Group #24906200 — subgroups

Subgroups — #24906200

Order 10624.

210 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19128258	C₂	1	1	yes	maximal, characteristic
2	#19127725	C₂	1	1	yes	maximal, characteristic
3	#19127448	C₂	1	2	yes	maximal, complemented
4	#19127446	C₂	1	2	yes	maximal, complemented
5	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	—	2	1	no	—
7	#14581792	C₂²	1	1	yes	characteristic
8	#14581792	C₂²	1	1	yes	characteristic
9	#14581784	C₄	1	2	yes	—
10	#14581771	—	2	1	no	—
11	#14581771	—	2	1	no	—
12	C₄×Dic₁₆₆	—	2	1	no	—
13	C₄×Dic₁₆₆	—	2	1	no	—
14	#14581751	C₄	1	2	yes	—
15	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
16	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×M₁₆×C₈₃	—	2	1	no	—
18	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₃	—	2	1	no	—
20	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×Dic₁₆₆	—	2	1	no	—
23	#13050574	—	4	2	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₄⋊Dic₈₃	—	2	1	no	—
26	C₄⋊Dic₈₃	—	2	1	no	—
27	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
28	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
29	Dic₈₃⋊C₄	—	2	2	no	—
30	Dic₈₃⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₈₃	—	2	2	no	—
32	C₄×Dic₈₃	—	2	2	no	—
33	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
36	M₁₆×C₈₃	—	4	2	no	—
37	C₂×C₈×C₈₃	—	2	2	no	—
38	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₄²×C₈₃	C₂³	1	1	yes	characteristic
42	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₈₃	—	2	2	no	—
55	C₂×Dic₈₃	—	2	2	no	—
56	Dic₁₆₆	—	4	1	no	—
57	Dic₁₆₆	—	4	1	no	—
58	Dic₁₆₆	—	4	1	no	—
59	Dic₁₆₆	—	4	1	no	—
60	C₈×C₈₃	—	4	2	no	—
61	C₈₃⋊C₈	—	4	2	no	—
62	C₂²×C₈₃	—	2	1	no	—
63	C₂²×C₈₃	—	2	1	no	—
64	C₂²×C₈₃	—	2	1	no	—
65	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	—	2	1	no	—
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	#101	1	2	yes	—
70	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₈₃	#137	1	1	yes	characteristic
73	C₄×C₈₃	#137	1	1	yes	characteristic
74	Dic₈₃	—	4	2	no	—
75	Dic₈₃	—	4	2	no	—
76	C₂×C₈₃	—	4	1	no	—
77	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
78	C₂×C₈₃	#417	1	1	yes	characteristic
79	C₂×C₈₃	#417	1	1	yes	characteristic
80	#1561	—	83	1	no	maximal, Sylow
81	C₈₃	#1561	1	1	yes	characteristic, complemented, Sylow
82	#533	—	83	1	no	—
83	#423	—	83	1	no	—
84	#423	D₈₃	1	1	yes	characteristic
85	#332	—	83	2	no	—
86	#332	—	83	2	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	166	1	no	—
89	C₄⋊Q₈	—	83	1	no	—
90	C₄⋊Q₈	—	83	1	no	—
91	#125	—	83	2	no	—
92	C₂²⋊Q₈	—	166	1	no	—
93	C₂²⋊Q₈	—	166	1	no	—
94	C₄×Q₈	—	166	1	no	—
95	C₄×Q₈	—	166	1	no	—
96	#117	—	83	2	no	—
97	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	83	2	no	—
100	C₄⋊C₈	—	166	1	no	—
101	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
102	C₂×Q₈	—	166	1	no	—
103	C₂×Q₈	—	166	1	no	—
104	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
105	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
107	M₁₆	—	4	2	no	—
108	M₁₆	—	332	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	166	2	no	—
111	C₄⋊C₄	—	83	2	no	—
112	C₄⋊C₄	—	83	2	no	—
113	C₄⋊C₄	—	166	2	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₄⋊C₄	—	166	1	no	—
116	C₄⋊C₄	—	166	1	no	—
117	C₂²⋊C₄	—	166	2	no	—
118	C₄²	—	166	2	no	—
119	C₄²	C₄×D₈₃	1	1	yes	characteristic
120	C₄²	D₃₃₂	1	1	yes	characteristic
121	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
122	C₄²	C₂²×D₈₃	1	1	yes	characteristic
123	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
124	Q₈	—	332	1	no	—
125	Q₈	—	332	1	no	—
126	Q₈	—	332	1	no	—
127	Q₈	—	332	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	166	2	no	—
134	C₂×C₄	—	166	2	no	—
135	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₈	—	332	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	332	2	no	—
151	C₄	—	332	2	no	—
152	C₄	#14581710	1	2	yes	—
153	C₄	#14581818	1	1	yes	characteristic
154	C₄	#14581820	1	1	yes	characteristic
155	C₄	#14581823	1	1	yes	characteristic
156	C₄	#14581825	1	1	yes	characteristic
157	C₂	—	4	1	no	—
158	C₂	C₂×#14581710	1	1	yes	characteristic
159	C₂	#19127700	1	1	yes	characteristic
160	C₂	#19127703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

#19127725

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal, complemented

#19127446

C₂

yes

maximal, complemented

#423×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

#14581792

C₂²

yes

characteristic

#14581792

C₂²

yes

characteristic

#14581784

C₄

yes

—

#14581771

—

#14581771

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

#14581751

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂×M₁₆×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#101

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#533

—

#423

—

#423

D₈₃

yes

characteristic

#332

—

#332

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

166

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

166

—

101

C₄⋊C₈

C₂×D₈₃

yes

—

102

C₂×Q₈

—

166

—

103

C₂×Q₈

—

166

—

104

C₂²×C₄

C₄×D₈₃

yes

characteristic

105

C₂²×C₄

D₃₃₂

yes

characteristic

106

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

107

M₁₆

—

108

M₁₆

—

332

—

109

C₂×C₈

—

110

C₂×C₈

—

166

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

166

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

—

166

—

116

C₄⋊C₄

—

166

—

117

C₂²⋊C₄

—

166

—

118

C₄²

—

166

—

119

C₄²

C₄×D₈₃

yes

characteristic

120

C₄²

D₃₃₂

yes

characteristic

121

C₄²

C₂²⋊D₈₃

yes

characteristic

122

C₄²

C₂²×D₈₃

yes

characteristic

123

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

124

Q₈

—

332

—

125

Q₈

—

332

—

126

Q₈

—

332

—

127

Q₈

—

332

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

166

—

134

C₂×C₄

—

166

—

135

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

138

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

140

C₂×C₄

C₂×D₃₃₂

yes

characteristic

141

C₈

—

142

C₈

—

332

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

332

—

151

C₄

—

332

—

152

C₄

#14581710

yes

—

153

C₄

#14581818

yes

characteristic

154

C₄

#14581820

yes

characteristic

155

C₄

#14581823

yes

characteristic

156

C₄

#14581825

yes

characteristic

157

C₂

—

158

C₂

C₂×#14581710

yes

characteristic

159

C₂

#19127700

yes

characteristic

160

C₂

#19127703

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1561
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #19127448	C₂	semidirect
4. #19127446	C₂	semidirect
81. C₈₃	#1561	semidirect

H ≅ G/N

type

3. #19127448

C₂

semidirect

4. #19127446

C₂

semidirect

81. C₈₃

#1561

semidirect

Subgroups — #24906200

Conjugacy classes of subgroups

Complements