Group #16854579 — subgroups

Subgroups — #16854579

Order 3968.

456 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 221 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	#14521527	C₂	1	2	yes	maximal
2	#14521156	C₂	1	4	yes	maximal, complemented
3	Q₁₆×Dic₃₁	C₂	1	4	yes	maximal, complemented
4	#14521087	C₂	1	1	yes	maximal, characteristic
5	#14521058	C₂	1	2	yes	maximal
6	#14521004	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₃₁	C₄	1	8	yes	complemented
10	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	Q₈×Dic₃₁	—	2	4	no	—
14	Q₈×Dic₃₁	C₂²	1	8	yes	—
15	#11758503	—	2	4	no	—
16	#11758503	C₂²	1	2	yes	—
17	#11758503	C₂²	1	1	yes	characteristic
18	C₂×#92474	C₂²	1	1	yes	characteristic
19	#11758423	C₂²	1	8	yes	—
20	#11758406	C₂²	1	2	yes	—
21	#11758405	C₂²	1	2	yes	—
22	#11758403	C₂²	1	2	yes	—
23	C₈×Dic₃₁	C₂²	1	2	yes	—
24	C₂×Q₈×C₃₁	—	2	2	no	—
25	C₂×Q₈×C₃₁	—	2	4	no	—
26	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₃₁	—	2	2	no	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
31	Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
32	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₃₁	—	2	2	no	—
36	C₂²⋊Dic₃₁	D₄	1	2	yes	—
37	C₄⋊Dic₃₁	—	2	8	no	—
38	C₄⋊Dic₃₁	D₄	1	2	yes	—
39	C₄⋊Dic₃₁	C₂³	1	4	yes	—
40	C₄×Dic₃₁	—	2	2	no	—
41	C₄×Dic₃₁	—	2	8	no	—
42	C₄×Dic₃₁	C₂³	1	2	yes	—
43	#92474	—	2	2	no	—
44	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
45	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₃₁	—	2	8	no	—
47	Q₈×C₃₁	—	2	4	no	—
48	Q₈×C₃₁	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₃₁	—	2	8	no	—
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	#41	1	2	yes	—
54	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₃₁	—	2	4	no	—
56	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
57	C₈×C₃₁	C₂²×C₄	1	2	yes	—
58	C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₃₁⋊C₈	—	2	2	no	—
60	#2950	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	—	2	2	no	—
62	C₂²×C₃₁	C₄×D₄	1	2	yes	—
63	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₃₁	—	2	8	no	—
65	C₄×C₃₁	C₄×D₄	1	2	yes	—
66	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
68	Dic₃₁	—	2	4	no	—
69	Dic₃₁	—	4	4	no	—
70	C₂²×Q₁₆	D₃₁	1	1	yes	characteristic
71	#519	—	31	2	no	—
72	#441	—	31	4	no	—
73	C₄×Q₁₆	—	31	4	no	—
74	#427	—	31	1	no	—
75	#419	—	31	2	no	—
76	#405	—	31	1	no	—
77	C₂×C₃₁	—	2	2	no	—
78	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
79	C₂×C₃₁	#578	1	2	yes	—
80	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
81	C₂×Q₁₆	Dic₃₁	1	8	yes	complemented
82	C₂×Q₁₆	C₂×D₃₁	1	2	yes	—
83	C₂×Q₁₆	C₂×D₃₁	1	2	yes	—
84	C₂×M₁₆	—	31	1	no	—
85	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
86	C₄×Q₈	—	31	8	no	—
87	C₄×Q₈	—	62	4	no	—
88	#117	—	31	1	no	—
89	#117	—	31	2	no	—
90	#117	—	62	4	no	—
91	#107	—	31	2	no	—
92	#106	—	31	2	no	—
93	#103	—	31	8	no	—
94	#97	—	31	2	no	—
95	C₄×C₈	—	31	2	no	—
96	C₃₁	#2950	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
100	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
103	Q₁₆	C₂×Dic₃₁	1	8	yes	—
104	Q₁₆	C₂×Dic₃₁	1	8	yes	—
105	M₁₆	—	62	2	no	—
106	C₂×C₈	—	31	2	no	—
107	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
108	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
110	C₄⋊C₄	—	31	2	no	—
111	C₄⋊C₄	—	31	4	no	—
112	C₄⋊C₄	—	62	8	no	—
113	C₂²⋊C₄	—	31	2	no	—
114	C₂²⋊C₄	—	62	2	no	—
115	C₄²	—	31	2	no	—
116	C₄²	—	62	8	no	—
117	C₄²	—	62	2	no	—
118	C₂³	D₄×D₃₁	1	1	yes	characteristic
119	Q₈	—	2	8	no	—
120	Q₈	—	2	4	no	—
121	Q₈	C₂²×Dic₃₁	1	8	yes	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	31	4	no	—
125	C₂×C₄	—	62	4	no	—
126	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
127	C₂×C₄	#92497	1	2	yes	—
128	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
129	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
130	C₈	—	62	2	no	—
131	C₈	C₂²×Dic₃₁	1	2	yes	—
132	C₈	C₂²×Dic₃₁	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	D₄×Dic₃₁	1	2	yes	—
135	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	62	4	no	—
138	C₄	—	124	4	no	—
139	C₄	D₄×Dic₃₁	1	2	yes	—
140	C₄	C₂×#92497	1	1	yes	characteristic
141	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
144	C₂	#14521862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521527

C₂

yes

maximal

#14521156

C₂

yes

maximal, complemented

Q₁₆×Dic₃₁

C₂

yes

maximal, complemented

#14521087

C₂

yes

maximal, characteristic

#14521058

C₂

yes

maximal

#14521004

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂×Q₁₆×C₃₁

C₄

yes

complemented

C₂×Q₁₆×C₃₁

C₂²

yes

—

C₂×Q₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

Q₈×Dic₃₁

—

Q₈×Dic₃₁

C₂²

yes

—

#11758503

—

#11758503

C₂²

yes

—

#11758503

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

#11758423

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

#11758403

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

Q₁₆×C₃₁

C₂×C₄

yes

—

Q₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂³

yes

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂²×C₄

yes

—

C₈×C₃₁

C₂²×C₄

yes

—

C₃₁⋊C₈

—

#2950

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×D₄

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂²×Q₁₆

D₃₁

yes

characteristic

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄×D₄

yes

characteristic

C₂×C₃₁

#578

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂×Q₁₆

Dic₃₁

yes

complemented

C₂×Q₁₆

C₂×D₃₁

yes

—

C₂×Q₁₆

C₂×D₃₁

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

—

—

—

—

—

—

—

C₄×C₈

—

C₃₁

#2950

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

100

C₂×Q₈

C₂²×D₃₁

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₃₁

yes

characteristic

103

Q₁₆

C₂×Dic₃₁

yes

—

104

Q₁₆

C₂×Dic₃₁

yes

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

C₂×Dic₃₁

yes

—

108

C₂×C₈

C₂²×D₃₁

yes

characteristic

109

C₂×C₈

C₂²×D₃₁

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₂³

D₄×D₃₁

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

C₂²×Dic₃₁

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₃₁

yes

characteristic

127

C₂×C₄

#92497

yes

—

128

C₂×C₄

C₂²×Dic₃₁

yes

—

129

C₂×C₄

C₂³×D₃₁

yes

characteristic

130

C₈

—

131

C₈

C₂²×Dic₃₁

yes

—

132

C₈

C₂²×Dic₃₁

yes

—

133

C₂²

—

134

C₂²

D₄×Dic₃₁

yes

—

135

C₂²

C₂×D₄×D₃₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

124

—

139

C₄

D₄×Dic₃₁

yes

—

140

C₄

C₂×#92497

yes

characteristic

141

C₄

C₂³×Dic₃₁

yes

characteristic

142

C₂

—

143

C₂

C₂×D₄×Dic₃₁

yes

characteristic

144

C₂

#14521862

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×Q₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×Q₁₆	D₃₁
hypocenter	C₃₁	#2950
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₃₁

commutator

C₄×C₃₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂²×Q₁₆×C₃₁

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂×C₄×C₃₁

C₂⁴

hypercenter

C₂²×Q₁₆

D₃₁

hypocenter

C₃₁

#2950

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #14521156	C₂	semidirect
3. Q₁₆×Dic₃₁	C₂	semidirect
9. C₂×Q₁₆×C₃₁	C₄	semidirect
81. C₂×Q₁₆	Dic₃₁	semidirect
96. C₃₁	#2950	semidirect

H ≅ G/N

type

2. #14521156

C₂

semidirect

3. Q₁₆×Dic₃₁

C₂

semidirect

9. C₂×Q₁₆×C₃₁

C₄

semidirect

81. C₂×Q₁₆

Dic₃₁

semidirect

96. C₃₁

#2950

semidirect

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

subgroup	index
#14521527	2
#14521156	2
Q₁₆×Dic₃₁	2
#14521087	2
#14521058	2
#14521004	2
C₂²×Q₁₆×C₃₁	2
#2950	31

subgroup

index

Q₁₆×Dic₃₁

C₂²×Q₁₆×C₃₁

Subgroups — #16854579

Conjugacy classes of subgroups

Complements

Maximal subgroups