Group #24916331 — subgroups

Subgroups — #24916331

Order 10624.

456 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 221 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	#19128179	C₂	1	2	yes	maximal
2	#19127808	C₂	1	4	yes	maximal, complemented
3	Q₁₆×Dic₈₃	C₂	1	4	yes	maximal, complemented
4	#19127739	C₂	1	1	yes	maximal, characteristic
5	#19127710	C₂	1	2	yes	maximal
6	#19127656	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	1	yes	characteristic
9	Q₈×Dic₈₃	—	2	4	no	—
10	Q₈×Dic₈₃	C₂²	1	8	yes	—
11	#14581751	—	2	4	no	—
12	#14581751	C₂²	1	2	yes	—
13	#14581751	C₂²	1	1	yes	characteristic
14	#14581726	C₂²	1	2	yes	—
15	#14581724	C₂²	1	2	yes	—
16	#14581716	C₂²	1	8	yes	—
17	#14581702	C₂²	1	2	yes	—
18	C₈×Dic₈₃	C₂²	1	2	yes	—
19	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₈₃	C₄	1	8	yes	complemented
21	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#13050574	—	2	2	no	—
25	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
26	C₄⋊Dic₈₃	—	2	8	no	—
27	C₄⋊Dic₈₃	D₄	1	2	yes	—
28	C₄⋊Dic₈₃	C₂³	1	4	yes	—
29	C₂²⋊Dic₈₃	—	2	2	no	—
30	C₂²⋊Dic₈₃	D₄	1	2	yes	—
31	C₄×Dic₈₃	—	2	2	no	—
32	C₄×Dic₈₃	—	2	8	no	—
33	C₄×Dic₈₃	C₂³	1	2	yes	—
34	C₂×Q₈×C₈₃	—	2	2	no	—
35	C₂×Q₈×C₈₃	—	2	4	no	—
36	C₂×Q₈×C₈₃	C₂×C₄	1	8	yes	—
37	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₈₃	—	2	2	no	—
39	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
40	Q₁₆×C₈₃	C₂×C₄	1	8	yes	—
41	Q₁₆×C₈₃	C₂×C₄	1	8	yes	—
42	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
43	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₈₃	—	2	8	no	—
47	Q₈×C₈₃	—	2	4	no	—
48	Q₈×C₈₃	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₈₃	—	2	8	no	—
50	C₂×C₄×C₈₃	—	2	4	no	—
51	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#41	1	2	yes	—
54	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₈₃	—	2	4	no	—
56	C₂×Dic₈₃	C₂×D₄	1	4	yes	—
57	C₈×C₈₃	C₂²×C₄	1	2	yes	—
58	C₈×C₈₃	C₂²×C₄	1	2	yes	—
59	C₈₃⋊C₈	—	2	2	no	—
60	C₂²×C₈₃	—	2	2	no	—
61	C₂²×C₈₃	C₄×D₄	1	2	yes	—
62	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₈₃	—	2	8	no	—
64	C₄×C₈₃	C₄×D₄	1	2	yes	—
65	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
67	Dic₈₃	—	2	4	no	—
68	Dic₈₃	—	4	4	no	—
69	C₂×C₈₃	—	2	2	no	—
70	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₈₃	#578	1	2	yes	—
72	#2950	—	83	1	no	maximal, Sylow
73	C₈₃	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	D₈₃	1	1	yes	characteristic
75	#519	—	83	2	no	—
76	#441	—	83	4	no	—
77	C₄×Q₁₆	—	83	4	no	—
78	#427	—	83	1	no	—
79	#419	—	83	2	no	—
80	#405	—	83	1	no	—
81	C₂²×Q₈	C₂×D₈₃	1	2	yes	—
82	C₂×Q₁₆	Dic₈₃	1	8	yes	complemented
83	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
84	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
85	C₂×M₁₆	—	83	1	no	—
86	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
87	C₄×Q₈	—	83	8	no	—
88	C₄×Q₈	—	166	4	no	—
89	#117	—	83	1	no	—
90	#117	—	83	2	no	—
91	#117	—	166	4	no	—
92	#107	—	83	2	no	—
93	#106	—	83	2	no	—
94	#103	—	83	8	no	—
95	#97	—	83	2	no	—
96	C₄×C₈	—	83	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×Dic₈₃	1	8	yes	—
100	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
103	Q₁₆	C₂×Dic₈₃	1	8	yes	—
104	Q₁₆	C₂×Dic₈₃	1	8	yes	—
105	M₁₆	—	166	2	no	—
106	C₂×C₈	—	83	2	no	—
107	C₂×C₈	C₂×Dic₈₃	1	4	yes	—
108	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
110	C₄⋊C₄	—	83	2	no	—
111	C₄⋊C₄	—	83	4	no	—
112	C₄⋊C₄	—	166	8	no	—
113	C₂²⋊C₄	—	83	2	no	—
114	C₂²⋊C₄	—	166	2	no	—
115	C₄²	—	83	2	no	—
116	C₄²	—	166	8	no	—
117	C₄²	—	166	2	no	—
118	C₂³	D₄×D₈₃	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	8	no	—
121	Q₈	C₂²×Dic₈₃	1	8	yes	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	83	4	no	—
125	C₂×C₄	—	166	4	no	—
126	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
127	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
128	C₂×C₄	#13050589	1	2	yes	—
129	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
130	C₈	—	166	2	no	—
131	C₈	C₂²×Dic₈₃	1	2	yes	—
132	C₈	C₂²×Dic₈₃	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	D₄×Dic₈₃	1	2	yes	—
135	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	332	4	no	—
138	C₄	—	166	4	no	—
139	C₄	D₄×Dic₈₃	1	2	yes	—
140	C₄	C₂³×Dic₈₃	1	1	yes	characteristic
141	C₄	C₂×#13050589	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×D₄×Dic₈₃	1	1	yes	characteristic
144	C₂	#19128514	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128179

C₂

yes

maximal

#19127808

C₂

yes

maximal, complemented

Q₁₆×Dic₈₃

C₂

yes

maximal, complemented

#19127739

C₂

yes

maximal, characteristic

#19127710

C₂

yes

maximal

#19127656

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

Q₈×Dic₈₃

—

Q₈×Dic₈₃

C₂²

yes

—

#14581751

—

#14581751

C₂²

yes

—

#14581751

C₂²

yes

characteristic

#14581726

C₂²

yes

—

#14581724

C₂²

yes

—

#14581716

C₂²

yes

—

#14581702

C₂²

yes

—

C₈×Dic₈₃

C₂²

yes

—

C₂²×Q₈×C₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

C₄

yes

complemented

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

C₂²⋊Dic₈₃

—

C₂²⋊Dic₈₃

D₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂³

yes

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂×Q₈×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

Q₁₆×C₈₃

C₂×C₄

yes

—

Q₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₈×C₈₃

C₂²×C₄

yes

—

C₈×C₈₃

C₂²×C₄

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

C₂×#41

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄×D₄

yes

characteristic

C₂×C₈₃

#578

yes

—

#2950

—

maximal, Sylow

C₈₃

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₈₃

yes

characteristic

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

—

C₂²×Q₈

C₂×D₈₃

yes

—

C₂×Q₁₆

Dic₈₃

yes

complemented

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

166

—

#117

—

#117

—

#117

—

166

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₈₃

yes

—

100

C₂×Q₈

C₂²×D₈₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₈₃

yes

characteristic

103

Q₁₆

C₂×Dic₈₃

yes

—

104

Q₁₆

C₂×Dic₈₃

yes

—

105

M₁₆

—

166

—

106

C₂×C₈

—

107

C₂×C₈

C₂×Dic₈₃

yes

—

108

C₂×C₈

C₂²×D₈₃

yes

characteristic

109

C₂×C₈

C₂²×D₈₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

166

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

166

—

115

C₄²

—

116

C₄²

—

166

—

117

C₄²

—

166

—

118

C₂³

D₄×D₈₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

C₂²×Dic₈₃

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

166

—

126

C₂×C₄

C₂²×Dic₈₃

yes

—

127

C₂×C₄

D₄×D₈₃

yes

characteristic

128

C₂×C₄

#13050589

yes

—

129

C₂×C₄

C₂³×D₈₃

yes

characteristic

130

C₈

—

166

—

131

C₈

C₂²×Dic₈₃

yes

—

132

C₈

C₂²×Dic₈₃

yes

—

133

C₂²

—

134

C₂²

D₄×Dic₈₃

yes

—

135

C₂²

C₂×D₄×D₈₃

yes

characteristic

136

C₄

—

137

C₄

—

332

—

138

C₄

—

166

—

139

C₄

D₄×Dic₈₃

yes

—

140

C₄

C₂³×Dic₈₃

yes

characteristic

141

C₄

C₂×#13050589

yes

characteristic

142

C₂

—

143

C₂

C₂×D₄×Dic₈₃

yes

characteristic

144

C₂

#19128514

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator	C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂²×Q₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂²×Q₁₆	D₈₃
hypocenter	C₈₃	#2950
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator

C₄×C₈₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₂²×Q₁₆×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂×C₄×C₈₃

C₂⁴

hypercenter

C₂²×Q₁₆

D₈₃

hypocenter

C₈₃

#2950

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #19127808	C₂	semidirect
3. Q₁₆×Dic₈₃	C₂	semidirect
20. C₂×Q₁₆×C₈₃	C₄	semidirect
73. C₈₃	#2950	semidirect
82. C₂×Q₁₆	Dic₈₃	semidirect

H ≅ G/N

type

2. #19127808

C₂

semidirect

3. Q₁₆×Dic₈₃

C₂

semidirect

20. C₂×Q₁₆×C₈₃

C₄

semidirect

73. C₈₃

#2950

semidirect

82. C₂×Q₁₆

Dic₈₃

semidirect

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

subgroup	index
#19128179	2
#19127808	2
Q₁₆×Dic₈₃	2
#19127739	2
#19127710	2
#19127656	2
C₂²×Q₁₆×C₈₃	2
#2950	83

subgroup

index

Q₁₆×Dic₈₃

C₂²×Q₁₆×C₈₃

Subgroups — #24916331

Conjugacy classes of subgroups

Complements

Maximal subgroups