Group #24907677 — subgroups

Subgroups — #24907677

Order 10624.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19128179	C₂	1	1	yes	maximal, characteristic
2	#19127741	C₂	1	2	yes	maximal
3	C₂×#14581711	C₂	1	1	yes	maximal, characteristic
4	#19127461	C₂	1	2	yes	maximal
5	#391×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	Q₈×Dic₈₃	—	2	2	no	—
8	Q₈×Dic₈₃	—	2	2	no	—
9	#14581751	—	2	2	no	—
10	#14581751	C₂²	1	2	yes	—
11	#14581725	—	2	2	no	—
12	#14581723	—	2	2	no	—
13	#14581711	C₄	1	4	yes	complemented
14	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	#95×C₈₃	—	2	2	no	—
18	#13050574	—	2	2	no	—
19	#13050574	C₂×C₄	1	4	yes	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₈₃	—	2	2	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	C₄⋊Dic₈₃	D₄	1	2	yes	—
24	C₄⋊Dic₈₃	D₄	1	2	yes	—
25	C₂²⋊Dic₈₃	—	2	2	no	—
26	C₄×Dic₈₃	—	2	4	no	—
27	C₄×Dic₈₃	—	2	2	no	—
28	C₂×Q₈×C₈₃	—	2	2	no	—
29	C₂×Q₈×C₈₃	—	2	2	no	—
30	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₈₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₈₃	—	2	1	no	—
33	C₂²×C₄×C₈₃	—	2	1	no	—
34	C₂²×C₄×C₈₃	D₄	1	2	yes	—
35	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₃	—	2	2	no	—
37	C₄²×C₈₃	—	2	2	no	—
38	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₈₃	—	2	4	no	—
40	Q₈×C₈₃	—	2	4	no	—
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	4	1	no	—
45	C₂×C₄×C₈₃	—	4	1	no	—
46	C₂×C₄×C₈₃	—	4	2	no	—
47	C₂×C₄×C₈₃	—	4	1	no	—
48	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	#41	1	2	yes	—
53	C₂×Dic₈₃	—	2	2	no	—
54	C₂×Dic₈₃	—	2	2	no	—
55	C₈₃⋊C₈	—	2	4	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₈₃	#120	1	1	yes	characteristic
60	C₂²×C₈₃	#123	1	1	yes	characteristic
61	C₄×C₈₃	—	2	4	no	—
62	C₄×C₈₃	—	4	2	no	—
63	C₄×C₈₃	—	8	1	no	—
64	C₄×C₈₃	C₄×D₄	1	2	yes	—
65	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
66	Dic₈₃	—	4	2	no	—
67	Dic₈₃	—	4	2	no	—
68	C₂×C₈₃	—	2	1	no	—
69	C₂×C₈₃	—	2	1	no	—
70	C₂×C₈₃	#385	1	1	yes	characteristic
71	C₂×C₈₃	#454	1	1	yes	characteristic
72	C₂×C₈₃	#456	1	1	yes	characteristic
73	#1857	—	83	1	no	maximal, Sylow
74	C₈₃	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	—	83	1	no	—
76	#428	—	83	2	no	—
77	C₂×#101	—	83	1	no	—
78	#391	D₈₃	1	1	yes	characteristic
79	#339	—	83	2	no	—
80	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
81	C₂×M₁₆	—	83	2	no	—
82	C₄×Q₈	—	166	2	no	—
83	C₄×Q₈	—	166	2	no	—
84	#117	—	83	2	no	—
85	#117	—	166	2	no	—
86	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
88	#107	—	166	2	no	—
89	#106	—	166	2	no	—
90	#101	—	83	4	no	—
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₁₆₆	1	2	yes	—
95	C₂×Q₈	C₄×D₈₃	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
100	M₁₆	—	83	4	no	—
101	M₁₆	—	166	2	no	—
102	C₂×C₈	—	166	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	83	2	no	—
105	C₄⋊C₄	—	83	2	no	—
106	C₄⋊C₄	—	166	2	no	—
107	C₄⋊C₄	—	166	2	no	—
108	C₂²⋊C₄	—	166	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	166	4	no	—
111	C₄²	—	166	2	no	—
112	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	166	2	no	—
123	C₂×C₄	—	166	2	no	—
124	C₂×C₄	Dic₈₃⋊C₄	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₈₃	1	2	yes	—
127	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
128	C₂×C₄	#13050589	1	2	yes	—
129	C₈	—	166	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
133	C₂²	#14581760	1	1	yes	characteristic
134	C₂²	#14581773	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	332	2	no	—
139	C₄	—	332	2	no	—
140	C₄	(C₂²⋊D₈₃)⋊C₄	1	2	yes	—
141	C₄	#14581769	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19127573	1	1	yes	characteristic
145	C₂	#19127873	1	1	yes	characteristic
146	C₂	#19127883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128179

C₂

yes

maximal, characteristic

#19127741

C₂

yes

maximal

C₂×#14581711

C₂

yes

maximal, characteristic

#19127461

C₂

yes

maximal

#391×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

Q₈×Dic₈₃

—

Q₈×Dic₈₃

—

#14581751

—

#14581751

C₂²

yes

—

#14581725

—

#14581723

—

#14581711

C₄

yes

complemented

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

—

#13050574

—

#13050574

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

D₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂×Q₈×C₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×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

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

#339

—

C₂²×Q₈

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

#117

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

#107

—

166

—

#106

—

166

—

#101

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₆₆

yes

—

C₂×Q₈

C₄×D₈₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

100

M₁₆

—

101

M₁₆

—

166

—

102

C₂×C₈

—

166

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

—

166

—

108

C₂²⋊C₄

—

166

—

109

C₄²

—

110

C₄²

—

166

—

111

C₄²

—

166

—

112

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

166

—

123

C₂×C₄

—

166

—

124

C₂×C₄

Dic₈₃⋊C₄

yes

—

125

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

126

C₂×C₄

D₄×D₈₃

yes

—

127

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

128

C₂×C₄

#13050589

yes

—

129

C₈

—

166

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

133

C₂²

#14581760

yes

characteristic

134

C₂²

#14581773

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

332

—

139

C₄

—

332

—

140

C₄

(C₂²⋊D₈₃)⋊C₄

yes

—

141

C₄

#14581769

yes

—

142

C₂

—

143

C₂

—

144

C₂

#19127573

yes

characteristic

145

C₂

#19127873

yes

characteristic

146

C₂

#19127883

yes

characteristic

	subgroup	G/H
center	C₂²	#14581760
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#391×C₈₃	C₂
socle	C₂²×C₈₃	#120
prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#391	D₈₃
hypocenter	C₈₃	#1857
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#14581760

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#391×C₈₃

C₂

socle

C₂²×C₈₃

#120

prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
13. #14581711	C₄	semidirect
74. C₈₃	#1857	semidirect

H ≅ G/N

type

13. #14581711

C₄

semidirect

74. C₈₃

#1857

semidirect

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

subgroup	index
#19128179	2
#19127741	2
C₂×#14581711	2
#19127461	2
#391×C₈₃	2
#1857	83

subgroup

index

#391×C₈₃

Subgroups — #24907677

Conjugacy classes of subgroups

Complements

Maximal subgroups