Group #16845801 — subgroups

Subgroups — #16845801

Order 3968.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14520909	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	—	2	1	no	—
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	C₃₁⋊#95	—	2	1	no	—
15	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
16	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	—	2	4	no	—
18	C₂²×C₄×C₃₁	—	2	4	no	—
19	C₂²×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₄×C₃₁	—	4	4	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	D₄	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₁	—	2	2	no	—
26	C₄⋊C₄×C₃₁	—	2	4	no	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²⋊Dic₃₁	—	4	2	no	—
30	C₄⋊Dic₃₁	—	2	4	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂×C₃₁⋊C₈	—	2	4	no	—
33	C₂³×C₃₁	—	2	2	no	—
34	C₂³×C₃₁	—	2	2	no	—
35	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₃₁	SD₁₆	1	2	yes	—
37	C₂³×C₃₁	Q₁₆	1	2	yes	—
38	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	4	8	no	—
44	C₂×C₄×C₃₁	—	4	4	no	—
45	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#41	1	4	yes	—
49	C₂×Dic₃₁	—	4	2	no	—
50	C₂×Dic₃₁	—	4	2	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₃₁⋊C₈	—	4	2	no	—
53	#1824	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	—	2	4	no	—
57	C₂²×C₃₁	—	2	4	no	—
58	C₂²×C₃₁	#103	1	4	yes	—
59	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₁	#123	1	2	yes	—
61	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₃₁	—	2	2	no	—
65	C₄×C₃₁	—	4	4	no	—
66	C₄×C₃₁	#117	1	2	yes	—
67	C₄×C₃₁	#123	1	2	yes	—
68	Dic₃₁	—	8	2	no	—
69	C₂²×C₄⋊C₄	D₃₁	1	1	yes	characteristic
70	C₂×C₂²⋊C₈	—	31	1	no	—
71	#380	—	31	1	no	—
72	#340	—	31	4	no	—
73	C₂×C₃₁	—	2	4	no	—
74	C₂×C₃₁	#381	1	1	yes	characteristic
75	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
76	C₂×C₃₁	#418	1	1	yes	characteristic
77	C₂×C₃₁	#481	1	2	yes	—
78	C₂×C₃₁	#484	1	2	yes	—
79	C₂³×C₄	—	2	1	no	—
80	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
81	C₂²×C₈	—	31	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	—	31	2	no	—
85	C₂×C₄⋊C₄	Dic₃₁	1	4	yes	complemented
86	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
87	C₂×C₂²⋊C₄	—	62	1	no	—
88	C₂²⋊C₈	—	62	2	no	—
89	#95	—	62	1	no	—
90	C₃₁	#1824	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
101	C₂×C₈	—	62	4	no	—
102	C₂×C₈	—	62	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	62	4	no	—
106	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
107	C₂²⋊C₄	—	124	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#92481	1	2	yes	—
111	C₂³	#92482	1	2	yes	—
112	C₂³	C₂²⋊Dic₃₁	1	2	yes	—
113	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	124	2	no	—
121	C₂×C₄	—	124	2	no	—
122	C₂×C₄	—	124	2	no	—
123	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
124	C₂×C₄	#92495	1	4	yes	—
125	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
127	C₈	—	124	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#11758423	1	4	yes	—
133	C₂²	#11758506	1	2	yes	—
134	C₂²	#11758509	1	2	yes	—
135	C₂²	C₂×#92481	1	1	yes	characteristic
136	C₂²	C₂×#92482	1	1	yes	characteristic
137	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	248	2	no	—
141	C₄	#11758503	1	2	yes	—
142	C₄	#11758506	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#14520912	1	1	yes	characteristic
145	C₂	C₂×#11758423	1	1	yes	characteristic
146	C₂	#14521054	1	1	yes	characteristic
147	C₂	#14521367	1	2	yes	—
148	C₂	#14521384	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

#14520909

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

—

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁⋊#95

—

C₂²⋊(C₃₁⋊C₈)

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

—

C₂³×C₃₁

SD₁₆

yes

—

C₂³×C₃₁

Q₁₆

yes

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1824

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#103

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#123

yes

—

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#123

yes

—

Dic₃₁

—

C₂²×C₄⋊C₄

D₃₁

yes

characteristic

C₂×C₂²⋊C₈

—

#380

—

#340

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#484

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₃₁

#1824

yes

characteristic, complemented, Sylow

C₂⁴

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

—

100

C₂²×C₄

C₂²×D₃₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×Dic₃₁

yes

—

107

C₂²⋊C₄

—

124

—

108

C₂³

—

109

C₂³

—

110

C₂³

#92481

yes

—

111

C₂³

#92482

yes

—

112

C₂³

C₂²⋊Dic₃₁

yes

—

113

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

124

—

121

C₂×C₄

—

124

—

122

C₂×C₄

—

124

—

123

C₂×C₄

C₂²⋊Dic₃₁

yes

—

124

C₂×C₄

#92495

yes

—

125

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

127

C₈

—

124

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#11758423

yes

—

133

C₂²

#11758506

yes

—

134

C₂²

#11758509

yes

—

135

C₂²

C₂×#92481

yes

characteristic

136

C₂²

C₂×#92482

yes

characteristic

137

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

248

—

141

C₄

#11758503

yes

—

142

C₄

#11758506

yes

—

143

C₂

—

144

C₂

#14520912

yes

characteristic

145

C₂

C₂×#11758423

yes

characteristic

146

C₂

#14521054

yes

characteristic

147

C₂

#14521367

yes

—

148

C₂

#14521384

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×C₄⋊C₄×C₃₁	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₃₁
hypocenter	C₃₁	#1824
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂²×C₄⋊C₄×C₃₁

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂⁴×C₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂²×C₄⋊C₄

D₃₁

hypocenter

C₃₁

#1824

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #14520811	C₂	semidirect
9. C₂×C₄⋊C₄×C₃₁	C₄	semidirect
85. C₂×C₄⋊C₄	Dic₃₁	semidirect
90. C₃₁	#1824	semidirect

H ≅ G/N

type

3. #14520811

C₂

semidirect

9. C₂×C₄⋊C₄×C₃₁

C₄

semidirect

85. C₂×C₄⋊C₄

Dic₃₁

semidirect

90. C₃₁

#1824

semidirect

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

subgroup	index
C₂×C₂²⋊(C₃₁⋊C₈)	2
#14520909	2
#14520811	2
C₂²×C₄⋊C₄×C₃₁	2
#1824	31

subgroup

index

C₂×C₂²⋊(C₃₁⋊C₈)

#14520909

#14520811

C₂²×C₄⋊C₄×C₃₁

#1824

Subgroups — #16845801

Conjugacy classes of subgroups

Complements

Maximal subgroups