Group #24908378 — subgroups

Subgroups — #24908378

Order 10624.

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

normal subgroups only

This group has 276 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	C₂²×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581728	C₂	1	1	yes	maximal, characteristic
3	#19127709	C₂	1	2	yes	maximal
4	C₂×#14581710	C₂	1	2	yes	maximal
5	#405×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	C₂×Dic₃₃₂	—	2	2	no	—
8	C₂×Dic₃₃₂	—	2	1	no	—
9	C₂×Dic₃₃₂	—	2	1	no	—
10	C₂×Dic₃₃₂	C₄	1	4	yes	complemented
11	C₂×#13050574	C₂²	1	2	yes	—
12	#14581728	—	2	2	no	—
13	#14581715	—	2	4	no	—
14	#14581710	—	2	4	no	—
15	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#117×C₈₃	C₂²	1	1	yes	characteristic
18	#97×C₈₃	—	2	1	no	—
19	C₄×C₈×C₈₃	—	2	1	no	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	—	2	2	no	—
22	C₂×Dic₁₆₆	—	2	2	no	—
23	C₂×Dic₁₆₆	—	2	2	no	—
24	C₂×Dic₁₆₆	—	2	2	no	—
25	C₂×Dic₁₆₆	C₂×C₄	1	4	yes	—
26	C₂²×Dic₈₃	—	2	2	no	—
27	Dic₃₃₂	—	2	4	no	—
28	Dic₃₃₂	—	2	4	no	—
29	#13050574	—	2	2	no	—
30	#13050574	—	2	2	no	—
31	C₂×C₈₃⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₈₃	—	2	1	no	—
34	M₁₆×C₈₃	D₄	1	2	yes	—
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₈₃	D₄	1	2	yes	—
38	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₈₃	D₄	1	2	yes	—
41	C₂²⋊C₄×C₈₃	—	2	1	no	—
42	C₄²×C₈₃	—	2	1	no	—
43	C₂³×C₈₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₈₃	—	2	2	no	—
45	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
51	C₂×Dic₈₃	—	2	4	no	—
52	C₂×Dic₈₃	—	4	2	no	—
53	C₂×Dic₈₃	—	4	2	no	—
54	Dic₁₆₆	—	2	4	no	—
55	Dic₁₆₆	—	2	4	no	—
56	Dic₁₆₆	—	4	2	no	—
57	Dic₁₆₆	—	4	2	no	—
58	C₈×C₈₃	—	2	2	no	—
59	C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
60	C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
61	C₈₃⋊C₈	—	4	2	no	—
62	C₂²×C₈₃	—	2	2	no	—
63	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₈₃	#121	1	1	yes	characteristic
65	C₂²×C₈₃	#124	1	1	yes	characteristic
66	C₄×C₈₃	—	4	2	no	—
67	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
69	C₄×C₈₃	#121	1	1	yes	characteristic
70	C₄×C₈₃	#127	1	1	yes	characteristic
71	Dic₈₃	—	4	4	no	—
72	C₂×C₈₃	—	2	1	no	—
73	C₂×C₈₃	—	2	1	no	—
74	C₂×C₈₃	#390	1	1	yes	characteristic
75	C₂×C₈₃	#473	1	2	yes	—
76	#1979	—	83	1	no	maximal, Sylow
77	C₈₃	#1979	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	83	1	no	—
79	C₂×#108	—	83	1	no	—
80	#419	—	83	2	no	—
81	C₂×#101	—	83	2	no	—
82	#405	D₈₃	1	1	yes	characteristic
83	C₂²×Q₈	—	83	2	no	—
84	C₂×Q₁₆	—	83	4	no	—
85	C₂×Q₁₆	—	166	2	no	—
86	C₂×Q₁₆	—	166	1	no	—
87	C₂×Q₁₆	—	166	1	no	—
88	C₂×M₁₆	—	83	2	no	—
89	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
91	#117	C₂×D₈₃	1	1	yes	characteristic
92	#108	—	166	2	no	—
93	#103	—	166	4	no	—
94	#101	—	166	4	no	—
95	#97	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₂×Q₈	—	83	4	no	—
98	C₂×Q₈	—	166	2	no	—
99	C₂×Q₈	—	166	2	no	—
100	C₂×Q₈	—	166	2	no	—
101	C₂×Q₈	—	166	2	no	—
102	C₂×Q₈	—	166	2	no	—
103	C₂²×C₄	—	166	2	no	—
104	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
105	Q₁₆	—	166	4	no	—
106	Q₁₆	—	166	4	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	166	2	no	—
109	M₁₆	—	166	2	no	—
110	M₁₆	D₃₃₂	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	166	2	no	—
113	C₂×C₈	C₄×D₈₃	1	2	yes	—
114	C₂×C₈	D₃₃₂	1	2	yes	—
115	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
117	C₄⋊C₄	D₃₃₂	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#13050590	1	1	yes	characteristic
121	Q₈	—	332	2	no	—
122	Q₈	—	332	2	no	—
123	Q₈	—	166	4	no	—
124	Q₈	—	166	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	332	2	no	—
127	C₂×C₄	—	332	2	no	—
128	C₂×C₄	—	166	4	no	—
129	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
134	C₂×C₄	#13050590	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	332	2	no	—
137	C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×D₃₃₂	1	1	yes	characteristic
141	C₂²	#14581766	1	1	yes	characteristic
142	C₂²	#14581780	1	1	yes	characteristic
143	C₄	—	4	2	no	—
144	C₄	—	332	4	no	—
145	C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
146	C₄	C₄×D₃₃₂	1	1	yes	characteristic
147	C₄	#14581766	1	1	yes	characteristic
148	C₄	#14581789	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#19127603	1	1	yes	characteristic
152	C₂	#19127976	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₃₂

C₂

yes

maximal, characteristic

C₂×#14581728

C₂

yes

maximal, characteristic

#19127709

C₂

yes

maximal

C₂×#14581710

C₂

yes

maximal

#405×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

C₄

yes

complemented

C₂×#13050574

C₂²

yes

—

#14581728

—

#14581715

—

#14581710

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊C₄

yes

—

C₈×C₈₃

C₂²⋊C₄

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#390

yes

characteristic

C₂×C₈₃

#473

yes

—

#1979

—

maximal, Sylow

C₈₃

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

—

—

—

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

characteristic

#108

—

166

—

#103

—

166

—

#101

—

166

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

100

C₂×Q₈

—

166

—

101

C₂×Q₈

—

166

—

102

C₂×Q₈

—

166

—

103

C₂²×C₄

—

166

—

104

C₂²×C₄

C₂²×D₈₃

yes

characteristic

105

Q₁₆

—

166

—

106

Q₁₆

—

166

—

107

M₁₆

—

108

M₁₆

—

166

—

109

M₁₆

—

166

—

110

M₁₆

D₃₃₂

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

166

—

113

C₂×C₈

C₄×D₈₃

yes

—

114

C₂×C₈

D₃₃₂

yes

—

115

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

116

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

117

C₄⋊C₄

D₃₃₂

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#13050590

yes

characteristic

121

Q₈

—

332

—

122

Q₈

—

332

—

123

Q₈

—

166

—

124

Q₈

—

166

—

125

C₂×C₄

—

126

C₂×C₄

—

332

—

127

C₂×C₄

—

332

—

128

C₂×C₄

—

166

—

129

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

131

C₂×C₄

C₂×D₃₃₂

yes

characteristic

132

C₂×C₄

C₂×D₃₃₂

yes

characteristic

133

C₂×C₄

C₂×D₃₃₂

yes

characteristic

134

C₂×C₄

#13050590

yes

characteristic

135

C₈

—

136

C₈

—

332

—

137

C₈

(C₂×D₈₃)⋊C₄

yes

—

138

C₈

(C₂×D₈₃)⋊C₄

yes

—

139

C₂²

—

140

C₂²

C₄×D₃₃₂

yes

characteristic

141

C₂²

#14581766

yes

characteristic

142

C₂²

#14581780

yes

characteristic

143

C₄

—

144

C₄

—

332

—

145

C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

146

C₄

C₄×D₃₃₂

yes

characteristic

147

C₄

#14581766

yes

characteristic

148

C₄

#14581789

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#19127603

yes

characteristic

152

C₂

#19127976

yes

—

	subgroup	G/H
center	C₂²	#14581766
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#405	D₈₃
hypocenter	C₈₃	#1979
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#14581766

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#405×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
10. C₂×Dic₃₃₂	C₄	semidirect
77. C₈₃	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₃₃₂

C₄

semidirect

77. C₈₃

#1979

semidirect

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

subgroup	index
C₂²×Dic₃₃₂	2
C₂×#14581728	2
#19127709	2
C₂×#14581710	2
#405×C₈₃	2
#1979	83

subgroup

index

C₂²×Dic₃₃₂

#405×C₈₃

Subgroups — #24908378

Conjugacy classes of subgroups

Complements

Maximal subgroups