Group #16847760 — subgroups

Subgroups — #16847760

Order 3968.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₂₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758417	C₂	1	1	yes	maximal, characteristic
3	#14521427	C₂	1	1	yes	maximal, characteristic
4	#14521143	C₂	1	1	yes	maximal, characteristic
5	Dic₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#14520852	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	#11758521	—	2	1	no	—
10	C₂×#92482	—	2	1	no	—
11	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
13	#11758464	—	2	1	no	—
14	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#107	C₂²	1	1	yes	characteristic
16	#11758417	—	2	2	no	—
17	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
19	#11758398	—	2	1	no	—
20	Dic₂₄₈	—	2	2	no	—
21	C₂×Q₈×C₃₁	—	2	1	no	—
22	C₂×Dic₆₂	—	2	1	no	—
23	C₂×Dic₆₂	—	2	1	no	—
24	Q₁₆×C₃₁	—	2	1	no	—
25	Q₁₆×C₃₁	—	2	1	no	—
26	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₃₁	—	2	1	no	—
28	#92482	—	4	2	no	—
29	Dic₃₁⋊C₄	—	4	1	no	—
30	Dic₃₁⋊C₄	—	4	1	no	—
31	Dic₃₁⋊C₄	—	4	1	no	—
32	C₄×Dic₃₁	—	4	1	no	—
33	C₄×Dic₃₁	D₄	1	1	yes	characteristic
34	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
35	Dic₁₂₄	—	2	1	no	—
36	Dic₁₂₄	—	2	1	no	—
37	Dic₁₂₄	—	2	1	no	—
38	Dic₁₂₄	D₄	1	2	yes	—
39	C₁₆×C₃₁	—	2	1	no	—
40	C₃₁⋊C₁₆	—	2	1	no	—
41	Q₈×C₃₁	—	4	1	no	—
42	Q₈×C₃₁	—	4	1	no	—
43	C₂×C₄×C₃₁	—	4	1	no	—
44	C₂×C₄×C₃₁	—	4	1	no	—
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃₁	—	4	1	no	—
47	C₂×Dic₃₁	—	4	1	no	—
48	C₂×Dic₃₁	D₈	1	1	yes	characteristic
49	C₂×Dic₃₁	D₈	1	1	yes	characteristic
50	Dic₆₂	—	2	2	no	—
51	Dic₆₂	—	4	1	no	—
52	Dic₆₂	—	4	1	no	—
53	Dic₆₂	—	4	1	no	—
54	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₃₁	#41	1	1	yes	characteristic
56	C₃₁⋊C₈	—	2	1	no	—
57	#2209	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
59	C₄×C₃₁	—	8	1	no	—
60	C₄×C₃₁	—	8	1	no	—
61	C₄×C₃₁	#121	1	1	yes	characteristic
62	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Dic₃₁	—	2	1	no	—
64	Dic₃₁	—	4	2	no	—
65	Dic₃₁	—	8	1	no	—
66	Dic₃₁	Q₃₂	1	2	yes	complemented
67	C₂×Q₃₂	—	31	1	no	—
68	C₂×Q₃₂	—	31	1	no	—
69	#494	—	31	1	no	—
70	C₄×Q₁₆	—	31	1	no	—
71	C₄⋊C₁₆	—	31	1	no	—
72	#358	—	31	1	no	—
73	#358	D₃₁	1	1	yes	characteristic
74	C₂×C₃₁	#459	1	1	yes	characteristic
75	C₂×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
76	C₂×C₃₁	#510	1	1	yes	characteristic
77	C₂×Q₁₆	—	31	1	no	—
78	C₂×Q₁₆	—	31	1	no	—
79	C₂×Q₁₆	—	62	1	no	—
80	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
81	C₄⋊Q₈	—	62	1	no	—
82	C₄×Q₈	—	62	1	no	—
83	Q₃₂	—	62	2	no	—
84	Q₃₂	—	62	2	no	—
85	C₂×C₁₆	—	31	1	no	—
86	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
87	#107	C₂×D₃₁	1	1	yes	characteristic
88	#103	—	62	1	no	—
89	C₄×C₈	—	31	1	no	—
90	C₃₁	#2209	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	62	1	no	—
93	C₂×Q₈	—	62	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	31	2	no	—
97	Q₁₆	—	62	1	no	—
98	Q₁₆	—	62	1	no	—
99	Q₁₆	—	62	1	no	—
100	Q₁₆	—	124	2	no	—
101	C₂×C₈	—	31	1	no	—
102	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	124	1	no	—
105	C₄⋊C₄	—	124	1	no	—
106	C₄⋊C₄	—	124	1	no	—
107	C₄²	—	31	1	no	—
108	C₄²	—	124	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	62	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	62	2	no	—
114	Q₈	—	124	1	no	—
115	Q₈	—	124	1	no	—
116	Q₈	—	124	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	31	1	no	—
120	C₂×C₄	—	31	1	no	—
121	C₂×C₄	—	124	1	no	—
122	C₂×C₄	—	124	1	no	—
123	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
124	C₈	—	62	1	no	—
125	C₈	#92495	1	1	yes	characteristic
126	C₈	D₄×D₃₁	1	1	yes	characteristic
127	C₂²	D₈×D₃₁	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	31	2	no	—
131	C₄	—	62	1	no	—
132	C₄	—	124	2	no	—
133	C₄	—	248	1	no	—
134	C₄	#11758461	1	1	yes	characteristic
135	C₄	#11758491	1	1	yes	characteristic
136	C₂	#14521243	1	1	yes	characteristic
137	C₂	Q₃₂×D₃₁	1	1	yes	characteristic
138	C₂	#14521488	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₄₈

C₂

yes

maximal, characteristic

C₂×#11758417

C₂

yes

maximal, characteristic

#14521427

C₂

yes

maximal, characteristic

#14521143

C₂

yes

maximal, characteristic

Dic₃₁⋊C₁₆

C₂

yes

maximal, characteristic

#14520852

C₂

yes

maximal, characteristic

C₃₁×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₃₁

C₂²

yes

characteristic

#11758521

—

C₂×#92482

—

C₂×Dic₁₂₄

C₂²

yes

characteristic

C₂×Dic₁₂₄

C₂²

yes

characteristic

#11758464

—

C₂×C₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#107

C₂²

yes

characteristic

#11758417

—

C₈×Dic₃₁

C₂²

yes

characteristic

C₂×C₃₁⋊C₁₆

C₂²

yes

characteristic

#11758398

—

Dic₂₄₈

—

C₂×Q₈×C₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

#92482

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

D₄

yes

—

C₁₆×C₃₁

—

C₃₁⋊C₁₆

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

D₈

yes

characteristic

C₂×Dic₃₁

D₈

yes

characteristic

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

C₂×D₄

yes

characteristic

C₈×C₃₁

#41

yes

characteristic

C₃₁⋊C₈

—

#2209

—

maximal, Sylow

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

Q₃₂

yes

complemented

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

D₃₁

yes

characteristic

C₂×C₃₁

#459

yes

characteristic

C₂×C₃₁

C₂×Q₃₂

yes

characteristic

C₂×C₃₁

#510

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₁

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₁

yes

characteristic

#107

C₂×D₃₁

yes

characteristic

#103

—

C₄×C₈

—

C₃₁

#2209

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

124

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₃₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

124

—

105

C₄⋊C₄

—

124

—

106

C₄⋊C₄

—

124

—

107

C₄²

—

108

C₄²

—

124

—

109

C₁₆

—

110

C₁₆

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

124

—

115

Q₈

—

124

—

116

Q₈

—

124

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

124

—

122

C₂×C₄

—

124

—

123

C₂×C₄

D₄×D₃₁

yes

characteristic

124

C₈

—

125

C₈

#92495

yes

characteristic

126

C₈

D₄×D₃₁

yes

characteristic

127

C₂²

D₈×D₃₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

124

—

133

C₄

—

248

—

134

C₄

#11758461

yes

characteristic

135

C₄

#11758491

yes

characteristic

136

C₂

#14521243

yes

characteristic

137

C₂

Q₃₂×D₃₁

yes

characteristic

138

C₂

#14521488

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₁
commutator, cosocle	C₂×C₈×C₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#358	C₂
socle, prime radical	C₂²×C₃₁	C₂×D₈
hypercenter	#358	D₃₁
hypocenter	C₃₁	#2209
LCS	G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₃₁

commutator, cosocle

C₂×C₈×C₃₁

C₂³

Frattini

C₂×C₈

C₂²×D₃₁

Fitting

C₃₁×#358

C₂

socle, prime radical

C₂²×C₃₁

C₂×D₈

D₃₁

C₃₁

G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
66. Dic₃₁	Q₃₂	semidirect
90. C₃₁	#2209	semidirect

H ≅ G/N

type

66. Dic₃₁

Q₃₂

semidirect

90. C₃₁

#2209

semidirect

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

subgroup	index
C₂×Dic₂₄₈	2
C₂×#11758417	2
#14521427	2
#14521143	2
Dic₃₁⋊C₁₆	2
#14520852	2
C₃₁×#358	2
#2209	31

subgroup

index

C₂×Dic₂₄₈

Dic₃₁⋊C₁₆

C₃₁×#358

Subgroups — #16847760

Conjugacy classes of subgroups

Complements

Maximal subgroups