Group #24905786 — subgroups

Subgroups — #24905786

Order 10624.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#19127754	C₂	1	1	yes	maximal, characteristic
3	#19127426	C₂	1	4	yes	maximal
4	#431×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	C₄	1	1	yes	characteristic
6	C₄×Dic₁₆₆	—	2	1	no	—
7	C₄×Dic₁₆₆	—	2	1	no	—
8	C₄×Dic₁₆₆	—	2	1	no	—
9	C₄×Dic₁₆₆	—	2	1	no	—
10	C₂×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
11	C₂×Dic₈₃⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
13	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
15	C₄×C₈₃⋊C₈	—	2	1	no	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₃	—	2	1	no	—
18	C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
19	C₄×C₈×C₈₃	—	2	1	no	—
20	C₂×Dic₁₆₆	—	2	1	no	—
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×Dic₁₆₆	—	2	2	no	—
23	C₂×Dic₁₆₆	C₈	1	4	yes	complemented
24	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
25	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
26	C₂×C₈₃⋊C₈	—	2	2	no	—
27	C₄⋊Dic₈₃	—	2	1	no	—
28	C₄⋊Dic₈₃	—	2	1	no	—
29	Dic₈₃⋊C₄	—	2	2	no	—
30	Dic₈₃⋊C₄	—	2	2	no	—
31	C₄×Dic₈₃	—	2	2	no	—
32	C₄×Dic₈₃	—	2	2	no	—
33	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₈₃	—	2	2	no	—
37	C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₈₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	2	2	no	—
54	C₂×Dic₈₃	—	2	2	no	—
55	C₂×Dic₈₃	—	4	2	no	—
56	C₂×Dic₈₃	C₂×C₈	1	4	yes	—
57	Dic₁₆₆	—	2	4	no	—
58	Dic₁₆₆	—	2	4	no	—
59	C₈×C₈₃	—	4	2	no	—
60	C₈₃⋊C₈	—	4	2	no	—
61	C₂²×C₈₃	—	2	1	no	—
62	C₂²×C₈₃	—	2	1	no	—
63	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈₃	C₂²×C₈	1	1	yes	characteristic
65	C₂²×C₈₃	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	—	2	1	no	—
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
70	C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
71	C₄×C₈₃	#101	1	2	yes	—
72	Dic₈₃	—	2	4	no	—
73	Dic₈₃	—	4	2	no	—
74	C₂×C₈₃	—	2	1	no	—
75	C₂×C₈₃	—	2	1	no	—
76	C₂×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₈₃	#410	1	1	yes	characteristic
78	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
79	#1461	—	83	1	no	maximal, Sylow
80	C₈₃	#1461	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄×Q₈	—	83	1	no	—
82	#431	—	83	1	no	—
83	#431	D₈₃	1	1	yes	characteristic
84	#324	—	83	4	no	—
85	C₂²×Q₈	—	83	1	no	—
86	C₄×Q₈	—	166	1	no	—
87	C₄×Q₈	—	166	1	no	—
88	C₄×Q₈	—	166	1	no	—
89	C₄×Q₈	—	166	1	no	—
90	C₂×C₄⋊C₄	—	83	1	no	—
91	C₂×C₄⋊C₄	—	83	2	no	—
92	C₂×C₄²	—	83	2	no	—
93	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	166	1	no	—
96	C₂²⋊C₈	—	83	2	no	—
97	C₂²⋊C₈	C₂×D₈₃	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	166	1	no	—
100	C₂×Q₈	—	83	4	no	—
101	C₂×Q₈	—	166	1	no	—
102	C₂×Q₈	—	166	1	no	—
103	C₂×Q₈	—	166	2	no	—
104	C₂²×C₄	—	83	2	no	—
105	C₂²×C₄	—	83	2	no	—
106	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	166	2	no	—
111	C₄⋊C₄	—	166	1	no	—
112	C₄⋊C₄	—	166	1	no	—
113	C₄⋊C₄	—	166	2	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₄²	—	166	2	no	—
116	C₄²	—	166	2	no	—
117	C₄²	D₃₃₂	1	1	yes	characteristic
118	C₄²	D₃₃₂	1	1	yes	characteristic
119	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
122	Q₈	—	166	4	no	—
123	Q₈	—	166	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	332	2	no	—
127	C₂×C₄	—	83	4	no	—
128	C₂×C₄	—	166	2	no	—
129	C₂×C₄	—	166	2	no	—
130	C₂×C₄	—	166	2	no	—
131	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₈×D₈₃	1	2	yes	—
136	C₂×C₄	#13050573	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	332	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₈₃	1	1	yes	characteristic
145	C₂²	C₂×#13050573	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	332	2	no	—
150	C₄	—	166	4	no	—
151	C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
152	C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
153	C₄	#14581710	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
157	C₂	#19127674	1	1	yes	characteristic
158	C₂	C₂×#14581710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₆₆

C₂

yes

maximal, characteristic

#19127754

C₂

yes

maximal, characteristic

#19127426

C₂

yes

maximal

#431×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₄

yes

characteristic

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

C₂×C₄⋊Dic₈₃

C₄

yes

characteristic

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₈

yes

complemented

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×C₈

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²×C₈

yes

characteristic

C₂²×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊C₈

yes

—

C₄×C₈₃

C₂²⋊C₈

yes

—

C₄×C₈₃

#101

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₂²⋊C₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

—

#431

D₈₃

yes

characteristic

#324

—

C₂²×Q₈

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₈₃

yes

—

C₄×C₈

—

C₄×C₈

—

166

—

100

C₂×Q₈

—

101

C₂×Q₈

—

166

—

102

C₂×Q₈

—

166

—

103

C₂×Q₈

—

166

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₈₃

yes

characteristic

107

C₂²×C₄

C₄×D₈₃

yes

characteristic

108

C₂²×C₄

C₂²×D₈₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

166

—

111

C₄⋊C₄

—

166

—

112

C₄⋊C₄

—

166

—

113

C₄⋊C₄

—

166

—

114

C₄⋊C₄

—

166

—

115

C₄²

—

166

—

116

C₄²

—

166

—

117

C₄²

D₃₃₂

yes

characteristic

118

C₄²

D₃₃₂

yes

characteristic

119

C₄²

C₂²⋊D₈₃

yes

characteristic

120

C₄²

C₂²⋊D₈₃

yes

characteristic

121

C₂³

C₂×C₄×D₈₃

yes

characteristic

122

Q₈

—

166

—

123

Q₈

—

166

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

332

—

127

C₂×C₄

—

128

C₂×C₄

—

166

—

129

C₂×C₄

—

166

—

130

C₂×C₄

—

166

—

131

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

135

C₂×C₄

C₈×D₈₃

yes

—

136

C₂×C₄

#13050573

yes

—

137

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

138

C₂×C₄

C₂×D₃₃₂

yes

characteristic

139

C₈

—

140

C₈

—

332

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

144

C₂²

C₂×C₈×D₈₃

yes

characteristic

145

C₂²

C₂×#13050573

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

332

—

150

C₄

—

166

—

151

C₄

(C₂×D₈₃)⋊C₈

yes

—

152

C₄

(C₂×D₈₃)⋊C₈

yes

—

153

C₄

#14581710

yes

—

154

C₂

—

155

C₂

—

156

C₂

C₂×(C₂×D₈₃)⋊C₈

yes

characteristic

157

C₂

#19127674

yes

characteristic

158

C₂

C₂×#14581710

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₈₃)⋊C₄
commutator	C₂²×C₈₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#431×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#431	D₈₃
hypocenter	C₈₃	#1461
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₈₃)⋊C₄

commutator

C₂²×C₈₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#431×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
23. C₂×Dic₁₆₆	C₈	semidirect
80. C₈₃	#1461	semidirect

H ≅ G/N

type

23. C₂×Dic₁₆₆

C₈

semidirect

80. C₈₃

#1461

semidirect

Subgroups — #24905786

Conjugacy classes of subgroups

Complements