Group #16845332 — subgroups

Subgroups — #16845332

Order 3968.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14521452	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521447	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521421	C₂	1	1	yes	maximal, characteristic
4	#14521421	C₂	1	1	yes	maximal, characteristic
5	#14521399	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#486	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₂×#92480	—	2	2	no	—
11	C₂×#92480	—	2	2	no	—
12	#11758446	C₂²	1	1	yes	characteristic
13	#11758446	C₂²	1	1	yes	characteristic
14	C₃₁×#102	—	2	2	no	—
15	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	#11758423	—	2	2	no	—
17	#11758405	—	2	2	no	—
18	#11758395	—	2	2	no	—
19	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₁	—	2	1	no	—
22	C₂×D₄×C₃₁	—	2	1	no	—
23	C₂×D₄×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	—	2	1	no	—
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×Dic₆₂	—	2	1	no	—
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	#92480	—	4	4	no	—
32	#92480	—	4	4	no	—
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
38	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
39	C₂³×C₃₁	—	4	1	no	—
40	Q₈×C₃₁	—	4	2	no	—
41	D₄×C₃₁	—	4	2	no	—
42	D₄×C₃₁	—	4	2	no	—
43	D₄×C₃₁	—	4	2	no	—
44	C₂×C₄×C₃₁	—	4	1	no	—
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₃₁	—	4	1	no	—
49	C₂×Dic₃₁	—	4	1	no	—
50	Dic₆₂	—	4	2	no	—
51	Dic₆₂	—	4	2	no	—
52	C₈×C₃₁	—	2	2	no	—
53	C₃₁⋊C₈	—	2	2	no	—
54	C₃₁⋊C₈	SD₁₆	1	4	yes	complemented
55	#1714	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	8	1	no	—
57	C₂²×C₃₁	—	8	1	no	—
58	C₂²×C₃₁	—	8	1	no	—
59	C₂²×C₃₁	#127	1	1	yes	characteristic
60	C₄×C₃₁	—	8	1	no	—
61	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
62	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₃₁	#137	1	2	yes	—
64	Dic₃₁	—	8	1	no	—
65	Dic₃₁	—	8	1	no	—
66	#500	—	31	1	no	—
67	#498	—	31	1	no	—
68	#492	—	31	1	no	—
69	#492	—	31	1	no	—
70	#487	—	31	1	no	—
71	#486	D₃₁	1	1	yes	characteristic
72	#322	—	31	1	no	—
73	C₂×C₃₁	—	16	1	no	—
74	C₂×C₃₁	#492	1	1	yes	characteristic
75	C₂×C₃₁	#496	1	1	yes	characteristic
76	C₂×C₃₁	#497	1	1	yes	characteristic
77	C₂×SD₁₆	—	62	2	no	—
78	C₂×SD₁₆	—	62	2	no	—
79	C₄⋊Q₈	—	31	1	no	—
80	C₄⋊Q₈	—	31	1	no	—
81	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
82	#127	C₂×D₃₁	1	1	yes	characteristic
83	#107	—	62	2	no	—
84	#106	—	62	2	no	—
85	#103	—	62	2	no	—
86	#102	—	2	2	no	—
87	C₄×C₈	—	31	1	no	—
88	C₄×C₈	—	31	1	no	—
89	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
90	C₃₁	#1714	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	62	1	no	—
93	C₂×Q₈	—	62	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	124	4	no	—
98	SD₁₆	—	124	4	no	—
99	C₂×C₈	—	31	2	no	—
100	C₂×C₈	—	31	2	no	—
101	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	62	1	no	—
105	C₄⋊C₄	—	62	1	no	—
106	C₄⋊C₄	—	62	1	no	—
107	C₄⋊C₄	—	62	1	no	—
108	C₄²	C₂²×D₃₁	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	124	2	no	—
112	Q₈	—	124	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	124	1	no	—
118	C₂×C₄	—	124	1	no	—
119	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	31	4	no	—
124	C₈	—	62	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#11758517	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	248	1	no	—
131	C₄	—	248	1	no	—
132	C₄	#11758488	1	2	yes	—
133	C₄	SD₁₆×D₃₁	1	2	yes	—
134	C₄	#11758528	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#14521420	1	1	yes	characteristic
137	C₂	#14521437	1	1	yes	characteristic
138	C₂	#14521444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521452

C₂

yes

maximal, characteristic, complemented

#14521447

C₂

yes

maximal, characteristic, complemented

#14521421

C₂

yes

maximal, characteristic

#14521421

C₂

yes

maximal, characteristic

#14521399

C₂

yes

maximal, characteristic, complemented

#14520771

C₂

yes

maximal, characteristic, complemented

C₃₁×#486

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

—

—

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×#102

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92480

—

#92480

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

SD₁₆

yes

complemented

#1714

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#127

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic

#322

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

C₂×D₃₁

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1714

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

124

—

SD₁₆

—

124

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₃₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²×D₃₁

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

124

—

112

Q₈

—

124

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

124

—

118

C₂×C₄

—

124

—

119

C₂×C₄

D₄×D₃₁

yes

characteristic

120

C₂×C₄

D₄×D₃₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#11758517

yes

characteristic

129

C₄

—

130

C₄

—

248

—

131

C₄

—

248

—

132

C₄

#11758488

yes

—

133

C₄

SD₁₆×D₃₁

yes

—

134

C₄

#11758528

yes

—

135

C₂

—

136

C₂

#14521420

yes

characteristic

137

C₂

#14521437

yes

characteristic

138

C₂

#14521444

yes

characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#486	C₂
socle	C₂²×C₃₁	#127
prime radical	C₃₁×#127	C₂²
hypercenter	#486	D₃₁
hypocenter	C₃₁	#1714
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#486

C₂

socle

C₂²×C₃₁

#127

prime radical

C₃₁×#127

C₂²

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #14521452	C₂	semidirect
2. #14521447	C₂	semidirect
5. #14521399	C₂	semidirect
6. #14520771	C₂	semidirect
54. C₃₁⋊C₈	SD₁₆	semidirect
90. C₃₁	#1714	semidirect

H ≅ G/N

type

1. #14521452

C₂

semidirect

2. #14521447

C₂

semidirect

5. #14521399

C₂

semidirect

6. #14520771

C₂

semidirect

54. C₃₁⋊C₈

SD₁₆

semidirect

90. C₃₁

#1714

semidirect

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

subgroup	index
#14521452	2
#14521447	2
#14521421	2
#14521399	2
#14520771	2
C₃₁×#486	2
#1714	31

subgroup

index

C₃₁×#486

Subgroups — #16845332

Conjugacy classes of subgroups

Complements

Maximal subgroups