Group #24990373 — subgroups

Subgroups — #24990373

Order 11008.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#19134755	C₂	1	1	yes	maximal, characteristic
2	#19134722	C₂	1	4	yes	maximal
3	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063259	—	2	2	no	—
6	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
7	#14582769	C₂²	1	2	yes	—
8	#14582661	—	2	8	no	—
9	C₈×C₄₃⋊C₈	C₄	1	4	yes	complemented
10	C₂×#107×C₄₃	—	2	2	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#384	C₂²	1	2	yes	—
13	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₄₃	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	—	2	2	no	—
16	#13063259	—	4	4	no	—
17	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₄₃⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₄₃	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
22	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
23	#107×C₄₃	—	4	4	no	—
24	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₃	—	4	2	no	—
27	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	4	2	no	—
30	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₄₃	D₈	1	4	yes	—
32	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
33	C₄⋊C₄×C₄₃	—	4	4	no	—
34	C₄⋊C₄×C₄₃	—	4	8	no	—
35	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₄₃	—	4	4	no	—
39	C₄⋊Dic₄₃	—	4	8	no	—
40	C₂×C₄₃⋊C₈	—	2	4	no	—
41	C₂×C₄₃⋊C₈	—	2	4	no	—
42	C₂×C₄₃⋊C₈	C₄⋊C₄	1	8	yes	—
43	C₂³×C₄₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₄₃	—	8	2	no	—
45	C₂×C₄×C₄₃	—	8	4	no	—
46	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₄₃	#135	1	4	yes	—
53	C₂×C₄×C₄₃	#135	1	4	yes	—
54	C₂×Dic₄₃	—	8	2	no	—
55	C₂×Dic₄₃	—	8	4	no	—
56	C₈×C₄₃	#107	1	8	yes	—
57	C₄₃⋊C₈	—	2	8	no	—
58	#8816	—	43	1	no	maximal, Sylow
59	C₂²×C₄₃	#384	1	1	yes	characteristic
60	C₂²×C₄₃	#493	1	1	yes	characteristic
61	C₂²×C₄₃	#494	1	1	yes	characteristic
62	C₂²×C₄₃	#495	1	2	yes	—
63	C₂²×C₄₃	#495	1	2	yes	—
64	C₄×C₄₃	—	16	2	no	—
65	C₄×C₄₃	C₂×#107	1	2	yes	—
66	C₄×C₄₃	#427	1	4	yes	—
67	C₄×C₄₃	#499	1	4	yes	—
68	C₄×C₄₃	#500	1	2	yes	—
69	Dic₄₃	—	16	2	no	—
70	#1845	—	43	1	no	—
71	#1845	D₄₃	1	1	yes	characteristic
72	#1834	—	43	4	no	—
73	C₂×C₈²	—	43	1	no	—
74	C₂×C₄₃	#1709	1	1	yes	characteristic
75	C₂×C₄₃	#1709	1	1	yes	characteristic
76	C₂×C₄₃	#1710	1	2	yes	—
77	C₂×C₄₃	#1845	1	1	yes	characteristic
78	C₂×C₄₃	#1846	1	2	yes	—
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	86	2	no	—
81	C₂×C₄×C₈	—	43	2	no	—
82	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
83	#384	—	43	2	no	—
84	#384	C₂×D₄₃	1	2	yes	—
85	#340	—	86	8	no	—
86	C₈²	—	43	4	no	—
87	C₄₃	#8816	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	43	4	no	—
89	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	86	4	no	—
93	C₂×C₄⋊C₄	—	86	2	no	—
94	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	172	4	no	—
97	C₄×C₈	—	43	4	no	—
98	C₄×C₈	—	43	4	no	—
99	C₄×C₈	Dic₈₆	1	2	yes	—
100	C₄×C₈	C₄×D₄₃	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	172	2	no	—
103	C₂²×C₄	D₄×D₄₃	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
105	C₂×C₈	—	43	8	no	—
106	C₂×C₈	—	86	4	no	—
107	C₂×C₈	—	86	4	no	—
108	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
109	C₂×C₈	#10621882	1	4	yes	—
110	C₂×C₈	#10621885	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	172	4	no	—
114	C₄⋊C₄	—	172	8	no	—
115	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
117	C₄²	Q₈×D₄₃	1	2	yes	—
118	C₂³	#13063321	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	344	2	no	—
122	C₂×C₄	—	344	4	no	—
123	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
124	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#13063323	1	1	yes	characteristic
126	C₂×C₄	#13063324	1	2	yes	—
127	C₂×C₄	C₂×#10621882	1	2	yes	—
128	C₂×C₄	C₂×#10621885	1	2	yes	—
129	C₂×C₄	#13063344	1	4	yes	—
130	C₂×C₄	#13063348	1	4	yes	—
131	C₈	—	86	8	no	—
132	C₈	#13063258	1	8	yes	—
133	C₂²	#14582768	1	1	yes	characteristic
134	C₂²	#14583275	1	1	yes	characteristic
135	C₂²	#14583278	1	1	yes	characteristic
136	C₂²	#14583282	1	2	yes	—
137	C₂²	#14583284	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	688	2	no	—
140	C₄	C₂×#13063258	1	2	yes	—
141	C₄	#14582936	1	4	yes	—
142	C₄	#14583300	1	4	yes	—
143	C₄	#14583302	1	2	yes	—
144	C₂	#19134200	1	1	yes	characteristic
145	C₂	#19134202	1	1	yes	characteristic
146	C₂	#19134204	1	2	yes	—
147	C₂	#19134754	1	1	yes	characteristic
148	C₂	#19134758	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134755

C₂

yes

maximal, characteristic

#19134722

C₂

yes

maximal

C₂×C₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₄₃×#1845

C₂

yes

maximal, characteristic

C₂×#13063259

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

—

#14582661

—

C₈×C₄₃⋊C₈

C₄

yes

complemented

C₂×#107×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

C₂²

yes

—

C₂²×C₄₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#107

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#495

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#107

yes

—

C₄×C₄₃

#427

yes

—

C₄×C₄₃

#499

yes

—

C₄×C₄₃

#500

yes

—

Dic₄₃

—

#1845

—

#1845

D₄₃

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#1845

yes

characteristic

C₂×C₄₃

#1846

yes

—

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#384

—

#384

C₂×D₄₃

yes

—

#340

—

C₈²

—

C₄₃

#8816

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#107

—

#107

—

172

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₈₆

yes

—

100

C₄×C₈

C₄×D₄₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

172

—

103

C₂²×C₄

D₄×D₄₃

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

Dic₄₃⋊C₄

yes

—

109

C₂×C₈

#10621882

yes

—

110

C₂×C₈

#10621885

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

172

—

114

C₄⋊C₄

—

172

—

115

C₄²

C₂×Dic₈₆

yes

characteristic

116

C₄²

C₂×C₄×D₄₃

yes

characteristic

117

C₄²

Q₈×D₄₃

yes

—

118

C₂³

#13063321

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

344

—

122

C₂×C₄

—

344

—

123

C₂×C₄

C₄⋊C₄×D₄₃

yes

—

124

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

125

C₂×C₄

#13063323

yes

characteristic

126

C₂×C₄

#13063324

yes

—

127

C₂×C₄

C₂×#10621882

yes

—

128

C₂×C₄

C₂×#10621885

yes

—

129

C₂×C₄

#13063344

yes

—

130

C₂×C₄

#13063348

yes

—

131

C₈

—

132

C₈

#13063258

yes

—

133

C₂²

#14582768

yes

characteristic

134

C₂²

#14583275

yes

characteristic

135

C₂²

#14583278

yes

characteristic

136

C₂²

#14583282

yes

—

137

C₂²

#14583284

yes

—

138

C₄

—

139

C₄

—

688

—

140

C₄

C₂×#13063258

yes

—

141

C₄

#14582936

yes

—

142

C₄

#14583300

yes

—

143

C₄

#14583302

yes

—

144

C₂

#19134200

yes

characteristic

145

C₂

#19134202

yes

characteristic

146

C₂

#19134204

yes

—

147

C₂

#19134754

yes

characteristic

148

C₂

#19134758

yes

—

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1845	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1845	D₄₃
hypocenter	C₄₃	#8816
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1845

C₂

socle, prime radical

C₂³×C₄₃

#127

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
9. C₈×C₄₃⋊C₈	C₄	semidirect
87. C₄₃	#8816	semidirect

H ≅ G/N

type

9. C₈×C₄₃⋊C₈

C₄

semidirect

87. C₄₃

#8816

semidirect

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

subgroup	index
#19134755	2
#19134722	2
C₂×C₈×C₄₃⋊C₈	2
C₄₃×#1845	2
#8816	43

subgroup

index

#19134755

#19134722

C₂×C₈×C₄₃⋊C₈

C₄₃×#1845

#8816

Subgroups — #24990373

Conjugacy classes of subgroups

Complements

Maximal subgroups