Group #24909511 — subgroups

Subgroups — #24909511

Order 10624.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14581738	C₂	1	1	yes	maximal, characteristic
2	#19128079	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#19127506	C₂	1	1	yes	maximal, characteristic
6	#19127505	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
9	C₂×#13050580	—	2	1	no	—
10	#14581790	—	2	1	no	—
11	Q₈×Dic₈₃	—	2	1	no	—
12	#14581738	—	2	2	no	—
13	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14581726	C₂²	1	1	yes	characteristic
15	#14581716	—	2	1	no	—
16	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	Q₃₂×C₈₃	—	2	2	no	—
20	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₆₆	—	2	1	no	—
22	Dic₃₃₂	—	2	1	no	—
23	Dic₃₃₂	—	2	1	no	—
24	#13050580	—	4	2	no	—
25	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₈₃	—	2	1	no	—
27	C₄⋊Dic₈₃	—	4	1	no	—
28	Dic₈₃⋊C₄	—	4	1	no	—
29	Dic₈₃⋊C₄	—	4	1	no	—
30	C₄×Dic₈₃	—	4	1	no	—
31	C₄×Dic₈₃	D₄	1	1	yes	characteristic
32	C₈₃⋊C₁₆	—	2	1	no	—
33	C₂×Q₈×C₈₃	—	2	1	no	—
34	C₂×Q₈×C₈₃	—	2	1	no	—
35	Q₁₆×C₈₃	—	2	1	no	—
36	Q₁₆×C₈₃	—	2	1	no	—
37	Q₁₆×C₈₃	—	2	1	no	—
38	Q₁₆×C₈₃	D₄	1	2	yes	—
39	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	C₁₆×C₈₃	—	2	1	no	—
41	Q₈×C₈₃	—	2	2	no	—
42	Q₈×C₈₃	—	4	1	no	—
43	Q₈×C₈₃	—	4	1	no	—
44	Q₈×C₈₃	—	4	1	no	—
45	C₂×C₄×C₈₃	—	4	1	no	—
46	C₂×C₄×C₈₃	—	4	1	no	—
47	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₈₃	—	4	1	no	—
49	C₂×Dic₈₃	—	4	1	no	—
50	C₂×Dic₈₃	D₈	1	1	yes	characteristic
51	C₂×Dic₈₃	D₈	1	1	yes	characteristic
52	Dic₁₆₆	—	4	1	no	—
53	Dic₁₆₆	—	4	1	no	—
54	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₈₃	#41	1	1	yes	characteristic
56	C₈₃⋊C₈	—	2	1	no	—
57	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
58	C₄×C₈₃	—	4	2	no	—
59	C₄×C₈₃	—	8	1	no	—
60	C₄×C₈₃	#121	1	1	yes	characteristic
61	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₈₃	—	2	1	no	—
63	Dic₈₃	—	8	1	no	—
64	Dic₈₃	—	8	1	no	—
65	Dic₈₃	Q₃₂	1	2	yes	complemented
66	C₂×C₈₃	#459	1	1	yes	characteristic
67	C₂×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₈₃	#510	1	1	yes	characteristic
69	#2209	—	83	1	no	maximal, Sylow
70	C₈₃	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	83	1	no	—
72	C₂×Q₃₂	D₈₃	1	1	yes	characteristic
73	#494	—	83	1	no	—
74	C₄×Q₁₆	—	83	1	no	—
75	C₄⋊C₁₆	—	83	1	no	—
76	#358	—	83	1	no	—
77	#358	—	83	1	no	—
78	C₂×Q₁₆	—	83	1	no	—
79	C₂×Q₁₆	—	166	1	no	—
80	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
82	C₄⋊Q₈	—	166	1	no	—
83	C₄×Q₈	—	166	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	166	2	no	—
86	C₂×C₁₆	—	83	1	no	—
87	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
88	#107	—	83	1	no	—
89	#103	—	166	1	no	—
90	C₄×C₈	—	83	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	166	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	332	2	no	—
98	Q₁₆	—	166	1	no	—
99	Q₁₆	—	166	1	no	—
100	Q₁₆	C₂²⋊D₈₃	1	2	yes	—
101	C₂×C₈	—	83	1	no	—
102	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
103	C₄⋊C₄	—	332	1	no	—
104	C₄⋊C₄	—	332	1	no	—
105	C₄⋊C₄	—	332	1	no	—
106	C₄⋊C₄	—	166	1	no	—
107	C₄²	—	332	1	no	—
108	C₄²	—	83	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	166	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	332	1	no	—
116	Q₈	—	332	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	332	1	no	—
120	C₂×C₄	—	332	1	no	—
121	C₂×C₄	—	83	1	no	—
122	C₂×C₄	—	83	1	no	—
123	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
124	C₈	—	166	1	no	—
125	C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
126	C₈	#13050589	1	1	yes	characteristic
127	C₂²	D₈×D₈₃	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	83	2	no	—
131	C₄	—	664	1	no	—
132	C₄	—	664	1	no	—
133	C₄	—	166	1	no	—
134	C₄	#14581764	1	1	yes	characteristic
135	C₄	#14581816	1	1	yes	characteristic
136	C₂	#19127894	1	1	yes	characteristic
137	C₂	Q₃₂×D₈₃	1	1	yes	characteristic
138	C₂	#19128143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581738

C₂

yes

maximal, characteristic

#19128079

C₂

yes

maximal, characteristic

Q₁₆×Dic₈₃

C₂

yes

maximal, characteristic

Dic₈₃⋊C₁₆

C₂

yes

maximal, characteristic

#19127506

C₂

yes

maximal, characteristic

#19127505

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

characteristic

C₂×#13050580

—

#14581790

—

Q₈×Dic₈₃

—

#14581738

—

C₂×C₈₃⋊C₁₆

C₂²

yes

characteristic

#14581726

C₂²

yes

characteristic

#14581716

—

C₈×Dic₈₃

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

characteristic

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050580

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

characteristic

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₁₆×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

D₈

yes

characteristic

C₂×Dic₈₃

D₈

yes

characteristic

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

C₂×D₄

yes

characteristic

C₈×C₈₃

#41

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

Q₃₂

yes

complemented

C₂×C₈₃

#459

yes

characteristic

C₂×C₈₃

C₂×Q₃₂

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

D₈₃

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

166

—

C₄×Q₈

—

166

—

Q₃₂

—

Q₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₃

yes

characteristic

#107

—

#103

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

332

—

Q₁₆

—

166

—

Q₁₆

—

166

—

100

Q₁₆

C₂²⋊D₈₃

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₈₃

yes

characteristic

103

C₄⋊C₄

—

332

—

104

C₄⋊C₄

—

332

—

105

C₄⋊C₄

—

332

—

106

C₄⋊C₄

—

166

—

107

C₄²

—

332

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

166

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

332

—

116

Q₈

—

332

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

332

—

120

C₂×C₄

—

332

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₈₃

yes

characteristic

124

C₈

—

166

—

125

C₈

C₂×C₂²⋊D₈₃

yes

characteristic

126

C₈

#13050589

yes

characteristic

127

C₂²

D₈×D₈₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

664

—

132

C₄

—

664

—

133

C₄

—

166

—

134

C₄

#14581764

yes

characteristic

135

C₄

#14581816

yes

characteristic

136

C₂

#19127894

yes

characteristic

137

C₂

Q₃₂×D₈₃

yes

characteristic

138

C₂

#19128143

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	C₂×Q₃₂×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₈₃
hypocenter	C₈₃	#2209
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₈₃

commutator, cosocle

C₂×C₈×C₈₃

C₂³

Frattini

C₂×C₈

C₂²×D₈₃

Fitting

C₂×Q₃₂×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₈

hypercenter

C₂×Q₃₂

D₈₃

hypocenter

C₈₃

#2209

LCS

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₈₃	Q₃₂	semidirect
70. C₈₃	#2209	semidirect

H ≅ G/N

type

65. Dic₈₃

Q₃₂

semidirect

70. C₈₃

#2209

semidirect

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

subgroup	index
C₂×#14581738	2
#19128079	2
Q₁₆×Dic₈₃	2
Dic₈₃⋊C₁₆	2
#19127506	2
#19127505	2
C₂×Q₃₂×C₈₃	2
#2209	83

subgroup

index

C₂×#14581738

#19128079

Q₁₆×Dic₈₃

Dic₈₃⋊C₁₆

#19127506

#19127505

C₂×Q₃₂×C₈₃

#2209

Subgroups — #24909511

Conjugacy classes of subgroups

Complements

Maximal subgroups