Group #24908751 — subgroups

Subgroups — #24908751

Order 10624.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19128501	C₂	1	1	yes	maximal, characteristic
2	#19127724	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14581718	C₂	1	2	yes	maximal, complemented
4	#19127683	C₂	1	2	yes	maximal, complemented
5	#338×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050590	C₂²	1	2	yes	—
7	#14581825	—	2	2	no	—
8	#14581820	—	2	2	no	—
9	#14581814	—	2	2	no	—
10	C₂×#13050574	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₈	—	2	2	no	—
13	#14581718	—	2	2	no	—
14	#14581718	—	2	2	no	—
15	#14581710	—	2	2	no	—
16	#14581708	—	2	2	no	—
17	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	#13050590	—	2	2	no	—
20	#13050590	—	2	2	no	—
21	#13050590	—	2	2	no	—
22	#13050590	—	2	2	no	—
23	C₂×Dic₁₆₆	D₄	1	2	yes	—
24	C₂×D₃₃₂	D₄	1	2	yes	—
25	C₂×C₂²⋊D₈₃	—	2	2	no	—
26	C₂²×Dic₈₃	—	2	1	no	—
27	C₂×C₄×D₈₃	—	2	2	no	—
28	Dic₃₃₂	—	2	2	no	—
29	#13050579	—	2	4	no	—
30	D₆₆₄	—	2	2	no	—
31	#13050574	—	2	2	no	—
32	C₂×C₈₃⋊C₈	—	2	1	no	—
33	C₄×Dic₈₃	—	2	1	no	—
34	C₄×Dic₈₃	D₄	1	1	yes	characteristic
35	C₄×Dic₈₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₈₃	—	2	2	no	—
38	M₁₆×C₈₃	—	2	2	no	—
39	M₁₆×C₈₃	—	2	2	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	D₄	1	2	yes	—
42	C₂³×C₈₃	#41	1	1	yes	characteristic
43	C₂²×D₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	#41	1	2	yes	—
48	C₂²⋊D₈₃	—	4	2	no	—
49	C₂²⋊D₈₃	—	4	2	no	—
50	C₂×Dic₈₃	—	2	1	no	—
51	C₂×Dic₈₃	—	2	1	no	—
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	4	2	no	—
54	D₃₃₂	—	2	2	no	—
55	D₃₃₂	—	2	2	no	—
56	C₄×D₈₃	—	4	2	no	—
57	C₄×D₈₃	—	4	2	no	—
58	Dic₁₆₆	—	2	2	no	—
59	Dic₁₆₆	—	2	2	no	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	—	4	2	no	—
62	C₈₃⋊C₈	—	4	1	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	#121	1	1	yes	characteristic
65	C₂²×C₈₃	#123	1	2	yes	—
66	C₂×D₈₃	—	4	2	no	—
67	C₂×D₈₃	—	8	2	no	—
68	C₄×C₈₃	#120	1	1	yes	characteristic
69	C₄×C₈₃	#121	1	2	yes	—
70	C₄×C₈₃	#124	1	1	yes	characteristic
71	Dic₈₃	—	2	2	no	—
72	Dic₈₃	—	4	1	no	—
73	Dic₈₃	—	4	2	no	—
74	C₂×C₈₃	—	2	1	no	—
75	C₂×C₈₃	—	2	2	no	—
76	C₂×C₈₃	#394	1	1	yes	characteristic
77	D₈₃	—	8	2	no	—
78	#2050	—	83	1	no	maximal, Sylow
79	C₈₃	#2050	1	1	yes	characteristic, complemented, Sylow
80	#575	—	83	1	no	—
81	#423	—	83	1	no	—
82	C₂×#104	—	83	2	no	—
83	#413	—	83	2	no	—
84	#338	D₈₃	1	1	yes	characteristic, complemented
85	C₂×#41	—	83	2	no	—
86	#137	—	166	2	no	—
87	AGL(1,ℤ/8)	—	166	2	no	—
88	#135	—	166	2	no	—
89	C₂×M₁₆	—	83	1	no	—
90	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
91	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
92	C₂×C₄²	—	83	1	no	—
93	C₄⋊C₈	—	166	2	no	—
94	#104	—	166	2	no	—
95	#104	—	166	2	no	—
96	#101	—	166	2	no	—
97	#100	—	166	2	no	—
98	#41	—	166	2	no	—
99	#41	—	166	2	no	—
100	#41	—	166	2	no	—
101	#41	—	166	2	no	—
102	C₂×Q₈	—	83	2	no	—
103	C₂×D₄	—	83	2	no	—
104	C₂×D₄	—	166	2	no	—
105	C₂²×C₄	—	166	1	no	—
106	C₂²×C₄	—	166	2	no	—
107	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
108	Q₁₆	—	166	2	no	—
109	SD₁₆	—	166	4	no	—
110	D₈	—	166	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	166	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	D₃₃₂	1	2	yes	—
118	C₄²	—	83	1	no	—
119	C₄²	—	83	1	no	—
120	C₄²	—	166	1	no	—
121	C₂³	—	166	2	no	—
122	C₂³	#13050591	1	1	yes	characteristic
123	Q₈	—	166	2	no	—
124	Q₈	—	166	2	no	—
125	D₄	—	332	2	no	—
126	D₄	—	332	2	no	—
127	D₄	—	166	2	no	—
128	D₄	—	166	2	no	—
129	C₂×C₄	—	332	2	no	—
130	C₂×C₄	—	332	2	no	—
131	C₂×C₄	—	332	2	no	—
132	C₂×C₄	—	166	1	no	—
133	C₂×C₄	—	166	2	no	—
134	C₂×C₄	—	166	1	no	—
135	C₂×C₄	D₄×D₈₃	1	2	yes	—
136	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
138	C₂×C₄	#13050590	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	332	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	332	2	no	—
144	C₂²	—	664	2	no	—
145	C₂²	#14581767	1	1	yes	characteristic
146	C₂²	#14581777	1	2	yes	—
147	C₄	—	332	1	no	—
148	C₄	—	332	2	no	—
149	C₄	—	166	2	no	—
150	C₄	#14581762	1	1	yes	characteristic
151	C₄	#14581765	1	2	yes	—
152	C₄	#14581780	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	664	2	no	—
156	C₂	#19127619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128501

C₂

yes

maximal, characteristic

#19127724

C₂

yes

maximal, characteristic, complemented

C₂×#14581718

C₂

yes

maximal, complemented

#19127683

C₂

yes

maximal, complemented

#338×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

—

—

—

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

—

—

—

—

C₂×Dic₁₆₆

D₄

yes

—

C₂×D₃₃₂

D₄

yes

—

C₂×C₂²⋊D₈₃

—

C₂²×Dic₈₃

—

C₂×C₄×D₈₃

—

Dic₃₃₂

—

#13050579

—

D₆₆₄

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂²⋊D₈₃

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#121

yes

—

C₄×C₈₃

#124

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#394

yes

characteristic

D₈₃

—

#2050

—

maximal, Sylow

C₈₃

#2050

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×#41

—

#137

—

166

—

AGL(1,ℤ/8)

—

166

—

#135

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

166

—

#104

—

166

—

#104

—

166

—

#101

—

166

—

#100

—

166

—

#41

—

166

—

#41

—

166

—

100

#41

—

166

—

101

#41

—

166

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

166

—

105

C₂²×C₄

—

166

—

106

C₂²×C₄

—

166

—

107

C₂²×C₄

C₂²×D₈₃

yes

characteristic

108

Q₁₆

—

166

—

109

SD₁₆

—

166

—

110

D₈

—

166

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

166

—

115

C₂×C₈

—

116

C₂×C₈

—

166

—

117

C₂×C₈

D₃₃₂

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

166

—

121

C₂³

—

166

—

122

C₂³

#13050591

yes

characteristic

123

Q₈

—

166

—

124

Q₈

—

166

—

125

D₄

—

332

—

126

D₄

—

332

—

127

D₄

—

166

—

128

D₄

—

166

—

129

C₂×C₄

—

332

—

130

C₂×C₄

—

332

—

131

C₂×C₄

—

332

—

132

C₂×C₄

—

166

—

133

C₂×C₄

—

166

—

134

C₂×C₄

—

166

—

135

C₂×C₄

D₄×D₈₃

yes

—

136

C₂×C₄

D₄×D₈₃

yes

characteristic

137

C₂×C₄

C₂×D₃₃₂

yes

characteristic

138

C₂×C₄

#13050590

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

332

—

142

C₂²

—

143

C₂²

—

332

—

144

C₂²

—

664

—

145

C₂²

#14581767

yes

characteristic

146

C₂²

#14581777

yes

—

147

C₄

—

332

—

148

C₄

—

332

—

149

C₄

—

166

—

150

C₄

#14581762

yes

characteristic

151

C₄

#14581765

yes

—

152

C₄

#14581780

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

664

—

156

C₂

#19127619

yes

characteristic

	subgroup	G/H
center	C₄	#14581780
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#338×C₈₃	C₂
socle	C₂×C₈₃	#394
prime radical	#19128501	C₂
hypercenter	#338	D₈₃
hypocenter	C₈₃	#2050
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

subgroup

G/H

center

C₄

#14581780

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#338×C₈₃

C₂

C₂×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

N	H ≅ G/N	type
2. #19127724	C₂	semidirect
3. C₂×#14581718	C₂	semidirect
4. #19127683	C₂	semidirect
5. #338×C₈₃	C₂	semidirect
79. C₈₃	#2050	semidirect
84. #338	D₈₃	semidirect

H ≅ G/N

type

2. #19127724

C₂

semidirect

3. C₂×#14581718

C₂

semidirect

4. #19127683

C₂

semidirect

5. #338×C₈₃

C₂

semidirect

79. C₈₃

#2050

semidirect

84. #338

D₈₃

semidirect

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

subgroup	index
#19128501	2
#19127724	2
C₂×#14581718	2
#19127683	2
#338×C₈₃	2
#2050	83

subgroup

index

#338×C₈₃

Subgroups — #24908751

Conjugacy classes of subgroups

Complements

Maximal subgroups