Group #31912869 — subgroups

Subgroups — #31912869

Order 18176.

306 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 151 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×#17985828	C₂	1	2	yes	maximal
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
4	#22586433	C₂	1	2	yes	maximal
5	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14556064	C₂²	1	2	yes	—
9	C₄×#11765849	—	2	1	no	—
10	#17985828	C₄	1	8	yes	complemented
11	#17985775	C₄	1	4	yes	—
12	#17985764	—	2	1	no	—
13	C₂×#108×C₇₁	C₂²	1	2	yes	—
14	C₄×M₁₆×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#341×C₇₁	—	2	1	no	—
17	C₂×#11765849	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	#14556064	—	2	2	no	—
23	#14556064	C₂×C₄	1	4	yes	—
24	#14556039	—	4	1	no	—
25	C₄×C₇₁⋊C₈	—	4	1	no	—
26	C₂×M₁₆×C₇₁	—	2	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	#108×C₇₁	—	2	2	no	—
31	#108×C₇₁	C₂×C₄	1	4	yes	—
32	#97×C₇₁	—	4	1	no	—
33	C₄×C₈×C₇₁	—	4	1	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	Q₈	1	2	yes	—
36	#11765849	—	2	4	no	—
37	#11765849	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₇₁⋊C₁₆	—	2	2	no	—
40	C₇₁⋊C₁₆	C₄²	1	4	yes	—
41	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₇₁	—	2	4	no	—
44	M₁₆×C₇₁	—	4	2	no	—
45	C₂×C₈×C₇₁	—	2	2	no	—
46	C₂×C₈×C₇₁	—	4	2	no	—
47	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	#41	1	2	yes	—
52	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
53	C₄²×C₇₁	D₈	1	1	yes	characteristic
54	C₄²×C₇₁	D₈	1	1	yes	characteristic
55	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	#107	1	2	yes	—
58	C₂×C₄×C₇₁	#107	1	2	yes	—
59	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
61	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#135	1	2	yes	—
63	C₈×C₇₁	—	2	2	no	—
64	C₈×C₇₁	—	4	4	no	—
65	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₇₁	#117	1	1	yes	characteristic
67	C₈×C₇₁	C₄×Q₈	1	2	yes	—
68	C₇₁⋊C₈	—	4	4	no	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₂²×C₇₁	C₄×Q₁₆	1	2	yes	—
72	C₄×C₇₁	—	2	2	no	—
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₇₁	#427	1	1	yes	characteristic
76	C₄×C₇₁	C₄×D₈	1	2	yes	—
77	#12676	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	2	2	no	—
79	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
80	C₂×C₇₁	#2161	1	2	yes	—
81	C₂×#368	—	71	2	no	—
82	C₂×#346	—	71	1	no	—
83	C₄×#108	—	71	1	no	—
84	C₄×#108	D₇₁	1	1	yes	characteristic
85	#1381	—	71	2	no	—
86	C₇₁	#12676	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	71	1	no	—
88	C₂²×C₁₆	—	71	1	no	—
89	C₂×#108	—	71	2	no	—
90	C₂×#108	C₂×D₇₁	1	2	yes	—
91	C₄×M₁₆	—	2	1	no	—
92	C₄×M₁₆	—	142	1	no	—
93	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
94	#368	—	71	8	no	—
95	#346	—	71	4	no	—
96	#341	—	2	1	no	—
97	#341	—	142	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	142	2	no	—
100	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂×C₁₆	—	71	2	no	—
104	C₂×C₁₆	—	71	4	no	—
105	C₂×C₁₆	—	71	2	no	—
106	C₂×C₁₆	—	142	2	no	—
107	#108	—	2	2	no	—
108	#108	—	71	4	no	—
109	#108	—	142	2	no	—
110	#108	C₄×D₇₁	1	4	yes	—
111	#97	—	4	1	no	—
112	#97	—	284	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	284	1	no	—
115	C₄×C₈	Dic₁₄₂	1	2	yes	—
116	C₄×C₈	C₄×D₇₁	1	2	yes	—
117	C₂²×C₄	#11765855	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	284	2	no	—
122	M₁₆	—	142	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	284	2	no	—
126	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
127	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
129	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
130	C₂×C₈	#11765865	1	2	yes	—
131	C₄²	Dic₇₁⋊C₄	1	2	yes	—
132	C₄²	#11765850	1	1	yes	characteristic
133	C₄²	#11765850	1	1	yes	characteristic
134	C₁₆	—	71	4	no	—
135	C₁₆	—	142	2	no	—
136	C₂³	C₂×#11765855	1	1	yes	characteristic
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#14556061	1	2	yes	—
139	C₂×C₄	#14556061	1	2	yes	—
140	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
142	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
143	C₂×C₄	#14556148	1	2	yes	—
144	C₈	—	2	2	no	—
145	C₈	—	4	4	no	—
146	C₈	—	284	4	no	—
147	C₈	C₄²×D₇₁	1	1	yes	characteristic
148	C₈	#14556085	1	1	yes	characteristic
149	C₈	C₄×Dic₁₄₂	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	C₂×#14556061	1	1	yes	characteristic
152	C₂²	C₄×#11765855	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
156	C₄	#17986035	1	1	yes	characteristic
157	C₄	C₄×#11765850	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	C₄×#14556061	1	1	yes	characteristic
160	C₂	#22590134	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985828

C₂

yes

maximal

C₂×#17985775

C₂

yes

maximal, characteristic

C₄×#14556064

C₂

yes

maximal, characteristic

#22586433

C₂

yes

maximal

C₄×#108×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

C₄×#11765849

—

#17985828

C₄

yes

complemented

#17985775

C₄

yes

—

#17985764

—

C₂×#108×C₇₁

C₂²

yes

—

C₄×M₁₆×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#341×C₇₁

—

C₂×#11765849

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

#108×C₇₁

—

#108×C₇₁

C₂×C₄

yes

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₇₁

Q₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

C₄×Q₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#427

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

—

#12676

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₄×#107

yes

characteristic

C₂×C₇₁

#2161

yes

—

—

—

—

D₇₁

yes

characteristic

#1381

—

C₇₁

#12676

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

C₂×D₇₁

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

—

—

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

100

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

101

C₂²×C₈

C₂²×D₇₁

yes

characteristic

102

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

142

—

107

#108

—

108

#108

—

109

#108

—

142

—

110

#108

C₄×D₇₁

yes

—

111

#97

—

112

#97

—

284

—

113

C₄×C₈

—

114

C₄×C₈

—

284

—

115

C₄×C₈

Dic₁₄₂

yes

—

116

C₄×C₈

C₄×D₇₁

yes

—

117

C₂²×C₄

#11765855

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

284

—

122

M₁₆

—

142

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

284

—

126

C₂×C₈

Dic₇₁⋊C₄

yes

—

127

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

128

C₂×C₈

C₂×C₄×D₇₁

yes

—

129

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

130

C₂×C₈

#11765865

yes

—

131

C₄²

Dic₇₁⋊C₄

yes

—

132

C₄²

#11765850

yes

characteristic

133

C₄²

#11765850

yes

characteristic

134

C₁₆

—

135

C₁₆

—

142

—

136

C₂³

C₂×#11765855

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

#14556061

yes

—

139

C₂×C₄

#14556061

yes

—

140

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

141

C₂×C₄

C₄×C₂²⋊D₇₁

yes

—

142

C₂×C₄

C₂×#11765850

yes

characteristic

143

C₂×C₄

#14556148

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

284

—

147

C₈

C₄²×D₇₁

yes

characteristic

148

C₈

#14556085

yes

characteristic

149

C₈

C₄×Dic₁₄₂

yes

—

150

C₂²

—

151

C₂²

C₂×#14556061

yes

characteristic

152

C₂²

C₄×#11765855

yes

—

153

C₄

—

154

C₄

—

155

C₄

C₄×Dic₇₁⋊C₄

yes

characteristic

156

C₄

#17986035

yes

characteristic

157

C₄

C₄×#11765850

yes

—

158

C₂

—

159

C₂

C₄×#14556061

yes

characteristic

160

C₂

#22590134

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	#12676
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

C₂×#11765850

commutator

C₈×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₄×#108×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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
10. #17985828	C₄	semidirect
86. C₇₁	#12676	semidirect

H ≅ G/N

type

10. #17985828

C₄

semidirect

86. C₇₁

#12676

semidirect

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

subgroup	index
C₂×#17985828	2
C₂×#17985775	2
C₄×#14556064	2
#22586433	2
C₄×#108×C₇₁	2
#12676	71

subgroup

index

C₄×#108×C₇₁

Subgroups — #31912869

Conjugacy classes of subgroups

Complements

Maximal subgroups