Group #24906170 — subgroups

Subgroups — #24906170

Order 10624.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19128256	C₂	1	1	yes	maximal, characteristic
2	#19127757	C₂	1	1	yes	maximal, characteristic
3	#19127753	C₂	1	1	yes	maximal, characteristic
4	#19127449	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127449	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127447	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581792	C₂²	1	1	yes	characteristic
9	#14581783	—	2	1	no	—
10	#14581772	C₄	1	2	yes	—
11	#14581768	—	2	1	no	—
12	Q₈×Dic₈₃	—	2	1	no	—
13	#14581757	C₄	1	2	yes	—
14	#14581750	—	2	1	no	—
15	C₄²×D₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₈₃)⋊C₈	—	2	1	no	—
19	(C₂×D₈₃)⋊C₈	—	2	1	no	—
20	#14581702	—	2	1	no	—
21	C₈×Dic₈₃	—	2	1	no	—
22	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
29	C₂×C₈₃⋊C₈	—	2	1	no	—
30	C₂×C₈₃⋊C₈	—	2	1	no	—
31	C₄⋊Dic₈₃	—	2	1	no	—
32	C₄⋊Dic₈₃	—	2	1	no	—
33	C₄⋊Dic₈₃	—	2	2	no	—
34	Dic₈₃⋊C₄	—	2	2	no	—
35	(C₂×D₈₃)⋊C₄	—	2	2	no	—
36	C₄×Dic₈₃	—	2	2	no	—
37	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₈₃	D₄	1	1	yes	characteristic
39	C₄×Dic₈₃	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	—	2	1	no	—
42	C₂×C₈×C₈₃	—	2	1	no	—
43	C₄⋊C₄×C₈₃	—	2	1	no	—
44	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
45	C₄²×C₈₃	C₂³	1	1	yes	characteristic
46	C₂²×D₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₈₃	—	4	1	no	—
48	Q₈×C₈₃	—	4	1	no	—
49	C₂×C₄×C₈₃	—	2	2	no	—
50	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₈₃	—	2	2	no	—
54	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₈₃	—	2	1	no	—
58	C₄×D₈₃	—	2	1	no	—
59	C₄×D₈₃	—	2	1	no	—
60	C₄×D₈₃	—	2	1	no	—
61	C₄×D₈₃	—	4	1	no	—
62	Dic₁₆₆	—	4	2	no	—
63	C₈×C₈₃	—	4	1	no	—
64	C₈×C₈₃	—	4	1	no	—
65	C₈₃⋊C₈	—	4	1	no	—
66	C₈₃⋊C₈	—	4	1	no	—
67	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₈₃	—	2	1	no	—
69	C₂×D₈₃	—	2	1	no	—
70	C₂×D₈₃	—	2	1	no	—
71	C₄×C₈₃	—	2	1	no	—
72	C₄×C₈₃	—	4	2	no	—
73	C₄×C₈₃	#135	1	1	yes	characteristic
74	C₄×C₈₃	#135	1	1	yes	characteristic
75	C₄×C₈₃	#137	1	1	yes	characteristic
76	C₄×C₈₃	#137	1	1	yes	characteristic
77	Dic₈₃	—	2	1	no	—
78	Dic₈₃	—	2	1	no	—
79	Dic₈₃	—	2	1	no	—
80	Dic₈₃	—	4	2	no	—
81	C₂×C₈₃	#413	1	1	yes	characteristic
82	C₂×C₈₃	#416	1	1	yes	characteristic
83	C₂×C₈₃	#419	1	1	yes	characteristic
84	D₈₃	—	4	1	no	—
85	#1554	—	83	1	no	maximal, Sylow
86	C₈₃	#1554	1	1	yes	characteristic, complemented, Sylow
87	#533	—	83	1	no	—
88	#432	—	83	1	no	—
89	#431	—	83	1	no	—
90	#333	—	83	1	no	—
91	#333	—	83	1	no	—
92	#332	—	83	1	no	—
93	#332	D₈₃	1	1	yes	characteristic, complemented
94	C₄⋊Q₈	—	83	1	no	—
95	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
96	#125	—	166	1	no	—
97	C₂²⋊Q₈	—	83	2	no	—
98	C₂²⋊Q₈	—	166	1	no	—
99	C₄×Q₈	—	83	2	no	—
100	C₄×Q₈	—	166	1	no	—
101	#117	—	166	1	no	—
102	C₂×C₄²	—	83	1	no	—
103	C₄⋊C₈	—	83	1	no	—
104	C₄⋊C₈	—	83	1	no	—
105	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
107	C₂²⋊C₈	—	166	1	no	—
108	C₂²⋊C₈	—	166	1	no	—
109	#97	—	166	1	no	—
110	C₄×C₈	—	166	1	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	83	2	no	—
113	C₂²×C₄	—	83	1	no	—
114	C₂²×C₄	—	83	1	no	—
115	C₂²×C₄	—	83	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	166	1	no	—
119	C₂×C₈	—	166	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	166	2	no	—
122	C₄⋊C₄	—	166	2	no	—
123	C₄⋊C₄	—	166	1	no	—
124	C₄⋊C₄	—	166	1	no	—
125	C₄⋊C₄	C₄×D₈₃	1	2	yes	—
126	C₂²⋊C₄	—	166	2	no	—
127	C₄²	—	83	1	no	—
128	C₄²	—	83	1	no	—
129	C₄²	—	83	1	no	—
130	C₄²	—	166	2	no	—
131	C₄²	C₂²×D₈₃	1	1	yes	characteristic
132	C₂³	—	83	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	332	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	332	1	no	—
138	C₂×C₄	—	83	1	no	—
139	C₂×C₄	—	83	1	no	—
140	C₂×C₄	—	83	1	no	—
141	C₂×C₄	—	166	2	no	—
142	C₂×C₄	—	166	1	no	—
143	C₂×C₄	—	166	1	no	—
144	C₂×C₄	—	166	1	no	—
145	C₂×C₄	—	166	1	no	—
146	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	332	1	no	—
152	C₈	—	332	1	no	—
153	C₂²	—	166	1	no	—
154	C₂²	—	166	1	no	—
155	C₂²	—	166	1	no	—
156	C₂²	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	332	2	no	—
160	C₄	—	166	1	no	—
161	C₄	—	166	1	no	—
162	C₄	—	166	1	no	—
163	C₄	#14581811	1	1	yes	characteristic
164	C₄	#14581813	1	1	yes	characteristic
165	C₄	#14581821	1	1	yes	characteristic
166	C₄	#14581824	1	1	yes	characteristic
167	C₂	—	332	1	no	—
168	C₂	#19127684	1	1	yes	characteristic
169	C₂	#19127694	1	1	yes	characteristic
170	C₂	#19127708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128256

C₂

yes

maximal, characteristic

#19127757

C₂

yes

maximal, characteristic

#19127753

C₂

yes

maximal, characteristic

#19127449

C₂

yes

maximal, characteristic, complemented

#19127449

C₂

yes

maximal, characteristic, complemented

#19127447

C₂

yes

maximal, characteristic, complemented

#332×C₈₃

C₂

yes

maximal, characteristic, complemented

#14581792

C₂²

yes

characteristic

#14581783

—

#14581772

C₄

yes

—

#14581768

—

Q₈×Dic₈₃

—

#14581757

C₄

yes

—

#14581750

—

C₄²×D₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

(C₂×D₈₃)⋊C₈

—

(C₂×D₈₃)⋊C₈

—

#14581702

—

C₈×Dic₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂×C₄×D₈₃

C₂×C₄

yes

characteristic

C₂×C₄×D₈₃

D₄

yes

characteristic

C₂×C₄×D₈₃

D₄

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

C₂²⋊C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₃

C₂²⋊C₄

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1554

—

maximal, Sylow

C₈₃

#1554

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₈₃

yes

characteristic, complemented

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

#125

—

166

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

100

C₄×Q₈

—

166

—

101

#117

—

166

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

C₂×D₈₃

yes

characteristic

106

C₄⋊C₈

C₂×D₈₃

yes

characteristic

107

C₂²⋊C₈

—

166

—

108

C₂²⋊C₈

—

166

—

109

#97

—

166

—

110

C₄×C₈

—

166

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

166

—

119

C₂×C₈

—

166

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

166

—

122

C₄⋊C₄

—

166

—

123

C₄⋊C₄

—

166

—

124

C₄⋊C₄

—

166

—

125

C₄⋊C₄

C₄×D₈₃

yes

—

126

C₂²⋊C₄

—

166

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

166

—

131

C₄²

C₂²×D₈₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

332

—

136

C₂×C₄

—

137

C₂×C₄

—

332

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

166

—

142

C₂×C₄

—

166

—

143

C₂×C₄

—

166

—

144

C₂×C₄

—

166

—

145

C₂×C₄

—

166

—

146

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

147

C₂×C₄

D₄×D₈₃

yes

characteristic

148

C₂×C₄

D₄×D₈₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

332

—

152

C₈

—

332

—

153

C₂²

—

166

—

154

C₂²

—

166

—

155

C₂²

—

166

—

156

C₂²

C₂²⋊C₄×D₈₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

332

—

160

C₄

—

166

—

161

C₄

—

166

—

162

C₄

—

166

—

163

C₄

#14581811

yes

characteristic

164

C₄

#14581813

yes

characteristic

165

C₄

#14581821

yes

characteristic

166

C₄

#14581824

yes

characteristic

167

C₂

—

332

—

168

C₂

#19127684

yes

characteristic

169

C₂

#19127694

yes

characteristic

170

C₂

#19127708

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#332×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#332	D₈₃
hypocenter	C₈₃	#1554
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#332×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #19127449	C₂	semidirect
5. #19127449	C₂	semidirect
6. #19127447	C₂	semidirect
7. #332×C₈₃	C₂	semidirect
86. C₈₃	#1554	semidirect
93. #332	D₈₃	semidirect

H ≅ G/N

type

4. #19127449

C₂

semidirect

5. #19127449

C₂

semidirect

6. #19127447

C₂

semidirect

7. #332×C₈₃

C₂

semidirect

86. C₈₃

#1554

semidirect

93. #332

D₈₃

semidirect

Subgroups — #24906170

Conjugacy classes of subgroups

Complements