Group #31916045 — subgroups

Subgroups — #31916045

Order 18176.

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

normal subgroups only

This group has 334 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×#17986401	C₂	1	1	yes	maximal, characteristic
2	#22590096	C₂	1	1	yes	maximal, characteristic
3	#22586450	C₂	1	4	yes	maximal
4	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556120	—	2	1	no	—
6	C₂×#14556068	C₂²	1	2	yes	—
7	#17986401	—	2	2	no	—
8	#17986401	C₄	1	4	yes	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556060	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
12	C₂×#108×C₇₁	C₂²	1	2	yes	—
13	C₄×M₁₆×C₇₁	—	2	1	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#341×C₇₁	—	2	1	no	—
16	#14556120	—	2	4	no	—
17	#14556120	—	4	1	no	—
18	#14556120	—	4	1	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×Dic₇₁⋊C₄	—	4	2	no	—
21	#14556068	—	2	4	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	#14556062	—	2	2	no	—
24	#14556062	C₂×C₄	1	4	yes	—
25	#14556060	—	2	4	no	—
26	C₂×M₁₆×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
30	#108×C₇₁	—	2	2	no	—
31	#108×C₇₁	Q₈	1	4	yes	—
32	#97×C₇₁	—	4	1	no	—
33	C₄×C₈×C₇₁	—	4	1	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂²×Dic₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	—	2	4	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	Dic₇₁⋊C₄	—	4	4	no	—
40	Dic₇₁⋊C₄	—	4	4	no	—
41	C₇₁⋊C₁₆	—	4	2	no	—
42	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₇₁	—	2	4	no	—
45	M₁₆×C₇₁	—	4	2	no	—
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₁	D₈	1	1	yes	characteristic
55	C₄²×C₇₁	D₈	1	1	yes	characteristic
56	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	4	2	no	—
58	C₂×C₄×C₇₁	#102	1	2	yes	—
59	C₂×C₄×C₇₁	#102	1	2	yes	—
60	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#137	1	2	yes	—
64	C₂×Dic₇₁	—	4	4	no	—
65	C₂×Dic₇₁	—	8	4	no	—
66	C₈×C₇₁	—	2	2	no	—
67	C₈×C₇₁	—	4	4	no	—
68	C₈×C₇₁	C₄×Q₈	1	2	yes	—
69	C₈×C₇₁	#124	1	1	yes	characteristic
70	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²×C₇₁	—	2	2	no	—
72	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
73	C₂²×C₇₁	#475	1	2	yes	—
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	#391	1	1	yes	characteristic
77	C₄×C₇₁	#419	1	1	yes	characteristic
78	C₄×C₇₁	#474	1	2	yes	—
79	Dic₇₁	—	8	4	no	—
80	#13280	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	2	no	—
82	C₂×C₇₁	#1988	1	1	yes	characteristic
83	C₂×C₇₁	#2148	1	2	yes	—
84	C₂×#499	—	71	1	no	—
85	#2150	—	71	1	no	—
86	C₄×#108	D₇₁	1	1	yes	characteristic
87	#1387	—	71	4	no	—
88	C₇₁	#13280	1	1	yes	characteristic, complemented, Sylow
89	C₂×#125	—	142	1	no	—
90	C₂×M₃₂	—	71	2	no	—
91	#499	—	71	4	no	—
92	#499	—	142	2	no	—
93	C₂×#108	C₂×D₇₁	1	2	yes	—
94	C₂×#107	—	71	2	no	—
95	C₂×#106	—	142	1	no	—
96	C₄×M₁₆	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
98	C₄⋊C₁₆	—	142	2	no	—
99	#341	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	#125	—	284	1	no	—
104	#125	—	284	1	no	—
105	#125	—	142	4	no	—
106	C₂×C₄⋊C₄	—	284	2	no	—
107	C₂×C₄⋊C₄	—	142	2	no	—
108	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
109	M₃₂	—	142	4	no	—
110	C₂×C₁₆	—	142	2	no	—
111	#108	—	2	2	no	—
112	#108	Dic₁₄₂	1	4	yes	—
113	#107	—	71	4	no	—
114	#107	—	142	2	no	—
115	#106	—	142	4	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₄×D₇₁	1	2	yes	—
119	C₄×C₈	D₂₈₄	1	2	yes	—
120	C₂²×C₄	—	284	2	no	—
121	C₂²×C₄	#11765852	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
131	C₂×C₈	#11765865	1	2	yes	—
132	C₄⋊C₄	—	284	4	no	—
133	C₄⋊C₄	—	284	4	no	—
134	C₄⋊C₄	—	284	2	no	—
135	C₄⋊C₄	—	142	4	no	—
136	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
137	C₄²	#11765850	1	1	yes	characteristic
138	C₄²	#11765850	1	1	yes	characteristic
139	C₁₆	—	284	2	no	—
140	C₂³	C₂×#11765852	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	284	4	no	—
143	C₂×C₄	—	568	4	no	—
144	C₂×C₄	#14556048	1	2	yes	—
145	C₂×C₄	#14556048	1	2	yes	—
146	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#14556107	1	2	yes	—
148	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
149	C₂×C₄	#14556158	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Dic₁₄₂	1	2	yes	—
153	C₈	#14556116	1	1	yes	characteristic
154	C₈	#14556128	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14556048	1	1	yes	characteristic
157	C₂²	#17986284	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	568	4	no	—
161	C₄	#17985905	1	1	yes	characteristic
162	C₄	#17986007	1	1	yes	characteristic
163	C₄	#17986279	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#22589292	1	1	yes	characteristic
166	C₂	#22590087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

C₂

yes

maximal, characteristic

#22590096

C₂

yes

maximal, characteristic

#22586450

C₂

yes

maximal

C₄×#108×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556120

—

C₂×#14556068

C₂²

yes

—

#17986401

—

#17986401

C₄

yes

—

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

C₄⋊(C₇₁⋊C₁₆)

—

C₂×#108×C₇₁

C₂²

yes

—

C₄×M₁₆×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#341×C₇₁

—

#14556120

—

#14556120

—

#14556120

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556062

—

#14556062

C₂×C₄

yes

—

#14556060

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

#108×C₇₁

—

#108×C₇₁

Q₈

yes

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#102

yes

characteristic

C₂²×C₇₁

#475

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#13280

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1988

yes

characteristic

C₂×C₇₁

#2148

yes

—

C₂×#499

—

#2150

—

C₄×#108

D₇₁

yes

characteristic

#1387

—

C₇₁

#13280

yes

characteristic, complemented, Sylow

C₂×#125

—

142

—

C₂×M₃₂

—

#499

—

#499

—

142

—

C₂×#108

C₂×D₇₁

yes

—

C₂×#107

—

C₂×#106

—

142

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

#341

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

102

C₂²×C₈

C₂²×D₇₁

yes

characteristic

103

#125

—

284

—

104

#125

—

284

—

105

#125

—

142

—

106

C₂×C₄⋊C₄

—

284

—

107

C₂×C₄⋊C₄

—

142

—

108

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

109

M₃₂

—

142

—

110

C₂×C₁₆

—

142

—

111

#108

—

112

#108

Dic₁₄₂

yes

—

113

#107

—

114

#107

—

142

—

115

#106

—

142

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

C₄×D₇₁

yes

—

119

C₄×C₈

D₂₈₄

yes

—

120

C₂²×C₄

—

284

—

121

C₂²×C₄

#11765852

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

128

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

129

C₂×C₈

C₂×D₂₈₄

yes

characteristic

130

C₂×C₈

C₂×Dic₁₄₂

yes

—

131

C₂×C₈

#11765865

yes

—

132

C₄⋊C₄

—

284

—

133

C₄⋊C₄

—

284

—

134

C₄⋊C₄

—

284

—

135

C₄⋊C₄

—

142

—

136

C₄²

(C₂×D₇₁)⋊C₄

yes

—

137

C₄²

#11765850

yes

characteristic

138

C₄²

#11765850

yes

characteristic

139

C₁₆

—

284

—

140

C₂³

C₂×#11765852

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

284

—

143

C₂×C₄

—

568

—

144

C₂×C₄

#14556048

yes

—

145

C₂×C₄

#14556048

yes

—

146

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

147

C₂×C₄

#14556107

yes

—

148

C₂×C₄

C₂×#11765850

yes

characteristic

149

C₂×C₄

#14556158

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Dic₁₄₂

yes

—

153

C₈

#14556116

yes

characteristic

154

C₈

#14556128

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#14556048

yes

characteristic

157

C₂²

#17986284

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

568

—

161

C₄

#17985905

yes

characteristic

162

C₄

#17986007

yes

characteristic

163

C₄

#17986279

yes

—

164

C₂

—

165

C₂

#22589292

yes

characteristic

166

C₂

#22590087

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556048
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	#13280
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#14556048

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₄×#108×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#102

prime radical

C₂³×C₇₁

C₂×SD₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
88. C₇₁	#13280	semidirect

H ≅ G/N

type

88. C₇₁

#13280

semidirect

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

subgroup	index
C₂×#17986401	2
#22590096	2
#22586450	2
C₄×#108×C₇₁	2
#13280	71

subgroup

index

C₄×#108×C₇₁

Subgroups — #31916045

Conjugacy classes of subgroups

Complements

Maximal subgroups