Group #31884384 — subgroups

Subgroups — #31884384

Order 18176.

238 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 115 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×#17985775	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×#14556064	C₄	1	2	yes	—
6	C₄×#11765849	—	2	1	no	—
7	#17985775	—	2	1	no	—
8	#17985775	C₄	1	2	yes	—
9	#17985764	—	2	1	no	—
10	#431×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#335×C₇₁	—	2	1	no	—
13	#334×C₇₁	C₄	1	2	yes	—
14	C₂×#11765849	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	—	2	4	no	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	#14556064	—	2	2	no	—
18	#14556064	C₈	1	4	yes	complemented
19	#14556039	—	4	1	no	—
20	C₄×C₇₁⋊C₈	—	4	1	no	—
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	2	no	—
25	C₂²⋊C₈×C₇₁	—	4	1	no	—
26	C₄×C₈×C₇₁	—	4	1	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
31	#11765849	—	2	4	no	—
32	#11765849	—	4	2	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₇₁⋊C₁₆	—	2	4	no	—
35	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	2	2	no	—
39	C₂×C₈×C₇₁	—	4	2	no	—
40	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
46	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₁	D₈	1	1	yes	characteristic
49	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂³×C₇₁	#103	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	4	1	no	—
53	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#104	1	2	yes	—
59	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	8	2	no	—
64	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
65	C₈×C₇₁	#97	1	1	yes	characteristic
66	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
67	C₇₁⋊C₈	—	4	4	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	—	2	1	no	—
70	C₂²×C₇₁	Q₈⋊C₈	1	2	yes	—
71	C₂²×C₇₁	#340	1	1	yes	characteristic
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
77	C₄×C₇₁	#336	1	1	yes	characteristic
78	C₄×C₇₁	#339	1	1	yes	characteristic
79	#7177	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	2	2	no	—
81	C₂×C₇₁	#1279	1	1	yes	characteristic
82	C₂×C₇₁	#1388	1	1	yes	characteristic
83	C₂×C₇₁	#1389	1	1	yes	characteristic
84	C₂×#346	—	71	1	no	—
85	C₄×#108	—	71	1	no	—
86	#1564	D₇₁	1	1	yes	characteristic
87	C₇₁	#7177	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	142	1	no	—
89	#431	—	2	1	no	—
90	C₂×#108	—	71	2	no	—
91	C₄×M₁₆	—	142	1	no	—
92	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
93	#346	—	71	2	no	—
94	#346	—	142	1	no	—
95	#341	—	142	1	no	—
96	#335	—	2	1	no	—
97	#334	Dic₇₁	1	2	yes	—
98	C₂×M₁₆	—	142	2	no	—
99	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
100	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂×C₁₆	—	142	4	no	—
103	C₂×C₁₆	—	142	2	no	—
104	#108	—	71	4	no	—
105	#108	—	142	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	4	1	no	—
108	#97	—	284	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	—	284	1	no	—
111	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
112	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	#11765853	1	1	yes	characteristic
117	C₂²×C₄	#11765855	1	1	yes	characteristic
118	M₁₆	—	284	2	no	—
119	M₁₆	—	142	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	284	2	no	—
123	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
124	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
125	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
127	C₂×C₈	C₈×D₇₁	1	2	yes	—
128	C₂×C₈	#11765848	1	2	yes	—
129	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
130	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
131	C₄²	#11765850	1	1	yes	characteristic
132	C₄²	#11765852	1	1	yes	characteristic
133	C₁₆	—	142	4	no	—
134	C₂³	#14556051	1	1	yes	characteristic
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
139	C₂×C₄	#14556048	1	1	yes	characteristic
140	C₂×C₄	#14556049	1	1	yes	characteristic
141	C₂×C₄	#14556052	1	1	yes	characteristic
142	C₂×C₄	#14556054	1	2	yes	—
143	C₂×C₄	#14556059	1	1	yes	characteristic
144	C₂×C₄	#14556061	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	8	2	no	—
148	C₈	—	284	4	no	—
149	C₈	C₈×Dic₇₁	1	1	yes	characteristic
150	C₈	#14556038	1	1	yes	characteristic
151	C₈	Dic₇₁⋊C₈	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#17985729	1	2	yes	—
155	C₂²	#17985761	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	#17985726	1	2	yes	—
161	C₄	#17985753	1	1	yes	characteristic
162	C₄	#17985760	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	#22586179	1	1	yes	characteristic
165	C₂	#22586453	1	1	yes	characteristic
166	C₂	#22586456	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985775

C₂

yes

maximal, characteristic

C₄×#14556064

C₂

yes

maximal, characteristic

C₇₁×#1564

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

—

C₂×#14556064

C₄

yes

—

C₄×#11765849

—

#17985775

—

#17985775

C₄

yes

—

#17985764

—

#431×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#335×C₇₁

—

#334×C₇₁

C₄

yes

—

C₂×#11765849

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

C₈

yes

complemented

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

SD₁₆

yes

characteristic

C₂³×C₇₁

#103

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

characteristic

C₈×C₇₁

#97

yes

characteristic

C₈×C₇₁

C₄⋊C₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

Q₈⋊C₈

yes

—

C₂²×C₇₁

#340

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

D₄⋊C₈

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

C₇₁

#7177

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

142

—

#431

—

C₂×#108

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#346

—

#346

—

142

—

#341

—

142

—

#335

—

#334

Dic₇₁

yes

—

C₂×M₁₆

—

142

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

100

C₂²×C₈

D₂₈₄

yes

characteristic

101

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

102

C₂×C₁₆

—

142

—

103

C₂×C₁₆

—

142

—

104

#108

—

105

#108

—

142

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

284

—

109

C₄×C₈

—

110

C₄×C₈

—

284

—

111

C₄×C₈

Dic₁₄₂

yes

characteristic

112

C₄×C₈

C₄×D₇₁

yes

characteristic

113

C₄×C₈

C₂×Dic₇₁

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

115

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

116

C₂²×C₄

#11765853

yes

characteristic

117

C₂²×C₄

#11765855

yes

characteristic

118

M₁₆

—

284

—

119

M₁₆

—

142

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

284

—

123

C₂×C₈

C₄×Dic₇₁

yes

characteristic

124

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

125

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

126

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

127

C₂×C₈

C₈×D₇₁

yes

—

128

C₂×C₈

#11765848

yes

—

129

C₄²

C₂²⋊Dic₇₁

yes

characteristic

130

C₄²

Dic₇₁⋊C₄

yes

characteristic

131

C₄²

#11765850

yes

characteristic

132

C₄²

#11765852

yes

characteristic

133

C₁₆

—

142

—

134

C₂³

#14556051

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₇₁⋊#95

yes

characteristic

138

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

139

C₂×C₄

#14556048

yes

characteristic

140

C₂×C₄

#14556049

yes

characteristic

141

C₂×C₄

#14556052

yes

characteristic

142

C₂×C₄

#14556054

yes

—

143

C₂×C₄

#14556059

yes

characteristic

144

C₂×C₄

#14556061

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

284

—

149

C₈

C₈×Dic₇₁

yes

characteristic

150

C₈

#14556038

yes

characteristic

151

C₈

Dic₇₁⋊C₈

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

#17985729

yes

—

155

C₂²

#17985761

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#17985726

yes

—

161

C₄

#17985753

yes

characteristic

162

C₄

#17985760

yes

characteristic

163

C₂

—

164

C₂

#22586179

yes

characteristic

165

C₂

#22586453

yes

characteristic

166

C₂

#22586456

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556048
commutator	C₈×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1564	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂³×C₇₁	#103
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1564	D₇₁
hypocenter	C₇₁	#7177
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂×C₄

#14556048

commutator

C₈×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₇₁×#1564

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
18. #14556064	C₈	semidirect
87. C₇₁	#7177	semidirect

H ≅ G/N

type

18. #14556064

C₈

semidirect

87. C₇₁

#7177

semidirect

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

subgroup	index
C₂×#17985775	2
C₄×#14556064	2
C₇₁×#1564	2
#7177	71

subgroup

index

C₂×#17985775

C₄×#14556064

C₇₁×#1564

#7177

Subgroups — #31884384

Conjugacy classes of subgroups

Complements

Maximal subgroups