Group #16846261 — subgroups

Subgroups — #16846261

Order 3968.

308 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14521816	C₂	1	1	yes	maximal, characteristic
2	#14521620	C₂	1	1	yes	maximal, characteristic
3	C₂×#11758412	C₂	1	2	yes	maximal, complemented
4	#14520820	C₂	1	2	yes	maximal, complemented
5	C₃₁×#413	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
9	#11758525	—	2	4	no	—
10	#11758521	—	2	1	no	—
11	#11758513	—	2	1	no	—
12	#11758512	C₄	1	4	yes	—
13	#11758503	C₂²	1	1	yes	characteristic
14	#11758501	—	2	2	no	—
15	C₂×#92474	—	2	2	no	—
16	C₂×#92474	C₂²	1	2	yes	—
17	C₃₁×#101	—	2	1	no	—
18	C₃₁×#100	—	2	1	no	—
19	#11758412	—	2	4	no	—
20	#41×C₃₁	—	2	2	no	—
21	#41×C₃₁	—	2	2	no	—
22	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	D₄	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
28	C₂×Dic₆₂	—	2	2	no	—
29	C₂×Dic₆₂	—	2	1	no	—
30	C₂×Dic₆₂	—	2	1	no	—
31	C₂×Dic₆₂	—	2	2	no	—
32	M₁₆×C₃₁	—	2	2	no	—
33	M₁₆×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂²⋊Dic₃₁	—	2	4	no	—
36	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₃₁	—	2	1	no	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	C₄×Dic₃₁	—	2	1	no	—
40	#92474	—	2	4	no	—
41	#92474	—	2	2	no	—
42	#92474	D₄	1	4	yes	—
43	C₂×C₃₁⋊C₈	—	2	4	no	—
44	C₂×C₃₁⋊C₈	—	2	2	no	—
45	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₃₁	—	2	2	no	—
48	D₄×C₃₁	—	2	4	no	—
49	D₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	#41	1	2	yes	—
57	C₂×Dic₃₁	—	2	2	no	—
58	C₂×Dic₃₁	—	2	2	no	—
59	C₂×Dic₃₁	—	2	4	no	—
60	Dic₆₂	—	4	1	no	—
61	Dic₆₂	—	4	2	no	—
62	Dic₆₂	—	4	1	no	—
63	C₈×C₃₁	—	4	2	no	—
64	C₃₁⋊C₈	—	2	4	no	—
65	#1915	—	31	1	no	maximal, Sylow
66	C₂²×C₃₁	—	2	2	no	—
67	C₂²×C₃₁	—	4	1	no	—
68	C₂²×C₃₁	—	4	2	no	—
69	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃₁	C₄×D₄	1	2	yes	—
71	C₄×C₃₁	—	2	2	no	—
72	C₄×C₃₁	#120	1	1	yes	characteristic
73	C₄×C₃₁	#121	1	2	yes	—
74	C₄×C₃₁	#123	1	1	yes	characteristic
75	Dic₃₁	—	4	2	no	—
76	Dic₃₁	—	4	2	no	—
77	#568	—	31	1	no	—
78	#536	—	31	1	no	—
79	#413	D₃₁	1	1	yes	characteristic
80	C₂×#101	—	31	2	no	—
81	#344	—	31	2	no	—
82	C₂×C₃₁	—	2	1	no	—
83	C₂×C₃₁	—	2	2	no	—
84	C₂×C₃₁	—	4	2	no	—
85	C₂×C₃₁	#386	1	1	yes	characteristic
86	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
87	C₂²×Q₈	—	31	1	no	—
88	#131	—	62	4	no	—
89	C₂×M₁₆	—	31	2	no	—
90	C₂×M₁₆	—	62	2	no	—
91	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
92	C₄⋊Q₈	—	62	1	no	—
93	#124	—	62	1	no	—
94	#123	—	31	4	no	—
95	C₂²⋊Q₈	—	62	2	no	—
96	#117	—	31	1	no	—
97	#101	—	2	1	no	—
98	#101	—	62	4	no	—
99	#100	—	2	1	no	—
100	C₃₁	#1915	1	1	yes	characteristic, complemented, Sylow
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	62	1	no	—
104	C₂×Q₈	—	62	2	no	—
105	C₂×Q₈	—	62	2	no	—
106	C₂×Q₈	—	62	1	no	—
107	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
108	C₂×D₄	C₄×D₃₁	1	2	yes	—
109	C₂×D₄	C₂²⋊D₃₁	1	1	yes	characteristic
110	C₂²×C₄	—	31	2	no	—
111	C₂²×C₄	D₁₂₄	1	2	yes	—
112	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	31	4	no	—
116	M₁₆	—	62	4	no	—
117	M₁₆	—	62	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	62	4	no	—
120	C₂×C₈	—	62	2	no	—
121	C₄⋊C₄	—	62	4	no	—
122	C₄⋊C₄	—	62	1	no	—
123	C₂²⋊C₄	—	31	2	no	—
124	C₂²⋊C₄	—	62	4	no	—
125	C₄²	—	62	1	no	—
126	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
127	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	124	1	no	—
130	Q₈	—	124	1	no	—
131	Q₈	—	124	2	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	62	4	no	—
137	C₂×C₄	—	62	2	no	—
138	C₂×C₄	—	62	2	no	—
139	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
141	C₂×C₄	D₄×D₃₁	1	2	yes	—
142	C₂×C₄	#92497	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	62	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	(C₂²⋊D₃₁)⋊C₄	1	2	yes	—
150	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	124	2	no	—
153	C₄	—	124	2	no	—
154	C₄	#11758459	1	1	yes	characteristic
155	C₄	#11758463	1	1	yes	characteristic
156	C₄	#11758516	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#14520926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521816

C₂

yes

maximal, characteristic

#14521620

C₂

yes

maximal, characteristic

C₂×#11758412

C₂

yes

maximal, complemented

#14520820

C₂

yes

maximal, complemented

C₃₁×#413

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

—

—

—

C₄

yes

—

#11758503

C₂²

yes

—

—

C₂²

yes

—

C₃₁×#101

—

C₃₁×#100

—

#11758412

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂×D₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

#92474

—

#92474

—

#92474

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²⋊C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1915

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

—

C₄×C₃₁

#120

yes

characteristic

C₄×C₃₁

#121

yes

—

C₄×C₃₁

#123

yes

characteristic

Dic₃₁

—

Dic₃₁

—

#568

—

#536

—

#413

D₃₁

yes

characteristic

C₂×#101

—

#344

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#386

yes

characteristic

C₂×#41

C₂×D₃₁

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

100

C₃₁

#1915

yes

characteristic, complemented, Sylow

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₃₁

yes

characteristic

108

C₂×D₄

C₄×D₃₁

yes

—

109

C₂×D₄

C₂²⋊D₃₁

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

D₁₂₄

yes

—

112

C₂²×C₄

C₂²×D₃₁

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₂³

(C₂×D₃₁)⋊C₄

yes

—

127

C₂³

C₂×C₄×D₃₁

yes

characteristic

128

Q₈

—

129

Q₈

—

124

—

130

Q₈

—

124

—

131

Q₈

—

124

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

140

C₂×C₄

C₂×D₁₂₄

yes

characteristic

141

C₂×C₄

D₄×D₃₁

yes

—

142

C₂×C₄

#92497

yes

—

143

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

(C₂²⋊D₃₁)⋊C₄

yes

—

150

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

151

C₄

—

152

C₄

—

124

—

153

C₄

—

124

—

154

C₄

#11758459

yes

characteristic

155

C₄

#11758463

yes

characteristic

156

C₄

#11758516

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#14520926

yes

characteristic

	subgroup	G/H
center	C₂	#14520926
commutator	C₂³×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#413	C₂
socle	C₂×C₃₁	#386
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#413	D₃₁
hypocenter	C₃₁	#1915
LCS	G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₂

#14520926

commutator

C₂³×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#413

C₂

socle

C₂×C₃₁

#386

prime radical

C₂×#41×C₃₁

C₂²

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. C₂×#11758412	C₂	semidirect
4. #14520820	C₂	semidirect
100. C₃₁	#1915	semidirect

H ≅ G/N

type

3. C₂×#11758412

C₂

semidirect

4. #14520820

C₂

semidirect

100. C₃₁

#1915

semidirect

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

subgroup	index
#14521816	2
#14521620	2
C₂×#11758412	2
#14520820	2
C₃₁×#413	2
#1915	31

subgroup

index

C₃₁×#413

Subgroups — #16846261

Conjugacy classes of subgroups

Complements

Maximal subgroups