Group #24905933 — subgroups

Subgroups — #24905933

Order 10624.

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

normal subgroups only

This group has 246 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	#19128251	C₂	1	1	yes	maximal, characteristic
2	#19127758	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127430	C₂	1	4	yes	maximal, complemented
4	#431×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13050590	—	2	1	no	—
6	#14581792	C₄	1	1	yes	characteristic
7	#14581789	C₄	1	1	yes	characteristic
8	#14581780	—	2	1	no	—
9	#14581766	—	2	1	no	—
10	#14581766	C₄	1	2	yes	—
11	C₄×D₃₃₂	—	2	1	no	—
12	C₄×D₃₃₂	C₂²	1	2	yes	—
13	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
15	#14581703	—	2	1	no	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₃	—	2	1	no	—
18	C₂²⋊C₈×C₈₃	—	2	1	no	—
19	C₄×C₈×C₈₃	C₂²	1	2	yes	—
20	#13050590	—	4	4	no	—
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×D₃₃₂	—	2	1	no	—
23	C₂×D₃₃₂	—	2	1	no	—
24	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊D₈₃	—	2	2	no	—
26	C₂×C₄×D₈₃	—	2	2	no	—
27	C₂×C₈₃⋊C₈	—	2	2	no	—
28	C₄⋊Dic₈₃	—	2	1	no	—
29	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
30	(C₂×D₈₃)⋊C₄	—	2	2	no	—
31	(C₂×D₈₃)⋊C₄	—	2	2	no	—
32	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₃	—	2	2	no	—
36	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×D₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	4	1	no	—
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊D₈₃	—	4	4	no	—
52	C₂×Dic₈₃	—	2	2	no	—
53	D₃₃₂	—	2	4	no	—
54	D₃₃₂	—	4	2	no	—
55	D₃₃₂	—	4	2	no	—
56	C₄×D₈₃	—	4	4	no	—
57	Dic₁₆₆	—	4	2	no	—
58	C₈×C₈₃	—	4	2	no	—
59	C₈₃⋊C₈	—	4	2	no	—
60	C₂²×C₈₃	—	2	1	no	—
61	C₂²×C₈₃	—	2	1	no	—
62	C₂²×C₈₃	—	2	1	no	—
63	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₈₃	—	4	2	no	—
65	C₂×D₈₃	—	4	2	no	—
66	C₂×D₈₃	—	4	2	no	—
67	C₄×C₈₃	—	2	1	no	—
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	—	2	1	no	—
70	C₄×C₈₃	—	2	1	no	—
71	C₄×C₈₃	#131	1	2	yes	—
72	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
73	Dic₈₃	—	4	2	no	—
74	C₂×C₈₃	—	4	1	no	—
75	C₂×C₈₃	#408	1	1	yes	characteristic
76	C₂×C₈₃	#418	1	1	yes	characteristic
77	C₂×C₈₃	#421	1	1	yes	characteristic
78	D₈₃	—	8	2	no	—
79	#1496	—	83	1	no	maximal, Sylow
80	C₈₃	#1496	1	1	yes	characteristic, complemented, Sylow
81	#532	—	83	1	no	—
82	#432	—	83	1	no	—
83	#431	D₈₃	1	1	yes	characteristic, complemented
84	D₄⋊C₈	—	83	4	no	—
85	C₂×#41	—	166	1	no	—
86	C₄⋊Q₈	—	83	1	no	—
87	#127	—	83	1	no	—
88	#124	—	166	1	no	—
89	#121	—	83	2	no	—
90	#121	—	166	1	no	—
91	C₄×D₄	—	83	2	no	—
92	C₄×D₄	—	166	1	no	—
93	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	83	2	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	166	1	no	—
98	#97	—	166	1	no	—
99	C₄×C₈	C₂×D₈₃	1	2	yes	—
100	#41	—	332	4	no	—
101	C₂×Q₈	—	166	1	no	—
102	C₂×D₄	—	83	2	no	—
103	C₂×D₄	—	166	2	no	—
104	C₂×D₄	—	166	1	no	—
105	C₂×D₄	—	166	1	no	—
106	C₂²×C₄	—	166	2	no	—
107	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
108	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	166	2	no	—
112	C₄⋊C₄	—	83	2	no	—
113	C₄⋊C₄	—	166	1	no	—
114	C₂²⋊C₄	—	166	2	no	—
115	C₂²⋊C₄	—	166	2	no	—
116	C₄²	C₄×D₈₃	1	1	yes	characteristic
117	C₄²	D₃₃₂	1	1	yes	characteristic
118	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
119	C₄²	C₂²×D₈₃	1	1	yes	characteristic
120	C₂³	—	166	2	no	—
121	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
122	Q₈	—	332	2	no	—
123	D₄	—	332	2	no	—
124	D₄	—	332	2	no	—
125	D₄	—	332	4	no	—
126	D₄	—	166	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	332	4	no	—
131	C₂×C₄	—	166	2	no	—
132	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	332	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	332	2	no	—
144	C₂²	—	332	2	no	—
145	C₂²	—	332	2	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	—	2	1	no	—
151	C₄	—	332	2	no	—
152	C₄	#14581798	1	2	yes	—
153	C₄	#14581820	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	664	2	no	—
156	C₂	#19127667	1	1	yes	characteristic
157	C₂	#19127707	1	1	yes	characteristic
158	C₂	#19127719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128251

C₂

yes

maximal, characteristic

#19127758

C₂

yes

maximal, characteristic, complemented

#19127430

C₂

yes

maximal, complemented

#431×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

—

#14581792

C₄

yes

characteristic

#14581789

C₄

yes

—

—

C₄

yes

—

C₄×D₃₃₂

—

C₄×D₃₃₂

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

—

#13050590

—

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

C₂×C₄

yes

—

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

(C₂×D₈₃)⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

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

C₂²×D₈₃

—

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

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#131

yes

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1496

—

maximal, Sylow

C₈₃

#1496

yes

characteristic, complemented, Sylow

#532

—

#432

—

#431

D₈₃

yes

characteristic, complemented

D₄⋊C₈

—

C₂×#41

—

166

—

C₄⋊Q₈

—

#127

—

#124

—

166

—

#121

—

#121

—

166

—

C₄×D₄

—

C₄×D₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#97

—

166

—

C₄×C₈

C₂×D₈₃

yes

—

100

#41

—

332

—

101

C₂×Q₈

—

166

—

102

C₂×D₄

—

103

C₂×D₄

—

166

—

104

C₂×D₄

—

166

—

105

C₂×D₄

—

166

—

106

C₂²×C₄

—

166

—

107

C₂²×C₄

C₄×D₈₃

yes

characteristic

108

C₂²×C₄

D₃₃₂

yes

characteristic

109

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

166

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

166

—

114

C₂²⋊C₄

—

166

—

115

C₂²⋊C₄

—

166

—

116

C₄²

C₄×D₈₃

yes

characteristic

117

C₄²

D₃₃₂

yes

characteristic

118

C₄²

C₂²⋊D₈₃

yes

characteristic

119

C₄²

C₂²×D₈₃

yes

characteristic

120

C₂³

—

166

—

121

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

122

Q₈

—

332

—

123

D₄

—

332

—

124

D₄

—

332

—

125

D₄

—

332

—

126

D₄

—

166

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

332

—

131

C₂×C₄

—

166

—

132

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

137

C₂×C₄

C₂×D₃₃₂

yes

characteristic

138

C₈

—

139

C₈

—

332

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

332

—

144

C₂²

—

332

—

145

C₂²

—

332

—

146

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

332

—

152

C₄

#14581798

yes

—

153

C₄

#14581820

yes

—

154

C₂

—

155

C₂

—

664

—

156

C₂

#19127667

yes

characteristic

157

C₂

#19127707

yes

characteristic

158

C₂

#19127719

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#431×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	#19128251	C₂
cosocle	C₄²×C₈₃	C₂³
hypercenter	#431	D₈₃
hypocenter	C₈₃	#1496
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#431×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

#19128251

C₂

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. #19127758	C₂	semidirect
3. #19127430	C₂	semidirect
4. #431×C₈₃	C₂	semidirect
80. C₈₃	#1496	semidirect
83. #431	D₈₃	semidirect

H ≅ G/N

type

2. #19127758

C₂

semidirect

3. #19127430

C₂

semidirect

4. #431×C₈₃

C₂

semidirect

80. C₈₃

#1496

semidirect

83. #431

D₈₃

semidirect

Subgroups — #24905933

Conjugacy classes of subgroups

Complements