Group #24908210 — subgroups

Subgroups — #24908210

Order 10624.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14581727	C₂	1	1	yes	maximal, characteristic
2	#19127737	C₂	1	1	yes	maximal, characteristic
3	#19127729	C₂	1	1	yes	maximal, characteristic
4	#19127728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#19127461	C₂	1	1	yes	maximal, characteristic
7	#339×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	#14581751	C₂²	1	1	yes	characteristic
12	#14581727	—	2	2	no	—
13	#14581725	C₄	1	2	yes	complemented
14	#14581723	C₄	1	2	yes	complemented
15	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
16	Dic₈₃⋊C₈	—	2	2	no	—
17	C₄×C₈₃⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	#117×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
21	#13050574	—	2	1	no	—
22	#13050574	—	2	1	no	—
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
30	C₂×C₈₃⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₈₃	—	2	1	no	—
33	C₄×Dic₈₃	—	2	1	no	—
34	C₂²×C₄×C₈₃	—	2	1	no	—
35	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₈₃	—	2	1	no	—
37	M₁₆×C₈₃	—	2	1	no	—
38	C₂×C₈×C₈₃	—	2	1	no	—
39	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₈₃	—	2	1	no	—
41	C₄²×C₈₃	—	2	1	no	—
42	C₄²×C₈₃	—	2	1	no	—
43	C₄²×C₈₃	Q₈	1	2	yes	—
44	C₂³×C₈₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₃	—	2	4	no	—
46	C₂×C₄×C₈₃	—	2	2	no	—
47	C₂×C₄×C₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₂×Dic₈₃	—	2	2	no	—
55	C₈×C₈₃	—	4	1	no	—
56	C₈₃⋊C₈	—	2	2	no	—
57	C₈₃⋊C₈	—	4	1	no	—
58	C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₈₃	—	2	2	no	—
61	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₈₃	#125	1	1	yes	characteristic
64	C₄×C₈₃	—	2	4	no	—
65	C₄×C₈₃	—	4	2	no	—
66	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₈₃	#121	1	1	yes	characteristic
69	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₈₃	—	4	2	no	—
71	C₂×C₈₃	—	2	1	no	—
72	C₂×C₈₃	—	2	1	no	—
73	C₂×C₈₃	#389	1	1	yes	characteristic
74	C₂×C₈₃	#443	1	2	yes	—
75	#1945	—	83	1	no	maximal, Sylow
76	C₈₃	#1945	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	83	1	no	—
78	#427	—	83	1	no	—
79	#424	—	83	1	no	—
80	#424	—	83	1	no	—
81	C₂×C₄×C₈	—	83	1	no	—
82	#339	—	83	1	no	—
83	#339	D₈₃	1	1	yes	characteristic
84	C₂×M₁₆	—	83	1	no	—
85	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
86	C₂²×C₈	—	83	1	no	—
87	C₂²×C₈	—	83	1	no	—
88	#117	—	83	1	no	—
89	#117	C₂×D₈₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
91	#108	—	166	2	no	—
92	#107	—	83	2	no	—
93	#106	—	83	2	no	—
94	C₄⋊C₈	—	166	2	no	—
95	C₄⋊C₈	—	166	2	no	—
96	C₄×C₈	—	166	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	166	1	no	—
102	M₁₆	—	166	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	83	1	no	—
105	C₂×C₈	—	83	1	no	—
106	C₂×C₈	—	83	2	no	—
107	C₂×C₈	—	83	2	no	—
108	C₂×C₈	—	83	2	no	—
109	C₂×C₈	—	166	1	no	—
110	C₂×C₈	—	166	1	no	—
111	C₂×C₈	—	166	1	no	—
112	C₄⋊C₄	—	83	2	no	—
113	C₄⋊C₄	C₄×D₈₃	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	166	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	166	1	no	—
119	C₄²	Dic₁₆₆	1	2	yes	—
120	C₂³	#13050589	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	166	2	no	—
125	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
130	C₂×C₄	#13050590	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	332	1	no	—
133	C₈	—	83	2	no	—
134	C₈	—	83	2	no	—
135	C₈	—	166	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
138	C₂²	#14581769	1	1	yes	characteristic
139	C₂²	#14581782	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	332	2	no	—
143	C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
144	C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
145	C₄	#14581765	1	1	yes	characteristic
146	C₄	#14581791	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#19127598	1	1	yes	characteristic
150	C₂	#19127815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581727

C₂

yes

maximal, characteristic

#19127737

C₂

yes

maximal, characteristic

#19127729

C₂

yes

maximal, characteristic

#19127728

C₂

yes

maximal, characteristic

C₂×C₄×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#19127461

C₂

yes

maximal, characteristic

#339×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

#14581727

—

#14581725

C₄

yes

complemented

#14581723

C₄

yes

complemented

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

Q₈

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

Q₈

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

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×Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₄⋊C₄

yes

—

C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

#125

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

#121

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#389

yes

characteristic

C₂×C₈₃

#443

yes

—

#1945

—

maximal, Sylow

C₈₃

#1945

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄×C₈

—

#339

—

#339

D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

#108

—

166

—

#107

—

#106

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

C₄×C₈

—

166

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

166

—

102

M₁₆

—

166

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

166

—

110

C₂×C₈

—

166

—

111

C₂×C₈

—

166

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₈₃

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

166

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

166

—

119

C₄²

Dic₁₆₆

yes

—

120

C₂³

#13050589

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

166

—

125

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

126

C₂×C₄

D₄×D₈₃

yes

characteristic

127

C₂×C₄

D₄×D₈₃

yes

characteristic

128

C₂×C₄

Q₈×D₈₃

yes

characteristic

129

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

130

C₂×C₄

#13050590

yes

characteristic

131

C₈

—

132

C₈

—

332

—

133

C₈

—

134

C₈

—

135

C₈

—

166

—

136

C₂²

—

137

C₂²

(C₂²⋊D₈₃)⋊C₄

yes

characteristic

138

C₂²

#14581769

yes

characteristic

139

C₂²

#14581782

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

332

—

143

C₄

C₄⋊C₄×D₈₃

yes

characteristic

144

C₄

C₄×Dic₁₆₆

yes

characteristic

145

C₄

#14581765

yes

characteristic

146

C₄

#14581791

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#19127598

yes

characteristic

150

C₂

#19127815

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#339×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#339	D₈₃
hypocenter	C₈₃	#1945
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#339×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂³×C₈₃

#41

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #14581725	C₄	semidirect
14. #14581723	C₄	semidirect
76. C₈₃	#1945	semidirect

H ≅ G/N

type

13. #14581725

C₄

semidirect

14. #14581723

C₄

semidirect

76. C₈₃

#1945

semidirect

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

subgroup	index
C₂×#14581727	2
#19127737	2
#19127729	2
#19127728	2
C₂×C₄×C₈₃⋊C₈	2
#19127461	2
#339×C₈₃	2
#1945	83

subgroup

index

C₂×C₄×C₈₃⋊C₈

#339×C₈₃

Subgroups — #24908210

Conjugacy classes of subgroups

Complements

Maximal subgroups