Group #24908939 — subgroups

Subgroups — #24908939

Order 10624.

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

normal subgroups only

This group has 230 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	C₂×#14581723	C₂	1	2	yes	maximal
2	#19127571	C₂	1	1	yes	maximal, characteristic
3	#19127463	C₂	1	2	yes	maximal
4	#19127456	C₂	1	1	yes	maximal, characteristic
5	#389×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	—	2	1	no	—
8	C₂×C₄⋊Dic₈₃	—	2	1	no	—
9	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	#14581723	—	2	4	no	—
11	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	#95×C₈₃	—	2	1	no	—
16	C₂²×Dic₈₃	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₈₃⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₈₃	—	2	2	no	—
21	C₄⋊Dic₈₃	—	2	2	no	—
22	C₄⋊Dic₈₃	—	4	2	no	—
23	C₄⋊Dic₈₃	—	4	2	no	—
24	C₂²×C₄×C₈₃	—	2	1	no	—
25	C₂²×C₄×C₈₃	—	2	1	no	—
26	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₃	—	2	1	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₄⋊C₄×C₈₃	—	4	2	no	—
32	C₄⋊C₄×C₈₃	D₄	1	2	yes	—
33	C₄²×C₈₃	—	2	2	no	—
34	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	—	2	1	no	—
36	C₂×C₄×C₈₃	—	2	1	no	—
37	C₂×C₄×C₈₃	—	2	1	no	—
38	C₂×C₄×C₈₃	—	2	1	no	—
39	C₂×C₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	—	4	1	no	—
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	—	4	1	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	—	4	1	no	—
45	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₈₃	#41	1	2	yes	—
50	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
51	C₂×Dic₈₃	—	4	2	no	—
52	C₂×Dic₈₃	—	4	2	no	—
53	C₂×Dic₈₃	—	4	2	no	—
54	C₈₃⋊C₈	—	2	4	no	—
55	C₂²×C₈₃	#121	1	1	yes	characteristic
56	C₂²×C₈₃	#123	1	1	yes	characteristic
57	C₂²×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₈₃	#137	1	1	yes	characteristic
62	C₄×C₈₃	—	2	2	no	—
63	C₄×C₈₃	—	2	2	no	—
64	C₄×C₈₃	—	4	2	no	—
65	C₄×C₈₃	—	8	1	no	—
66	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₈₃	#125	1	2	yes	—
68	Dic₈₃	—	8	2	no	—
69	C₂×C₈₃	#398	1	1	yes	characteristic
70	C₂×C₈₃	#460	1	1	yes	characteristic
71	C₂×C₈₃	#461	1	1	yes	characteristic
72	C₂×C₈₃	#481	1	1	yes	characteristic
73	C₂×C₈₃	#483	1	1	yes	characteristic
74	C₂×C₈₃	#498	1	1	yes	characteristic
75	C₂×C₈₃	#501	1	1	yes	characteristic
76	#2084	—	83	1	no	maximal, Sylow
77	C₈₃	#2084	1	1	yes	characteristic, complemented, Sylow
78	C₂×#106	—	83	2	no	—
79	#389	D₈₃	1	1	yes	characteristic
80	#384	—	83	1	no	—
81	#340	—	83	2	no	—
82	#336	—	83	1	no	—
83	C₂²×C₈	—	83	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	83	2	no	—
86	C₂×C₄⋊C₄	—	166	1	no	—
87	C₂×C₄⋊C₄	—	166	1	no	—
88	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
91	#106	—	166	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	166	2	no	—
96	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
99	C₂×C₈	—	83	4	no	—
100	C₂×C₈	—	166	2	no	—
101	C₂×C₈	—	166	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	332	2	no	—
106	C₄⋊C₄	—	332	2	no	—
107	C₄⋊C₄	—	166	2	no	—
108	C₄⋊C₄	—	166	2	no	—
109	C₄⋊C₄	D₃₃₂	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	332	2	no	—
123	C₂×C₄	—	332	2	no	—
124	C₂×C₄	—	332	2	no	—
125	C₂×C₄	#13050577	1	2	yes	—
126	C₂×C₄	#13050578	1	2	yes	—
127	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₈₃	1	2	yes	—
129	C₂×C₄	#13050589	1	2	yes	—
130	C₂×C₄	#13050590	1	1	yes	characteristic
131	C₈	—	166	4	no	—
132	C₂²	#14581766	1	1	yes	characteristic
133	C₂²	#14581773	1	1	yes	characteristic
134	C₂²	#14581790	1	1	yes	characteristic
135	C₂²	C₂×#13050577	1	1	yes	characteristic
136	C₂²	C₂×#13050578	1	1	yes	characteristic
137	C₂²	#14581817	1	1	yes	characteristic
138	C₂²	#14581822	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	664	2	no	—
144	C₄	#14581772	1	2	yes	—
145	C₄	#14581782	1	2	yes	—
146	C₂	#19127631	1	1	yes	characteristic
147	C₂	#19127901	1	1	yes	characteristic
148	C₂	#19127906	1	1	yes	characteristic
149	C₂	#19128017	1	1	yes	characteristic
150	C₂	#19128029	1	1	yes	characteristic
151	C₂	#19128097	1	1	yes	characteristic
152	C₂	#19128111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581723

C₂

yes

maximal

#19127571

C₂

yes

maximal, characteristic

#19127463

C₂

yes

maximal

#19127456

C₂

yes

maximal, characteristic

#389×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581723

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

Q₈

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

D₄

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₄⋊Q₈

yes

characteristic

C₂²×C₈₃

C₂×SD₁₆

yes

characteristic

C₂²×C₈₃

C₂×SD₁₆

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#125

yes

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#106

—

#389

D₈₃

yes

—

—

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

#106

—

166

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

166

—

101

C₂×C₈

—

166

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

332

—

106

C₄⋊C₄

—

332

—

107

C₄⋊C₄

—

166

—

108

C₄⋊C₄

—

166

—

109

C₄⋊C₄

D₃₃₂

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

332

—

123

C₂×C₄

—

332

—

124

C₂×C₄

—

332

—

125

C₂×C₄

#13050577

yes

—

126

C₂×C₄

#13050578

yes

—

127

C₂×C₄

C₂×D₃₃₂

yes

characteristic

128

C₂×C₄

Q₈×D₈₃

yes

—

129

C₂×C₄

#13050589

yes

—

130

C₂×C₄

#13050590

yes

characteristic

131

C₈

—

166

—

132

C₂²

#14581766

yes

characteristic

133

C₂²

#14581773

yes

characteristic

134

C₂²

#14581790

yes

characteristic

135

C₂²

C₂×#13050577

yes

characteristic

136

C₂²

C₂×#13050578

yes

characteristic

137

C₂²

#14581817

yes

characteristic

138

C₂²

#14581822

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

664

—

144

C₄

#14581772

yes

—

145

C₄

#14581782

yes

—

146

C₂

#19127631

yes

characteristic

147

C₂

#19127901

yes

characteristic

148

C₂

#19127906

yes

characteristic

149

C₂

#19128017

yes

characteristic

150

C₂

#19128029

yes

characteristic

151

C₂

#19128097

yes

characteristic

152

C₂

#19128111

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#389×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#389	D₈₃
hypocenter	C₈₃	#2084
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#389×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
77. C₈₃	#2084	semidirect

H ≅ G/N

type

77. C₈₃

#2084

semidirect

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

subgroup	index
C₂×#14581723	2
#19127571	2
#19127463	2
#19127456	2
#389×C₈₃	2
#2084	83

subgroup

index

#389×C₈₃

Subgroups — #24908939

Conjugacy classes of subgroups

Complements

Maximal subgroups