Group #16846666 — subgroups

Subgroups — #16846666

Order 3968.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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
2	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520812	C₂	1	2	yes	maximal
4	#14520812	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	—	2	1	no	—
12	C₂×Dic₃₁⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₃₁	—	2	1	no	—
14	#11758447	—	2	2	no	—
15	#11758447	C₄	1	4	yes	complemented
16	C₃₁×C₄⋊C₈	—	2	2	no	—
17	C₃₁⋊#95	—	2	1	no	—
18	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₁	—	2	2	no	—
22	C₂²×Dic₃₁	—	2	1	no	—
23	C₂²×Dic₃₁	—	2	1	no	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	4	no	—
26	C₄²×C₃₁	C₂×C₄	1	2	yes	—
27	C₄²×C₃₁	D₄	1	2	yes	—
28	C₄⋊Dic₃₁	—	2	1	no	—
29	C₄⋊Dic₃₁	—	2	1	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₃₁	Q₈	1	4	yes	—
34	Dic₃₁⋊C₄	—	2	4	no	—
35	Dic₃₁⋊C₄	—	2	4	no	—
36	C₄×Dic₃₁	—	4	2	no	—
37	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	1	no	—
40	C₂×C₄×C₃₁	—	2	1	no	—
41	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₂×Dic₃₁	—	2	2	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	4	2	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₂×Dic₃₁	—	4	2	no	—
53	C₂×Dic₃₁	—	4	1	no	—
54	C₂×Dic₃₁	—	4	1	no	—
55	C₈×C₃₁	—	4	2	no	—
56	#1989	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
59	C₂²×C₃₁	#135	1	2	yes	—
60	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₃₁	#137	1	1	yes	characteristic
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	2	2	no	—
64	C₄×C₃₁	C₄×Q₈	1	2	yes	—
65	C₄×C₃₁	#124	1	1	yes	characteristic
66	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₃₁	—	4	2	no	—
68	Dic₃₁	—	4	2	no	—
69	Dic₃₁	—	4	4	no	—
70	C₂×#125	—	31	1	no	—
71	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	31	1	no	—
73	#340	—	31	2	no	—
74	#340	—	31	2	no	—
75	C₂×C₃₁	#391	1	1	yes	characteristic
76	C₂×C₃₁	#416	1	1	yes	characteristic
77	C₂×C₃₁	#417	1	1	yes	characteristic
78	C₂×C₃₁	#478	1	2	yes	—
79	C₂×C₃₁	#479	1	2	yes	—
80	C₂²×C₈	C₂×D₃₁	1	2	yes	—
81	#125	—	31	4	no	—
82	#125	—	62	2	no	—
83	C₂×C₄⋊C₄	—	31	1	no	—
84	C₂×C₄⋊C₄	—	31	1	no	—
85	C₂×C₄⋊C₄	—	31	2	no	—
86	C₂×C₄⋊C₄	—	62	1	no	—
87	C₂×C₄⋊C₄	—	62	1	no	—
88	C₂×C₄²	—	62	1	no	—
89	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#95	—	62	1	no	—
92	C₃₁	#1989	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	62	2	no	—
94	C₂²×C₄	—	62	1	no	—
95	C₂²×C₄	—	62	1	no	—
96	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
97	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	31	2	no	—
102	C₄⋊C₄	—	31	2	no	—
103	C₄⋊C₄	—	31	4	no	—
104	C₄⋊C₄	—	62	4	no	—
105	C₄⋊C₄	—	62	4	no	—
106	C₄⋊C₄	—	62	2	no	—
107	C₄⋊C₄	—	62	1	no	—
108	C₄⋊C₄	—	62	1	no	—
109	C₄²	—	124	2	no	—
110	C₄²	C₄×D₃₁	1	2	yes	—
111	C₄²	C₂²⋊D₃₁	1	2	yes	—
112	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	—	62	2	no	—
118	C₂×C₄	—	62	2	no	—
119	C₂×C₄	—	124	2	no	—
120	C₂×C₄	—	124	2	no	—
121	C₂×C₄	—	124	2	no	—
122	C₂×C₄	—	124	1	no	—
123	C₂×C₄	—	124	1	no	—
124	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
125	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
127	C₂×C₄	#92497	1	2	yes	—
128	C₂×C₄	Q₈×D₃₁	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₂²	#11758474	1	2	yes	—
132	C₂²	#11758481	1	2	yes	—
133	C₂²	#11758485	1	1	yes	characteristic
134	C₂²	#11758486	1	1	yes	characteristic
135	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	124	4	no	—
139	C₄	—	124	2	no	—
140	C₄	—	124	2	no	—
141	C₄	#11758464	1	2	yes	—
142	C₄	#11758513	1	1	yes	characteristic
143	C₄	#11758521	1	1	yes	characteristic
144	C₂	#14520954	1	1	yes	characteristic
145	C₂	#14521046	1	1	yes	characteristic
146	C₂	#14521051	1	1	yes	characteristic
147	C₂	#14521350	1	2	yes	—
148	C₂	#14521357	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic

#14520812

C₂

yes

maximal

#14520812

C₂

yes

maximal

C₂×C₃₁×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂×Dic₃₁⋊C₄

—

C₂×C₄×Dic₃₁

—

#11758447

—

#11758447

C₄

yes

complemented

C₃₁×C₄⋊C₈

—

C₃₁⋊#95

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

Q₈

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

#1989

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#125

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#479

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#95

—

C₃₁

#1989

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

124

—

110

C₄²

C₄×D₃₁

yes

—

111

C₄²

C₂²⋊D₃₁

yes

—

112

C₂³

C₂×D₁₂₄

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

124

—

120

C₂×C₄

—

124

—

121

C₂×C₄

—

124

—

122

C₂×C₄

—

124

—

123

C₂×C₄

—

124

—

124

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

125

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

127

C₂×C₄

#92497

yes

—

128

C₂×C₄

Q₈×D₃₁

yes

—

129

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

130

C₈

—

131

C₂²

#11758474

yes

—

132

C₂²

#11758481

yes

—

133

C₂²

#11758485

yes

characteristic

134

C₂²

#11758486

yes

characteristic

135

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

124

—

139

C₄

—

124

—

140

C₄

—

124

—

141

C₄

#11758464

yes

—

142

C₄

#11758513

yes

characteristic

143

C₄

#11758521

yes

characteristic

144

C₂

#14520954

yes

characteristic

145

C₂

#14521046

yes

characteristic

146

C₂

#14521051

yes

characteristic

147

C₂

#14521350

yes

—

148

C₂

#14521357

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	#1989
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

hypocenter

C₃₁

#1989

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. #11758447	C₄	semidirect
92. C₃₁	#1989	semidirect

H ≅ G/N

type

15. #11758447

C₄

semidirect

92. C₃₁

#1989

semidirect

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

subgroup	index
C₂×#11758447	2
C₄×C₄⋊Dic₃₁	2
#14520812	2
C₂×C₃₁×C₄⋊C₈	2
#1989	31

subgroup

index

C₂×#11758447

C₄×C₄⋊Dic₃₁

#14520812

C₂×C₃₁×C₄⋊C₈

#1989

Subgroups — #16846666

Conjugacy classes of subgroups

Complements

Maximal subgroups