Group #24920467 — subgroups

Subgroups — #24920467

Order 10624.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19128426	C₂	1	2	yes	maximal
2	#19128079	C₂	1	2	yes	maximal
3	#19127909	C₂	1	4	yes	maximal
4	#19127908	C₂	1	2	yes	maximal
5	C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₈₃	C₂	1	2	yes	maximal
8	#494×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
10	C₂×#13050580	C₂²	1	4	yes	—
11	#14581790	C₂²	1	4	yes	—
12	#14581783	—	2	2	no	—
13	#14581782	—	2	4	no	—
14	Q₈×Dic₈₃	C₂²	1	4	yes	—
15	C₄×Dic₁₆₆	C₂²	1	2	yes	—
16	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
17	#14581757	—	2	2	no	—
18	#14581726	C₂²	1	1	yes	characteristic
19	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₈₃⋊C₈	C₂²	1	2	yes	—
21	#14581717	C₂²	1	2	yes	—
22	#14581716	C₂²	1	4	yes	—
23	C₈×Dic₈₃	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
27	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
29	C₂×Dic₁₆₆	C₂³	1	2	yes	—
30	Dic₃₃₂	—	2	2	no	—
31	#13050580	—	4	4	no	—
32	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
33	C₄⋊Dic₈₃	—	2	4	no	—
34	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
35	C₄⋊Dic₈₃	C₂³	1	2	yes	—
36	Dic₈₃⋊C₄	—	2	2	no	—
37	Dic₈₃⋊C₄	—	2	4	no	—
38	Dic₈₃⋊C₄	—	2	4	no	—
39	Dic₈₃⋊C₄	D₄	1	2	yes	—
40	C₄×Dic₈₃	—	2	2	no	—
41	C₄×Dic₈₃	—	2	4	no	—
42	C₄×Dic₈₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
44	Q₁₆×C₈₃	—	2	4	no	—
45	Q₁₆×C₈₃	—	4	2	no	—
46	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₈₃	—	2	2	no	—
49	C₄⋊C₄×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	C₂³	1	1	yes	characteristic
51	Q₈×C₈₃	—	2	4	no	—
52	Q₈×C₈₃	—	2	4	no	—
53	C₂×C₄×C₈₃	—	2	4	no	—
54	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₈₃	—	2	2	no	—
58	C₂×Dic₈₃	—	2	2	no	—
59	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
60	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
61	Dic₁₆₆	—	2	2	no	—
62	Dic₁₆₆	—	2	2	no	—
63	Dic₁₆₆	Q₁₆	1	4	yes	complemented
64	C₈×C₈₃	—	2	1	no	—
65	C₈×C₈₃	#41	1	2	yes	—
66	C₈₃⋊C₈	—	2	2	no	—
67	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	—	4	4	no	—
70	C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
71	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
72	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
73	Dic₈₃	—	2	2	no	—
74	Dic₈₃	—	4	2	no	—
75	Dic₈₃	—	4	2	no	—
76	Dic₈₃	C₂×Q₁₆	1	4	yes	—
77	C₂×C₈₃	#550	1	1	yes	characteristic
78	C₂×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
79	C₂×C₈₃	#578	1	1	yes	characteristic
80	#3395	—	83	1	no	maximal, Sylow
81	C₈₃	#3395	1	1	yes	characteristic, complemented, Sylow
82	#557	—	83	2	no	—
83	#494	—	83	2	no	—
84	#494	D₈₃	1	1	yes	characteristic
85	#462	—	83	4	no	—
86	#462	—	83	2	no	—
87	C₈×Q₈	—	83	1	no	—
88	C₄×Q₁₆	—	83	2	no	—
89	C₄×Q₁₆	—	83	1	no	—
90	C₂×Q₁₆	—	83	4	no	—
91	C₂×Q₁₆	—	83	1	no	—
92	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
93	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
94	C₄⋊Q₈	—	83	4	no	—
95	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
96	#125	—	166	4	no	—
97	#125	—	166	2	no	—
98	C₄×Q₈	—	83	1	no	—
99	C₄×Q₈	—	83	4	no	—
100	C₄×Q₈	—	83	2	no	—
101	C₄×Q₈	—	166	2	no	—
102	#107	—	83	1	no	—
103	C₄⋊C₈	—	83	2	no	—
104	C₄⋊C₈	—	83	1	no	—
105	#103	—	83	4	no	—
106	#103	—	83	2	no	—
107	C₄×C₈	—	83	2	no	—
108	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
109	C₂×Q₈	—	83	1	no	—
110	C₂×Q₈	—	83	2	no	—
111	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
112	Q₁₆	—	2	4	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	332	4	no	—
115	Q₁₆	—	166	2	no	—
116	C₂×C₈	—	83	2	no	—
117	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	83	2	no	—
122	C₄⋊C₄	—	83	1	no	—
123	C₄⋊C₄	—	83	2	no	—
124	C₄⋊C₄	—	166	4	no	—
125	C₄⋊C₄	—	166	4	no	—
126	C₄⋊C₄	—	166	4	no	—
127	C₄⋊C₄	—	166	2	no	—
128	C₄²	—	83	2	no	—
129	C₄²	—	166	4	no	—
130	C₄²	—	166	2	no	—
131	C₄²	C₂²×D₈₃	1	1	yes	characteristic
132	Q₈	—	2	4	no	—
133	Q₈	—	2	4	no	—
134	Q₈	—	83	4	no	—
135	Q₈	—	166	2	no	—
136	Q₈	—	166	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	83	2	no	—
139	C₂×C₄	—	83	2	no	—
140	C₂×C₄	—	166	2	no	—
141	C₂×C₄	—	166	2	no	—
142	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
143	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	166	2	no	—
147	C₈	#13050589	1	2	yes	—
148	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	4	no	—
151	C₄	—	332	2	no	—
152	C₄	—	332	2	no	—
153	C₄	—	83	4	no	—
154	C₄	—	166	2	no	—
155	C₄	Q₁₆×D₈₃	1	2	yes	—
156	C₄	C₂×#13050589	1	1	yes	characteristic
157	C₄	#14581837	1	1	yes	characteristic
158	C₂	#19128371	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
160	C₂	#19128514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128426

C₂

yes

maximal

#19128079

C₂

yes

maximal

#19127909

C₂

yes

maximal

#19127908

C₂

yes

maximal

C₈×Dic₁₆₆

C₂

yes

maximal, characteristic

C₄×Dic₃₃₂

C₂

yes

maximal, characteristic

Q₁₆×Dic₈₃

C₂

yes

maximal

#494×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

characteristic

C₂×#13050580

C₂²

yes

—

#14581790

C₂²

yes

—

#14581783

—

#14581782

—

Q₈×Dic₈₃

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581757

—

#14581726

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

C₂²

yes

—

#14581717

C₂²

yes

—

#14581716

C₂²

yes

—

C₈×Dic₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

D₄

yes

characteristic

C₂×Dic₁₆₆

C₂³

yes

—

Dic₃₃₂

—

#13050580

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

D₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂³

yes

—

C₂×Q₈×C₈₃

C₂³

yes

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×D₄

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

Q₁₆

yes

complemented

C₈×C₈₃

—

C₈×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

C₂×#41

yes

characteristic

C₄×C₈₃

H(5,𝔽₂)

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

C₂×Q₁₆

yes

—

C₂×C₈₃

#550

yes

characteristic

C₂×C₈₃

C₂²×Q₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#557

—

#494

—

#494

D₈₃

yes

characteristic

#462

—

#462

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

—

#125

—

166

—

#125

—

166

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

166

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₈₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²×D₈₃

yes

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

332

—

115

Q₁₆

—

166

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₈₃

yes

characteristic

118

C₂×C₈

C₂²×D₈₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

166

—

125

C₄⋊C₄

—

166

—

126

C₄⋊C₄

—

166

—

127

C₄⋊C₄

—

166

—

128

C₄²

—

129

C₄²

—

166

—

130

C₄²

—

166

—

131

C₄²

C₂²×D₈₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

166

—

136

Q₈

—

166

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

166

—

141

C₂×C₄

—

166

—

142

C₂×C₄

D₄×D₈₃

yes

characteristic

143

C₂×C₄

D₄×D₈₃

yes

characteristic

144

C₂×C₄

C₂³×D₈₃

yes

characteristic

145

C₈

—

146

C₈

—

166

—

147

C₈

#13050589

yes

—

148

C₂²

C₂×D₄×D₈₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

332

—

152

C₄

—

332

—

153

C₄

—

154

C₄

—

166

—

155

C₄

Q₁₆×D₈₃

yes

—

156

C₄

C₂×#13050589

yes

characteristic

157

C₄

#14581837

yes

characteristic

158

C₂

#19128371

yes

characteristic

159

C₂

C₂×Q₁₆×D₈₃

yes

characteristic

160

C₂

#19128514

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#494×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂²×D₄
hypercenter	#494	D₈₃
hypocenter	C₈₃	#3395
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#494×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂²×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
63. Dic₁₆₆	Q₁₆	semidirect
81. C₈₃	#3395	semidirect

H ≅ G/N

type

63. Dic₁₆₆

Q₁₆

semidirect

81. C₈₃

#3395

semidirect

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

subgroup	index
#19128426	2
#19128079	2
#19127909	2
#19127908	2
C₈×Dic₁₆₆	2
C₄×Dic₃₃₂	2
Q₁₆×Dic₈₃	2
#494×C₈₃	2
#3395	83

subgroup

index

C₈×Dic₁₆₆

C₄×Dic₃₃₂

Q₁₆×Dic₈₃

#494×C₈₃

#3395

Subgroups — #24920467

Conjugacy classes of subgroups

Complements

Maximal subgroups