	#17985828	1	2	yes	—
155	C₂	—	4	1	no	—
156	C₂	#22586198	1	1	yes	characteristic
157	C₂	#22586433	1	1	yes	characteristic
158	C₂	#22586440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590139

C₂

yes

maximal, characteristic

#22587096

C₂

yes

maximal, characteristic

C₇₁×#1837

C₂

yes

maximal, characteristic

#17986023

—

C₄⋊(C₇₁⋊C₁₆)

—

C₂²⋊(C₇₁⋊C₁₆)

—

C₄×C₇₁⋊C₁₆

C₄

yes

—

#17985750

—

#17985750

C₄

yes

—

#423×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#341×C₇₁

—

#341×C₇₁

C₄

yes

—

C₂×#11765849

—

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

Q₁₆

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

SD₁₆

yes

characteristic

C₂³×C₇₁

#103

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#340

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₄×C₇₁

#368

yes

—

#7219

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1566

—

C₇₁

#7219

yes

characteristic, complemented, Sylow

#423

—

#423

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

C₂²⋊C₁₆

—

142

—

C₄×C₁₆

—

#341

—

#341

Dic₇₁

yes

—

#334

—

#334

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

284

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

—

142

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

284

—

103

C₄⋊C₈

—

142

—

104

C₄×C₈

Dic₁₄₂

yes

characteristic

105

C₄×C₈

C₄×D₇₁

yes

characteristic

106

C₄×C₈

C₄×D₇₁

yes

characteristic

107

C₄×C₈

D₂₈₄

yes

characteristic

108

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

109

C₂²×C₄

#11765852

yes

characteristic

110

C₂²×C₄

#11765855

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

568

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

284

—

118

C₂×C₈

C₄×Dic₇₁

yes

characteristic

119

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

120

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

121

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

122

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

123

C₄²

Dic₇₁⋊C₄

yes

characteristic

124

C₄²

#11765850

yes

characteristic

125

C₄²

#11765853

yes

characteristic

126

C₁₆

—

284

—

127

C₂³

#14556052

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₇₁⋊#95

yes

characteristic

133

C₂×C₄

#14556048

yes

characteristic

134

C₂×C₄

#14556049

yes

characteristic

135

C₂×C₄

#14556051

yes

characteristic

136

C₂×C₄

#14556059

yes

characteristic

137

C₂×C₄

#14556061

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

568

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#17985761

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#17985755

yes

characteristic

152

C₄

#17985769

yes

characteristic

153

C₄

#17985823

yes

—

154

C₄

#17985828

yes

—

155

C₂

—

156

C₂

#22586198

yes

characteristic

157

C₂

#22586433

yes

characteristic

158

C₂

#22586440

yes

characteristic

	subgroup	G/H
center	C₂²	#17985761
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1837	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂³×C₇₁	#103
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1837	D₇₁
hypocenter	C₇₁	#7219
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1837

subgroup

G/H

center

C₂²

#17985761

commutator

C₂×C₈×C₇₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₇₁×#1837

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1837

N	H ≅ G/N	type
83. C₇₁	#7219	semidirect

H ≅ G/N

type

83. C₇₁

#7219

semidirect

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

subgroup	index
#22590139	2
#22587096	2
C₇₁×#1837	2
#7219	71

subgroup

index

#22590139

#22587096

C₇₁×#1837

#7219

Subgroups — #31884494

Conjugacy classes of subgroups

Complements

Maximal subgroups