Group #16846449 — subgroups

Subgroups — #16846449

Order 3968.

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

normal subgroups only

This group has 240 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	C₂×#11758406	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758405	C₂	1	1	yes	maximal, characteristic
3	#14520947	C₂	1	2	yes	maximal
4	#14520812	C₂	1	2	yes	maximal
5	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	—	2	2	no	—
12	C₃₁×#97	—	2	2	no	—
13	C₃₁⋊#95	—	2	2	no	—
14	#11758406	C₄	1	4	yes	complemented
15	#11758405	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₂²×Dic₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	1	no	—
21	C₂×C₈×C₃₁	—	2	1	no	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₁	D₄	1	2	yes	—
24	C₂×C₈×C₃₁	Q₈	1	2	yes	—
25	C₂×C₈×C₃₁	Q₈	1	2	yes	—
26	C₄²×C₃₁	—	2	1	no	—
27	C₄²×C₃₁	—	2	1	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
32	Dic₃₁⋊C₄	—	4	4	no	—
33	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	4	no	—
44	C₂×Dic₃₁	—	4	2	no	—
45	C₂×Dic₃₁	—	4	2	no	—
46	C₂×Dic₃₁	—	4	2	no	—
47	C₂×Dic₃₁	—	4	2	no	—
48	C₂×Dic₃₁	—	4	2	no	—
49	C₈×C₃₁	—	2	2	no	—
50	C₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
51	#1943	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₃₁	#121	1	1	yes	characteristic
54	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₃₁	#137	1	2	yes	—
57	C₄×C₃₁	—	4	2	no	—
58	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₃₁	#125	1	1	yes	characteristic
62	Dic₃₁	—	4	4	no	—
63	Dic₃₁	—	8	2	no	—
64	C₂×#107	—	31	1	no	—
65	C₂×#106	—	31	1	no	—
66	C₂×#97	D₃₁	1	1	yes	characteristic
67	#389	—	31	2	no	—
68	#340	—	31	2	no	—
69	C₂×C₃₁	#389	1	1	yes	characteristic
70	C₂×C₃₁	#441	1	1	yes	characteristic
71	C₂×C₃₁	#442	1	1	yes	characteristic
72	C₂×C₃₁	#468	1	2	yes	—
73	C₂×C₃₁	#501	1	2	yes	—
74	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	31	2	no	—
78	C₂×C₄⋊C₄	—	62	2	no	—
79	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
80	#107	—	31	4	no	—
81	#106	—	62	2	no	—
82	#97	—	2	2	no	—
83	#95	—	62	2	no	—
84	C₃₁	#1943	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	62	2	no	—
86	C₂²×C₄	—	62	2	no	—
87	C₂²×C₄	D₁₂₄	1	2	yes	—
88	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	Dic₆₂	1	2	yes	—
92	C₂×C₈	Dic₆₂	1	2	yes	—
93	C₂×C₈	C₄×D₃₁	1	2	yes	—
94	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
95	C₄⋊C₄	—	31	4	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₄⋊C₄	—	62	2	no	—
98	C₄⋊C₄	—	62	2	no	—
99	C₄⋊C₄	—	124	4	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	—	124	2	no	—
108	C₂×C₄	—	124	2	no	—
109	C₂×C₄	—	124	2	no	—
110	C₂×C₄	—	124	2	no	—
111	C₂×C₄	—	124	2	no	—
112	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
115	C₂×C₄	#92495	1	1	yes	characteristic
116	C₂×C₄	#92495	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	Dic₃₁⋊C₄	1	4	yes	complemented
120	C₂²	#11758446	1	1	yes	characteristic
121	C₂²	C₄×D₁₂₄	1	1	yes	characteristic
122	C₂²	#11758485	1	2	yes	—
123	C₂²	#11758486	1	2	yes	—
124	C₂²	#11758507	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	124	4	no	—
127	C₄	—	248	2	no	—
128	C₄	C₄×Dic₆₂	1	1	yes	characteristic
129	C₄	#11758447	1	1	yes	characteristic
130	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
131	C₄	#11758501	1	1	yes	characteristic
132	C₂	#14520947	1	1	yes	characteristic
133	C₂	#14521157	1	1	yes	characteristic
134	C₂	#14521162	1	1	yes	characteristic
135	C₂	#14521295	1	2	yes	—
136	C₂	#14521458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758406

C₂

yes

maximal, characteristic

C₂×#11758405

C₂

yes

maximal, characteristic

#14520947

C₂

yes

maximal

#14520812

C₂

yes

maximal

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₃₁×#97

—

C₃₁⋊#95

—

#11758406

C₄

yes

complemented

#11758405

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

complemented

#1943

—

maximal, Sylow

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

#121

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₃₁

#125

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₃₁

yes

characteristic

#389

—

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#501

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#1943

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₁₂₄

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

124

—

108

C₂×C₄

—

124

—

109

C₂×C₄

—

124

—

110

C₂×C₄

—

124

—

111

C₂×C₄

—

124

—

112

C₂×C₄

C₂×Dic₆₂

yes

characteristic

113

C₂×C₄

C₂×Dic₆₂

yes

characteristic

114

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

115

C₂×C₄

#92495

yes

characteristic

116

C₂×C₄

#92495

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

118

C₈

—

119

C₈

Dic₃₁⋊C₄

yes

complemented

120

C₂²

#11758446

yes

characteristic

121

C₂²

C₄×D₁₂₄

yes

characteristic

122

C₂²

#11758485

yes

—

123

C₂²

#11758486

yes

—

124

C₂²

#11758507

yes

characteristic

125

C₄

—

126

C₄

—

124

—

127

C₄

—

248

—

128

C₄

C₄×Dic₆₂

yes

characteristic

129

C₄

#11758447

yes

characteristic

130

C₄

C₂×Dic₃₁⋊C₄

yes

characteristic

131

C₄

#11758501

yes

characteristic

132

C₂

#14520947

yes

characteristic

133

C₂

#14521157

yes

characteristic

134

C₂

#14521162

yes

characteristic

135

C₂

#14521295

yes

—

136

C₂

#14521458

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#97	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#1943
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#97

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×#97

D₃₁

hypocenter

C₃₁

#1943

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
14. #11758406	C₄	semidirect
50. C₈×C₃₁	C₄⋊C₄	semidirect
84. C₃₁	#1943	semidirect
119. C₈	Dic₃₁⋊C₄	semidirect

H ≅ G/N

type

14. #11758406

C₄

semidirect

50. C₈×C₃₁

C₄⋊C₄

semidirect

84. C₃₁

#1943

semidirect

119. C₈

Dic₃₁⋊C₄

semidirect

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

subgroup	index
C₂×#11758406	2
C₂×#11758405	2
#14520947	2
#14520812	2
C₂×C₃₁×#97	2
#1943	31

subgroup

index

C₂×C₃₁×#97

Subgroups — #16846449

Conjugacy classes of subgroups

Complements

Maximal subgroups