Group #31909169 — subgroups

Subgroups — #31909169

Order 18176.

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

normal subgroups only

This group has 242 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	#22590172	C₂	1	2	yes	maximal, complemented
2	#22586410	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	—	2	2	no	—
7	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
8	C₂²⋊(C₇₁⋊C₁₆)	—	2	2	no	—
9	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×D₈×C₇₁	C₄	1	1	yes	characteristic
12	C₈×D₄×C₇₁	C₂²	1	2	yes	—
13	#335×C₇₁	C₄	1	1	yes	characteristic
14	D₄⋊C₈×C₇₁	C₄	1	2	yes	—
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	#14556068	—	4	4	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	1	no	—
19	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
21	C₂×D₈×C₇₁	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	—	2	2	no	—
23	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
24	#107×C₇₁	C₈	1	1	yes	characteristic
25	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
26	#102×C₇₁	C₈	1	2	yes	—
27	C₂²⋊C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₇₁⋊C₁₆	—	2	2	no	—
32	C₇₁⋊C₁₆	—	2	1	no	—
33	C₇₁⋊C₁₆	—	4	2	no	—
34	C₇₁⋊C₁₆	D₈	1	2	yes	complemented
35	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₇₁	—	2	2	no	—
37	D₈×C₇₁	—	2	2	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
47	C₂²⋊C₄×C₇₁	—	2	2	no	—
48	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₇₁	—	2	2	no	—
50	D₄×C₇₁	—	2	2	no	—
51	D₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	4	2	no	—
54	C₂×C₄×C₇₁	—	4	2	no	—
55	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
58	C₈×C₇₁	—	2	2	no	—
59	C₈×C₇₁	—	2	1	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	4	2	no	—
62	C₈×C₇₁	M₃₂	1	2	yes	complemented
63	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
64	C₈×C₇₁	#135	1	1	yes	characteristic
65	C₂²×C₇₁	—	4	2	no	—
66	C₂²×C₇₁	—	4	2	no	—
67	C₂²×C₇₁	—	4	2	no	—
68	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	4	2	no	—
71	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
72	C₄×C₇₁	#443	1	1	yes	characteristic
73	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
74	C₄×C₇₁	#504	1	1	yes	characteristic
75	#12039	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	8	2	no	—
77	C₂×C₇₁	C₈×D₈	1	1	yes	characteristic
78	C₂×C₇₁	#2169	1	1	yes	characteristic
79	C₂×C₇₁	#2170	1	1	yes	characteristic
80	#2169	—	71	2	no	—
81	C₈×D₈	D₇₁	1	1	yes	characteristic
82	#1370	—	71	1	no	—
83	D₄⋊C₁₆	—	71	2	no	—
84	C₈×C₁₆	—	71	1	no	—
85	C₇₁	#12039	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	142	2	no	—
87	C₄×D₈	Dic₇₁	1	1	yes	characteristic
88	C₈×D₄	C₂×D₇₁	1	2	yes	—
89	C₄⋊C₁₆	—	71	2	no	—
90	C₂²⋊C₁₆	—	142	2	no	—
91	C₄×C₁₆	—	71	1	no	—
92	C₄×C₁₆	—	71	1	no	—
93	#335	Dic₇₁	1	1	yes	characteristic
94	D₄⋊C₈	Dic₇₁	1	2	yes	—
95	C₈²	C₂×D₇₁	1	1	yes	characteristic
96	C₂×D₈	C₇₁⋊C₈	1	1	yes	characteristic
97	C₂²×C₈	—	2	2	no	—
98	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
99	M₃₂	—	284	4	no	—
100	C₂×C₁₆	—	71	1	no	—
101	C₂×C₁₆	—	71	1	no	—
102	C₂×C₁₆	—	142	2	no	—
103	C₂×C₁₆	—	142	1	no	—
104	#107	C₇₁⋊C₈	1	1	yes	characteristic
105	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
106	#102	C₇₁⋊C₈	1	2	yes	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
110	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
112	C₂²×C₄	—	2	2	no	—
113	D₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
119	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
120	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
121	C₂×C₈	#11765864	1	1	yes	characteristic
122	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
125	C₁₆	—	284	2	no	—
126	C₁₆	—	71	2	no	—
127	C₁₆	—	142	2	no	—
128	C₁₆	—	142	1	no	—
129	C₂³	—	2	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
137	C₂×C₄	#14556135	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	4	2	no	—
142	C₈	#14556068	1	2	yes	complemented
143	C₈	D₈×D₇₁	1	1	yes	characteristic
144	C₈	#14556147	1	1	yes	characteristic
145	C₂²	—	4	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	2	no	—
148	C₂²	D₄×C₇₁⋊C₈	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	D₈×Dic₇₁	1	1	yes	characteristic
152	C₄	#17986116	1	1	yes	characteristic
153	C₄	C₂×#14556068	1	1	yes	characteristic
154	C₄	#17986416	1	1	yes	characteristic
155	C₂	—	8	2	no	—
156	C₂	D₈×C₇₁⋊C₈	1	1	yes	characteristic
157	C₂	#22590172	1	1	yes	characteristic
158	C₂	#22590177	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590172

C₂

yes

maximal, complemented

#22586410

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, complemented

C₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₁₆)

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₄×D₈×C₇₁

C₄

yes

characteristic

C₈×D₄×C₇₁

C₂²

yes

—

#335×C₇₁

C₄

yes

characteristic

D₄⋊C₈×C₇₁

C₄

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₂×D₈×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

—

C₄×D₄×C₇₁

C₂×C₄

yes

—

#107×C₇₁

C₈

yes

characteristic

C₄⋊C₈×C₇₁

C₂×C₄

yes

—

#102×C₇₁

C₈

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

D₈

yes

complemented

C₂×D₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

C₂×C₈

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

#131

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₃₂

yes

complemented

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₈×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

#443

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₈×D₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₈×D₈

D₇₁

yes

characteristic

#1370

—

D₄⋊C₁₆

—

C₈×C₁₆

—

C₇₁

#12039

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

142

—

C₄×D₈

Dic₇₁

yes

characteristic

C₈×D₄

C₂×D₇₁

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

142

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₇₁

yes

characteristic

D₄⋊C₈

Dic₇₁

yes

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×D₈

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×Dic₇₁

yes

—

M₃₂

—

284

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

142

—

103

C₂×C₁₆

—

142

—

104

#107

C₇₁⋊C₈

yes

characteristic

105

C₄⋊C₈

C₂×Dic₇₁

yes

—

106

#102

C₇₁⋊C₈

yes

—

107

C₂²⋊C₈

—

108

C₄×C₈

C₂×Dic₇₁

yes

characteristic

109

C₄×C₈

C₂×Dic₇₁

yes

characteristic

110

C₄×C₈

C₂²×D₇₁

yes

characteristic

111

C₂×D₄

C₂×C₇₁⋊C₈

yes

—

112

C₂²×C₄

—

113

D₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

119

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

120

C₂×C₈

D₄×D₇₁

yes

characteristic

121

C₂×C₈

#11765864

yes

characteristic

122

C₄⋊C₄

C₂×C₇₁⋊C₈

yes

—

123

C₂²⋊C₄

—

124

C₄²

C₂²×Dic₇₁

yes

characteristic

125

C₁₆

—

284

—

126

C₁₆

—

127

C₁₆

—

142

—

128

C₁₆

—

142

—

129

C₂³

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

D₄×Dic₇₁

yes

characteristic

136

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

137

C₂×C₄

#14556135

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#14556068

yes

complemented

143

C₈

D₈×D₇₁

yes

characteristic

144

C₈

#14556147

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

D₄×C₇₁⋊C₈

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

D₈×Dic₇₁

yes

characteristic

152

C₄

#17986116

yes

characteristic

153

C₄

C₂×#14556068

yes

characteristic

154

C₄

#17986416

yes

characteristic

155

C₂

—

156

C₂

D₈×C₇₁⋊C₈

yes

characteristic

157

C₂

#22590172

yes

characteristic

158

C₂

#22590177

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₈×D₈×C₇₁	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×D₈×C₇₁	C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₈×D₈	D₇₁
hypocenter	C₇₁	#12039
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

subgroup

G/H

center

C₂×C₈

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₈×D₈×C₇₁

C₂

socle

C₂²×C₇₁

C₈×D₄

prime radical

C₂×D₈×C₇₁

C₈

cosocle

C₄×C₈×C₇₁

C₂³

hypercenter

C₈×D₈

D₇₁

hypocenter

C₇₁

#12039

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

N	H ≅ G/N	type
1. #22590172	C₂	semidirect
2. #22586410	C₂	semidirect
3. D₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. C₈×C₇₁⋊C₁₆	C₂	semidirect
34. C₇₁⋊C₁₆	D₈	semidirect
62. C₈×C₇₁	M₃₂	semidirect
85. C₇₁	#12039	semidirect
142. C₈	#14556068	semidirect

H ≅ G/N

type

1. #22590172

C₂

semidirect

2. #22586410

C₂

semidirect

3. D₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. C₈×C₇₁⋊C₁₆

C₂

semidirect

34. C₇₁⋊C₁₆

D₈

semidirect

62. C₈×C₇₁

M₃₂

semidirect

85. C₇₁

#12039

semidirect

142. C₈

#14556068

semidirect

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

subgroup	index
#22590172	2
#22586410	2
D₄⋊(C₇₁⋊C₁₆)	2
C₈×C₇₁⋊C₁₆	2
C₈×D₈×C₇₁	2
#12039	71

subgroup

index

#22590172

#22586410

D₄⋊(C₇₁⋊C₁₆)

C₈×C₇₁⋊C₁₆

C₈×D₈×C₇₁

#12039

Subgroups — #31909169

Conjugacy classes of subgroups

Complements

Maximal subgroups