Group #31918035 — subgroups

Subgroups — #31918035

Order 18176.

190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 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×#17985839	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22590198	C₂	1	1	yes	maximal, characteristic
4	#22586539	C₂	1	1	yes	maximal, characteristic
5	#22586522	C₂	1	1	yes	maximal, characteristic
6	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₇₁	—	2	1	no	—
10	#17986093	—	2	1	no	—
11	#17985839	C₄	1	4	yes	complemented
12	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
13	#17985825	C₂²	1	1	yes	characteristic
14	#17985803	—	2	1	no	—
15	#17985802	—	2	1	no	—
16	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
19	#366×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₈₄	—	2	1	no	—
21	Q₈×Dic₇₁	—	4	1	no	—
22	#14556093	—	4	1	no	—
23	Dic₅₆₈	—	2	1	no	—
24	Dic₅₆₈	C₂×C₄	1	2	yes	—
25	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
26	#14556062	—	2	1	no	—
27	#14556052	—	4	1	no	—
28	#14556051	—	4	1	no	—
29	C₈×Dic₇₁	D₄	1	1	yes	characteristic
30	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₇₁	—	2	1	no	—
32	Q₃₂×C₇₁	—	2	1	no	—
33	Q₃₂×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
35	#107×C₇₁	—	2	1	no	—
36	C₃₂×C₇₁	—	2	1	no	—
37	C₂×Dic₁₄₂	—	4	1	no	—
38	Dic₂₈₄	—	2	2	no	—
39	Dic₂₈₄	—	4	1	no	—
40	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
41	C₄⋊Dic₇₁	—	4	1	no	—
42	C₄⋊Dic₇₁	—	8	1	no	—
43	Dic₇₁⋊C₄	—	8	1	no	—
44	C₄×Dic₇₁	—	8	1	no	—
45	C₄×Dic₇₁	—	8	1	no	—
46	C₄×Dic₇₁	D₈	1	1	yes	characteristic
47	C₇₁⋊C₁₆	—	2	1	no	—
48	C₂×Q₈×C₇₁	—	4	1	no	—
49	Q₁₆×C₇₁	—	2	2	no	—
50	Q₁₆×C₇₁	—	4	1	no	—
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₇₁	—	4	1	no	—
53	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₇₁	#41	1	1	yes	characteristic
55	Q₈×C₇₁	—	4	2	no	—
56	Q₈×C₇₁	—	8	1	no	—
57	C₂×C₄×C₇₁	—	8	1	no	—
58	C₂×C₄×C₇₁	—	8	1	no	—
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₇₁	—	8	1	no	—
61	C₂×Dic₇₁	—	8	1	no	—
62	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
63	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
64	Dic₁₄₂	—	4	2	no	—
65	Dic₁₄₂	—	8	1	no	—
66	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
67	C₈×C₇₁	#135	1	1	yes	characteristic
68	C₇₁⋊C₈	—	2	1	no	—
69	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₇₁	—	8	2	no	—
71	C₄×C₇₁	—	16	1	no	—
72	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
73	C₄×C₇₁	#508	1	1	yes	characteristic
74	Dic₇₁	—	2	1	no	—
75	Dic₇₁	—	8	2	no	—
76	Dic₇₁	—	16	1	no	—
77	Dic₇₁	Q₆₄	1	2	yes	complemented
78	C₄×Q₆₄	—	71	1	no	maximal, Sylow
79	C₂×C₇₁	C₄×D₁₆	1	1	yes	characteristic
80	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
81	C₂×C₇₁	#2262	1	1	yes	characteristic
82	C₂×Q₆₄	—	71	1	no	—
83	C₄×Q₃₂	—	71	1	no	—
84	C₄×Q₃₂	—	71	1	no	—
85	#1423	—	71	1	no	—
86	#1416	—	71	1	no	—
87	#1416	D₇₁	1	1	yes	characteristic
88	C₄×C₃₂	—	71	1	no	—
89	C₇₁	C₄×Q₆₄	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₃₂	—	71	1	no	—
91	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
92	C₄×Q₁₆	—	142	1	no	—
93	C₄×Q₁₆	—	142	1	no	—
94	Q₆₄	—	71	4	no	—
95	C₂×C₃₂	—	71	1	no	—
96	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
97	#366	—	71	1	no	—
98	#366	C₂×D₇₁	1	1	yes	characteristic
99	#358	—	142	1	no	—
100	#358	—	142	1	no	—
101	C₄×C₁₆	—	71	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	142	1	no	—
104	C₄×Q₈	—	284	1	no	—
105	C₄×Q₈	—	284	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	71	2	no	—
108	Q₃₂	—	142	1	no	—
109	Q₃₂	C₄×D₇₁	1	2	yes	—
110	C₂×C₁₆	—	71	1	no	—
111	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	142	1	no	—
114	#103	—	284	1	no	—
115	#103	—	284	1	no	—
116	C₄×C₈	—	71	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	71	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	284	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	284	1	no	—
124	Q₁₆	—	142	2	no	—
125	C₂×C₈	—	71	1	no	—
126	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	284	1	no	—
129	C₄⋊C₄	—	568	1	no	—
130	C₄⋊C₄	—	568	1	no	—
131	C₄²	—	568	1	no	—
132	C₄²	—	568	1	no	—
133	C₄²	—	71	1	no	—
134	C₁₆	—	142	1	no	—
135	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
136	C₁₆	#11765864	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	284	2	no	—
140	Q₈	—	568	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	568	1	no	—
144	C₂×C₄	—	568	1	no	—
145	C₂×C₄	—	71	1	no	—
146	C₂×C₄	—	71	1	no	—
147	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
148	C₈	—	142	1	no	—
149	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
150	C₈	#14556149	1	1	yes	characteristic
151	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	568	2	no	—
155	C₄	—	71	2	no	—
156	C₄	—	1136	1	no	—
157	C₄	—	142	1	no	—
158	C₄	#17986079	1	1	yes	characteristic
159	C₄	#17986430	1	1	yes	characteristic
160	C₂	#22590184	1	1	yes	characteristic
161	C₂	Q₆₄×D₇₁	1	1	yes	characteristic
162	C₂	#22590648	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985839

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₁

C₂

yes

maximal, characteristic

#22590198

C₂

yes

maximal, characteristic

#22586539

C₂

yes

maximal, characteristic

#22586522

C₂

yes

maximal, characteristic

C₃₂×Dic₇₁

C₂

yes

maximal, characteristic

C₇₁×#1416

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₂²

yes

characteristic

Q₁₆×Dic₇₁

—

#17986093

—

#17985839

C₄

yes

complemented

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985825

C₂²

yes

characteristic

#17985803

—

#17985802

—

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#366×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

Q₈×Dic₇₁

—

#14556093

—

Dic₅₆₈

—

Dic₅₆₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

C₂×C₄

yes

—

C₂×Q₁₆×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

C₃₂×C₇₁

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

D₁₆

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

Q₆₄

yes

complemented

C₄×Q₆₄

—

maximal, Sylow

C₂×C₇₁

C₄×D₁₆

yes

characteristic

C₂×C₇₁

C₂×Q₆₄

yes

characteristic

C₂×C₇₁

#2262

yes

characteristic

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1423

—

#1416

—

#1416

D₇₁

yes

characteristic

C₄×C₃₂

—

C₇₁

C₄×Q₆₄

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₇₁

yes

characteristic

C₄×Q₁₆

—

142

—

C₄×Q₁₆

—

142

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#366

—

#366

C₂×D₇₁

yes

characteristic

#358

—

142

—

100

#358

—

142

—

101

C₄×C₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

142

—

104

C₄×Q₈

—

284

—

105

C₄×Q₈

—

284

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

142

—

109

Q₃₂

C₄×D₇₁

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

112

#107

—

113

#107

—

142

—

114

#103

—

284

—

115

#103

—

284

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

284

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

284

—

124

Q₁₆

—

142

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₇₁

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

284

—

129

C₄⋊C₄

—

568

—

130

C₄⋊C₄

—

568

—

131

C₄²

—

568

—

132

C₄²

—

568

—

133

C₄²

—

134

C₁₆

—

142

—

135

C₁₆

C₂×C₄×D₇₁

yes

characteristic

136

C₁₆

#11765864

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

284

—

140

Q₈

—

568

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

568

—

144

C₂×C₄

—

568

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

D₈×D₇₁

yes

characteristic

148

C₈

—

142

—

149

C₈

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

150

C₈

#14556149

yes

characteristic

151

C₂²

D₁₆×D₇₁

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

568

—

155

C₄

—

156

C₄

—

1136

—

157

C₄

—

142

—

158

C₄

#17986079

yes

characteristic

159

C₄

#17986430

yes

characteristic

160

C₂

#22590184

yes

characteristic

161

C₂

Q₆₄×D₇₁

yes

characteristic

162

C₂

#22590648

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1416	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1416	D₇₁
hypocenter	C₇₁	C₄×Q₆₄
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator

C₁₆×C₇₁

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1416

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

#1416

D₇₁

hypocenter

C₇₁

C₄×Q₆₄

LCS

G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #17985839	C₄	semidirect
77. Dic₇₁	Q₆₄	semidirect
89. C₇₁	C₄×Q₆₄	semidirect

H ≅ G/N

type

11. #17985839

C₄

semidirect

77. Dic₇₁

Q₆₄

semidirect

89. C₇₁

C₄×Q₆₄

semidirect

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

subgroup	index
C₂×#17985839	2
Q₃₂×Dic₇₁	2
#22590198	2
#22586539	2
#22586522	2
C₃₂×Dic₇₁	2
C₇₁×#1416	2
C₄×Q₆₄	71

subgroup

index

Q₃₂×Dic₇₁

C₃₂×Dic₇₁

C₇₁×#1416

C₄×Q₆₄

Subgroups — #31918035

Conjugacy classes of subgroups

Complements

Maximal subgroups