Group #16845228 — subgroups

Subgroups — #16845228

Order 3968.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14521452	C₂	1	1	yes	maximal, characteristic
2	#14521260	C₂	1	1	yes	maximal, characteristic
3	#14521256	C₂	1	1	yes	maximal, characteristic
4	#14521176	C₂	1	1	yes	maximal, characteristic
5	#14520799	C₂	1	1	yes	maximal, characteristic
6	#14520794	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#332	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#92482	—	2	1	no	—
10	C₂×Dic₁₂₄	—	2	1	no	—
11	#11758446	C₂²	1	1	yes	characteristic
12	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#11758423	—	2	1	no	—
16	#11758404	—	2	1	no	—
17	#11758403	—	2	1	no	—
18	Dic₃₁⋊C₈	—	2	1	no	—
19	#11758395	—	2	1	no	—
20	#11758395	—	2	1	no	—
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	—	2	1	no	—
24	C₂×Dic₆₂	—	2	1	no	—
25	C₂×Dic₆₂	D₄	1	1	yes	characteristic
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₂×C₈×C₃₁	—	2	1	no	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	#92482	—	4	2	no	—
32	C₄⋊Dic₃₁	—	2	1	no	—
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
35	Dic₃₁⋊C₄	—	2	1	no	—
36	C₄×Dic₃₁	—	2	1	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
40	Dic₁₂₄	—	4	2	no	—
41	Q₈×C₃₁	—	4	1	no	—
42	Q₈×C₃₁	—	4	1	no	—
43	C₂×C₄×C₃₁	—	4	1	no	—
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
47	C₂×Dic₃₁	—	2	1	no	—
48	C₂×Dic₃₁	—	2	1	no	—
49	C₂×Dic₃₁	—	4	1	no	—
50	Dic₆₂	—	2	1	no	—
51	Dic₆₂	—	2	1	no	—
52	Dic₆₂	—	4	1	no	—
53	Dic₆₂	—	4	1	no	—
54	C₈×C₃₁	—	4	1	no	—
55	C₈×C₃₁	—	4	1	no	—
56	C₃₁⋊C₈	—	2	1	no	—
57	C₃₁⋊C₈	—	4	1	no	—
58	C₃₁⋊C₈	Q₁₆	1	2	yes	complemented
59	#1694	—	31	1	no	maximal, Sylow
60	C₂²×C₃₁	#121	1	1	yes	characteristic
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	8	1	no	—
63	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₄×C₃₁	#137	1	1	yes	characteristic
67	Dic₃₁	—	4	1	no	—
68	Dic₃₁	—	4	1	no	—
69	Dic₃₁	—	8	1	no	—
70	#500	—	31	1	no	—
71	#462	—	31	1	no	—
72	#462	—	31	1	no	—
73	#446	—	31	1	no	—
74	#334	—	31	1	no	—
75	#332	—	31	1	no	—
76	#332	D₃₁	1	1	yes	characteristic
77	C₂×C₃₁	#462	1	1	yes	characteristic
78	C₂×C₃₁	#469	1	1	yes	characteristic
79	C₂×C₃₁	#473	1	1	yes	characteristic
80	C₂×Q₁₆	—	62	1	no	—
81	C₂×Q₁₆	—	62	1	no	—
82	C₄⋊Q₈	—	31	1	no	—
83	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
84	C₄×Q₈	—	31	1	no	—
85	#107	—	62	1	no	—
86	#107	—	62	1	no	—
87	C₄⋊C₈	—	31	1	no	—
88	C₄⋊C₈	—	62	1	no	—
89	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
90	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
91	#103	—	62	1	no	—
92	#103	—	62	1	no	—
93	#97	—	62	1	no	—
94	C₄×C₈	—	31	1	no	—
95	C₃₁	#1694	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	31	1	no	—
98	C₂×Q₈	—	62	1	no	—
99	Q₁₆	—	124	2	no	—
100	Q₁₆	—	124	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	31	1	no	—
104	C₂×C₈	—	31	1	no	—
105	C₂×C₈	—	62	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	31	1	no	—
109	C₄⋊C₄	—	62	1	no	—
110	C₄⋊C₄	—	62	1	no	—
111	C₄⋊C₄	—	62	1	no	—
112	C₄²	—	62	1	no	—
113	C₄²	C₂²×D₃₁	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	62	1	no	—
117	Q₈	—	62	1	no	—
118	Q₈	—	124	1	no	—
119	Q₈	—	124	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	62	1	no	—
122	C₂×C₄	—	62	1	no	—
123	C₂×C₄	—	124	1	no	—
124	C₂×C₄	#92495	1	1	yes	characteristic
125	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
126	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	31	2	no	—
130	C₈	—	62	1	no	—
131	C₈	—	124	1	no	—
132	C₂²	#11758461	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	124	1	no	—
136	C₄	—	124	1	no	—
137	C₄	—	248	1	no	—
138	C₄	#11758488	1	1	yes	characteristic
139	C₄	#11758491	1	1	yes	characteristic
140	C₄	#11758492	1	1	yes	characteristic
141	C₄	Q₁₆×D₃₁	1	1	yes	characteristic
142	C₂	#14521258	1	1	yes	characteristic
143	C₂	#14521301	1	1	yes	characteristic
144	C₂	#14521323	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521452

C₂

yes

maximal, characteristic

#14521260

C₂

yes

maximal, characteristic

#14521256

C₂

yes

maximal, characteristic

#14521176

C₂

yes

maximal, characteristic

#14520799

C₂

yes

maximal, characteristic

#14520794

C₂

yes

maximal, characteristic

C₃₁×#332

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#92482

—

C₂×Dic₁₂₄

—

#11758446

C₂²

yes

characteristic

C₄×Dic₆₂

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₂²

yes

—

—

—

Dic₃₁⋊C₈

—

#11758395

—

#11758395

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

D₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

characteristic

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

Dic₁₂₄

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

Q₁₆

yes

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

—

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

characteristic

C₄⋊C₈

C₂×D₃₁

yes

—

—

—

C₄×C₈

—

C₃₁

#1694

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

124

—

100

Q₁₆

—

124

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂²×D₃₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

124

—

119

Q₈

—

124

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

—

124

C₂×C₄

#92495

yes

characteristic

125

C₂×C₄

D₄×D₃₁

yes

characteristic

126

C₂×C₄

D₄×D₃₁

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

124

—

132

C₂²

#11758461

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

124

—

136

C₄

—

124

—

137

C₄

—

248

—

138

C₄

#11758488

yes

characteristic

139

C₄

#11758491

yes

characteristic

140

C₄

#11758492

yes

characteristic

141

C₄

Q₁₆×D₃₁

yes

characteristic

142

C₂

#14521258

yes

characteristic

143

C₂

#14521301

yes

characteristic

144

C₂

#14521323

yes

characteristic

	subgroup	G/H
center	C₂²	#11758461
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#332	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#332	D₃₁
hypocenter	C₃₁	#1694
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#11758461

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#332

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. C₃₁⋊C₈	Q₁₆	semidirect
95. C₃₁	#1694	semidirect

H ≅ G/N

type

58. C₃₁⋊C₈

Q₁₆

semidirect

95. C₃₁

#1694

semidirect

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

subgroup	index
#14521452	2
#14521260	2
#14521256	2
#14521176	2
#14520799	2
#14520794	2
C₃₁×#332	2
#1694	31

subgroup

index

C₃₁×#332

Subgroups — #16845228

Conjugacy classes of subgroups

Complements

Maximal subgroups