Group #31913840 — subgroups

Subgroups — #31913840

Order 18176.

282 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 119 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×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#22588510	C₂	1	2	yes	maximal
3	#22586434	C₂	1	1	yes	maximal, characteristic
4	#22586433	C₂	1	2	yes	maximal
5	C₇₁×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	C₂×#14556063	—	2	2	no	—
9	#17986022	—	2	2	no	—
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₇₁	C₄	1	2	yes	—
12	C₁₆×Dic₇₁	C₄	1	2	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	2	yes	—
15	C₂×#11765849	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
20	#14556064	—	2	2	no	—
21	#14556064	—	2	2	no	—
22	#14556063	—	4	4	no	—
23	Dic₇₁⋊C₈	—	4	4	no	—
24	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
27	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	—	2	2	no	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	#108×C₇₁	—	2	2	no	—
33	#108×C₇₁	—	2	2	no	—
34	C₂²×Dic₇₁	Q₁₆	1	2	yes	—
35	#11765849	—	4	2	no	—
36	#11765849	—	4	2	no	—
37	C₂×C₇₁⋊C₈	—	4	2	no	—
38	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₇₁⋊C₈	SD₁₆	1	4	yes	—
41	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₁	D₈	1	2	yes	—
44	C₇₁⋊C₁₆	—	2	2	no	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₇₁	—	4	2	no	—
47	M₁₆×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	—	4	2	no	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	#41	1	2	yes	—
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₁₆×C₇₁	—	2	2	no	—
54	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#124	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
59	C₂×Dic₇₁	—	2	4	no	—
60	C₂×Dic₇₁	#107	1	2	yes	—
61	C₂×Dic₇₁	#107	1	2	yes	—
62	C₈×C₇₁	—	8	2	no	—
63	C₈×C₇₁	#117	1	1	yes	characteristic
64	C₈×C₇₁	#117	1	1	yes	characteristic
65	C₈×C₇₁	#125	1	2	yes	—
66	C₇₁⋊C₈	—	8	2	no	—
67	C₇₁⋊C₈	#106	1	2	yes	—
68	C₇₁⋊C₈	#106	1	2	yes	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₂²×C₇₁	#487	1	2	yes	—
72	C₄×C₇₁	#382	1	1	yes	characteristic
73	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
74	C₄×C₇₁	#486	1	2	yes	—
75	Dic₇₁	—	2	2	no	—
76	Dic₇₁	#368	1	4	yes	—
77	#12891	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	2	2	no	—
79	C₂×C₇₁	#1834	1	1	yes	characteristic
80	C₂×C₇₁	C₂×#368	1	2	yes	—
81	C₂×C₄×C₁₆	—	71	1	no	—
82	#1848	—	71	2	no	—
83	#1381	—	71	1	no	—
84	#1381	—	71	2	no	—
85	#1381	D₇₁	1	1	yes	characteristic
86	C₇₁	#12891	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	71	1	no	—
88	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
89	C₂×#108	—	71	2	no	—
90	C₂×#108	—	142	2	no	—
91	C₂×#108	C₂×D₇₁	1	2	yes	—
92	#423	—	142	2	no	—
93	C₂×C₄×C₈	—	71	1	no	—
94	C₄×C₁₆	—	71	2	no	—
95	C₄×C₁₆	—	71	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	142	2	no	—
98	C₂²×C₈	—	71	1	no	—
99	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
100	C₂×C₄²	—	71	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	71	2	no	—
103	C₂×C₁₆	—	142	2	no	—
104	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	284	4	no	—
108	#108	—	142	2	no	—
109	#108	—	142	2	no	—
110	C₄⋊C₈	—	284	4	no	—
111	C₄×C₈	—	71	1	no	—
112	C₄×C₈	—	71	1	no	—
113	C₄×C₈	—	71	1	no	—
114	C₄×C₈	—	71	1	no	—
115	C₂²×C₄	—	71	2	no	—
116	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	284	2	no	—
120	M₁₆	—	284	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	284	2	no	—
123	C₂×C₈	—	71	1	no	—
124	C₂×C₈	—	71	1	no	—
125	C₂×C₈	—	71	4	no	—
126	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
127	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
128	C₂×C₈	#11765864	1	2	yes	—
129	C₂×C₈	#11765865	1	2	yes	—
130	C₄²	—	71	1	no	—
131	C₄²	—	71	1	no	—
132	C₄²	—	71	2	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	142	2	no	—
135	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
136	C₂×C₄	—	71	2	no	—
137	C₂×C₄	—	71	2	no	—
138	C₂×C₄	—	142	4	no	—
139	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
140	C₂×C₄	#14556114	1	2	yes	—
141	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
142	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	568	2	no	—
145	C₈	—	71	2	no	—
146	C₈	—	71	2	no	—
147	C₈	#14556084	1	1	yes	characteristic
148	C₈	#14556085	1	1	yes	characteristic
149	C₈	#14556118	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#107×D₇₁	1	1	yes	characteristic
152	C₂²	#17986348	1	2	yes	—
153	C₄	—	71	4	no	—
154	C₄	—	142	2	no	—
155	C₄	#17985863	1	1	yes	characteristic
156	C₄	#106×D₇₁	1	1	yes	characteristic
157	C₄	#17986344	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#22588466	1	1	yes	characteristic
160	C₂	#368×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#22588510

C₂

yes

maximal

#22586434

C₂

yes

maximal, characteristic

#22586433

C₂

yes

maximal

C₇₁×#1381

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

C₂×#14556063

—

#17986022

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

C₁₆×Dic₇₁

C₄

yes

—

C₁₆×Dic₇₁

C₄

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556064

—

#14556064

—

#14556063

—

Dic₇₁⋊C₈

—

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

—

C₂²×Dic₇₁

Q₁₆

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

D₈

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

C₂×Dic₇₁

#107

yes

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

#106

yes

—

C₇₁⋊C₈

#106

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#487

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

#368

yes

—

#12891

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1834

yes

characteristic

C₂×C₇₁

C₂×#368

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

C₂×#108

—

142

—

C₂×#108

C₂×D₇₁

yes

—

#423

—

142

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₁

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

142

—

104

C₂×C₁₆

C₄×D₇₁

yes

—

105

#108

—

106

#108

—

107

#108

—

284

—

108

#108

—

142

—

109

#108

—

142

—

110

C₄⋊C₈

—

284

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₄×D₇₁

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

284

—

120

M₁₆

—

284

—

121

C₂×C₈

—

122

C₂×C₈

—

284

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

127

C₂×C₈

Q₈×D₇₁

yes

characteristic

128

C₂×C₈

#11765864

yes

—

129

C₂×C₈

#11765865

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₁₆

—

134

C₁₆

—

142

—

135

C₂³

Q₁₆×D₇₁

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

142

—

139

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

140

C₂×C₄

#14556114

yes

—

141

C₂×C₄

D₈×D₇₁

yes

characteristic

142

C₂×C₄

SD₁₆×D₇₁

yes

—

143

C₈

—

144

C₈

—

568

—

145

C₈

—

146

C₈

—

147

C₈

#14556084

yes

characteristic

148

C₈

#14556085

yes

characteristic

149

C₈

#14556118

yes

—

150

C₂²

—

151

C₂²

#107×D₇₁

yes

characteristic

152

C₂²

#17986348

yes

—

153

C₄

—

154

C₄

—

142

—

155

C₄

#17985863

yes

characteristic

156

C₄

#106×D₇₁

yes

characteristic

157

C₄

#17986344

yes

—

158

C₂

—

159

C₂

#22588466

yes

characteristic

160

C₂

#368×D₇₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1381	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1381	D₇₁
hypocenter	C₇₁	#12891
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1381

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
86. C₇₁	#12891	semidirect

H ≅ G/N

type

86. C₇₁

#12891

semidirect

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

subgroup	index
C₂×C₁₆×Dic₇₁	2
#22588510	2
#22586434	2
#22586433	2
C₇₁×#1381	2
#12891	71

subgroup

index

C₂×C₁₆×Dic₇₁

C₇₁×#1381

Subgroups — #31913840

Conjugacy classes of subgroups

Complements

Maximal subgroups