Group #24906232 — subgroups

Subgroups — #24906232

Order 10624.

168 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19127752	C₂	1	1	yes	maximal, characteristic
2	#19127723	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127452	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127451	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127451	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050573	—	2	1	no	—
9	C₂×C₈×D₈₃	—	2	1	no	—
10	C₄²×D₈₃	C₂²	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₈₃⋊C₈	—	2	1	no	—
14	Dic₈₃⋊C₈	—	2	1	no	—
15	Dic₈₃⋊C₈	C₄	1	1	yes	characteristic
16	Dic₈₃⋊C₈	C₄	1	1	yes	characteristic
17	(C₂×D₈₃)⋊C₈	C₄	1	1	yes	characteristic
18	(C₂×D₈₃)⋊C₈	C₄	1	1	yes	characteristic
19	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₈₃	Q₈	1	1	yes	characteristic
26	#13050573	—	4	2	no	—
27	C₂×C₈₃⋊C₈	—	2	1	no	—
28	C₂×C₈₃⋊C₈	—	2	1	no	—
29	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₈×D₈₃	—	4	2	no	—
32	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₈₃	D₄	1	1	yes	characteristic
34	C₄×Dic₈₃	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	C₂²×D₈₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄×D₈₃	—	2	1	no	—
48	C₄×D₈₃	—	2	1	no	—
49	C₄×D₈₃	—	4	1	no	—
50	C₄×D₈₃	SD₁₆	1	2	yes	—
51	C₄×D₈₃	SD₁₆	1	2	yes	—
52	C₈×C₈₃	—	2	1	no	—
53	C₈×C₈₃	—	2	1	no	—
54	C₈×C₈₃	—	4	1	no	—
55	C₈×C₈₃	—	4	1	no	—
56	C₈₃⋊C₈	—	2	1	no	—
57	C₈₃⋊C₈	—	2	1	no	—
58	C₈₃⋊C₈	—	4	1	no	—
59	C₈₃⋊C₈	—	4	1	no	—
60	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×D₈₃	—	2	1	no	—
62	C₂×D₈₃	—	2	1	no	—
63	C₂×D₈₃	#106	1	2	yes	—
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	#131	1	1	yes	characteristic
66	C₄×C₈₃	#131	1	1	yes	characteristic
67	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
69	Dic₈₃	—	2	1	no	—
70	Dic₈₃	—	2	1	no	—
71	Dic₈₃	#106	1	2	yes	—
72	C₂×C₈₃	#424	1	1	yes	characteristic
73	C₂×C₈₃	C₂×#106	1	1	yes	characteristic
74	C₂×C₈₃	#430	1	1	yes	characteristic
75	D₈₃	—	2	2	no	—
76	#1570	—	83	1	no	maximal, Sylow
77	C₈₃	#1570	1	1	yes	characteristic, complemented, Sylow
78	#431	—	83	1	no	—
79	#423	—	83	1	no	—
80	C₂×C₄⋊C₈	—	83	1	no	—
81	#334	—	83	1	no	—
82	#334	—	83	1	no	—
83	#334	—	83	1	no	—
84	#334	D₈₃	1	1	yes	characteristic, complemented
85	C₂×M₁₆	—	166	1	no	—
86	C₂²×C₈	—	166	1	no	—
87	C₂×C₄²	—	83	1	no	—
88	C₄⋊C₈	—	83	1	no	—
89	C₄⋊C₈	—	83	1	no	—
90	C₄⋊C₈	—	83	1	no	—
91	C₄⋊C₈	—	83	1	no	—
92	C₄⋊C₈	—	166	1	no	—
93	C₄⋊C₈	—	166	1	no	—
94	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
95	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
96	C₂²⋊C₈	—	83	1	no	—
97	C₂²⋊C₈	—	83	1	no	—
98	C₄×C₈	—	83	1	no	—
99	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
100	C₂²×C₄	—	83	1	no	—
101	C₂²×C₄	—	83	1	no	—
102	C₂²×C₄	—	83	1	no	—
103	M₁₆	—	332	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	332	2	no	—
107	C₂×C₈	—	83	1	no	—
108	C₂×C₈	—	83	1	no	—
109	C₂×C₈	—	166	1	no	—
110	C₂×C₈	—	166	1	no	—
111	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
112	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
113	C₄²	—	83	1	no	—
114	C₄²	—	83	1	no	—
115	C₄²	—	83	1	no	—
116	C₄²	C₂²×D₈₃	1	1	yes	characteristic
117	C₂³	—	83	1	no	—
118	C₂×C₄	—	332	1	no	—
119	C₂×C₄	—	83	1	no	—
120	C₂×C₄	—	83	2	no	—
121	C₂×C₄	—	83	2	no	—
122	C₂×C₄	—	83	1	no	—
123	C₂×C₄	—	83	1	no	—
124	C₂×C₄	—	166	1	no	—
125	C₂×C₄	—	166	1	no	—
126	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	2	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	332	1	no	—
134	C₈	—	332	1	no	—
135	C₈	—	166	1	no	—
136	C₈	—	166	1	no	—
137	C₂²	—	83	2	no	—
138	C₂²	—	166	1	no	—
139	C₂²	—	166	1	no	—
140	C₂²	C₄⋊C₄×D₈₃	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	83	2	no	—
143	C₄	—	166	1	no	—
144	C₄	—	166	1	no	—
145	C₄	#14581798	1	1	yes	characteristic
146	C₄	#14581800	1	1	yes	characteristic
147	C₄	SD₁₆×D₈₃	1	1	yes	characteristic
148	C₄	SD₁₆×D₈₃	1	1	yes	characteristic
149	C₂	—	166	2	no	—
150	C₂	#19127726	1	1	yes	characteristic
151	C₂	#106×D₈₃	1	1	yes	characteristic
152	C₂	#19127748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127752

C₂

yes

maximal, characteristic

#19127723

C₂

yes

maximal, characteristic

C₄⋊C₈×D₈₃

C₂

yes

maximal, characteristic

#19127452

C₂

yes

maximal, characteristic, complemented

#19127451

C₂

yes

maximal, characteristic, complemented

#19127451

C₂

yes

maximal, characteristic, complemented

#334×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050573

—

C₂×C₈×D₈₃

—

C₄²×D₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

Dic₈₃⋊C₈

—

Dic₈₃⋊C₈

C₄

yes

characteristic

Dic₈₃⋊C₈

C₄

yes

characteristic

(C₂×D₈₃)⋊C₈

C₄

yes

characteristic

(C₂×D₈₃)⋊C₈

C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄×D₈₃

C₂×C₄

yes

characteristic

C₂×C₄×D₈₃

D₄

yes

characteristic

C₂×C₄×D₈₃

Q₈

yes

characteristic

#13050573

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₈×D₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₄×Dic₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×Dic₈₃

C₄⋊C₄

yes

characteristic

C₂×Dic₈₃

C₄⋊C₄

yes

characteristic

C₂×Dic₈₃

C₄⋊C₄

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

SD₁₆

yes

—

C₄×D₈₃

SD₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

#106

yes

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

#106

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1570

—

maximal, Sylow

C₈₃

#1570

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×M₁₆

—

166

—

C₂²×C₈

—

166

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

M₁₆

—

332

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

332

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

166

—

110

C₂×C₈

—

166

—

111

C₂×C₈

C₄×D₈₃

yes

characteristic

112

C₂×C₈

C₄×D₈₃

yes

characteristic

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₈₃

yes

characteristic

117

C₂³

—

118

C₂×C₄

—

332

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

166

—

125

C₂×C₄

—

166

—

126

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

127

C₂×C₄

D₄×D₈₃

yes

characteristic

128

C₂×C₄

Q₈×D₈₃

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

332

—

134

C₈

—

332

—

135

C₈

—

166

—

136

C₈

—

166

—

137

C₂²

—

138

C₂²

—

166

—

139

C₂²

—

166

—

140

C₂²

C₄⋊C₄×D₈₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

166

—

144

C₄

—

166

—

145

C₄

#14581798

yes

characteristic

146

C₄

#14581800

yes

characteristic

147

C₄

SD₁₆×D₈₃

yes

characteristic

148

C₄

SD₁₆×D₈₃

yes

characteristic

149

C₂

—

166

—

150

C₂

#19127726

yes

characteristic

151

C₂

#106×D₈₃

yes

characteristic

152

C₂

#19127748

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#334×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂²×D₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#334	D₈₃
hypocenter	C₈₃	#1570
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#334×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂²×D₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
4. #19127452	C₂	semidirect
5. #19127451	C₂	semidirect
6. #19127451	C₂	semidirect
7. #334×C₈₃	C₂	semidirect
77. C₈₃	#1570	semidirect
84. #334	D₈₃	semidirect

H ≅ G/N

type

4. #19127452

C₂

semidirect

5. #19127451

C₂

semidirect

6. #19127451

C₂

semidirect

7. #334×C₈₃

C₂

semidirect

77. C₈₃

#1570

semidirect

84. #334

D₈₃

semidirect

Subgroups — #24906232

Conjugacy classes of subgroups

Complements