Group #16846669 — subgroups

Subgroups — #16846669

Order 3968.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	2	yes	maximal
4	#14520811	C₂	1	2	yes	maximal
5	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#125	—	2	2	no	—
7	C₃₁×#125	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
9	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Dic₃₁	—	2	1	no	—
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₃₁⋊#95	—	2	1	no	—
17	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
18	C₂²×C₄×C₃₁	—	2	1	no	—
19	C₂²×C₄×C₃₁	—	2	1	no	—
20	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₄²×C₃₁	D₄	1	2	yes	—
32	C₄⋊Dic₃₁	—	2	2	no	—
33	C₄⋊Dic₃₁	Q₈	1	4	yes	—
34	C₄×Dic₃₁	—	4	2	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂×C₃₁⋊C₈	—	2	4	no	—
37	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	1	no	—
40	C₂×C₄×C₃₁	—	2	1	no	—
41	C₂×C₄×C₃₁	—	2	2	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	—	4	2	no	—
44	C₂×C₄×C₃₁	—	4	1	no	—
45	C₂×C₄×C₃₁	—	4	1	no	—
46	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₃₁	#41	1	2	yes	—
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	—	4	2	no	—
54	C₂×Dic₃₁	—	4	2	no	—
55	C₃₁⋊C₈	—	4	2	no	—
56	#1989	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
59	C₂²×C₃₁	#135	1	2	yes	—
60	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₃₁	#137	1	1	yes	characteristic
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	2	2	no	—
64	C₄×C₃₁	—	4	2	no	—
65	C₄×C₃₁	—	4	2	no	—
66	C₄×C₃₁	C₄×Q₈	1	2	yes	—
67	C₄×C₃₁	#124	1	1	yes	characteristic
68	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₃₁	—	4	4	no	—
70	C₂×#125	D₃₁	1	1	yes	characteristic
71	C₂×C₄⋊C₈	—	31	1	no	—
72	C₄×C₄⋊C₄	—	31	1	no	—
73	#340	—	31	2	no	—
74	#340	—	31	2	no	—
75	C₂×C₃₁	#391	1	1	yes	characteristic
76	C₂×C₃₁	#416	1	1	yes	characteristic
77	C₂×C₃₁	#417	1	1	yes	characteristic
78	C₂×C₃₁	#478	1	2	yes	—
79	C₂×C₃₁	#479	1	2	yes	—
80	C₂²×C₈	—	31	2	no	—
81	#125	—	2	2	no	—
82	#125	Dic₃₁	1	4	yes	complemented
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	31	2	no	—
86	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
88	C₂×C₄²	—	62	1	no	—
89	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
90	C₄⋊C₈	—	62	2	no	—
91	#95	—	62	1	no	—
92	C₃₁	#1989	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	62	2	no	—
96	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	C₂×C₈	—	62	4	no	—
100	C₂×C₈	—	62	2	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	31	4	no	—
106	C₄⋊C₄	—	62	2	no	—
107	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
108	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
109	C₄²	—	124	2	no	—
110	C₄²	C₂×Dic₃₁	1	2	yes	—
111	C₄²	C₂²⋊D₃₁	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	62	4	no	—
122	C₂×C₄	—	124	2	no	—
123	C₂×C₄	—	124	2	no	—
124	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
125	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
126	C₂×C₄	Q₈×D₃₁	1	2	yes	—
127	C₂×C₄	#92499	1	2	yes	—
128	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
130	C₈	—	124	2	no	—
131	C₂²	#11758523	1	2	yes	—
132	C₂²	#11758526	1	1	yes	characteristic
133	C₂²	#11758527	1	2	yes	—
134	C₂²	#11758528	1	1	yes	characteristic
135	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	124	4	no	—
141	C₄	#11758521	1	1	yes	characteristic
142	C₄	Q₈×Dic₃₁	1	2	yes	—
143	C₄	#11758524	1	1	yes	characteristic
144	C₂	#14520956	1	1	yes	characteristic
145	C₂	#14521045	1	1	yes	characteristic
146	C₂	#14521050	1	1	yes	characteristic
147	C₂	#14521346	1	2	yes	—
148	C₂	#14521353	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

#14520811

C₂

yes

maximal

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₃₁×#125

—

C₃₁×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁⋊#95

—

C₄⋊(C₃₁⋊C₈)

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

Q₈

yes

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1989

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

C₂×#125

D₃₁

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#479

yes

—

C₂²×C₈

—

#125

—

#125

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#95

—

C₃₁

#1989

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₃₁

yes

—

108

C₄⋊C₄

C₂×Dic₃₁

yes

—

109

C₄²

—

124

—

110

C₄²

C₂×Dic₃₁

yes

—

111

C₄²

C₂²⋊D₃₁

yes

—

112

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

124

—

123

C₂×C₄

—

124

—

124

C₂×C₄

C₂²⋊Dic₃₁

yes

—

125

C₂×C₄

C₂²⋊Dic₃₁

yes

—

126

C₂×C₄

Q₈×D₃₁

yes

—

127

C₂×C₄

#92499

yes

—

128

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

130

C₈

—

124

—

131

C₂²

#11758523

yes

—

132

C₂²

#11758526

yes

characteristic

133

C₂²

#11758527

yes

—

134

C₂²

#11758528

yes

characteristic

135

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

124

—

141

C₄

#11758521

yes

characteristic

142

C₄

Q₈×Dic₃₁

yes

—

143

C₄

#11758524

yes

characteristic

144

C₂

#14520956

yes

characteristic

145

C₂

#14521045

yes

characteristic

146

C₂

#14521050

yes

characteristic

147

C₂

#14521346

yes

—

148

C₂

#14521353

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#1989
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
7. C₃₁×#125	C₄	semidirect
82. #125	Dic₃₁	semidirect
92. C₃₁	#1989	semidirect

H ≅ G/N

type

7. C₃₁×#125

C₄

semidirect

82. #125

Dic₃₁

semidirect

92. C₃₁

#1989

semidirect

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

subgroup	index
C₂×C₄⋊(C₃₁⋊C₈)	2
C₄×C₄⋊Dic₃₁	2
#14520811	2
C₂×C₃₁×#125	2
#1989	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₄×C₄⋊Dic₃₁

#14520811

C₂×C₃₁×#125

#1989

Subgroups — #16846669

Conjugacy classes of subgroups

Complements

Maximal subgroups