Group #16847274 — subgroups

Subgroups — #16847274

Order 3968.

214 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14520919	C₂	1	1	yes	maximal, characteristic
2	#14520811	C₂	1	2	yes	maximal
3	#14520811	C₂	1	2	yes	maximal
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#389	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	—	2	1	no	—
11	C₂×C₄⋊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×C₄×C₃₁	—	2	1	no	—
16	C₂²×C₄×C₃₁	—	2	1	no	—
17	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	—	2	2	no	—
21	C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄⋊C₄×C₃₁	—	4	2	no	—
24	C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
25	C₄²×C₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	—	2	4	no	—
27	C₄⋊Dic₃₁	—	4	2	no	—
28	C₄⋊Dic₃₁	—	4	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	1	no	—
34	C₂×C₄×C₃₁	—	2	1	no	—
35	C₂×C₄×C₃₁	—	2	1	no	—
36	C₂×C₄×C₃₁	—	2	1	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	4	1	no	—
39	C₂×C₄×C₃₁	—	4	1	no	—
40	C₂×C₄×C₃₁	—	4	1	no	—
41	C₂×C₄×C₃₁	—	4	1	no	—
42	C₂×C₄×C₃₁	—	4	1	no	—
43	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₃₁	#41	1	2	yes	—
48	C₂×C₄×C₃₁	#41	1	2	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	#2100	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₃₁	#123	1	1	yes	characteristic
56	C₂²×C₃₁	#124	1	1	yes	characteristic
57	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	4	2	no	—
64	C₄×C₃₁	—	8	1	no	—
65	C₄×C₃₁	#125	1	2	yes	—
66	C₄×C₃₁	#126	1	2	yes	—
67	Dic₃₁	—	8	2	no	—
68	#389	D₃₁	1	1	yes	characteristic
69	#384	—	31	1	no	—
70	#340	—	31	2	no	—
71	#340	—	31	2	no	—
72	#336	—	31	1	no	—
73	C₂×C₃₁	#400	1	1	yes	characteristic
74	C₂×C₃₁	#476	1	1	yes	characteristic
75	C₂×C₃₁	#477	1	1	yes	characteristic
76	C₂×C₃₁	#481	1	1	yes	characteristic
77	C₂×C₃₁	#484	1	1	yes	characteristic
78	C₂×C₃₁	#487	1	1	yes	characteristic
79	C₂×C₃₁	#490	1	1	yes	characteristic
80	C₂²×C₈	—	31	2	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	31	2	no	—
83	C₂×C₄⋊C₄	—	62	1	no	—
84	C₂×C₄⋊C₄	—	62	1	no	—
85	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
88	#95	—	2	1	no	—
89	C₃₁	#2100	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	62	2	no	—
93	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
96	C₂×C₈	—	62	2	no	—
97	C₂×C₈	—	62	2	no	—
98	C₂×C₈	—	62	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄⋊C₄	—	124	2	no	—
104	C₄⋊C₄	—	124	2	no	—
105	C₄⋊C₄	Dic₆₂	1	2	yes	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	124	2	no	—
119	C₂×C₄	—	124	2	no	—
120	C₂×C₄	—	124	2	no	—
121	C₂×C₄	#92480	1	2	yes	—
122	C₂×C₄	#92482	1	2	yes	—
123	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
124	C₂×C₄	#92495	1	1	yes	characteristic
125	C₂×C₄	#92497	1	2	yes	—
126	C₂×C₄	#92499	1	2	yes	—
127	C₈	—	124	2	no	—
128	C₂²	#11758501	1	1	yes	characteristic
129	C₂²	C₂×#92480	1	1	yes	characteristic
130	C₂²	#11758512	1	1	yes	characteristic
131	C₂²	C₂×#92482	1	1	yes	characteristic
132	C₂²	#11758524	1	1	yes	characteristic
133	C₂²	#11758526	1	1	yes	characteristic
134	C₂²	#11758526	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	8	1	no	—
139	C₄	—	248	2	no	—
140	C₄	#11758467	1	2	yes	—
141	C₄	#11758475	1	2	yes	—
142	C₂	#14520988	1	1	yes	characteristic
143	C₂	#14521339	1	1	yes	characteristic
144	C₂	#14521344	1	1	yes	characteristic
145	C₂	#14521365	1	1	yes	characteristic
146	C₂	#14521382	1	1	yes	characteristic
147	C₂	#14521401	1	1	yes	characteristic
148	C₂	#14521415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14520919

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

#14520811

C₂

yes

maximal

#14520804

C₂

yes

maximal, characteristic

C₃₁×#389

C₂

yes

maximal, characteristic

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

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×#95

—

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

Q₈

yes

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

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

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#2100

—

maximal, Sylow

C₂²×C₃₁

C₂²⋊Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#125

yes

—

C₄×C₃₁

#126

yes

—

Dic₃₁

—

#389

D₃₁

yes

—

—

—

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

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×D₃₁

yes

characteristic

#95

—

C₃₁

#2100

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

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

124

—

104

C₄⋊C₄

—

124

—

105

C₄⋊C₄

Dic₆₂

yes

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

—

124

—

120

C₂×C₄

—

124

—

121

C₂×C₄

#92480

yes

—

122

C₂×C₄

#92482

yes

—

123

C₂×C₄

C₂×Dic₆₂

yes

characteristic

124

C₂×C₄

#92495

yes

characteristic

125

C₂×C₄

#92497

yes

—

126

C₂×C₄

#92499

yes

—

127

C₈

—

124

—

128

C₂²

#11758501

yes

characteristic

129

C₂²

C₂×#92480

yes

characteristic

130

C₂²

#11758512

yes

characteristic

131

C₂²

C₂×#92482

yes

characteristic

132

C₂²

#11758524

yes

characteristic

133

C₂²

#11758526

yes

characteristic

134

C₂²

#11758526

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

248

—

140

C₄

#11758467

yes

—

141

C₄

#11758475

yes

—

142

C₂

#14520988

yes

characteristic

143

C₂

#14521339

yes

characteristic

144

C₂

#14521344

yes

characteristic

145

C₂

#14521365

yes

characteristic

146

C₂

#14521382

yes

characteristic

147

C₂

#14521401

yes

characteristic

148

C₂

#14521415

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#389	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#389	D₃₁
hypocenter	C₃₁	#2100
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#389

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
89. C₃₁	#2100	semidirect

H ≅ G/N

type

89. C₃₁

#2100

semidirect

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

subgroup	index
#14520919	2
#14520811	2
#14520804	2
C₃₁×#389	2
#2100	31

subgroup

index

C₃₁×#389

Subgroups — #16847274

Conjugacy classes of subgroups

Complements

Maximal subgroups