Group #24906956 — subgroups

Subgroups — #24906956

Order 10624.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19128080	C₂	1	1	yes	maximal, characteristic
2	#19127987	C₂	1	1	yes	maximal, characteristic
3	#19127983	C₂	1	1	yes	maximal, characteristic
4	#19127828	C₂	1	1	yes	maximal, characteristic
5	#19127451	C₂	1	1	yes	maximal, characteristic
6	#19127448	C₂	1	1	yes	maximal, characteristic
7	#332×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050580	—	2	1	no	—
9	C₂×#13050580	—	2	1	no	—
10	#14581792	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
12	#14581724	—	2	1	no	—
13	#14581723	—	2	1	no	—
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₈₃⋊C₈	—	2	1	no	—
16	#14581717	—	2	1	no	—
17	#14581715	—	2	1	no	—
18	#14581702	—	2	1	no	—
19	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	—	2	1	no	—
24	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
25	#13050580	—	4	2	no	—
26	#13050580	—	4	2	no	—
27	C₂×C₈₃⋊C₈	—	2	1	no	—
28	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₈₃	—	2	1	no	—
31	C₄⋊Dic₈₃	—	2	1	no	—
32	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
33	Dic₈₃⋊C₄	—	2	1	no	—
34	C₄×Dic₈₃	—	2	1	no	—
35	C₂×Q₈×C₈₃	—	2	1	no	—
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₂×C₈×C₈₃	—	2	1	no	—
38	C₄⋊C₄×C₈₃	—	2	1	no	—
39	C₄⋊C₄×C₈₃	—	2	1	no	—
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	Q₈×C₈₃	—	4	1	no	—
42	Q₈×C₈₃	—	4	1	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂×Dic₈₃	—	2	1	no	—
48	C₂×Dic₈₃	—	2	1	no	—
49	C₂×Dic₈₃	—	4	1	no	—
50	Dic₁₆₆	—	2	1	no	—
51	Dic₁₆₆	—	2	1	no	—
52	Dic₁₆₆	—	4	1	no	—
53	Dic₁₆₆	—	4	1	no	—
54	C₈×C₈₃	—	4	1	no	—
55	C₈×C₈₃	—	4	1	no	—
56	C₈₃⋊C₈	—	2	1	no	—
57	C₈₃⋊C₈	—	4	1	no	—
58	C₈₃⋊C₈	SD₁₆	1	2	yes	—
59	C₂²×C₈₃	#121	1	1	yes	characteristic
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	8	1	no	—
62	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₈₃	#137	1	1	yes	characteristic
65	C₄×C₈₃	#137	1	1	yes	characteristic
66	Dic₈₃	—	4	1	no	—
67	Dic₈₃	—	4	1	no	—
68	Dic₈₃	—	8	1	no	—
69	C₂×C₈₃	#460	1	1	yes	characteristic
70	C₂×C₈₃	#470	1	1	yes	characteristic
71	C₂×C₈₃	#473	1	1	yes	characteristic
72	#1690	—	83	1	no	maximal, Sylow
73	C₈₃	#1690	1	1	yes	characteristic, complemented, Sylow
74	#494	—	83	1	no	—
75	#475	—	83	1	no	—
76	#475	—	83	1	no	—
77	#446	—	83	1	no	—
78	#334	—	83	1	no	—
79	#332	—	83	1	no	—
80	#332	D₈₃	1	1	yes	characteristic
81	C₂×Q₁₆	—	166	1	no	—
82	C₂×Q₁₆	—	166	1	no	—
83	C₄⋊Q₈	—	83	1	no	—
84	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
85	C₄×Q₈	—	83	1	no	—
86	#106	—	166	1	no	—
87	#106	—	166	1	no	—
88	C₄⋊C₈	—	83	1	no	—
89	C₄⋊C₈	—	166	1	no	—
90	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
92	#103	—	166	1	no	—
93	#103	—	166	1	no	—
94	#97	—	166	1	no	—
95	C₄×C₈	—	83	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	83	1	no	—
98	C₂×Q₈	—	166	1	no	—
99	Q₁₆	—	332	2	no	—
100	Q₁₆	—	332	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	83	1	no	—
104	C₂×C₈	—	83	1	no	—
105	C₂×C₈	—	166	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	83	1	no	—
109	C₄⋊C₄	—	166	1	no	—
110	C₄⋊C₄	—	166	1	no	—
111	C₄⋊C₄	—	166	1	no	—
112	C₄²	—	166	1	no	—
113	C₄²	C₂²×D₈₃	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	332	1	no	—
117	Q₈	—	332	1	no	—
118	Q₈	—	166	1	no	—
119	Q₈	—	166	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	332	1	no	—
122	C₂×C₄	—	166	1	no	—
123	C₂×C₄	—	166	1	no	—
124	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
125	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
126	C₂×C₄	#13050590	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	332	1	no	—
130	C₈	—	83	2	no	—
131	C₈	—	166	1	no	—
132	C₂²	#14581765	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	332	1	no	—
136	C₄	—	332	1	no	—
137	C₄	—	664	1	no	—
138	C₄	SD₁₆×D₈₃	1	1	yes	characteristic
139	C₄	#14581816	1	1	yes	characteristic
140	C₄	#14581821	1	1	yes	characteristic
141	C₄	#14581824	1	1	yes	characteristic
142	C₂	#19127900	1	1	yes	characteristic
143	C₂	#19127958	1	1	yes	characteristic
144	C₂	#19127975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128080

C₂

yes

maximal, characteristic

#19127987

C₂

yes

maximal, characteristic

#19127983

C₂

yes

maximal, characteristic

#19127828

C₂

yes

maximal, characteristic

#19127451

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal, characteristic

#332×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050580

—

C₂×#13050580

—

#14581792

C₂²

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581724

—

#14581723

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581717

—

#14581715

—

#14581702

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

characteristic

#13050580

—

#13050580

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

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

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

SD₁₆

yes

—

C₂²×C₈₃

#121

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×SD₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

#106

—

166

—

#106

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₈

C₂×D₈₃

yes

characteristic

#103

—

166

—

#103

—

166

—

#97

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

Q₁₆

—

332

—

100

Q₁₆

—

332

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

166

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

166

—

110

C₄⋊C₄

—

166

—

111

C₄⋊C₄

—

166

—

112

C₄²

—

166

—

113

C₄²

C₂²×D₈₃

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

332

—

117

Q₈

—

332

—

118

Q₈

—

166

—

119

Q₈

—

166

—

120

C₂×C₄

—

121

C₂×C₄

—

332

—

122

C₂×C₄

—

166

—

123

C₂×C₄

—

166

—

124

C₂×C₄

D₄×D₈₃

yes

characteristic

125

C₂×C₄

D₄×D₈₃

yes

characteristic

126

C₂×C₄

#13050590

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

332

—

130

C₈

—

131

C₈

—

166

—

132

C₂²

#14581765

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

332

—

136

C₄

—

332

—

137

C₄

—

664

—

138

C₄

SD₁₆×D₈₃

yes

characteristic

139

C₄

#14581816

yes

characteristic

140

C₄

#14581821

yes

characteristic

141

C₄

#14581824

yes

characteristic

142

C₂

#19127900

yes

characteristic

143

C₂

#19127958

yes

characteristic

144

C₂

#19127975

yes

characteristic

	subgroup	G/H
center	C₂²	#14581765
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#332×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#121
hypercenter	#332	D₈₃
hypocenter	C₈₃	#1690
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14581765

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#332×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
73. C₈₃	#1690	semidirect

H ≅ G/N

type

73. C₈₃

#1690

semidirect

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

subgroup	index
#19128080	2
#19127987	2
#19127983	2
#19127828	2
#19127451	2
#19127448	2
#332×C₈₃	2
#1690	83

subgroup

index

#332×C₈₃

Subgroups — #24906956

Conjugacy classes of subgroups

Complements

Maximal subgroups