Group #16845739 — subgroups

Subgroups — #16845739

Order 3968.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758395	C₂	1	2	yes	maximal, complemented
3	#14520906	C₂	1	2	yes	maximal
4	#14520812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×(C₂×D₃₁)⋊C₄	—	2	2	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₃₁	—	2	4	no	—
13	C₂×C₈×D₃₁	C₄	1	4	yes	complemented
14	C₃₁⋊#95	—	2	2	no	—
15	#11758395	—	2	4	no	—
16	C₂³×D₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	—	2	2	no	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₁	—	2	2	no	—
20	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
21	C₂×C₄×D₃₁	—	2	2	no	—
22	C₂×C₄×D₃₁	—	2	2	no	—
23	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₃₁	Q₈	1	2	yes	—
25	C₂×C₈×C₃₁	—	2	2	no	—
26	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	(C₂×D₃₁)⋊C₄	—	4	4	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	—	2	2	no	—
32	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
34	C₈×D₃₁	—	2	8	no	—
35	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₃₁	—	2	2	no	—
37	C₂²×D₃₁	—	2	2	no	—
38	C₂²×D₃₁	C₄⋊C₄	1	2	yes	—
39	C₂²×D₃₁	D₈	1	2	yes	—
40	C₂²×D₃₁	Q₁₆	1	2	yes	—
41	C₂×C₄×C₃₁	—	4	2	no	—
42	C₂×C₄×C₃₁	—	4	2	no	—
43	C₂×C₄×C₃₁	—	4	2	no	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₃₁	#41	1	2	yes	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	4	2	no	—
50	C₂×Dic₃₁	—	4	2	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
53	C₄×D₃₁	—	2	2	no	—
54	C₄×D₃₁	—	2	2	no	—
55	C₄×D₃₁	—	2	4	no	—
56	C₈×C₃₁	—	2	2	no	—
57	C₃₁⋊C₈	C₂²⋊C₄	1	4	yes	complemented
58	#1808	—	31	1	no	maximal, Sylow
59	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₁	#121	1	2	yes	—
61	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
62	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₃₁	#135	1	2	yes	—
64	C₂×D₃₁	—	2	2	no	—
65	C₂×D₃₁	—	2	2	no	—
66	C₂×D₃₁	—	2	4	no	—
67	C₂×D₃₁	—	2	4	no	—
68	C₂×D₃₁	#107	1	4	yes	complemented
69	C₄×C₃₁	—	8	2	no	—
70	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₃₁	#117	1	1	yes	characteristic
72	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
73	Dic₃₁	—	2	2	no	—
74	Dic₃₁	—	8	2	no	—
75	C₂³×C₈	—	31	1	no	—
76	C₂×#107	—	31	2	no	—
77	#380	—	31	2	no	—
78	#340	—	31	1	no	—
79	#340	D₃₁	1	1	yes	characteristic, complemented
80	C₂×C₃₁	#380	1	1	yes	characteristic
81	C₂×C₃₁	C₂×#107	1	1	yes	characteristic
82	C₂×C₃₁	#427	1	1	yes	characteristic
83	C₂×C₃₁	#466	1	2	yes	—
84	C₂×C₃₁	#467	1	2	yes	—
85	D₃₁	—	2	4	no	—
86	C₂³×C₄	—	31	1	no	—
87	C₂²×C₈	—	31	1	no	—
88	C₂²×C₈	—	31	4	no	—
89	C₂²×C₈	—	62	4	no	—
90	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	31	2	no	—
92	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
93	C₂×C₂²⋊C₄	—	62	2	no	—
94	#107	—	62	4	no	—
95	#95	—	62	2	no	—
96	C₃₁	#1808	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	31	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	31	1	no	—
100	C₂²×C₄	—	31	2	no	—
101	C₂²×C₄	—	31	2	no	—
102	C₂²×C₄	—	62	2	no	—
103	C₂²×C₄	—	62	2	no	—
104	C₂²×C₄	—	62	2	no	—
105	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	31	2	no	—
108	C₂×C₈	—	31	4	no	—
109	C₂×C₈	—	62	8	no	—
110	C₂×C₈	C₄×D₃₁	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	62	2	no	—
114	C₄⋊C₄	—	62	2	no	—
115	C₂²⋊C₄	—	124	4	no	—
116	C₂³	—	31	2	no	—
117	C₂³	—	31	2	no	—
118	C₂³	—	31	2	no	—
119	C₂³	—	62	2	no	—
120	C₂³	—	62	2	no	—
121	C₂³	D₄×D₃₁	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	31	2	no	—
126	C₂×C₄	—	62	2	no	—
127	C₂×C₄	—	62	2	no	—
128	C₂×C₄	—	62	2	no	—
129	C₂×C₄	—	62	4	no	—
130	C₂×C₄	—	124	2	no	—
131	C₂×C₄	—	124	2	no	—
132	C₂×C₄	—	124	2	no	—
133	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
134	C₂×C₄	#92495	1	2	yes	—
135	C₂×C₄	D₄×D₃₁	1	2	yes	—
136	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	31	4	no	—
139	C₂²	—	31	4	no	—
140	C₂²	—	62	2	no	—
141	C₂²	—	62	4	no	—
142	C₂²	—	62	4	no	—
143	C₂²	—	62	2	no	—
144	C₂²	#11758461	1	2	yes	—
145	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
146	C₂²	D₈×D₃₁	1	1	yes	characteristic
147	C₂²	#11758493	1	2	yes	—
148	C₂²	Q₁₆×D₃₁	1	1	yes	characteristic
149	C₄	—	8	2	no	—
150	C₄	—	62	2	no	—
151	C₄	—	248	2	no	—
152	C₄	#11758449	1	1	yes	characteristic
153	C₄	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
154	C₄	#11758473	1	2	yes	—
155	C₂	—	62	4	no	—
156	C₂	#14520908	1	1	yes	characteristic
157	C₂	#107×D₃₁	1	1	yes	characteristic
158	C₂	#14521084	1	1	yes	characteristic
159	C₂	#14521284	1	2	yes	—
160	C₂	#14521289	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₃₁

C₂

yes

maximal, characteristic

C₂×#11758395

C₂

yes

maximal, complemented

#14520906

C₂

yes

maximal

#14520812

C₂

yes

maximal, characteristic, complemented

C₃₁×#340

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×(C₂×D₃₁)⋊C₄

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₃₁

—

C₂×C₈×D₃₁

C₄

yes

complemented

C₃₁⋊#95

—

#11758395

—

C₂³×D₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

characteristic

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

Q₈

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

(C₂×D₃₁)⋊C₄

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₈×D₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₄⋊C₄

yes

—

C₂²×D₃₁

D₈

yes

—

C₂²×D₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

C₂²⋊C₄

yes

complemented

#1808

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₂²×C₃₁

#135

yes

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#107

yes

complemented

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

C₂³×C₈

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#467

yes

—

D₃₁

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

#107

—

#95

—

C₃₁

#1808

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₃₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄×D₃₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

124

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

D₄×D₃₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

124

—

131

C₂×C₄

—

124

—

132

C₂×C₄

—

124

—

133

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

134

C₂×C₄

#92495

yes

—

135

C₂×C₄

D₄×D₃₁

yes

—

136

C₂×C₄

Q₈×D₃₁

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#11758461

yes

—

145

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

146

C₂²

D₈×D₃₁

yes

characteristic

147

C₂²

#11758493

yes

—

148

C₂²

Q₁₆×D₃₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

248

—

152

C₄

#11758449

yes

characteristic

153

C₄

C₂²⋊C₄×D₃₁

yes

characteristic

154

C₄

#11758473

yes

—

155

C₂

—

156

C₂

#14520908

yes

characteristic

157

C₂

#107×D₃₁

yes

characteristic

158

C₂

#14521084

yes

characteristic

159

C₂

#14521284

yes

—

160

C₂

#14521289

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#340	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂³×D₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#340	D₃₁
hypocenter	C₃₁	#1808
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#340

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂³×D₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#11758395	C₂	semidirect
4. #14520812	C₂	semidirect
5. C₃₁×#340	C₂	semidirect
13. C₂×C₈×D₃₁	C₄	semidirect
57. C₃₁⋊C₈	C₂²⋊C₄	semidirect
68. C₂×D₃₁	#107	semidirect
79. #340	D₃₁	semidirect
96. C₃₁	#1808	semidirect

H ≅ G/N

type

2. C₂×#11758395

C₂

semidirect

4. #14520812

C₂

semidirect

5. C₃₁×#340

C₂

semidirect

13. C₂×C₈×D₃₁

C₄

semidirect

57. C₃₁⋊C₈

C₂²⋊C₄

semidirect

68. C₂×D₃₁

#107

semidirect

79. #340

D₃₁

semidirect

96. C₃₁

#1808

semidirect

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

subgroup	index
C₂²×C₈×D₃₁	2
C₂×#11758395	2
#14520906	2
#14520812	2
C₃₁×#340	2
#1808	31

subgroup

index

C₂²×C₈×D₃₁

C₃₁×#340

Subgroups — #16845739

Conjugacy classes of subgroups

Complements

Maximal subgroups