Group #24906280 — subgroups

Subgroups — #24906280

Order 10624.

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

normal subgroups only

This group has 160 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	C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#13050580	C₂	1	1	yes	maximal, characteristic
3	#19127451	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#19127431	C₂	1	1	yes	maximal, characteristic
6	#19127423	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050580	C₄	1	1	yes	characteristic
9	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#14581725	C₄	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₈	—	2	1	no	—
13	#14581716	C₄	1	1	yes	characteristic
14	#14581715	C₄	1	1	yes	characteristic
15	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₈×Dic₈₃	—	2	1	no	—
18	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₃	—	2	1	no	—
20	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₈₃	—	2	1	no	—
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
24	#13050580	C₈	1	4	yes	complemented
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×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₃	—	2	1	no	—
34	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₃	—	2	1	no	—
37	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₈₃	—	2	1	no	—
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	Q₈×C₈₃	—	2	1	no	—
41	Q₈×C₈₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₈₃	—	2	1	no	—
43	C₂×C₄×C₈₃	—	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	Dic₁₆₆	—	2	1	no	—
50	Dic₁₆₆	C₂×C₈	1	2	yes	—
51	C₈×C₈₃	—	2	1	no	—
52	C₈×C₈₃	—	2	1	no	—
53	C₈×C₈₃	—	4	1	no	—
54	C₈₃⋊C₈	—	2	1	no	—
55	C₈₃⋊C₈	—	4	1	no	—
56	C₈₃⋊C₈	—	4	1	no	—
57	C₈₃⋊C₈	C₂×C₈	1	2	yes	—
58	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	4	1	no	—
62	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₈₃	#131	1	1	yes	characteristic
64	C₄×C₈₃	#137	1	1	yes	characteristic
65	C₄×C₈₃	#137	1	1	yes	characteristic
66	Dic₈₃	—	2	2	no	—
67	Dic₈₃	—	4	1	no	—
68	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
69	C₂×C₈₃	#441	1	1	yes	characteristic
70	C₂×C₈₃	#444	1	1	yes	characteristic
71	#1579	—	83	1	no	maximal, Sylow
72	C₈₃	#1579	1	1	yes	characteristic, complemented, Sylow
73	C₈×Q₈	—	83	1	no	—
74	C₈×Q₈	D₈₃	1	1	yes	characteristic
75	C₄×Q₁₆	—	83	1	no	—
76	#334	—	83	1	no	—
77	Q₈⋊C₈	—	83	1	no	—
78	Q₈⋊C₈	—	83	1	no	—
79	#322	—	83	1	no	—
80	C₂×Q₁₆	—	83	1	no	—
81	C₄×Q₈	—	83	1	no	—
82	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
83	#107	—	83	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	83	1	no	—
86	C₄⋊C₈	—	166	1	no	—
87	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
88	#103	—	83	1	no	—
89	#103	—	83	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	83	1	no	—
92	C₄×C₈	—	83	1	no	—
93	C₄×C₈	—	166	1	no	—
94	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
95	C₂×Q₈	—	83	1	no	—
96	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
97	Q₁₆	—	83	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	83	1	no	—
100	C₂×C₈	—	83	1	no	—
101	C₂×C₈	—	166	1	no	—
102	C₂×C₈	—	166	1	no	—
103	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	83	1	no	—
107	C₄⋊C₄	—	166	1	no	—
108	C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	166	1	no	—
111	C₄²	C₂²×D₈₃	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	83	2	no	—
114	Q₈	—	166	1	no	—
115	Q₈	C₈×D₈₃	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	166	1	no	—
119	C₂×C₄	—	166	1	no	—
120	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
122	C₂×C₄	#13050590	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	332	1	no	—
127	C₈	—	332	1	no	—
128	C₈	—	83	2	no	—
129	C₈	—	166	1	no	—
130	C₂²	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	1	no	—
134	C₄	—	332	1	no	—
135	C₄	—	166	2	no	—
136	C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
137	C₄	#14581798	1	1	yes	characteristic
138	C₄	#14581822	1	1	yes	characteristic
139	C₄	#14581823	1	1	yes	characteristic
140	C₂	C₈×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂	#19127806	1	1	yes	characteristic
142	C₂	#19127821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₆₆

C₂

yes

maximal, characteristic

C₄×#13050580

C₂

yes

maximal, characteristic

#19127451

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127431

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, characteristic

C₈×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050580

C₄

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581725

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581716

C₄

yes

characteristic

#14581715

C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₈₃

—

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

C₂×C₄

yes

characteristic

#13050580

C₈

yes

complemented

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×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

C₂×C₈

yes

—

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

C₂×C₈

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂×C₈

yes

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×C₈

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₈×D₄

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₈×Q₈

D₈₃

yes

characteristic

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₈₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

166

—

102

C₂×C₈

—

166

—

103

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

104

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

166

—

108

C₄⋊C₄

C₄×D₈₃

yes

characteristic

109

C₄²

—

110

C₄²

—

166

—

111

C₄²

C₂²×D₈₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

166

—

115

Q₈

C₈×D₈₃

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

166

—

119

C₂×C₄

—

166

—

120

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

122

C₂×C₄

#13050590

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

332

—

127

C₈

—

332

—

128

C₈

—

129

C₈

—

166

—

130

C₂²

C₄×C₂²⋊D₈₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

332

—

135

C₄

—

166

—

136

C₄

C₂×C₈×D₈₃

yes

characteristic

137

C₄

#14581798

yes

characteristic

138

C₄

#14581822

yes

characteristic

139

C₄

#14581823

yes

characteristic

140

C₂

C₈×C₂²⋊D₈₃

yes

characteristic

141

C₂

#19127806

yes

characteristic

142

C₂

#19127821

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	C₈×Q₈×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₈×Q₈	D₈₃
hypocenter	C₈₃	#1579
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

C₈×Q₈×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₈×Q₈

D₈₃

hypocenter

C₈₃

#1579

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
24. #13050580	C₈	semidirect
72. C₈₃	#1579	semidirect

H ≅ G/N

type

24. #13050580

C₈

semidirect

72. C₈₃

#1579

semidirect

Subgroups — #24906280

Conjugacy classes of subgroups

Complements