Group #31888615 — subgroups

Subgroups — #31888615

Order 18176.

390 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	2	yes	maximal
3	#22586632	C₂	1	1	yes	maximal, characteristic
4	#22586172	C₂	1	2	yes	maximal
5	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
11	#17985751	C₄	1	4	yes	complemented
12	#17985749	C₄	1	4	yes	complemented
13	#17985721	—	2	2	no	—
14	#431×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#322×C₇₁	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	—	2	2	no	—
20	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₈×Dic₇₁	—	2	4	no	—
27	C₈×Dic₇₁	—	2	4	no	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	D₄	1	2	yes	—
34	C₄×C₈×C₇₁	Q₈	1	2	yes	—
35	C₂²×Dic₇₁	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
39	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
40	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
42	C₄×Dic₇₁	—	2	8	no	—
43	C₄×Dic₇₁	—	2	2	no	—
44	C₄×Dic₇₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	#41	1	2	yes	—
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	4	no	—
50	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
51	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
52	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₇₁	#117	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
60	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
62	C₂×Dic₇₁	—	2	8	no	—
63	C₂×Dic₇₁	—	2	4	no	—
64	C₈×C₇₁	—	2	4	no	—
65	C₈×C₇₁	—	4	4	no	—
66	C₇₁⋊C₈	—	2	2	no	—
67	C₇₁⋊C₈	—	4	4	no	—
68	C₇₁⋊C₈	#97	1	2	yes	—
69	C₇₁⋊C₈	#97	1	2	yes	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₇₁	#432	1	2	yes	—
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
75	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
76	C₄×C₇₁	#423	1	2	yes	—
77	C₄×C₇₁	#443	1	4	yes	—
78	Dic₇₁	—	2	8	no	—
79	#8267	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	2	2	no	—
81	C₂×C₇₁	#1767	1	1	yes	characteristic
82	C₂×C₇₁	#1772	1	2	yes	—
83	C₄²×C₈	—	71	1	no	—
84	#1564	—	71	2	no	—
85	#1452	—	71	1	no	—
86	#1452	D₇₁	1	1	yes	characteristic
87	#1276	—	71	2	no	—
88	C₇₁	#8267	1	1	yes	characteristic, complemented, Sylow
89	#431	—	71	2	no	—
90	#431	C₂×D₇₁	1	2	yes	—
91	C₂×C₄×C₈	—	71	1	no	—
92	C₂×C₄×C₈	—	71	2	no	—
93	C₂×C₄×C₈	—	71	2	no	—
94	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
95	C₄³	—	71	1	no	—
96	#335	—	71	4	no	—
97	#334	—	71	4	no	—
98	#322	—	2	2	no	—
99	#322	—	142	2	no	—
100	C₂²×C₈	—	71	1	no	—
101	C₂²×C₈	—	71	1	no	—
102	C₂²×C₈	C₄×D₇₁	1	2	yes	—
103	C₂×C₄²	—	71	2	no	—
104	C₂×C₄²	—	142	2	no	—
105	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
106	C₄⋊C₈	—	71	4	no	—
107	C₄⋊C₈	C₄×D₇₁	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	142	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	71	2	no	—
112	C₄×C₈	—	71	2	no	—
113	C₄×C₈	—	142	4	no	—
114	C₄×C₈	—	142	4	no	—
115	C₄×C₈	—	142	2	no	—
116	C₄×C₈	Dic₁₄₂	1	2	yes	—
117	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
118	C₂²×C₄	—	142	2	no	—
119	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
120	C₂²×C₄	#11765865	1	2	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	71	1	no	—
125	C₂×C₈	—	71	1	no	—
126	C₂×C₈	—	71	4	no	—
127	C₂×C₈	—	71	2	no	—
128	C₂×C₈	—	142	4	no	—
129	C₂×C₈	—	142	2	no	—
130	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
131	C₄²	—	71	4	no	—
132	C₄²	—	142	8	no	—
133	C₄²	—	142	2	no	—
134	C₄²	C₂×C₄×D₇₁	1	2	yes	—
135	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
136	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
137	C₂³	#14556085	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	142	8	no	—
140	C₂×C₄	—	142	4	no	—
141	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
144	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
145	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
146	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	284	4	no	—
150	C₈	—	71	2	no	—
151	C₈	—	71	2	no	—
152	C₈	—	142	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
155	C₂²	#17986054	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	142	8	no	—
158	C₄	#97×D₇₁	1	1	yes	characteristic
159	C₄	#17985951	1	1	yes	characteristic
160	C₄	#17986022	1	2	yes	—
161	C₄	#17986114	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#22588194	1	1	yes	characteristic
164	C₂	#22588220	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#22587086

C₂

yes

maximal

#22586632

C₂

yes

maximal, characteristic

#22586172

C₂

yes

maximal

C₇₁×#1452

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

—

C₂×C₈×Dic₇₁

C₄

yes

—

C₄²×Dic₇₁

C₂²

yes

characteristic

#17985751

C₄

yes

complemented

#17985749

C₄

yes

complemented

#17985721

—

#431×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

C₂×C₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄²

yes

—

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

#97

yes

—

C₇₁⋊C₈

#97

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#432

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

#423

yes

—

C₄×C₇₁

#443

yes

—

Dic₇₁

—

#8267

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1767

yes

characteristic

C₂×C₇₁

#1772

yes

—

C₄²×C₈

—

#1564

—

#1452

—

#1452

D₇₁

yes

characteristic

#1276

—

C₇₁

#8267

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄³

—

—

—

—

—

142

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₄×D₇₁

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

142

—

105

C₂×C₄²

C₂²×D₇₁

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₇₁

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

142

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

142

—

114

C₄×C₈

—

142

—

115

C₄×C₈

—

142

—

116

C₄×C₈

Dic₁₄₂

yes

—

117

C₄×C₈

C₂²⋊D₇₁

yes

—

118

C₂²×C₄

—

142

—

119

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

120

C₂²×C₄

#11765865

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

142

—

129

C₂×C₈

—

142

—

130

C₂×C₈

Dic₇₁⋊C₄

yes

—

131

C₄²

—

132

C₄²

—

142

—

133

C₄²

—

142

—

134

C₄²

C₂×C₄×D₇₁

yes

—

135

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

136

C₄²

C₂×Dic₁₄₂

yes

characteristic

137

C₂³

#14556085

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

142

—

140

C₂×C₄

—

142

—

141

C₂×C₄

C₄²×D₇₁

yes

characteristic

142

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

143

C₂×C₄

C₄×C₂²⋊D₇₁

yes

—

144

C₂×C₄

C₄×Dic₁₄₂

yes

—

145

C₂×C₄

M₁₆×D₇₁

yes

—

146

C₂×C₄

M₁₆×D₇₁

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

284

—

150

C₈

—

151

C₈

—

152

C₈

—

142

—

153

C₂²

—

154

C₂²

C₄×Dic₇₁⋊C₄

yes

characteristic

155

C₂²

#17986054

yes

—

156

C₄

—

157

C₄

—

142

—

158

C₄

#97×D₇₁

yes

characteristic

159

C₄

#17985951

yes

characteristic

160

C₄

#17986022

yes

—

161

C₄

#17986114

yes

—

162

C₂

—

163

C₂

#22588194

yes

characteristic

164

C₂

#22588220

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1452	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1452	D₇₁
hypocenter	C₇₁	#8267
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1452

C₂

socle

C₂²×C₇₁

C₄×C₄⋊C₄

prime radical

C₂³×C₇₁

#117

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

N	H ≅ G/N	type
11. #17985751	C₄	semidirect
12. #17985749	C₄	semidirect
88. C₇₁	#8267	semidirect

H ≅ G/N

type

11. #17985751

C₄

semidirect

12. #17985749

C₄

semidirect

88. C₇₁

#8267

semidirect

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

subgroup	index
C₄×C₈×Dic₇₁	2
#22587086	2
#22586632	2
#22586172	2
C₇₁×#1452	2
#8267	71

subgroup

index

C₄×C₈×Dic₇₁

C₇₁×#1452

Subgroups — #31888615

Conjugacy classes of subgroups

Complements

Maximal subgroups