Group #24990294 — subgroups

Subgroups — #24990294

Order 11008.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#19134767	C₂	1	1	yes	maximal, characteristic
2	#19132887	C₂	1	1	yes	maximal, characteristic
3	#19132453	C₂	1	4	yes	maximal
4	C₄₃×#1835	C₂	1	1	yes	maximal, characteristic
5	#14582952	C₂²	1	2	yes	—
6	C₂×#13063261	—	2	2	no	—
7	#14582923	C₂²	1	2	yes	—
8	#14582621	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₄₃×#382	C₂²	1	2	yes	—
11	C₄₃×#340	—	2	2	no	—
12	C₂×#10621877	—	2	2	no	—
13	#13063261	—	4	4	no	—
14	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
17	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₄₃	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
20	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₃	Q₈	1	2	yes	—
23	#95×C₄₃	—	2	4	no	—
24	C₂²×C₄×C₄₃	—	4	2	no	—
25	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	#41	1	2	yes	—
27	C₂×C₈×C₄₃	—	2	2	no	—
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
30	C₄⋊C₄×C₄₃	—	4	2	no	—
31	C₄⋊C₄×C₄₃	—	4	2	no	—
32	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₄₃	#41	1	2	yes	—
35	#10621877	—	4	4	no	—
36	C₂×C₄₃⋊C₈	—	2	4	no	—
37	C₂×C₄₃⋊C₈	—	4	2	no	—
38	C₂³×C₄₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₄₃	—	4	2	no	—
40	C₂×C₄×C₄₃	—	8	4	no	—
41	C₂×C₄×C₄₃	—	8	2	no	—
42	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	#117	1	2	yes	—
44	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
45	C₂×C₄×C₄₃	#125	1	2	yes	—
46	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₄₃	#137	1	2	yes	—
48	C₈×C₄₃	—	2	4	no	—
49	C₄₃⋊C₈	—	4	4	no	—
50	C₄₃⋊C₈	—	8	2	no	—
51	#8794	—	43	1	no	maximal, Sylow
52	C₂²×C₄₃	—	2	2	no	—
53	C₂²×C₄₃	#382	1	1	yes	characteristic
54	C₂²×C₄₃	#489	1	2	yes	—
55	C₄×C₄₃	—	2	2	no	—
56	C₄×C₄₃	—	2	2	no	—
57	C₄×C₄₃	—	16	2	no	—
58	C₄×C₄₃	#428	1	2	yes	—
59	C₄×C₄₃	#490	1	1	yes	characteristic
60	C₄×C₄₃	#491	1	1	yes	characteristic
61	#1848	—	43	1	no	—
62	#1835	D₄₃	1	1	yes	characteristic
63	#1452	—	43	1	no	—
64	#1286	—	43	4	no	—
65	C₂×C₄₃	—	2	2	no	—
66	C₂×C₄₃	#1833	1	2	yes	—
67	C₂×C₄₃	#1836	1	1	yes	characteristic
68	#431	—	43	2	no	—
69	C₂×#108	—	86	2	no	—
70	#423	—	43	2	no	—
71	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
72	#382	C₂×D₄₃	1	2	yes	—
73	#340	—	2	2	no	—
74	#322	—	43	4	no	—
75	C₄₃	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	86	2	no	—
77	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
80	#108	—	172	4	no	—
81	C₄⋊C₈	—	86	2	no	—
82	C₄⋊C₈	—	86	4	no	—
83	C₂²⋊C₈	—	86	2	no	—
84	C₄×C₈	—	43	4	no	—
85	C₄×C₈	Dic₈₆	1	2	yes	—
86	C₄×C₈	C₄×D₄₃	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#10621900	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
91	M₁₆	—	172	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	86	4	no	—
95	C₂×C₈	—	172	2	no	—
96	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
100	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
101	C₄²	#10621902	1	2	yes	—
102	C₂³	#13063312	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#13063283	1	2	yes	—
108	C₂×C₄	#13063314	1	1	yes	characteristic
109	C₂×C₄	#13063316	1	2	yes	—
110	C₂×C₄	#13063350	1	2	yes	—
111	C₂×C₄	#13063355	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	344	2	no	—
114	C₈	—	172	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#14582764	1	1	yes	characteristic
117	C₂²	#14583258	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#14582942	1	2	yes	—
122	C₄	#14583265	1	1	yes	characteristic
123	C₄	#14583269	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#19134718	1	2	yes	—
126	C₂	#19134728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134767

C₂

yes

maximal, characteristic

#19132887

C₂

yes

maximal, characteristic

#19132453

C₂

yes

maximal

C₄₃×#1835

C₂

yes

maximal, characteristic

#14582952

C₂²

yes

—

C₂×#13063261

—

#14582923

C₂²

yes

—

#14582621

C₄

yes

complemented

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#382

C₂²

yes

—

C₄₃×#340

—

C₂×#10621877

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

#125

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#8794

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#382

yes

characteristic

C₂²×C₄₃

#489

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#428

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

D₄₃

yes

characteristic

#1452

—

#1286

—

C₂×C₄₃

—

C₂×C₄₃

#1833

yes

—

C₂×C₄₃

yes

—

—

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#382

C₂×D₄₃

yes

—

#340

—

#322

—

C₄₃

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#108

—

172

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₈₆

yes

—

C₄×C₈

C₄×D₄₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#10621900

yes

—

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

M₁₆

—

172

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

Dic₄₃⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₈₆

yes

characteristic

100

C₄²

C₂×C₄×D₄₃

yes

characteristic

101

C₄²

#10621902

yes

—

102

C₂³

#13063312

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

107

C₂×C₄

#13063283

yes

—

108

C₂×C₄

#13063314

yes

characteristic

109

C₂×C₄

#13063316

yes

—

110

C₂×C₄

#13063350

yes

—

111

C₂×C₄

#13063355

yes

—

112

C₈

—

113

C₈

—

344

—

114

C₈

—

172

—

115

C₂²

—

116

C₂²

#14582764

yes

characteristic

117

C₂²

#14583258

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#14582942

yes

—

122

C₄

#14583265

yes

characteristic

123

C₄

#14583269

yes

characteristic

124

C₂

—

125

C₂

#19134718

yes

—

126

C₂

#19134728

yes

characteristic

	subgroup	G/H
center	C₂²	#14582764
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1835	C₂
socle	C₂²×C₄₃	#382
prime radical	C₂³×C₄₃	#124
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1835	D₄₃
hypocenter	C₄₃	#8794
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#14582764

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1835

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #14582621	C₄	semidirect
75. C₄₃	#8794	semidirect

H ≅ G/N

type

8. #14582621

C₄

semidirect

75. C₄₃

#8794

semidirect

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

subgroup	index
#19134767	2
#19132887	2
#19132453	2
C₄₃×#1835	2
#8794	43

subgroup

index

C₄₃×#1835

Subgroups — #24990294

Conjugacy classes of subgroups

Complements

Maximal subgroups