Group #31909005 — subgroups

Subgroups — #31909005

Order 18176.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#346×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#22587085	C₂	1	2	yes	maximal
3	#22586406	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586405	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₁	C₄	1	1	yes	characteristic
8	#17986051	—	2	2	no	—
9	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
10	#17985775	C₂²	1	1	yes	characteristic
11	#17985774	C₄	1	1	yes	characteristic
12	#17985774	C₄	1	1	yes	characteristic
13	#17985751	—	2	2	no	—
14	#17985750	C₂²	1	2	yes	—
15	#346×C₇₁	C₂²	1	1	yes	characteristic
16	#334×C₇₁	C₂²	1	2	yes	—
17	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₇₁	Q₈	1	1	yes	characteristic
19	C₄²×D₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₁₆×D₇₁	—	2	2	no	—
23	C₁₆×D₇₁	C₈	1	4	yes	complemented
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	(C₂×D₇₁)⋊C₈	—	4	2	no	—
26	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₈×Dic₇₁	—	4	2	no	—
28	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₁	SD₁₆	1	2	yes	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₈×D₇₁	—	2	2	no	—
40	C₈×D₇₁	C₂×C₈	1	2	yes	—
41	C₈×D₇₁	M₁₆	1	2	yes	—
42	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₁	SD₁₆	1	2	yes	—
44	C₇₁⋊C₁₆	—	2	1	no	—
45	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₁	—	4	2	no	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₁	—	2	1	no	—
51	C₁₆×C₇₁	C₂×C₈	1	2	yes	—
52	C₂²×D₇₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
55	C₂×Dic₇₁	#106	1	2	yes	—
56	C₂×Dic₇₁	#106	1	1	yes	characteristic
57	C₄×D₇₁	—	4	2	no	—
58	C₄×D₇₁	C₄⋊C₈	1	2	yes	—
59	C₄×D₇₁	#108	1	2	yes	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	8	2	no	—
62	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
64	C₇₁⋊C₈	—	2	1	no	—
65	C₇₁⋊C₈	—	8	2	no	—
66	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
67	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
68	C₂×D₇₁	—	2	2	no	—
69	C₂×D₇₁	#334	1	2	yes	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
73	Dic₇₁	—	2	2	no	—
74	Dic₇₁	#334	1	2	yes	—
75	#11999	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
77	C₂×C₇₁	#2155	1	2	yes	—
78	D₇₁	—	2	2	no	—
79	C₂×#346	—	71	1	no	—
80	#1564	—	71	2	no	—
81	#1368	—	71	1	no	—
82	#1368	—	71	2	no	—
83	#1368	D₇₁	1	1	yes	characteristic, complemented
84	C₇₁	#11999	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	71	1	no	—
86	C₂²×C₁₆	—	71	1	no	—
87	#431	—	142	2	no	—
88	C₂×C₄×C₈	—	71	1	no	—
89	#346	—	71	1	no	—
90	#346	—	71	1	no	—
91	#346	—	71	1	no	—
92	#346	C₂×D₇₁	1	1	yes	characteristic
93	#335	—	142	2	no	—
94	#334	—	71	2	no	—
95	#334	C₂×D₇₁	1	2	yes	—
96	C₂²×C₈	—	71	1	no	—
97	C₂²×C₈	—	71	1	no	—
98	C₂×C₄²	—	71	1	no	—
99	C₂×C₁₆	—	71	1	no	—
100	C₂×C₁₆	—	71	4	no	—
101	C₂×C₁₆	—	71	1	no	—
102	C₂×C₁₆	—	142	2	no	—
103	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
104	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	142	2	no	—
107	C₂²⋊C₈	—	284	2	no	—
108	C₄×C₈	—	284	2	no	—
109	C₄×C₈	—	71	1	no	—
110	C₄×C₈	—	71	1	no	—
111	C₄×C₈	—	71	1	no	—
112	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	—	71	2	no	—
114	C₂²×C₄	—	71	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	284	2	no	—
117	C₂×C₈	—	71	1	no	—
118	C₂×C₈	—	71	2	no	—
119	C₂×C₈	—	71	2	no	—
120	C₂×C₈	—	71	1	no	—
121	C₂×C₈	—	142	2	no	—
122	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
124	C₄²	—	71	1	no	—
125	C₄²	—	71	2	no	—
126	C₄²	D₄×D₇₁	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	71	2	no	—
129	C₁₆	—	142	1	no	—
130	C₁₆	C₈×D₇₁	1	2	yes	—
131	C₂³	—	71	1	no	—
132	C₂×C₄	—	284	2	no	—
133	C₂×C₄	—	71	1	no	—
134	C₂×C₄	—	71	2	no	—
135	C₂×C₄	—	71	2	no	—
136	C₂×C₄	—	71	2	no	—
137	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
138	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	568	2	no	—
142	C₈	—	71	2	no	—
143	C₈	—	142	1	no	—
144	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
145	C₈	M₁₆×D₇₁	1	1	yes	characteristic
146	C₂²	—	71	2	no	—
147	C₂²	—	142	2	no	—
148	C₂²	#106×D₇₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	71	2	no	—
151	C₄	—	142	2	no	—
152	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
153	C₄	#108×D₇₁	1	1	yes	characteristic
154	C₂	—	142	2	no	—
155	C₂	#334×D₇₁	1	1	yes	characteristic
156	C₂	#22590111	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#346×D₇₁

C₂

yes

maximal, characteristic

#22587085

C₂

yes

maximal

#22586406

C₂

yes

maximal, characteristic, complemented

#22586405

C₂

yes

maximal, complemented

C₇₁×#1368

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₁

C₄

yes

characteristic

C₂×C₁₆×D₇₁

C₄

yes

characteristic

#17986051

—

C₄×C₈×D₇₁

C₂²

yes

characteristic

#17985775

C₂²

yes

characteristic

#17985774

C₄

yes

characteristic

#17985774

C₄

yes

characteristic

#17985751

—

#17985750

C₂²

yes

—

#346×C₇₁

C₂²

yes

characteristic

#334×C₇₁

C₂²

yes

—

C₂×C₈×D₇₁

C₂×C₄

yes

characteristic

C₂×C₈×D₇₁

Q₈

yes

characteristic

C₄²×D₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₇₁

—

C₁₆×D₇₁

C₈

yes

complemented

C₄⋊(C₇₁⋊C₈)

—

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₈×Dic₇₁

—

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄×D₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₁

SD₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

C₂×C₈

yes

—

C₈×D₇₁

M₁₆

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

SD₁₆

yes

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₂×C₈

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×C₈

yes

—

C₂²×D₇₁

#106

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×Dic₇₁

#106

yes

—

C₂×Dic₇₁

#106

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

C₄⋊C₈

yes

—

C₄×D₇₁

#108

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₂²×C₇₁

C₂×#106

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

#334

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₁

C₂×#108

yes

characteristic

Dic₇₁

—

Dic₇₁

#334

yes

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₇₁

#11999

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

142

—

C₂×C₄×C₈

—

—

—

—

C₂×D₇₁

yes

characteristic

#335

—

142

—

#334

—

#334

C₂×D₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

142

—

103

C₂×C₁₆

C₄×D₇₁

yes

characteristic

104

C₂×C₁₆

C₄×D₇₁

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

142

—

107

C₂²⋊C₈

—

284

—

108

C₄×C₈

—

284

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₇₁

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

284

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

142

—

122

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

123

C₂×C₈

Q₈×D₇₁

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

D₄×D₇₁

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

142

—

130

C₁₆

C₈×D₇₁

yes

—

131

C₂³

—

132

C₂×C₄

—

284

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

138

C₂×C₄

SD₁₆×D₇₁

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

568

—

142

C₈

—

143

C₈

—

142

—

144

C₈

C₂×C₈×D₇₁

yes

characteristic

145

C₈

M₁₆×D₇₁

yes

characteristic

146

C₂²

—

147

C₂²

—

142

—

148

C₂²

#106×D₇₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

142

—

152

C₄

C₄⋊C₈×D₇₁

yes

characteristic

153

C₄

#108×D₇₁

yes

characteristic

154

C₂

—

142

—

155

C₂

#334×D₇₁

yes

characteristic

156

C₂

#22590111

yes

—

	subgroup	G/H
center	C₂²	#106×D₇₁
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1368	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂²×D₇₁	#106
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1368	D₇₁
hypocenter	C₇₁	#11999
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₇₁

commutator

C₈×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1368

C₂

C₂²×C₇₁

C₂²×D₇₁

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #22586406	C₂	semidirect
4. #22586405	C₂	semidirect
5. C₇₁×#1368	C₂	semidirect
23. C₁₆×D₇₁	C₈	semidirect
83. #1368	D₇₁	semidirect
84. C₇₁	#11999	semidirect

H ≅ G/N

type

3. #22586406

C₂

semidirect

4. #22586405

C₂

semidirect

5. C₇₁×#1368

C₂

semidirect

23. C₁₆×D₇₁

C₈

semidirect

83. #1368

D₇₁

semidirect

84. C₇₁

#11999

semidirect

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

subgroup	index
#346×D₇₁	2
#22587085	2
#22586406	2
#22586405	2
C₇₁×#1368	2
#11999	71

subgroup

index

#346×D₇₁

C₇₁×#1368

Subgroups — #31909005

Conjugacy classes of subgroups

Complements

Maximal subgroups