Group #24907611 — subgroups

Subgroups — #24907611

Order 10624.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
2	#19127472	C₂	1	1	yes	maximal, characteristic
3	#19127462	C₂	1	1	yes	maximal, characteristic
4	#19127460	C₂	1	1	yes	maximal, characteristic
5	C₈₃⋊#338	C₂	1	1	yes	maximal, characteristic
6	#19127457	C₂	1	1	yes	maximal, characteristic
7	#344×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	1	yes	characteristic
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	#14581751	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
12	#14581702	C₄	1	2	yes	complemented
13	C₈×Dic₈₃	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	#117×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₃	—	2	1	no	—
18	#13050574	—	2	1	no	—
19	#13050574	—	2	1	no	—
20	#13050574	—	2	1	no	—
21	#13050574	—	2	1	no	—
22	C₂×C₈₃⋊C₈	—	2	1	no	—
23	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
24	C₄⋊Dic₈₃	—	2	1	no	—
25	C₂²⋊Dic₈₃	—	2	1	no	—
26	C₄×Dic₈₃	—	2	1	no	—
27	C₄×Dic₈₃	—	2	1	no	—
28	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₈₃	—	2	1	no	—
32	M₁₆×C₈₃	—	2	1	no	—
33	M₁₆×C₈₃	—	2	1	no	—
34	M₁₆×C₈₃	—	2	1	no	—
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₈₃	—	2	1	no	—
38	C₂²⋊C₄×C₈₃	—	2	1	no	—
39	C₄²×C₈₃	—	2	1	no	—
40	C₂³×C₈₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
48	C₂×Dic₈₃	—	2	1	no	—
49	C₂×Dic₈₃	—	2	1	no	—
50	C₂×Dic₈₃	—	4	2	no	—
51	C₂×Dic₈₃	—	4	1	no	—
52	C₈×C₈₃	—	2	2	no	—
53	C₈×C₈₃	—	4	1	no	—
54	C₈₃⋊C₈	—	2	2	no	—
55	C₈₃⋊C₈	—	4	1	no	—
56	C₂²×C₈₃	—	4	1	no	—
57	C₂²×C₈₃	#117	1	1	yes	characteristic
58	C₂²×C₈₃	#126	1	1	yes	characteristic
59	C₂²×C₈₃	#126	1	1	yes	characteristic
60	C₄×C₈₃	—	8	1	no	—
61	C₄×C₈₃	#117	1	1	yes	characteristic
62	C₄×C₈₃	#117	1	1	yes	characteristic
63	C₄×C₈₃	#126	1	1	yes	characteristic
64	C₄×C₈₃	#126	1	1	yes	characteristic
65	Dic₈₃	—	2	2	no	—
66	Dic₈₃	—	4	1	no	—
67	Dic₈₃	—	8	1	no	—
68	C₂×C₈₃	—	2	1	no	—
69	C₂×C₈₃	—	2	1	no	—
70	C₂×C₈₃	—	2	1	no	—
71	C₂×C₈₃	#383	1	1	yes	characteristic
72	#1841	—	83	1	no	maximal, Sylow
73	C₈₃	#1841	1	1	yes	characteristic, complemented, Sylow
74	C₄×M₁₆	—	83	1	no	—
75	#344	—	83	1	no	—
76	#344	D₈₃	1	1	yes	characteristic
77	#339	—	83	1	no	—
78	#339	—	83	1	no	—
79	#338	—	83	1	no	—
80	H(3,ℤ/4)	—	83	1	no	—
81	C₂×M₁₆	—	83	1	no	—
82	C₂×M₁₆	—	83	1	no	—
83	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
85	#117	—	83	1	no	—
86	#117	C₂×D₈₃	1	1	yes	characteristic
87	C₂×C₄²	—	83	1	no	—
88	#97	—	83	2	no	—
89	C₄×C₈	—	83	2	no	—
90	C₂²×C₄	—	166	1	no	—
91	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	166	1	no	—
97	M₁₆	—	166	1	no	—
98	M₁₆	—	166	1	no	—
99	M₁₆	—	166	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	83	2	no	—
102	C₂×C₈	—	166	1	no	—
103	C₂×C₈	C₄×D₈₃	1	2	yes	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	166	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	166	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	83	1	no	—
110	C₄²	—	83	1	no	—
111	C₄²	—	166	1	no	—
112	C₄²	—	166	1	no	—
113	C₂³	#13050590	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	332	2	no	—
116	C₂×C₄	—	332	1	no	—
117	C₂×C₄	—	166	1	no	—
118	C₂×C₄	—	166	1	no	—
119	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
120	C₂×C₄	#13050589	1	1	yes	characteristic
121	C₂×C₄	#13050589	1	1	yes	characteristic
122	C₂×C₄	#13050589	1	1	yes	characteristic
123	C₂×C₄	#13050590	1	1	yes	characteristic
124	C₂×C₄	#13050591	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	4	1	no	—
127	C₈	—	332	1	no	—
128	C₈	—	166	2	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#14581749	1	1	yes	characteristic
131	C₂²	#14581785	1	1	yes	characteristic
132	C₂²	#14581787	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	332	1	no	—
135	C₄	—	664	1	no	—
136	C₄	—	166	2	no	—
137	C₄	#14581748	1	1	yes	characteristic
138	C₄	#14581750	1	1	yes	characteristic
139	C₄	#14581785	1	1	yes	characteristic
140	C₄	#14581787	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19127568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₈₃

C₂

yes

maximal, characteristic

#19127472

C₂

yes

maximal, characteristic

#19127462

C₂

yes

maximal, characteristic

#19127460

C₂

yes

maximal, characteristic

C₈₃⋊#338

C₂

yes

maximal, characteristic

#19127457

C₂

yes

maximal, characteristic

#344×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581702

C₄

yes

complemented

C₈×Dic₈₃

C₄

yes

complemented

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

—

—

—

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

#344

—

#344

D₈₃

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂×M₁₆

C₂×D₈₃

yes

characteristic

#117

—

#117

C₂×D₈₃

yes

characteristic

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

—

166

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

166

—

103

C₂×C₈

C₄×D₈₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

166

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

166

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

166

—

112

C₄²

—

166

—

113

C₂³

#13050590

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

332

—

116

C₂×C₄

—

332

—

117

C₂×C₄

—

166

—

118

C₂×C₄

—

166

—

119

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

120

C₂×C₄

#13050589

yes

characteristic

121

C₂×C₄

#13050589

yes

characteristic

122

C₂×C₄

#13050589

yes

characteristic

123

C₂×C₄

#13050590

yes

characteristic

124

C₂×C₄

#13050591

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

332

—

128

C₈

—

166

—

129

C₂²

—

130

C₂²

#14581749

yes

characteristic

131

C₂²

#14581785

yes

characteristic

132

C₂²

#14581787

yes

characteristic

133

C₄

—

134

C₄

—

332

—

135

C₄

—

664

—

136

C₄

—

166

—

137

C₄

#14581748

yes

characteristic

138

C₄

#14581750

yes

characteristic

139

C₄

#14581785

yes

characteristic

140

C₄

#14581787

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#19127568

yes

characteristic

	subgroup	G/H
center	C₄	#14581748
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#344×C₈₃	C₂
socle	C₂×C₈₃	#383
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#344	D₈₃
hypocenter	C₈₃	#1841
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14581748

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#344×C₈₃

C₂

C₂×C₈₃

C₂³×C₈₃

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
12. #14581702	C₄	semidirect
13. C₈×Dic₈₃	C₄	semidirect
73. C₈₃	#1841	semidirect

H ≅ G/N

type

12. #14581702

C₄

semidirect

13. C₈×Dic₈₃

C₄

semidirect

73. C₈₃

#1841

semidirect

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

subgroup	index
M₁₆×Dic₈₃	2
#19127472	2
#19127462	2
#19127460	2
C₈₃⋊#338	2
#19127457	2
#344×C₈₃	2
#1841	83

subgroup

index

M₁₆×Dic₈₃

C₈₃⋊#338

#344×C₈₃

Subgroups — #24907611

Conjugacy classes of subgroups

Complements

Maximal subgroups