Group #31909168 — subgroups

Subgroups — #31909168

Order 18176.

242 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590171	C₂	1	2	yes	maximal, complemented
2	C₈×D₅₆₈	C₂	1	1	yes	maximal, characteristic
3	#22586294	C₂	1	2	yes	maximal, complemented
4	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	—	2	2	no	—
7	C₄×D₅₆₈	C₄	1	1	yes	characteristic
8	C₈×D₂₈₄	C₂²	1	2	yes	—
9	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17985752	C₄	1	1	yes	characteristic
13	#17985728	C₄	1	2	yes	—
14	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₅₆₈	C₈	1	1	yes	characteristic
17	C₂×C₈×D₇₁	—	2	2	no	—
18	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
19	#14556067	—	4	4	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
23	#14556062	C₈	1	1	yes	characteristic
24	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
25	#14556050	C₈	1	2	yes	—
26	(C₂×D₇₁)⋊C₈	—	2	2	no	—
27	C₂×C₁₆×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	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
32	C₂×C₄×D₇₁	—	2	2	no	—
33	D₅₆₈	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₈×D₇₁	—	4	2	no	—
36	C₈×D₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
38	(C₂×D₇₁)⋊C₄	—	2	2	no	—
39	C₇₁⋊C₁₆	—	2	2	no	—
40	C₇₁⋊C₁₆	—	2	1	no	—
41	C₇₁⋊C₁₆	D₈	1	2	yes	complemented
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
47	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₇₁	—	4	2	no	—
49	C₂²×D₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	2	no	—
54	D₂₈₄	—	2	2	no	—
55	D₂₈₄	—	2	2	no	—
56	C₄×D₇₁	—	4	2	no	—
57	C₄×D₇₁	—	4	2	no	—
58	C₈×C₇₁	—	2	2	no	—
59	C₈×C₇₁	—	2	1	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	M₃₂	1	2	yes	complemented
62	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
63	C₈×C₇₁	#135	1	1	yes	characteristic
64	C₇₁⋊C₈	—	4	2	no	—
65	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
66	C₂×D₇₁	—	4	2	no	—
67	C₂×D₇₁	—	4	2	no	—
68	C₂×D₇₁	—	4	2	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
71	C₄×C₇₁	#443	1	1	yes	characteristic
72	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
73	C₄×C₇₁	#504	1	1	yes	characteristic
74	Dic₇₁	—	4	2	no	—
75	#12039	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	C₈×D₈	1	1	yes	characteristic
77	C₂×C₇₁	#2169	1	1	yes	characteristic
78	C₂×C₇₁	#2170	1	1	yes	characteristic
79	D₇₁	—	8	2	no	—
80	#2169	—	71	2	no	—
81	C₈×D₈	—	71	1	no	—
82	#1370	D₇₁	1	1	yes	characteristic, complemented
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	—	71	1	no	—
88	C₈×D₄	—	71	2	no	—
89	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
90	C₂²⋊C₁₆	—	142	2	no	—
91	C₄×C₁₆	—	71	1	no	—
92	C₄×C₁₆	—	71	1	no	—
93	#335	—	71	1	no	—
94	D₄⋊C₈	—	71	2	no	—
95	C₈²	C₂×D₇₁	1	1	yes	characteristic
96	C₂×D₈	—	71	1	no	—
97	C₂²×C₈	—	142	2	no	—
98	C₄×D₄	—	71	2	no	—
99	M₃₂	—	284	4	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	71	1	no	—
102	C₂×C₁₆	—	71	1	no	—
103	C₂×C₁₆	—	142	1	no	—
104	#107	—	71	1	no	—
105	C₄⋊C₈	—	71	2	no	—
106	#102	—	71	2	no	—
107	C₂²⋊C₈	—	142	2	no	—
108	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
109	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
110	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	C₂×D₄	—	71	2	no	—
112	C₂²×C₄	—	142	2	no	—
113	D₈	—	142	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	284	2	no	—
116	C₂×C₈	—	284	2	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
119	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
120	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
121	C₂×C₈	#11765866	1	1	yes	characteristic
122	C₄⋊C₄	—	71	2	no	—
123	C₂²⋊C₄	—	142	2	no	—
124	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₁₆	—	4	2	no	—
126	C₁₆	—	71	2	no	—
127	C₁₆	—	142	2	no	—
128	C₁₆	—	142	1	no	—
129	C₂³	—	142	2	no	—
130	D₄	—	142	2	no	—
131	D₄	—	142	2	no	—
132	C₂×C₄	—	284	2	no	—
133	C₂×C₄	—	284	2	no	—
134	C₂×C₄	—	142	2	no	—
135	C₂×C₄	#14556090	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
137	C₂×C₄	#14556136	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	284	2	no	—
142	C₈	#14556067	1	2	yes	complemented
143	C₈	D₈×D₇₁	1	1	yes	characteristic
144	C₈	#14556151	1	1	yes	characteristic
145	C₂²	—	284	2	no	—
146	C₂²	—	284	2	no	—
147	C₂²	—	284	2	no	—
148	C₂²	#17986064	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	284	2	no	—
151	C₄	#17986081	1	1	yes	characteristic
152	C₄	#17986115	1	1	yes	characteristic
153	C₄	C₂×#14556067	1	1	yes	characteristic
154	C₄	#17986417	1	1	yes	characteristic
155	C₂	—	568	2	no	—
156	C₂	#22587134	1	1	yes	characteristic
157	C₂	#22590171	1	1	yes	characteristic
158	C₂	#22590176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590171

C₂

yes

maximal, complemented

C₈×D₅₆₈

C₂

yes

maximal, characteristic

#22586294

C₂

yes

maximal, complemented

C₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇₁×#1370

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

—

C₄×D₅₆₈

C₄

yes

characteristic

C₈×D₂₈₄

C₂²

yes

—

(C₂×D₇₁)⋊C₁₆

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985752

C₄

yes

characteristic

#17985728

C₄

yes

—

C₄⋊C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×D₅₆₈

C₈

yes

characteristic

C₂×C₈×D₇₁

—

C₄×D₂₈₄

C₂×C₄

yes

—

#14556067

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

#14556062

C₈

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

#14556050

C₈

yes

—

(C₂×D₇₁)⋊C₈

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×D₂₈₄

C₂×C₈

yes

—

C₂×C₄×D₇₁

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

C₂×C₈

yes

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

D₈

yes

complemented

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

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

#131

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₃₂

yes

complemented

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₈×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

#443

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

#504

yes

characteristic

Dic₇₁

—

#12039

—

maximal, Sylow

C₂×C₇₁

C₈×D₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#2169

—

C₈×D₈

—

#1370

D₇₁

yes

characteristic, complemented

D₄⋊C₁₆

—

C₈×C₁₆

—

C₇₁

#12039

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

142

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

C₂×D₇₁

yes

—

C₂²⋊C₁₆

—

142

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

142

—

C₄×D₄

—

M₃₂

—

284

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

142

—

104

#107

—

105

C₄⋊C₈

—

106

#102

—

107

C₂²⋊C₈

—

142

—

108

C₄×C₈

C₄×D₇₁

yes

characteristic

109

C₄×C₈

C₄×D₇₁

yes

characteristic

110

C₄×C₈

C₂²×D₇₁

yes

characteristic

111

C₂×D₄

—

112

C₂²×C₄

—

142

—

113

D₈

—

142

—

114

C₂×C₈

—

115

C₂×C₈

—

284

—

116

C₂×C₈

—

284

—

117

C₂×C₈

—

142

—

118

C₂×C₈

C₈×D₇₁

yes

characteristic

119

C₂×C₈

C₈×D₇₁

yes

characteristic

120

C₂×C₈

D₄×D₇₁

yes

characteristic

121

C₂×C₈

#11765866

yes

characteristic

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

142

—

124

C₄²

C₂×C₄×D₇₁

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

—

142

—

128

C₁₆

—

142

—

129

C₂³

—

142

—

130

D₄

—

142

—

131

D₄

—

142

—

132

C₂×C₄

—

284

—

133

C₂×C₄

—

284

—

134

C₂×C₄

—

142

—

135

C₂×C₄

#14556090

yes

characteristic

136

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

137

C₂×C₄

#14556136

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

284

—

142

C₈

#14556067

yes

complemented

143

C₈

D₈×D₇₁

yes

characteristic

144

C₈

#14556151

yes

characteristic

145

C₂²

—

284

—

146

C₂²

—

284

—

147

C₂²

—

284

—

148

C₂²

#17986064

yes

characteristic

149

C₄

—

150

C₄

—

284

—

151

C₄

#17986081

yes

characteristic

152

C₄

#17986115

yes

characteristic

153

C₄

C₂×#14556067

yes

characteristic

154

C₄

#17986417

yes

characteristic

155

C₂

—

568

—

156

C₂

#22587134

yes

characteristic

157

C₂

#22590171

yes

characteristic

158

C₂

#22590176

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1370	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×D₅₆₈	C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1370	D₇₁
hypocenter	C₇₁	#12039
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1370

C₂

socle

C₂²×C₇₁

C₈×D₄

prime radical

C₂×D₅₆₈

C₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
1. #22590171	C₂	semidirect
3. #22586294	C₂	semidirect
4. C₈×C₇₁⋊C₁₆	C₂	semidirect
5. C₇₁×#1370	C₂	semidirect
41. C₇₁⋊C₁₆	D₈	semidirect
61. C₈×C₇₁	M₃₂	semidirect
82. #1370	D₇₁	semidirect
85. C₇₁	#12039	semidirect
142. C₈	#14556067	semidirect

H ≅ G/N

type

1. #22590171

C₂

semidirect

3. #22586294

C₂

semidirect

4. C₈×C₇₁⋊C₁₆

C₂

semidirect

5. C₇₁×#1370

C₂

semidirect

41. C₇₁⋊C₁₆

D₈

semidirect

61. C₈×C₇₁

M₃₂

semidirect

82. #1370

D₇₁

semidirect

85. C₇₁

#12039

semidirect

142. C₈

#14556067

semidirect

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

subgroup	index
#22590171	2
C₈×D₅₆₈	2
#22586294	2
C₈×C₇₁⋊C₁₆	2
C₇₁×#1370	2
#12039	71

subgroup

index

#22590171

C₈×D₅₆₈

#22586294

C₈×C₇₁⋊C₁₆

C₇₁×#1370

#12039

Subgroups — #31909168

Conjugacy classes of subgroups

Complements

Maximal subgroups