Group #16845217 — subgroups

Subgroups — #16845217

Order 3968.

146 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14521447	C₂	1	1	yes	maximal, characteristic
2	#14521344	C₂	1	1	yes	maximal, characteristic
3	#14521259	C₂	1	1	yes	maximal, characteristic
4	#14520802	C₂	1	1	yes	maximal, characteristic
5	#14520797	C₂	1	1	yes	maximal, characteristic
6	#14520794	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#446	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92482	—	2	1	no	—
10	#11758446	C₂²	1	1	yes	characteristic
11	#11758446	C₂²	1	1	yes	characteristic
12	C₃₁×C₄⋊C₈	—	2	1	no	—
13	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃₁×#97	—	2	1	no	—
15	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	#11758423	—	2	1	no	—
17	#11758405	—	2	1	no	—
18	#11758405	—	2	1	no	—
19	#11758398	—	2	1	no	—
20	#11758395	—	2	1	no	—
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×Dic₆₂	—	2	1	no	—
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₄²×C₃₁	—	2	1	no	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	#92482	—	4	2	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	C₄⋊Dic₃₁	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	—	2	1	no	—
40	Q₈×C₃₁	—	2	1	no	—
41	Q₈×C₃₁	—	2	1	no	—
42	C₂×C₄×C₃₁	—	2	1	no	—
43	C₂×C₄×C₃₁	—	2	1	no	—
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
47	C₂×Dic₃₁	—	4	1	no	—
48	C₂×Dic₃₁	—	4	1	no	—
49	Dic₆₂	—	4	1	no	—
50	Dic₆₂	—	4	1	no	—
51	Dic₆₂	—	8	1	no	—
52	C₈×C₃₁	—	2	1	no	—
53	C₈×C₃₁	—	4	1	no	—
54	C₈×C₃₁	Q₁₆	1	2	yes	complemented
55	C₃₁⋊C₈	—	4	1	no	—
56	C₃₁⋊C₈	—	4	1	no	—
57	#1692	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	#121	1	1	yes	characteristic
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	4	1	no	—
61	C₄×C₃₁	—	4	1	no	—
62	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₃₁	#137	1	1	yes	characteristic
65	C₄×C₃₁	#137	1	1	yes	characteristic
66	Dic₃₁	—	8	1	no	—
67	Dic₃₁	—	8	1	no	—
68	#498	—	31	1	no	—
69	#477	—	31	1	no	—
70	#462	—	31	1	no	—
71	#446	D₃₁	1	1	yes	characteristic
72	#335	—	31	1	no	—
73	#333	—	31	1	no	—
74	#332	—	31	1	no	—
75	C₂×C₃₁	#462	1	1	yes	characteristic
76	C₂×C₃₁	#468	1	1	yes	characteristic
77	C₂×C₃₁	#471	1	1	yes	characteristic
78	C₂×Q₁₆	—	62	1	no	—
79	C₄⋊Q₈	—	31	1	no	—
80	C₄⋊Q₈	—	31	1	no	—
81	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
82	#107	—	62	1	no	—
83	#106	—	62	1	no	—
84	#106	—	62	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	31	1	no	—
87	C₄⋊C₈	—	31	1	no	—
88	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
89	#103	—	62	1	no	—
90	#103	—	62	1	no	—
91	#97	—	2	1	no	—
92	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
93	C₃₁	#1692	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	62	1	no	—
95	C₂×Q₈	—	62	1	no	—
96	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
97	Q₁₆	—	124	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	62	1	no	—
100	C₂×C₈	—	62	1	no	—
101	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
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	C₄⋊C₄	D₁₂₄	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₃₁	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	124	1	no	—
114	Q₈	—	124	1	no	—
115	Q₈	—	248	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	124	1	no	—
119	C₂×C₄	—	124	1	no	—
120	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
121	C₂×C₄	#92495	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	124	1	no	—
126	C₈	—	124	1	no	—
127	C₈	#92482	1	2	yes	complemented
128	C₂²	#11758507	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	248	1	no	—
133	C₄	—	248	1	no	—
134	C₄	#11758485	1	1	yes	characteristic
135	C₄	#11758486	1	1	yes	characteristic
136	C₄	C₂×#92482	1	1	yes	characteristic
137	C₄	#11758528	1	1	yes	characteristic
138	C₂	#14521259	1	1	yes	characteristic
139	C₂	#14521295	1	1	yes	characteristic
140	C₂	#14521312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521447

C₂

yes

maximal, characteristic

#14521344

C₂

yes

maximal, characteristic

#14521259

C₂

yes

maximal, characteristic

#14520802

C₂

yes

maximal, characteristic

#14520797

C₂

yes

maximal, characteristic

#14520794

C₂

yes

maximal, characteristic

C₃₁×#446

C₂

yes

maximal, characteristic

C₄×Q₈×C₃₁

C₂²

yes

characteristic

C₂×#92482

—

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₃₁×#97

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

Q₁₆

yes

complemented

C₃₁⋊C₈

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

—

—

—

D₃₁

yes

—

—

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₃₁

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

—

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1692

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂₄

yes

characteristic

Q₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

D₁₂₄

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₃₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

124

—

114

Q₈

—

124

—

115

Q₈

—

248

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

—

124

—

120

C₂×C₄

C₂×D₁₂₄

yes

characteristic

121

C₂×C₄

#92495

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

124

—

126

C₈

—

124

—

127

C₈

#92482

yes

complemented

128

C₂²

#11758507

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

248

—

133

C₄

—

248

—

134

C₄

#11758485

yes

characteristic

135

C₄

#11758486

yes

characteristic

136

C₄

C₂×#92482

yes

characteristic

137

C₄

#11758528

yes

characteristic

138

C₂

#14521259

yes

characteristic

139

C₂

#14521295

yes

characteristic

140

C₂

#14521312

yes

characteristic

	subgroup	G/H
center	C₂²	#11758507
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#446	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#446	D₃₁
hypocenter	C₃₁	#1692
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#11758507

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#446

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
54. C₈×C₃₁	Q₁₆	semidirect
93. C₃₁	#1692	semidirect
127. C₈	#92482	semidirect

H ≅ G/N

type

54. C₈×C₃₁

Q₁₆

semidirect

93. C₃₁

#1692

semidirect

127. C₈

#92482

semidirect

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

subgroup	index
#14521447	2
#14521344	2
#14521259	2
#14520802	2
#14520797	2
#14520794	2
C₃₁×#446	2
#1692	31

subgroup

index

C₃₁×#446

Subgroups — #16845217

Conjugacy classes of subgroups

Complements

Maximal subgroups