Group #16847273 — subgroups

Subgroups — #16847273

Order 3968.

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

normal subgroups only

This group has 214 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	#14520947	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×#384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
7	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₁⋊C₄	—	2	1	no	—
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	C₃₁⋊#95	—	2	1	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₁	—	2	1	no	—
20	C₂²×Dic₃₁	—	2	1	no	—
21	C₄⋊C₄×C₃₁	—	2	4	no	—
22	C₄⋊C₄×C₃₁	—	4	2	no	—
23	C₄⋊C₄×C₃₁	—	4	2	no	—
24	C₄²×C₃₁	—	2	2	no	—
25	C₄⋊Dic₃₁	—	2	1	no	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	Q₈	1	2	yes	—
28	Dic₃₁⋊C₄	—	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	—	4	2	no	—
38	C₂×C₄×C₃₁	—	4	2	no	—
39	C₂×C₄×C₃₁	—	4	2	no	—
40	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₃₁	#41	1	2	yes	—
44	C₂×C₄×C₃₁	#41	1	2	yes	—
45	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
46	C₂×Dic₃₁	—	2	2	no	—
47	C₂×Dic₃₁	—	4	1	no	—
48	C₂×Dic₃₁	—	4	1	no	—
49	C₂×Dic₃₁	—	4	1	no	—
50	C₂×Dic₃₁	—	4	1	no	—
51	C₂×Dic₃₁	—	4	1	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	—	8	2	no	—
64	C₄×C₃₁	#125	1	2	yes	—
65	C₄×C₃₁	#126	1	2	yes	—
66	Dic₃₁	—	4	2	no	—
67	Dic₃₁	—	8	1	no	—
68	#389	—	31	1	no	—
69	#384	D₃₁	1	1	yes	characteristic
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	—	2	1	no	—
83	C₂×C₄⋊C₄	—	31	1	no	—
84	C₂×C₄⋊C₄	—	31	1	no	—
85	C₂×C₄⋊C₄	—	62	1	no	—
86	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
87	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
88	#95	—	62	1	no	—
89	C₃₁	#2100	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	62	1	no	—
92	C₂²×C₄	—	62	1	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	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	31	2	no	—
103	C₄⋊C₄	—	62	1	no	—
104	C₄⋊C₄	—	62	2	no	—
105	C₄⋊C₄	—	124	2	no	—
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	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	62	2	no	—
116	C₂×C₄	—	124	1	no	—
117	C₂×C₄	—	124	1	no	—
118	C₂×C₄	—	124	1	no	—
119	C₂×C₄	—	124	1	no	—
120	C₂×C₄	—	124	1	no	—
121	C₂×C₄	#92481	1	2	yes	—
122	C₂×C₄	#92482	1	2	yes	—
123	C₂×C₄	#92495	1	2	yes	—
124	C₂×C₄	#92497	1	2	yes	—
125	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
126	C₂×C₄	#92499	1	1	yes	characteristic
127	C₈	—	124	2	no	—
128	C₂²	#11758506	1	1	yes	characteristic
129	C₂²	#11758509	1	1	yes	characteristic
130	C₂²	#11758509	1	1	yes	characteristic
131	C₂²	#11758513	1	1	yes	characteristic
132	C₂²	C₂×#92481	1	1	yes	characteristic
133	C₂²	C₂×#92482	1	1	yes	characteristic
134	C₂²	#11758523	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	124	2	no	—
139	C₄	—	248	1	no	—
140	C₄	#11758466	1	2	yes	—
141	C₄	#11758475	1	2	yes	—
142	C₂	#14520987	1	1	yes	characteristic
143	C₂	#14521336	1	1	yes	characteristic
144	C₂	#14521341	1	1	yes	characteristic
145	C₂	#14521367	1	1	yes	characteristic
146	C₂	#14521384	1	1	yes	characteristic
147	C₂	#14521399	1	1	yes	characteristic
148	C₂	#14521413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14520947

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

#14520811

C₂

yes

maximal

#14520804

C₂

yes

maximal, characteristic

C₃₁×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×Dic₃₁⋊C₄

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁⋊#95

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

Q₈

yes

—

Dic₃₁⋊C₄

—

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

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

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

#125

yes

—

C₄×C₃₁

#126

yes

—

Dic₃₁

—

Dic₃₁

—

#389

—

#384

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×C₄⋊C₄

C₂×D₃₁

yes

—

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

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

124

—

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

—

124

—

117

C₂×C₄

—

124

—

118

C₂×C₄

—

124

—

119

C₂×C₄

—

124

—

120

C₂×C₄

—

124

—

121

C₂×C₄

#92481

yes

—

122

C₂×C₄

#92482

yes

—

123

C₂×C₄

#92495

yes

—

124

C₂×C₄

#92497

yes

—

125

C₂×C₄

Q₈×D₃₁

yes

characteristic

126

C₂×C₄

#92499

yes

characteristic

127

C₈

—

124

—

128

C₂²

#11758506

yes

characteristic

129

C₂²

#11758509

yes

characteristic

130

C₂²

#11758509

yes

characteristic

131

C₂²

#11758513

yes

characteristic

132

C₂²

C₂×#92481

yes

characteristic

133

C₂²

C₂×#92482

yes

characteristic

134

C₂²

#11758523

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

124

—

139

C₄

—

248

—

140

C₄

#11758466

yes

—

141

C₄

#11758475

yes

—

142

C₂

#14520987

yes

characteristic

143

C₂

#14521336

yes

characteristic

144

C₂

#14521341

yes

characteristic

145

C₂

#14521367

yes

characteristic

146

C₂

#14521384

yes

characteristic

147

C₂

#14521399

yes

characteristic

148

C₂

#14521413

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#384	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#384	D₃₁
hypocenter	C₃₁	#2100
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#384

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

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
#14520947	2
#14520811	2
#14520804	2
C₃₁×#384	2
#2100	31

subgroup

index

C₃₁×#384

Subgroups — #16847273

Conjugacy classes of subgroups

Complements

Maximal subgroups