Group #31912870 — subgroups

Subgroups — #31912870

Order 18176.

306 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17985829	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₇₁	C₂	1	2	yes	maximal
4	#22586433	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	M₁₆×Dic₇₁	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985829	C₄	1	4	yes	complemented
11	#17985774	C₄	1	4	yes	—
12	#17985763	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	2	yes	—
15	#368×C₇₁	C₄	1	4	yes	complemented
16	C₂×#11765849	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	#14556064	—	2	2	no	—
23	#14556064	C₂×C₄	1	4	yes	—
24	#14556038	—	4	2	no	—
25	C₈×Dic₇₁	—	4	2	no	—
26	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
27	C₈×Dic₇₁	Q₈	1	2	yes	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	#108×C₇₁	—	2	2	no	—
34	#108×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₁	Q₁₆	1	2	yes	—
36	#11765849	—	2	4	no	—
37	#11765849	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
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	—	2	4	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	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
55	C₁₆×C₇₁	C₄²	1	4	yes	—
56	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
63	C₂×Dic₇₁	—	4	2	no	—
64	C₂×Dic₇₁	#107	1	4	yes	—
65	C₈×C₇₁	—	4	4	no	—
66	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₇₁	#117	1	1	yes	characteristic
68	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
70	C₇₁⋊C₈	—	2	2	no	—
71	C₇₁⋊C₈	—	4	4	no	—
72	C₂²×C₇₁	—	2	2	no	—
73	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
74	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
75	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
76	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₇₁	#427	1	1	yes	characteristic
78	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
79	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
80	Dic₇₁	—	2	4	no	—
81	#12676	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	—	2	1	no	—
84	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
85	C₂×C₇₁	#2161	1	2	yes	—
86	C₂×#368	—	71	1	no	—
87	C₂×#368	D₇₁	1	1	yes	characteristic
88	C₂×#346	—	71	1	no	—
89	C₄×#108	—	71	2	no	—
90	#1381	—	71	2	no	—
91	C₇₁	#12676	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₁₆	—	71	1	no	—
93	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
94	C₂×#108	—	71	2	no	—
95	C₂×#108	C₂×D₇₁	1	2	yes	—
96	C₄×M₁₆	—	142	2	no	—
97	C₂×C₄×C₈	—	71	1	no	—
98	#368	—	71	4	no	—
99	#368	Dic₇₁	1	4	yes	complemented
100	#346	—	71	4	no	—
101	#341	—	142	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	142	2	no	—
104	C₂²×C₈	—	71	1	no	—
105	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×C₄²	—	71	1	no	—
107	C₂×C₁₆	—	71	2	no	—
108	C₂×C₁₆	—	142	1	no	—
109	C₂×C₁₆	—	142	1	no	—
110	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
111	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	71	4	no	—
115	#108	—	142	2	no	—
116	#108	C₂×Dic₇₁	1	4	yes	—
117	#97	—	284	2	no	—
118	C₄×C₈	—	284	2	no	—
119	C₄×C₈	—	71	2	no	—
120	C₄×C₈	—	71	2	no	—
121	C₂²×C₄	—	71	2	no	—
122	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	284	2	no	—
126	M₁₆	—	142	4	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	284	2	no	—
129	C₂×C₈	—	71	2	no	—
130	C₂×C₈	—	142	1	no	—
131	C₂×C₈	—	142	1	no	—
132	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
136	C₂×C₈	#11765864	1	1	yes	characteristic
137	C₂×C₈	#11765866	1	1	yes	characteristic
138	C₄²	—	71	2	no	—
139	C₄²	—	71	2	no	—
140	C₁₆	—	142	2	no	—
141	C₁₆	C₄×Dic₇₁	1	4	yes	—
142	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
143	C₂×C₄	—	284	2	no	—
144	C₂×C₄	—	71	4	no	—
145	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
146	C₂×C₄	#14556090	1	1	yes	characteristic
147	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
148	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
149	C₂×C₄	#14556147	1	1	yes	characteristic
150	C₂×C₄	#14556151	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	284	4	no	—
153	C₈	—	142	2	no	—
154	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
155	C₈	#14556086	1	1	yes	characteristic
156	C₈	#14556093	1	1	yes	characteristic
157	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	#107×D₇₁	1	1	yes	characteristic
160	C₂²	#17986093	1	1	yes	characteristic
161	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
162	C₄	—	142	4	no	—
163	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
164	C₄	#17986036	1	1	yes	characteristic
165	C₄	#17986081	1	1	yes	characteristic
166	C₄	D₈×Dic₇₁	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#107×Dic₇₁	1	1	yes	characteristic
170	C₂	#22590133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal, characteristic

C₂×#17985774

C₂

yes

maximal, characteristic

#108×Dic₇₁

C₂

yes

maximal

#22586433

C₂

yes

maximal

C₂×#368×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

M₁₆×Dic₇₁

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985829

C₄

yes

complemented

#17985774

C₄

yes

—

#17985763

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

#368×C₇₁

C₄

yes

complemented

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556038

—

C₈×Dic₇₁

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

Q₁₆

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

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

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₁₆×C₇₁

C₄²

yes

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

C₄×Q₁₆

yes

characteristic

C₂²×C₇₁

C₄×Q₁₆

yes

characteristic

C₄×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#427

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

characteristic

Dic₇₁

—

#12676

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×#107

yes

characteristic

C₂×C₇₁

#2161

yes

—

C₂×#368

—

C₂×#368

D₇₁

yes

—

—

—

C₇₁

#12676

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₇₁

yes

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

—

#368

—

#368

Dic₇₁

yes

complemented

100

#346

—

101

#341

—

142

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

142

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×D₇₁

yes

characteristic

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

142

—

109

C₂×C₁₆

—

142

—

110

C₂×C₁₆

C₄×D₇₁

yes

—

111

C₂×C₁₆

C₄×D₇₁

yes

—

112

C₂×C₁₆

C₂×Dic₇₁

yes

—

113

#108

—

114

#108

—

115

#108

—

142

—

116

#108

C₂×Dic₇₁

yes

—

117

#97

—

284

—

118

C₄×C₈

—

284

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₂²×C₄

—

122

C₂²×C₄

D₄×D₇₁

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

284

—

126

M₁₆

—

142

—

127

C₂×C₈

—

128

C₂×C₈

—

284

—

129

C₂×C₈

—

130

C₂×C₈

—

142

—

131

C₂×C₈

—

142

—

132

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

133

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

134

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

135

C₂×C₈

Q₈×D₇₁

yes

characteristic

136

C₂×C₈

#11765864

yes

characteristic

137

C₂×C₈

#11765866

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₁₆

—

142

—

141

C₁₆

C₄×Dic₇₁

yes

—

142

C₂³

Q₁₆×D₇₁

yes

characteristic

143

C₂×C₄

—

284

—

144

C₂×C₄

—

145

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

146

C₂×C₄

#14556090

yes

characteristic

147

C₂×C₄

D₄×Dic₇₁

yes

characteristic

148

C₂×C₄

D₈×D₇₁

yes

characteristic

149

C₂×C₄

#14556147

yes

characteristic

150

C₂×C₄

#14556151

yes

characteristic

151

C₈

—

152

C₈

—

284

—

153

C₈

—

142

—

154

C₈

C₂×C₄×Dic₇₁

yes

characteristic

155

C₈

#14556086

yes

characteristic

156

C₈

#14556093

yes

characteristic

157

C₈

Q₈×Dic₇₁

yes

characteristic

158

C₂²

—

159

C₂²

#107×D₇₁

yes

characteristic

160

C₂²

#17986093

yes

characteristic

161

C₂²

Q₁₆×Dic₇₁

yes

characteristic

162

C₄

—

142

—

163

C₄

C₄⋊C₄×Dic₇₁

yes

characteristic

164

C₄

#17986036

yes

characteristic

165

C₄

#17986081

yes

characteristic

166

C₄

D₈×Dic₇₁

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#107×Dic₇₁

yes

characteristic

170

C₂

#22590133

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	#12676
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₈×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#368×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #17985829	C₄	semidirect
15. #368×C₇₁	C₄	semidirect
91. C₇₁	#12676	semidirect
99. #368	Dic₇₁	semidirect

H ≅ G/N

type

10. #17985829

C₄

semidirect

15. #368×C₇₁

C₄

semidirect

91. C₇₁

#12676

semidirect

99. #368

Dic₇₁

semidirect

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

subgroup	index
C₂×#17985829	2
C₂×#17985774	2
#108×Dic₇₁	2
#22586433	2
C₂×#368×C₇₁	2
#12676	71

subgroup

index

#108×Dic₇₁

C₂×#368×C₇₁

Subgroups — #31912870

Conjugacy classes of subgroups

Complements

Maximal subgroups