Group #24905285 — subgroups

Subgroups — #24905285

Order 10624.

158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19127599	C₂	1	1	yes	maximal, characteristic
2	#19127456	C₂	1	1	yes	maximal, characteristic
3	#336×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈₃⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
7	C₂×Dic₈₃⋊C₄	—	2	1	no	—
8	C₈₃⋊#95	—	2	1	no	—
9	C₂²×C₈×C₈₃	—	2	1	no	—
10	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
11	C₂²×Dic₈₃	—	2	1	no	—
12	C₂²×Dic₈₃	—	2	1	no	—
13	C₂×C₈₃⋊C₈	—	2	2	no	—
14	C₂×C₈₃⋊C₈	—	4	1	no	—
15	C₂×C₈₃⋊C₈	—	4	1	no	—
16	C₄⋊Dic₈₃	—	2	1	no	—
17	C₄⋊Dic₈₃	—	4	1	no	—
18	C₄⋊Dic₈₃	C₈	1	2	yes	complemented
19	Dic₈₃⋊C₄	—	4	1	no	—
20	Dic₈₃⋊C₄	—	4	1	no	—
21	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₃	—	2	2	no	—
25	C₂×C₈×C₈₃	—	4	1	no	—
26	C₂×C₈×C₈₃	—	4	1	no	—
27	C₄²×C₈₃	—	4	1	no	—
28	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	—	2	1	no	—
30	C₂×C₄×C₈₃	—	2	1	no	—
31	C₂×C₄×C₈₃	—	2	1	no	—
32	C₂×C₄×C₈₃	—	2	1	no	—
33	C₂×C₄×C₈₃	—	2	1	no	—
34	C₂×C₄×C₈₃	—	2	1	no	—
35	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₈₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	4	1	no	—
43	C₂×Dic₈₃	—	4	1	no	—
44	C₂×Dic₈₃	—	4	1	no	—
45	C₂×Dic₈₃	—	4	1	no	—
46	C₂×Dic₈₃	—	4	1	no	—
47	C₈×C₈₃	—	4	2	no	—
48	C₈₃⋊C₈	—	4	2	no	—
49	C₂²×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²×C₈₃	#99	1	1	yes	characteristic
51	C₂²×C₈₃	#101	1	1	yes	characteristic
52	C₂²×C₈₃	#102	1	1	yes	characteristic
53	C₂²×C₈₃	#103	1	1	yes	characteristic
54	C₂²×C₈₃	#104	1	1	yes	characteristic
55	C₂²×C₈₃	#104	1	1	yes	characteristic
56	C₄×C₈₃	—	2	1	no	—
57	C₄×C₈₃	—	2	1	no	—
58	C₄×C₈₃	—	2	1	no	—
59	C₄×C₈₃	—	2	1	no	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	2	1	no	—
62	Dic₈₃	—	4	2	no	—
63	Dic₈₃	—	8	1	no	—
64	C₂×C₈₃	#324	1	1	yes	characteristic
65	C₂×C₈₃	D₄⋊C₈	1	1	yes	characteristic
66	C₂×C₈₃	Q₈⋊C₈	1	1	yes	characteristic
67	C₂×C₈₃	#327	1	1	yes	characteristic
68	C₂×C₈₃	#328	1	1	yes	characteristic
69	C₂×C₈₃	#330	1	1	yes	characteristic
70	C₂×C₈₃	#332	1	1	yes	characteristic
71	#1271	—	83	1	no	maximal, Sylow
72	C₈₃	#1271	1	1	yes	characteristic, complemented, Sylow
73	#389	—	83	1	no	—
74	#336	—	83	1	no	—
75	#336	D₈₃	1	1	yes	characteristic
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	166	1	no	—
78	C₂×C₄⋊C₄	—	83	1	no	—
79	C₂×C₄⋊C₄	—	83	1	no	—
80	C₂×C₄⋊C₄	—	166	1	no	—
81	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
82	#95	—	166	1	no	—
83	C₂²×C₄	—	166	1	no	—
84	C₂²×C₄	—	166	1	no	—
85	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
86	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	332	1	no	—
92	C₂×C₈	—	332	1	no	—
93	C₂×C₈	—	166	2	no	—
94	C₄⋊C₄	—	332	1	no	—
95	C₄⋊C₄	—	332	1	no	—
96	C₄⋊C₄	—	332	1	no	—
97	C₄⋊C₄	—	83	2	no	—
98	C₄⋊C₄	—	166	1	no	—
99	C₄²	—	4	1	no	—
100	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	332	1	no	—
108	C₂×C₄	—	332	1	no	—
109	C₂×C₄	—	332	1	no	—
110	C₂×C₄	—	332	1	no	—
111	C₂×C₄	—	332	1	no	—
112	C₂×C₄	—	166	2	no	—
113	C₂×C₄	C₈×D₈₃	1	1	yes	characteristic
114	C₂×C₄	#13050573	1	1	yes	characteristic
115	C₂×C₄	D₆₆₄	1	1	yes	characteristic
116	C₂×C₄	#13050578	1	1	yes	characteristic
117	C₂×C₄	#13050579	1	1	yes	characteristic
118	C₂×C₄	#13050580	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	332	2	no	—
121	C₂²	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
122	C₂²	#14581706	1	1	yes	characteristic
123	C₂²	#14581710	1	1	yes	characteristic
124	C₂²	#14581714	1	1	yes	characteristic
125	C₂²	#14581715	1	1	yes	characteristic
126	C₂²	#14581718	1	1	yes	characteristic
127	C₂²	#14581720	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	332	2	no	—
135	C₄	—	664	1	no	—
136	C₂	#19127426	1	1	yes	characteristic
137	C₂	#19127430	1	1	yes	characteristic
138	C₂	#19127431	1	1	yes	characteristic
139	C₂	#19127436	1	1	yes	characteristic
140	C₂	#19127437	1	1	yes	characteristic
141	C₂	#19127442	1	1	yes	characteristic
142	C₂	#19127446	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127599

C₂

yes

maximal, characteristic

#19127456

C₂

yes

maximal, characteristic

#336×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

—

C₂×C₄⋊Dic₈₃

C₄

yes

characteristic

C₂×C₄⋊Dic₈₃

C₄

yes

characteristic

C₂×Dic₈₃⋊C₄

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₈

yes

complemented

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

characteristic

C₂×C₄×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

Q₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²⋊C₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#324

yes

characteristic

C₂×C₈₃

D₄⋊C₈

yes

characteristic

C₂×C₈₃

Q₈⋊C₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#389

—

#336

—

#336

D₈₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#95

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

332

—

C₂×C₈

—

166

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

—

100

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

332

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

332

—

110

C₂×C₄

—

332

—

111

C₂×C₄

—

332

—

112

C₂×C₄

—

166

—

113

C₂×C₄

C₈×D₈₃

yes

characteristic

114

C₂×C₄

#13050573

yes

characteristic

115

C₂×C₄

D₆₆₄

yes

characteristic

116

C₂×C₄

#13050578

yes

characteristic

117

C₂×C₄

#13050579

yes

characteristic

118

C₂×C₄

#13050580

yes

characteristic

119

C₈

—

120

C₈

—

332

—

121

C₂²

(C₂×D₈₃)⋊C₈

yes

characteristic

122

C₂²

#14581706

yes

characteristic

123

C₂²

#14581710

yes

characteristic

124

C₂²

#14581714

yes

characteristic

125

C₂²

#14581715

yes

characteristic

126

C₂²

#14581718

yes

characteristic

127

C₂²

#14581720

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

332

—

135

C₄

—

664

—

136

C₂

#19127426

yes

characteristic

137

C₂

#19127430

yes

characteristic

138

C₂

#19127431

yes

characteristic

139

C₂

#19127436

yes

characteristic

140

C₂

#19127437

yes

characteristic

141

C₂

#19127442

yes

characteristic

142

C₂

#19127446

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂×C₈
Frattini	C₂×C₄²	C₂×D₈₃
Fitting	#336×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂²
hypercenter	#336	D₈₃
hypocenter	C₈₃	#1271
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂×C₈

Frattini

C₂×C₄²

C₂×D₈₃

Fitting

#336×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
18. C₄⋊Dic₈₃	C₈	semidirect
72. C₈₃	#1271	semidirect

H ≅ G/N

type

18. C₄⋊Dic₈₃

C₈

semidirect

72. C₈₃

#1271

semidirect

Subgroups — #24905285

Conjugacy classes of subgroups

Complements