Group #16846857 — subgroups

Subgroups — #16846857

Order 3968.

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

normal subgroups only

This group has 294 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	C₂×#11758447	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11758420	C₂	1	2	yes	maximal
3	#14520811	C₂	1	2	yes	maximal, complemented
4	#14520804	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	—	2	2	no	—
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	#11758447	—	2	4	no	—
14	#11758420	—	2	4	no	—
15	C₂⁴×C₃₁	—	2	1	no	—
16	C₂×D₄×C₃₁	—	2	2	no	—
17	C₂×D₄×C₃₁	—	2	2	no	—
18	C₂×D₄×C₃₁	—	2	1	no	—
19	C₂×D₄×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₁	—	2	1	no	—
21	C₂²×C₄×C₃₁	D₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂²⋊C₄×C₃₁	—	4	4	no	—
26	C₄²×C₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	D₄	1	4	yes	—
29	Dic₃₁⋊C₄	—	2	8	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂×C₃₁⋊C₈	—	2	2	no	—
33	C₂³×C₃₁	—	2	2	no	—
34	C₂³×C₃₁	—	4	1	no	—
35	C₂³×C₃₁	—	4	1	no	—
36	C₂³×C₃₁	—	4	2	no	—
37	C₂³×C₃₁	—	4	2	no	—
38	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
39	D₄×C₃₁	—	2	4	no	—
40	D₄×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	—	2	2	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	—	2	2	no	—
44	C₂×C₄×C₃₁	—	4	1	no	—
45	C₂×C₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
49	C₂×C₄×C₃₁	#41	1	2	yes	—
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₂×Dic₃₁	—	4	2	no	—
53	C₃₁⋊C₈	—	4	2	no	—
54	#2025	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	4	2	no	—
56	C₂²×C₃₁	—	4	4	no	—
57	C₂²×C₃₁	—	8	2	no	—
58	C₂²×C₃₁	—	8	1	no	—
59	C₂²×C₃₁	—	8	1	no	—
60	C₂²×C₃₁	#120	1	1	yes	characteristic
61	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₃₁	#124	1	1	yes	characteristic
63	C₂²×C₃₁	#135	1	2	yes	—
64	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
65	C₄×C₃₁	—	4	2	no	—
66	C₄×C₃₁	—	8	1	no	—
67	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
68	C₄×C₃₁	#124	1	2	yes	—
69	Dic₃₁	—	4	4	no	—
70	C₂×#125	—	31	1	no	—
71	C₂×#102	—	31	2	no	—
72	#390	D₃₁	1	1	yes	characteristic
73	#340	—	31	2	no	—
74	#336	—	31	1	no	—
75	C₂×C₃₁	—	8	2	no	—
76	C₂×C₃₁	#393	1	1	yes	characteristic
77	C₂×C₃₁	#449	1	2	yes	—
78	C₂×C₃₁	#478	1	2	yes	—
79	C₂×C₃₁	#488	1	1	yes	characteristic
80	C₂×C₃₁	#490	1	1	yes	characteristic
81	C₂²×D₄	C₂×D₃₁	1	1	yes	characteristic
82	C₂²×C₈	—	31	2	no	—
83	#125	—	62	4	no	—
84	C₂×C₄⋊C₄	—	31	2	no	—
85	C₂×C₄⋊C₄	—	62	2	no	—
86	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
89	#102	—	62	4	no	—
90	C₃₁	#2025	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	Dic₆₂	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	62	2	no	—
98	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
100	C₂×C₈	—	62	2	no	—
101	C₂×C₈	—	62	2	no	—
102	C₂×C₈	—	62	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	31	4	no	—
105	C₄⋊C₄	—	62	8	no	—
106	C₄⋊C₄	—	62	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	62	4	no	—
123	C₂×C₄	—	124	2	no	—
124	C₂×C₄	—	124	2	no	—
125	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
126	C₂×C₄	#92495	1	1	yes	characteristic
127	C₂×C₄	D₄×D₃₁	1	2	yes	—
128	C₂×C₄	#92497	1	2	yes	—
129	C₈	—	124	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#11758501	1	1	yes	characteristic
136	C₂²	#11758509	1	2	yes	—
137	C₂²	#11758513	1	1	yes	characteristic
138	C₂²	#11758514	1	1	yes	characteristic
139	C₂²	#11758527	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	124	4	no	—
143	C₄	#11758455	1	2	yes	—
144	C₄	#11758462	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#14520961	1	1	yes	characteristic
147	C₂	#14521192	1	2	yes	—
148	C₂	#14521349	1	2	yes	—
149	C₂	#14521403	1	1	yes	characteristic
150	C₂	#14521413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic, complemented

C₂×#11758420

C₂

yes

maximal

#14520811

C₂

yes

maximal, complemented

#14520804

C₂

yes

maximal, characteristic, complemented

C₃₁×#390

C₂

yes

maximal, characteristic

C₂²×D₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758447

—

#11758420

—

C₂⁴×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

—

Dic₃₁⋊C₄

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#2025

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#120

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#124

yes

—

Dic₃₁

—

C₂×#125

—

C₂×#102

—

#390

D₃₁

yes

characteristic

#340

—

#336

—

C₂×C₃₁

—

C₂×C₃₁

#393

yes

characteristic

C₂×C₃₁

#449

yes

—

C₂×C₃₁

#478

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#102

—

C₃₁

#2025

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₆₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

—

124

C₂×C₄

—

124

—

125

C₂×C₄

C₂×Dic₆₂

yes

characteristic

126

C₂×C₄

#92495

yes

characteristic

127

C₂×C₄

D₄×D₃₁

yes

—

128

C₂×C₄

#92497

yes

—

129

C₈

—

124

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#11758501

yes

characteristic

136

C₂²

#11758509

yes

—

137

C₂²

#11758513

yes

characteristic

138

C₂²

#11758514

yes

characteristic

139

C₂²

#11758527

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

124

—

143

C₄

#11758455

yes

—

144

C₄

#11758462

yes

—

145

C₂

—

146

C₂

#14520961

yes

characteristic

147

C₂

#14521192

yes

—

148

C₂

#14521349

yes

—

149

C₂

#14521403

yes

characteristic

150

C₂

#14521413

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#390	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂²×D₄×C₃₁	C₂²
hypercenter	#390	D₃₁
hypocenter	C₃₁	#2025
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#390

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂²×D₄×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#11758447	C₂	semidirect
3. #14520811	C₂	semidirect
4. #14520804	C₂	semidirect
90. C₃₁	#2025	semidirect

H ≅ G/N

type

1. C₂×#11758447

C₂

semidirect

3. #14520811

C₂

semidirect

4. #14520804

C₂

semidirect

90. C₃₁

#2025

semidirect

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

subgroup	index
C₂×#11758447	2
C₂×#11758420	2
#14520811	2
#14520804	2
C₃₁×#390	2
#2025	31

subgroup

index

C₃₁×#390

Subgroups — #16846857

Conjugacy classes of subgroups

Complements

Maximal subgroups