Group #24906294 — subgroups

Subgroups — #24906294

Order 10624.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic
2	#19127780	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127454	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127430	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127430	C₂	1	1	yes	maximal, characteristic, complemented
7	#322×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₆₄	C₄	1	1	yes	characteristic
9	C₂×#13050573	—	2	1	no	—
10	C₂×C₈×D₈₃	—	2	1	no	—
11	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
12	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
13	#14581726	C₄	1	1	yes	characteristic
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
16	#14581714	C₄	1	1	yes	characteristic
17	#14581714	C₄	1	1	yes	characteristic
18	(C₂×D₈₃)⋊C₈	—	2	1	no	—
19	(C₂×D₈₃)⋊C₈	—	2	1	no	—
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×D₃₃₂	C₂×C₄	1	1	yes	characteristic
24	C₂×D₃₃₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₈₃	—	2	1	no	—
26	C₂×C₄×D₈₃	—	2	1	no	—
27	D₆₆₄	—	2	2	no	—
28	#13050573	—	4	2	no	—
29	C₂×C₈₃⋊C₈	—	2	1	no	—
30	C₂×C₈₃⋊C₈	—	2	1	no	—
31	C₈×D₈₃	—	4	2	no	—
32	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
34	(C₂×D₈₃)⋊C₄	—	2	1	no	—
35	(C₂×D₈₃)⋊C₄	—	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	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₄²×C₈₃	C₂³	1	1	yes	characteristic
42	C₂²×D₈₃	—	2	1	no	—
43	C₂²×D₈₃	—	2	1	no	—
44	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂×Dic₈₃	—	2	1	no	—
48	C₂×Dic₈₃	—	2	1	no	—
49	D₃₃₂	—	2	1	no	—
50	D₃₃₂	—	2	1	no	—
51	D₃₃₂	—	2	1	no	—
52	D₃₃₂	M₁₆	1	2	yes	complemented
53	C₄×D₈₃	—	4	1	no	—
54	C₄×D₈₃	—	4	1	no	—
55	C₄×D₈₃	—	4	1	no	—
56	C₄×D₈₃	—	4	1	no	—
57	C₈×C₈₃	—	2	1	no	—
58	C₈×C₈₃	—	2	1	no	—
59	C₈×C₈₃	—	2	1	no	—
60	C₈×C₈₃	—	4	1	no	—
61	C₈×C₈₃	D₈	1	2	yes	complemented
62	C₈₃⋊C₈	—	4	1	no	—
63	C₈₃⋊C₈	—	4	1	no	—
64	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
65	C₂×D₈₃	—	2	2	no	—
66	C₂×D₈₃	—	4	1	no	—
67	C₂×D₈₃	—	4	1	no	—
68	C₂×D₈₃	—	4	1	no	—
69	C₂×D₈₃	—	4	1	no	—
70	C₂×D₈₃	—	4	1	no	—
71	C₄×C₈₃	—	2	1	no	—
72	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₈₃	#131	1	1	yes	characteristic
74	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
75	C₄×C₈₃	#135	1	1	yes	characteristic
76	Dic₈₃	—	4	1	no	—
77	Dic₈₃	—	4	1	no	—
78	C₂×C₈₃	#435	1	1	yes	characteristic
79	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
80	C₂×C₈₃	#444	1	1	yes	characteristic
81	D₈₃	—	4	2	no	—
82	D₈₃	—	8	1	no	—
83	#1581	—	83	1	no	maximal, Sylow
84	C₈₃	#1581	1	1	yes	characteristic, complemented, Sylow
85	C₄×D₈	—	83	1	no	—
86	#436	—	83	1	no	—
87	C₈×D₄	—	83	1	no	—
88	#335	—	83	1	no	—
89	D₄⋊C₈	—	83	1	no	—
90	D₄⋊C₈	—	83	1	no	—
91	#322	D₈₃	1	1	yes	characteristic, complemented
92	C₂×D₈	—	83	1	no	—
93	C₂×M₁₆	—	166	1	no	—
94	C₂²×C₈	—	166	1	no	—
95	C₄×D₄	—	83	1	no	—
96	C₄×D₄	—	83	1	no	—
97	#107	—	83	1	no	—
98	C₄⋊C₈	—	83	1	no	—
99	C₄⋊C₈	—	83	1	no	—
100	#102	—	83	1	no	—
101	#102	—	83	1	no	—
102	C₂²⋊C₈	—	166	1	no	—
103	C₂²⋊C₈	—	166	1	no	—
104	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
105	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
106	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
107	C₂×D₄	—	83	1	no	—
108	C₂×D₄	—	83	1	no	—
109	C₂²×C₄	—	166	1	no	—
110	C₂²×C₄	—	166	1	no	—
111	D₈	—	166	2	no	—
112	M₁₆	—	332	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	332	2	no	—
115	C₂×C₈	—	166	1	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
118	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₂×C₈	D₃₃₂	1	1	yes	characteristic
120	C₂×C₈	D₃₃₂	1	1	yes	characteristic
121	C₄⋊C₄	—	83	1	no	—
122	C₄⋊C₄	—	83	1	no	—
123	C₂²⋊C₄	—	166	1	no	—
124	C₂²⋊C₄	—	166	1	no	—
125	C₄²	C₂²×D₈₃	1	1	yes	characteristic
126	C₂³	—	166	1	no	—
127	C₂³	—	166	1	no	—
128	D₄	—	83	2	no	—
129	D₄	—	166	1	no	—
130	D₄	—	166	1	no	—
131	D₄	—	166	1	no	—
132	C₂×C₄	—	332	1	no	—
133	C₂×C₄	—	332	1	no	—
134	C₂×C₄	—	332	1	no	—
135	C₂×C₄	—	332	1	no	—
136	C₂×C₄	—	166	1	no	—
137	C₂×C₄	—	166	1	no	—
138	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
140	C₂×C₄	#13050590	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	332	1	no	—
146	C₈	—	332	1	no	—
147	C₈	D₆₆₄	1	2	yes	complemented
148	C₂²	—	332	1	no	—
149	C₂²	—	332	1	no	—
150	C₂²	—	332	1	no	—
151	C₂²	—	332	1	no	—
152	C₂²	—	332	1	no	—
153	C₂²	—	166	2	no	—
154	C₂²	C₄×D₃₃₂	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	332	1	no	—
157	C₄	—	332	1	no	—
158	C₄	M₁₆×D₈₃	1	1	yes	characteristic
159	C₄	#14581800	1	1	yes	characteristic
160	C₄	C₂×D₆₆₄	1	1	yes	characteristic
161	C₄	#14581814	1	1	yes	characteristic
162	C₂	—	332	2	no	—
163	C₂	—	664	1	no	—
164	C₂	#19127774	1	1	yes	characteristic
165	C₂	C₄×D₆₆₄	1	1	yes	characteristic
166	C₂	#19127824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₆₆₄

C₂

yes

maximal, characteristic

#19127780

C₂

yes

maximal, characteristic, complemented

C₈×D₃₃₂

C₂

yes

maximal, characteristic, complemented

#19127454

C₂

yes

maximal, characteristic, complemented

#19127430

C₂

yes

maximal, characteristic, complemented

#19127430

C₂

yes

maximal, characteristic, complemented

#322×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×D₆₆₄

C₄

yes

characteristic

C₂×#13050573

—

C₂×C₈×D₈₃

—

C₄×D₃₃₂

C₂²

yes

characteristic

C₄×D₃₃₂

C₂²

yes

characteristic

#14581726

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

#14581714

C₄

yes

characteristic

#14581714

C₄

yes

characteristic

(C₂×D₈₃)⋊C₈

—

(C₂×D₈₃)⋊C₈

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×D₃₃₂

C₂×C₄

yes

characteristic

C₂×D₃₃₂

C₂×C₄

yes

characteristic

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

D₆₆₄

—

#13050573

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

characteristic

C₄⋊Dic₈₃

C₂×C₄

yes

characteristic

(C₂×D₈₃)⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

M₁₆

yes

complemented

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

D₈

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#435

yes

characteristic

C₂×C₈₃

C₄×D₈

yes

characteristic

C₂×C₈₃

#444

yes

characteristic

D₈₃

—

D₈₃

—

#1581

—

maximal, Sylow

C₈₃

#1581

yes

characteristic, complemented, Sylow

C₄×D₈

—

#436

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

D₄⋊C₈

—

#322

D₈₃

yes

characteristic, complemented

C₂×D₈

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

166

—

C₄×D₄

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#102

—

101

#102

—

102

C₂²⋊C₈

—

166

—

103

C₂²⋊C₈

—

166

—

104

C₄×C₈

C₂×D₈₃

yes

characteristic

105

C₄×C₈

C₂×D₈₃

yes

characteristic

106

C₄×C₈

C₂×D₈₃

yes

characteristic

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

166

—

110

C₂²×C₄

—

166

—

111

D₈

—

166

—

112

M₁₆

—

332

—

113

C₂×C₈

—

114

C₂×C₈

—

332

—

115

C₂×C₈

—

166

—

116

C₂×C₈

—

166

—

117

C₂×C₈

C₄×D₈₃

yes

characteristic

118

C₂×C₈

C₄×D₈₃

yes

characteristic

119

C₂×C₈

D₃₃₂

yes

characteristic

120

C₂×C₈

D₃₃₂

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

166

—

124

C₂²⋊C₄

—

166

—

125

C₄²

C₂²×D₈₃

yes

characteristic

126

C₂³

—

166

—

127

C₂³

—

166

—

128

D₄

—

129

D₄

—

166

—

130

D₄

—

166

—

131

D₄

—

166

—

132

C₂×C₄

—

332

—

133

C₂×C₄

—

332

—

134

C₂×C₄

—

332

—

135

C₂×C₄

—

332

—

136

C₂×C₄

—

166

—

137

C₂×C₄

—

166

—

138

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

139

C₂×C₄

C₂×D₃₃₂

yes

characteristic

140

C₂×C₄

#13050590

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

332

—

146

C₈

—

332

—

147

C₈

D₆₆₄

yes

complemented

148

C₂²

—

332

—

149

C₂²

—

332

—

150

C₂²

—

332

—

151

C₂²

—

332

—

152

C₂²

—

332

—

153

C₂²

—

166

—

154

C₂²

C₄×D₃₃₂

yes

characteristic

155

C₄

—

156

C₄

—

332

—

157

C₄

—

332

—

158

C₄

M₁₆×D₈₃

yes

characteristic

159

C₄

#14581800

yes

characteristic

160

C₄

C₂×D₆₆₄

yes

characteristic

161

C₄

#14581814

yes

characteristic

162

C₂

—

332

—

163

C₂

—

664

—

164

C₂

#19127774

yes

characteristic

165

C₂

C₄×D₆₆₄

yes

characteristic

166

C₂

#19127824

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#322×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₆₆₄	C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#322	D₈₃
hypocenter	C₈₃	#1581
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#322×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₆₆₄

C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. #19127780	C₂	semidirect
3. C₈×D₃₃₂	C₂	semidirect
4. #19127454	C₂	semidirect
5. #19127430	C₂	semidirect
6. #19127430	C₂	semidirect
7. #322×C₈₃	C₂	semidirect
52. D₃₃₂	M₁₆	semidirect
61. C₈×C₈₃	D₈	semidirect
84. C₈₃	#1581	semidirect
91. #322	D₈₃	semidirect
147. C₈	D₆₆₄	semidirect

H ≅ G/N

type

2. #19127780

C₂

semidirect

3. C₈×D₃₃₂

C₂

semidirect

4. #19127454

C₂

semidirect

5. #19127430

C₂

semidirect

6. #19127430

C₂

semidirect

7. #322×C₈₃

C₂

semidirect

52. D₃₃₂

M₁₆

semidirect

61. C₈×C₈₃

D₈

semidirect

84. C₈₃

#1581

semidirect

91. #322

D₈₃

semidirect

147. C₈

D₆₆₄

semidirect

Subgroups — #24906294

Conjugacy classes of subgroups

Complements