Group #16845797 — subgroups

Subgroups — #16845797

Order 3968.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14520909	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	—	2	1	no	—
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	C₃₁⋊#95	—	2	1	no	—
15	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
16	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	—	2	4	no	—
18	C₂²×C₄×C₃₁	—	2	4	no	—
19	C₂²×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₄×C₃₁	—	4	2	no	—
21	C₂²×C₄×C₃₁	—	4	2	no	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	D₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²⋊Dic₃₁	—	4	2	no	—
31	C₄⋊Dic₃₁	—	2	4	no	—
32	C₂×C₃₁⋊C₈	—	2	2	no	—
33	C₂×C₃₁⋊C₈	—	2	4	no	—
34	C₂³×C₃₁	—	2	2	no	—
35	C₂³×C₃₁	—	2	2	no	—
36	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₃₁	D₈	1	2	yes	—
38	C₂³×C₃₁	SD₁₆	1	2	yes	—
39	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	2	4	no	—
44	C₂×C₄×C₃₁	—	4	8	no	—
45	C₂×C₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	—	4	2	no	—
47	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	4	yes	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₂×Dic₃₁	—	4	2	no	—
53	C₂×Dic₃₁	—	4	2	no	—
54	C₃₁⋊C₈	—	4	2	no	—
55	#1823	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	—	2	2	no	—
58	C₂²×C₃₁	—	2	4	no	—
59	C₂²×C₃₁	—	2	4	no	—
60	C₂²×C₃₁	#102	1	4	yes	—
61	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃₁	#123	1	2	yes	—
63	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₃₁	#137	1	2	yes	—
66	C₄×C₃₁	—	2	2	no	—
67	C₄×C₃₁	—	4	4	no	—
68	C₄×C₃₁	#117	1	2	yes	—
69	C₄×C₃₁	#123	1	2	yes	—
70	Dic₃₁	—	8	2	no	—
71	C₂²×C₄⋊C₄	D₃₁	1	1	yes	characteristic
72	C₂×C₂²⋊C₈	—	31	1	no	—
73	#380	—	31	1	no	—
74	#340	—	31	4	no	—
75	C₂×C₃₁	—	2	4	no	—
76	C₂×C₃₁	#381	1	1	yes	characteristic
77	C₂×C₃₁	C₂×#102	1	1	yes	characteristic
78	C₂×C₃₁	#419	1	1	yes	characteristic
79	C₂×C₃₁	#480	1	2	yes	—
80	C₂×C₃₁	#483	1	2	yes	—
81	C₂³×C₄	—	2	1	no	—
82	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
83	C₂²×C₈	—	31	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	31	2	no	—
87	C₂×C₄⋊C₄	Dic₃₁	1	4	yes	complemented
88	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
89	C₂×C₂²⋊C₄	—	62	1	no	—
90	C₂²⋊C₈	—	62	2	no	—
91	#95	—	62	1	no	—
92	C₃₁	#1823	1	1	yes	characteristic, complemented, Sylow
93	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	62	2	no	—
100	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
103	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
104	C₂×C₈	—	62	4	no	—
105	C₂×C₈	—	62	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	62	4	no	—
109	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
110	C₂²⋊C₄	—	124	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	#92479	1	2	yes	—
114	C₂³	#92480	1	2	yes	—
115	C₂³	C₂²⋊Dic₃₁	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	124	2	no	—
125	C₂×C₄	—	124	2	no	—
126	C₂×C₄	—	124	2	no	—
127	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
128	C₂×C₄	#92495	1	4	yes	—
129	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
131	C₈	—	124	2	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#11758420	1	4	yes	—
137	C₂²	#11758506	1	2	yes	—
138	C₂²	C₂×#92479	1	1	yes	characteristic
139	C₂²	C₂×#92480	1	1	yes	characteristic
140	C₂²	#11758519	1	2	yes	—
141	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	—	248	2	no	—
145	C₄	#11758503	1	2	yes	—
146	C₄	#11758506	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#14520912	1	1	yes	characteristic
149	C₂	C₂×#11758420	1	1	yes	characteristic
150	C₂	#14521058	1	1	yes	characteristic
151	C₂	#14521362	1	2	yes	—
152	C₂	#14521379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

#14520909

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

—

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁⋊#95

—

C₂²⋊(C₃₁⋊C₈)

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

—

C₂³×C₃₁

D₈

yes

—

C₂³×C₃₁

SD₁₆

yes

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1823

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#102

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#123

yes

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#123

yes

—

Dic₃₁

—

C₂²×C₄⋊C₄

D₃₁

yes

characteristic

C₂×C₂²⋊C₈

—

#380

—

#340

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#483

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₃₁

#1823

yes

characteristic, complemented, Sylow

C₂⁴

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×Dic₃₁

yes

—

101

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

102

C₂²×C₄

C₂²⋊D₃₁

yes

—

103

C₂²×C₄

C₂²×D₃₁

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×Dic₃₁

yes

—

110

C₂²⋊C₄

—

124

—

111

C₂³

—

112

C₂³

—

113

C₂³

#92479

yes

—

114

C₂³

#92480

yes

—

115

C₂³

C₂²⋊Dic₃₁

yes

—

116

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

124

—

125

C₂×C₄

—

124

—

126

C₂×C₄

—

124

—

127

C₂×C₄

C₂²⋊Dic₃₁

yes

—

128

C₂×C₄

#92495

yes

—

129

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

131

C₈

—

124

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11758420

yes

—

137

C₂²

#11758506

yes

—

138

C₂²

C₂×#92479

yes

characteristic

139

C₂²

C₂×#92480

yes

characteristic

140

C₂²

#11758519

yes

—

141

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

248

—

145

C₄

#11758503

yes

—

146

C₄

#11758506

yes

—

147

C₂

—

148

C₂

#14520912

yes

characteristic

149

C₂

C₂×#11758420

yes

characteristic

150

C₂

#14521058

yes

characteristic

151

C₂

#14521362

yes

—

152

C₂

#14521379

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×C₄⋊C₄×C₃₁	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₃₁
hypocenter	C₃₁	#1823
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂²×C₄⋊C₄×C₃₁

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂⁴×C₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂²×C₄⋊C₄

D₃₁

hypocenter

C₃₁

#1823

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #14520811	C₂	semidirect
9. C₂×C₄⋊C₄×C₃₁	C₄	semidirect
87. C₂×C₄⋊C₄	Dic₃₁	semidirect
92. C₃₁	#1823	semidirect

H ≅ G/N

type

3. #14520811

C₂

semidirect

9. C₂×C₄⋊C₄×C₃₁

C₄

semidirect

87. C₂×C₄⋊C₄

Dic₃₁

semidirect

92. C₃₁

#1823

semidirect

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

subgroup	index
C₂×C₂²⋊(C₃₁⋊C₈)	2
#14520909	2
#14520811	2
C₂²×C₄⋊C₄×C₃₁	2
#1823	31

subgroup

index

C₂×C₂²⋊(C₃₁⋊C₈)

#14520909

#14520811

C₂²×C₄⋊C₄×C₃₁

#1823

Subgroups — #16845797

Conjugacy classes of subgroups

Complements

Maximal subgroups