Group #24906320 — subgroups

Subgroups — #24906320

Order 10624.

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

normal subgroups only

This group has 168 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	#19127831	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13050578	C₂	1	1	yes	maximal, characteristic
3	#19127453	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127431	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#19127423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050578	C₄	1	1	yes	characteristic
9	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#14581723	C₄	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₈	—	2	1	no	—
13	#14581715	C₄	1	1	yes	characteristic
14	#14581713	C₄	1	1	yes	characteristic
15	#14581702	—	2	1	no	—
16	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₃	—	2	1	no	—
19	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₈₃	—	2	1	no	—
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
24	#13050578	—	2	2	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	—	2	1	no	—
27	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
30	Dic₈₃⋊C₄	—	2	1	no	—
31	C₄×Dic₈₃	—	2	1	no	—
32	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₃	—	2	1	no	—
34	C₂×C₈×C₈₃	—	2	1	no	—
35	C₂×C₈×C₈₃	—	4	2	no	—
36	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₈₃	—	2	1	no	—
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	—	2	1	no	—
42	D₄×C₈₃	—	2	1	no	—
43	D₄×C₈₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	4	1	no	—
46	C₂×C₄×C₈₃	—	4	1	no	—
47	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	1	no	—
51	C₂×Dic₈₃	—	2	1	no	—
52	Dic₁₆₆	—	2	1	no	—
53	Dic₁₆₆	—	2	1	no	—
54	C₈×C₈₃	—	2	1	no	—
55	C₈×C₈₃	—	2	1	no	—
56	C₈×C₈₃	—	4	1	no	—
57	C₈₃⋊C₈	—	2	2	no	—
58	C₈₃⋊C₈	—	2	1	no	—
59	C₈₃⋊C₈	—	2	1	no	—
60	C₈₃⋊C₈	—	4	1	no	—
61	C₂²×C₈₃	—	2	2	no	—
62	C₂²×C₈₃	—	4	1	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₈₃	—	2	1	no	—
66	C₄×C₈₃	—	4	1	no	—
67	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₈₃	#131	1	1	yes	characteristic
69	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₈₃	#137	1	1	yes	characteristic
71	Dic₈₃	—	4	1	no	—
72	Dic₈₃	—	4	1	no	—
73	C₂×C₈₃	—	4	2	no	—
74	C₂×C₈₃	#435	1	1	yes	characteristic
75	C₂×C₈₃	#440	1	1	yes	characteristic
76	C₂×C₈₃	#443	1	1	yes	characteristic
77	#1585	—	83	1	no	maximal, Sylow
78	C₈₃	#1585	1	1	yes	characteristic, complemented, Sylow
79	#446	—	83	1	no	—
80	C₄×SD₁₆	—	83	1	no	—
81	C₈×D₄	D₈₃	1	1	yes	characteristic
82	#335	—	83	1	no	—
83	Q₈⋊C₈	—	83	1	no	—
84	D₄⋊C₈	—	83	1	no	—
85	#322	—	83	1	no	—
86	C₂×SD₁₆	—	83	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₄×Q₈	—	83	1	no	—
89	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
90	#106	—	83	1	no	—
91	C₄⋊C₈	—	83	1	no	—
92	C₄⋊C₈	—	166	1	no	—
93	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
94	#103	—	83	1	no	—
95	#102	—	83	1	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	166	1	no	—
98	C₄×C₈	—	83	1	no	—
99	C₄×C₈	—	83	1	no	—
100	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
101	C₂×Q₈	—	83	1	no	—
102	C₂×D₄	C₄×D₈₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	166	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	4	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	C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
113	C₄⋊C₄	—	83	1	no	—
114	C₄⋊C₄	—	166	1	no	—
115	C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	166	1	no	—
118	C₄²	C₂²×D₈₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	166	1	no	—
121	Q₈	—	166	1	no	—
122	D₄	—	2	1	no	—
123	D₄	#13050573	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	166	1	no	—
128	C₂×C₄	—	166	1	no	—
129	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₂×C₄	#13050590	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	332	1	no	—
136	C₈	—	166	2	no	—
137	C₈	—	166	1	no	—
138	C₈	—	166	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	332	1	no	—
146	C₄	—	332	1	no	—
147	C₄	C₂×#13050573	1	1	yes	characteristic
148	C₄	#14581798	1	1	yes	characteristic
149	C₄	#14581818	1	1	yes	characteristic
150	C₄	#14581822	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#19127772	1	1	yes	characteristic
153	C₂	#19127801	1	1	yes	characteristic
154	C₂	#19127816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127831

C₂

yes

maximal, characteristic, complemented

C₄×#13050578

C₂

yes

maximal, characteristic

#19127453

C₂

yes

maximal, characteristic, complemented

#19127431

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050578

C₄

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581723

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581715

C₄

yes

characteristic

#14581713

C₄

yes

characteristic

#14581702

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈×C₈₃

—

C₄×D₄×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

C₂×C₄

yes

characteristic

#13050578

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₈₃

C₂×C₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

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

#446

—

C₄×SD₁₆

—

C₈×D₄

D₈₃

yes

characteristic

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

C₂×D₈₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

#97

—

166

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₈₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

C₄×D₈₃

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

166

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

166

—

110

C₂×C₈

—

166

—

111

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

112

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

C₄×D₈₃

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

166

—

118

C₄²

C₂²×D₈₃

yes

characteristic

119

C₂³

—

120

Q₈

—

166

—

121

Q₈

—

166

—

122

D₄

—

123

D₄

#13050573

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

166

—

128

C₂×C₄

—

166

—

129

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

131

C₂×C₄

#13050590

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

332

—

136

C₈

—

166

—

137

C₈

—

166

—

138

C₈

—

166

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₄×C₂²⋊D₈₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

332

—

146

C₄

—

332

—

147

C₄

C₂×#13050573

yes

characteristic

148

C₄

#14581798

yes

characteristic

149

C₄

#14581818

yes

characteristic

150

C₄

#14581822

yes

characteristic

151

C₂

—

152

C₂

#19127772

yes

characteristic

153

C₂

#19127801

yes

characteristic

154

C₂

#19127816

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₈×D₄×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₄×C₈₃	C₂×C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₈×D₄	D₈₃
hypocenter	C₈₃	#1585
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₄

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₈×D₄×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₄×C₈₃

C₂×C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₈×D₄

D₈₃

hypocenter

C₈₃

#1585

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. #19127831	C₂	semidirect
3. #19127453	C₂	semidirect
4. #19127431	C₂	semidirect
6. #19127423	C₂	semidirect
78. C₈₃	#1585	semidirect

H ≅ G/N

type

1. #19127831

C₂

semidirect

3. #19127453

C₂

semidirect

4. #19127431

C₂

semidirect

6. #19127423

C₂

semidirect

78. C₈₃

#1585

semidirect

Subgroups — #24906320

Conjugacy classes of subgroups

Complements