Group #31916049 — subgroups

Subgroups — #31916049

Order 18176.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22586432	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556120	—	2	1	no	—
6	#17986401	—	2	2	no	—
7	#17986401	C₄	1	4	yes	complemented
8	C₂×#14556062	C₂²	1	2	yes	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556060	—	2	1	no	—
11	C₂×#14556060	—	2	1	no	—
12	#17985865	—	2	1	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₁	—	2	2	no	—
16	#14556120	—	2	4	no	—
17	#14556120	—	4	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×Dic₇₁⋊C₄	—	4	2	no	—
21	#14556062	—	2	2	no	—
22	#14556062	—	2	4	no	—
23	#14556062	C₂×C₄	1	4	yes	—
24	#14556060	—	2	4	no	—
25	#14556060	—	4	2	no	—
26	C₇₁⋊#95	—	4	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	C₂×C₁₆×C₇₁	—	2	2	no	—
31	C₂×C₁₆×C₇₁	—	2	2	no	—
32	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₁	D₄	1	2	yes	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₂²×Dic₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	—	2	4	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	C₄⋊Dic₇₁	—	4	2	no	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	Dic₇₁⋊C₄	—	4	8	no	—
42	C₂²×C₄×C₇₁	D₈	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₇₁	#41	1	2	yes	—
51	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₇₁	SD₁₆	1	2	yes	—
53	C₁₆×C₇₁	—	2	4	no	—
54	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	#102	1	4	yes	—
57	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₁	—	4	4	no	—
64	C₂×Dic₇₁	—	8	4	no	—
65	C₂×Dic₇₁	—	8	2	no	—
66	C₂×Dic₇₁	—	8	4	no	—
67	C₈×C₇₁	#103	1	4	yes	—
68	C₈×C₇₁	C₄×Q₈	1	2	yes	—
69	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
70	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
71	C₂²×C₇₁	#486	1	1	yes	characteristic
72	C₂²×C₇₁	#500	1	1	yes	characteristic
73	C₂²×C₇₁	#508	1	4	yes	—
74	C₄×C₇₁	—	2	4	no	—
75	C₄×C₇₁	#391	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
77	C₄×C₇₁	#487	1	1	yes	characteristic
78	C₄×C₇₁	#498	1	1	yes	characteristic
79	Dic₇₁	—	8	4	no	—
80	Dic₇₁	—	16	2	no	—
81	#13281	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	#1990	1	1	yes	characteristic
83	C₂×C₇₁	#2138	1	2	yes	—
84	C₂×C₇₁	#2242	1	2	yes	—
85	C₂×C₇₁	#2253	1	2	yes	—
86	C₂×#499	—	71	1	no	—
87	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
88	#1846	—	71	1	no	—
89	#1380	—	71	4	no	—
90	C₇₁	#13281	1	1	yes	characteristic, complemented, Sylow
91	C₂×#125	—	142	1	no	—
92	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
93	#499	—	71	4	no	—
94	#499	—	142	2	no	—
95	C₂×#107	—	71	2	no	—
96	C₂×#107	—	71	2	no	—
97	C₂×#106	—	142	1	no	—
98	C₂×#106	—	142	1	no	—
99	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
100	#382	—	142	1	no	—
101	C₄×C₁₆	—	2	2	no	—
102	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	#125	—	284	2	no	—
105	#125	—	142	4	no	—
106	C₂×C₄⋊C₄	—	284	2	no	—
107	C₂×C₄⋊C₄	—	142	2	no	—
108	C₂×C₄⋊C₄	—	142	2	no	—
109	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Dic₁₄₂	1	4	yes	—
113	#107	—	71	4	no	—
114	#107	—	142	4	no	—
115	#107	—	142	2	no	—
116	#106	—	284	2	no	—
117	#106	—	142	4	no	—
118	C₄×C₈	C₄×D₇₁	1	2	yes	—
119	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
120	#95	—	284	2	no	—
121	C₂²×C₄	—	284	2	no	—
122	C₂²×C₄	—	284	2	no	—
123	C₂²×C₄	D₅₆₈	1	2	yes	—
124	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
125	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
126	C₂×C₈	#11765854	1	2	yes	—
127	C₂×C₈	Dic₂₈₄	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
131	C₂×C₈	#11765865	1	2	yes	—
132	C₄⋊C₄	—	284	8	no	—
133	C₄⋊C₄	—	284	2	no	—
134	C₄⋊C₄	—	284	2	no	—
135	C₄⋊C₄	—	284	2	no	—
136	C₄⋊C₄	—	142	4	no	—
137	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
138	C₄²	#11765854	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	284	4	no	—
143	C₂×C₄	—	568	4	no	—
144	C₂×C₄	—	568	2	no	—
145	C₂×C₄	—	568	4	no	—
146	C₂×C₄	#14556050	1	4	yes	—
147	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#14556116	1	1	yes	characteristic
149	C₂×C₄	#14556128	1	1	yes	characteristic
150	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
151	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
152	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
153	C₈	#14556053	1	4	yes	—
154	C₈	C₄×Dic₁₄₂	1	2	yes	—
155	C₈	#14556107	1	2	yes	—
156	C₂²	C₂×#14556050	1	1	yes	characteristic
157	C₂²	#17986346	1	1	yes	characteristic
158	C₂²	#17986404	1	1	yes	characteristic
159	C₂²	#17986431	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	568	4	no	—
162	C₄	—	1136	2	no	—
163	C₄	#17985905	1	1	yes	characteristic
164	C₄	C₂×#14556053	1	1	yes	characteristic
165	C₄	#17986350	1	1	yes	characteristic
166	C₄	#17986397	1	1	yes	characteristic
167	C₂	#22589301	1	1	yes	characteristic
168	C₂	#22590048	1	2	yes	—
169	C₂	#22590572	1	2	yes	—
170	C₂	#22590619	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

C₂

yes

maximal, characteristic

#22588504

C₂

yes

maximal, characteristic

#22586432

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556120

—

#17986401

—

#17986401

C₄

yes

complemented

C₂×#14556062

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

C₂×#14556060

—

#17985865

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

—

#14556120

—

#14556120

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

#14556062

—

#14556062

—

#14556062

C₂×C₄

yes

—

#14556060

—

#14556060

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

Q₈

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

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

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2242

yes

—

C₂×C₇₁

#2253

yes

—

C₂×#499

—

C₂×C₄×C₁₆

D₇₁

yes

characteristic

#1846

—

#1380

—

C₇₁

#13281

yes

characteristic, complemented, Sylow

C₂×#125

—

142

—

C₂²×C₁₆

C₂×D₇₁

yes

—

#499

—

#499

—

142

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

142

—

C₂×#106

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

100

#382

—

142

—

101

C₄×C₁₆

—

102

C₂²×C₈

D₂₈₄

yes

characteristic

103

C₂²×C₈

C₂²×D₇₁

yes

characteristic

104

#125

—

284

—

105

#125

—

142

—

106

C₂×C₄⋊C₄

—

284

—

107

C₂×C₄⋊C₄

—

142

—

108

C₂×C₄⋊C₄

—

142

—

109

C₂×C₄²

D₂₈₄

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₁₄₂

yes

—

113

#107

—

114

#107

—

142

—

115

#107

—

142

—

116

#106

—

284

—

117

#106

—

142

—

118

C₄×C₈

C₄×D₇₁

yes

—

119

C₄×C₈

C₂²⋊D₇₁

yes

—

120

#95

—

284

—

121

C₂²×C₄

—

284

—

122

C₂²×C₄

—

284

—

123

C₂²×C₄

D₅₆₈

yes

—

124

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

125

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

126

C₂×C₈

#11765854

yes

—

127

C₂×C₈

Dic₂₈₄

yes

—

128

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₇₁

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

—

284

—

136

C₄⋊C₄

—

142

—

137

C₄²

(C₂×D₇₁)⋊C₄

yes

—

138

C₄²

#11765854

yes

—

139

C₁₆

—

140

C₂³

C₂×D₅₆₈

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

284

—

143

C₂×C₄

—

568

—

144

C₂×C₄

—

568

—

145

C₂×C₄

—

568

—

146

C₂×C₄

#14556050

yes

—

147

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

148

C₂×C₄

#14556116

yes

characteristic

149

C₂×C₄

#14556128

yes

characteristic

150

C₂×C₄

C₂×#11765854

yes

characteristic

151

C₂×C₄

C₂×#11765854

yes

characteristic

152

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

153

C₈

#14556053

yes

—

154

C₈

C₄×Dic₁₄₂

yes

—

155

C₈

#14556107

yes

—

156

C₂²

C₂×#14556050

yes

characteristic

157

C₂²

#17986346

yes

characteristic

158

C₂²

#17986404

yes

characteristic

159

C₂²

#17986431

yes

—

160

C₄

—

161

C₄

—

568

—

162

C₄

—

1136

—

163

C₄

#17985905

yes

characteristic

164

C₄

C₂×#14556053

yes

characteristic

165

C₄

#17986350

yes

characteristic

166

C₄

#17986397

yes

characteristic

167

C₂

#22589301

yes

characteristic

168

C₂

#22590048

yes

—

169

C₂

#22590572

yes

—

170

C₂

#22590619

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#13281
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×C₄×C₁₆×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₁

hypocenter

C₇₁

#13281

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
7. #17986401	C₄	semidirect
90. C₇₁	#13281	semidirect

H ≅ G/N

type

7. #17986401

C₄

semidirect

90. C₇₁

#13281

semidirect

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

subgroup	index
C₂×#17986401	2
#22588504	2
#22586432	2
C₂×C₄×C₁₆×C₇₁	2
#13281	71

subgroup

index

C₂×C₄×C₁₆×C₇₁

Subgroups — #31916049

Conjugacy classes of subgroups

Complements

Maximal subgroups