Group #24908996 — subgroups

Subgroups — #24908996

Order 10624.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#14581723	C₂	1	2	yes	maximal, complemented
2	C₂×#14581713	C₂	1	2	yes	maximal
3	#19127571	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127456	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	—	2	2	no	—
8	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
9	#14581723	—	2	4	no	—
10	#14581713	—	2	4	no	—
11	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₃	—	2	2	no	—
16	C₂×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₈₃	—	2	2	no	—
20	C₄⋊Dic₈₃	—	2	2	no	—
21	C₄⋊Dic₈₃	—	4	4	no	—
22	C₂⁴×C₈₃	—	2	1	no	—
23	C₂×D₄×C₈₃	—	2	2	no	—
24	C₂×D₄×C₈₃	—	2	2	no	—
25	C₂×D₄×C₈₃	—	2	1	no	—
26	C₂×D₄×C₈₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₈₃	—	2	1	no	—
28	C₂²×C₄×C₈₃	D₄	1	2	yes	—
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₂²⋊C₄×C₈₃	—	4	4	no	—
32	C₄²×C₈₃	—	2	2	no	—
33	C₂³×C₈₃	—	2	2	no	—
34	C₂³×C₈₃	—	4	1	no	—
35	C₂³×C₈₃	—	4	1	no	—
36	C₂³×C₈₃	—	4	2	no	—
37	C₂³×C₈₃	—	4	2	no	—
38	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
39	D₄×C₈₃	—	2	4	no	—
40	D₄×C₈₃	—	4	2	no	—
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	—	4	2	no	—
45	C₂×C₄×C₈₃	SD₁₆	1	4	yes	—
46	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₈₃	#41	1	2	yes	—
49	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
50	C₂×Dic₈₃	—	4	2	no	—
51	C₂×Dic₈₃	—	4	4	no	—
52	C₈₃⋊C₈	—	2	4	no	—
53	C₂²×C₈₃	—	4	2	no	—
54	C₂²×C₈₃	—	4	4	no	—
55	C₂²×C₈₃	—	8	2	no	—
56	C₂²×C₈₃	—	8	1	no	—
57	C₂²×C₈₃	—	8	1	no	—
58	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₈₃	#123	1	1	yes	characteristic
60	C₂²×C₈₃	#127	1	1	yes	characteristic
61	C₂²×C₈₃	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₈₃	—	2	4	no	—
64	C₄×C₈₃	—	8	1	no	—
65	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₈₃	#123	1	2	yes	—
67	Dic₈₃	—	8	2	no	—
68	C₂×C₈₃	—	8	2	no	—
69	C₂×C₈₃	#399	1	1	yes	characteristic
70	C₂×C₈₃	#476	1	2	yes	—
71	C₂×C₈₃	#481	1	2	yes	—
72	C₂×C₈₃	#492	1	1	yes	characteristic
73	C₂×C₈₃	#496	1	1	yes	characteristic
74	#2094	—	83	1	no	maximal, Sylow
75	C₈₃	#2094	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	83	2	no	—
77	C₂×#102	—	83	2	no	—
78	#390	D₈₃	1	1	yes	characteristic
79	#384	—	83	1	no	—
80	#336	—	83	1	no	—
81	C₂²×D₄	C₂×D₈₃	1	1	yes	characteristic
82	C₂²×C₈	—	83	2	no	—
83	C₂×C₄⋊C₄	—	83	2	no	—
84	C₂×C₄⋊C₄	—	166	2	no	—
85	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
88	#106	—	166	4	no	—
89	#102	—	166	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	Dic₁₆₆	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	166	2	no	—
97	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
99	C₂×C₈	—	83	4	no	—
100	C₂×C₈	—	166	2	no	—
101	C₂×C₈	—	166	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	332	4	no	—
104	C₄⋊C₄	—	166	2	no	—
105	C₄⋊C₄	—	166	2	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	332	2	no	—
121	C₂×C₄	—	332	4	no	—
122	C₂×C₄	#13050578	1	4	yes	—
123	C₂×C₄	D₄×D₈₃	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
125	C₂×C₄	#13050589	1	2	yes	—
126	C₂×C₄	#13050590	1	1	yes	characteristic
127	C₈	—	166	4	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	2	no	—
133	C₂²	#14581771	1	1	yes	characteristic
134	C₂²	#14581773	1	1	yes	characteristic
135	C₂²	#14581788	1	1	yes	characteristic
136	C₂²	C₂×#13050578	1	2	yes	—
137	C₂²	#14581817	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	664	2	no	—
141	C₄	#14581769	1	2	yes	—
142	C₄	#14581774	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#19127637	1	1	yes	characteristic
145	C₂	#19127991	1	2	yes	—
146	C₂	#19128017	1	2	yes	—
147	C₂	#19128073	1	1	yes	characteristic
148	C₂	#19128091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581723

C₂

yes

maximal, complemented

C₂×#14581713

C₂

yes

maximal

#19127571

C₂

yes

maximal, characteristic, complemented

#19127456

C₂

yes

maximal, characteristic, complemented

#390×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581723

—

#14581713

—

C₂²×D₄×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂⁴×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊Q₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₂×SD₁₆

yes

—

C₂²×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#123

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#399

yes

characteristic

C₂×C₈₃

#476

yes

—

C₂×C₈₃

#481

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#102

—

#390

D₈₃

yes

characteristic

#384

—

#336

—

C₂²×D₄

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#106

—

166

—

#102

—

166

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₆₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

166

—

101

C₂×C₈

—

166

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

332

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

332

—

121

C₂×C₄

—

332

—

122

C₂×C₄

#13050578

yes

—

123

C₂×C₄

D₄×D₈₃

yes

—

124

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

125

C₂×C₄

#13050589

yes

—

126

C₂×C₄

#13050590

yes

characteristic

127

C₈

—

166

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#14581771

yes

characteristic

134

C₂²

#14581773

yes

characteristic

135

C₂²

#14581788

yes

characteristic

136

C₂²

C₂×#13050578

yes

—

137

C₂²

#14581817

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

664

—

141

C₄

#14581769

yes

—

142

C₄

#14581774

yes

—

143

C₂

—

144

C₂

#19127637

yes

characteristic

145

C₂

#19127991

yes

—

146

C₂

#19128017

yes

—

147

C₂

#19128073

yes

characteristic

148

C₂

#19128091

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#390×C₈₃	C₂
socle	C₂³×C₈₃	C₂×D₄
prime radical	C₂²×D₄×C₈₃	C₂²
hypercenter	#390	D₈₃
hypocenter	C₈₃	#2094
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#390×C₈₃

C₂

socle

C₂³×C₈₃

C₂×D₄

prime radical

C₂²×D₄×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#14581723	C₂	semidirect
3. #19127571	C₂	semidirect
4. #19127456	C₂	semidirect
75. C₈₃	#2094	semidirect

H ≅ G/N

type

1. C₂×#14581723

C₂

semidirect

3. #19127571

C₂

semidirect

4. #19127456

C₂

semidirect

75. C₈₃

#2094

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14581723	2
C₂×#14581713	2
#19127571	2
#19127456	2
#390×C₈₃	2
#2094	83

subgroup

index

#390×C₈₃

Subgroups — #24908996

Conjugacy classes of subgroups

Complements

Maximal subgroups