Group #24983469 — subgroups

Subgroups — #24983469

Order 11008.

222 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 127 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	#19133343	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582952	—	2	2	no	—
4	#14582652	C₄	1	4	yes	—
5	#14582650	—	2	2	no	—
6	C₂²×C₁₆×C₄₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₄₃	—	2	2	no	—
9	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
11	C₄×C₄₃⋊C₈	—	4	2	no	—
12	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₄₃	—	2	2	no	—
16	C₂×C₁₆×C₄₃	—	2	2	no	—
17	C₂×C₁₆×C₄₃	C₈	1	4	yes	complemented
18	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₃	D₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
22	C₂×C₈×C₄₃	—	2	2	no	—
23	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
24	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
27	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
28	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₄₃	D₈	1	1	yes	characteristic
30	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
31	C₂×C₄₃⋊C₈	—	4	4	no	—
32	C₁₆×C₄₃	—	2	4	no	—
33	C₂³×C₄₃	#106	1	1	yes	characteristic
34	C₂×C₄×C₄₃	—	2	4	no	—
35	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
36	C₂×C₄×C₄₃	#102	1	2	yes	—
37	C₂×C₄×C₄₃	#103	1	2	yes	—
38	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
39	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
40	C₂×C₄×C₄₃	#108	1	2	yes	—
41	C₈×C₄₃	—	2	2	no	—
42	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
43	C₈×C₄₃	#97	1	1	yes	characteristic
44	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
45	C₄₃⋊C₈	—	8	4	no	—
46	#7155	—	43	1	no	maximal, Sylow
47	C₂²×C₄₃	—	2	2	no	—
48	C₂²×C₄₃	#334	1	2	yes	—
49	C₂²×C₄₃	#340	1	1	yes	characteristic
50	C₄×C₄₃	—	2	2	no	—
51	C₄×C₄₃	#335	1	2	yes	—
52	C₄×C₄₃	#336	1	1	yes	characteristic
53	C₄×C₄₃	#341	1	1	yes	characteristic
54	C₄×C₄₃	#359	1	4	yes	—
55	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic
56	#1564	—	43	2	no	—
57	C₂×C₄₃	—	2	2	no	—
58	C₂×C₄₃	#1277	1	1	yes	characteristic
59	C₂×C₄₃	#1382	1	2	yes	—
60	C₂²×C₁₆	Dic₄₃	1	2	yes	—
61	#431	—	86	2	no	—
62	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
63	C₄×C₁₆	—	2	2	no	—
64	#335	—	43	4	no	—
65	#334	—	86	2	no	—
66	C₄₃	#7155	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
68	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
69	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₄₃⋊C₈	1	4	yes	complemented
73	C₄⋊C₈	—	86	4	no	—
74	C₂²⋊C₈	—	172	2	no	—
75	C₄×C₈	—	172	2	no	—
76	C₄×C₈	C₄×D₄₃	1	2	yes	—
77	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
78	C₂²×C₄	#10621873	1	2	yes	—
79	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	172	4	no	—
82	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
83	C₂×C₈	#10621877	1	2	yes	—
84	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
85	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
86	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
87	C₄²	D₃₄₄	1	1	yes	characteristic
88	C₄²	Dic₁₇₂	1	1	yes	characteristic
89	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
90	C₁₆	—	2	4	no	—
91	C₂³	#13063257	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
94	C₂×C₄	#13063247	1	2	yes	—
95	C₂×C₄	#13063250	1	2	yes	—
96	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
97	C₂×C₄	#13063259	1	1	yes	characteristic
98	C₂×C₄	#13063261	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	344	4	no	—
101	C₈	C₄×C₄₃⋊C₈	1	1	yes	characteristic
102	C₈	#13063236	1	1	yes	characteristic
103	C₈	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	#14582650	1	2	yes	—
106	C₂²	#14582662	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	#14582652	1	2	yes	—
109	C₄	#14582654	1	1	yes	characteristic
110	C₄	#14582664	1	1	yes	characteristic
111	C₄	#14582707	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	#19132430	1	1	yes	characteristic
114	C₂	#19132691	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19133343

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₃

C₂

yes

maximal, characteristic

#14582952

—

#14582652

C₄

yes

—

#14582650

—

C₂²×C₁₆×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₈

yes

complemented

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

Q₁₆

yes

characteristic

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

#106

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#103

yes

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

#107

yes

characteristic

C₂×C₄×C₄₃

#108

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₄×C₈

yes

characteristic

C₈×C₄₃

#97

yes

characteristic

C₈×C₄₃

C₂²⋊C₈

yes

—

C₄₃⋊C₈

—

#7155

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#334

yes

—

C₂²×C₄₃

#340

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#335

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄×C₁₆

D₄₃

yes

characteristic

#1564

—

C₂×C₄₃

—

C₂×C₄₃

#1277

yes

characteristic

C₂×C₄₃

#1382

yes

—

C₂²×C₁₆

Dic₄₃

yes

—

#431

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄×C₁₆

—

#335

—

#334

—

C₄₃

#7155

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₈₆

yes

characteristic

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

C₂×C₄²

D₁₇₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄₃⋊C₈

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

172

—

C₄×C₈

—

172

—

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₂²×C₄

#10621873

yes

—

C₂²×C₄

C₄⋊Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

C₂×C₄₃⋊C₈

yes

—

C₂×C₈

#10621877

yes

—

C₂×C₈

C₄×Dic₄₃

yes

characteristic

C₂×C₈

Dic₄₃⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

C₄²

D₃₄₄

yes

characteristic

C₄²

Dic₁₇₂

yes

characteristic

C₄²

(C₂×D₄₃)⋊C₄

yes

—

C₁₆

—

C₂³

#13063257

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄₃⋊#95

yes

characteristic

C₂×C₄

#13063247

yes

—

C₂×C₄

#13063250

yes

—

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

—

C₂×C₄

#13063259

yes

characteristic

C₂×C₄

#13063261

yes

—

C₈

—

100

C₈

—

344

—

101

C₈

C₄×C₄₃⋊C₈

yes

characteristic

102

C₈

#13063236

yes

characteristic

103

C₈

C₂²⋊(C₄₃⋊C₈)

yes

—

104

C₂²

—

105

C₂²

#14582650

yes

—

106

C₂²

#14582662

yes

characteristic

107

C₄

—

108

C₄

#14582652

yes

—

109

C₄

#14582654

yes

characteristic

110

C₄

#14582664

yes

characteristic

111

C₄

#14582707

yes

—

112

C₂

—

113

C₂

#19132430

yes

characteristic

114

C₂

#19132691

yes

—

	subgroup	G/H
center	C₄²	D₃₄₄
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#106
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#7155
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄²

D₃₄₄

commutator

C₈×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₂×C₄×C₁₆×C₄₃

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₂×C₄×C₁₆

D₄₃

hypocenter

C₄₃

#7155

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
17. C₂×C₁₆×C₄₃	C₈	semidirect
66. C₄₃	#7155	semidirect
72. C₂×C₁₆	C₄₃⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₄₃

C₈

semidirect

66. C₄₃

#7155

semidirect

72. C₂×C₁₆

C₄₃⋊C₈

semidirect

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

subgroup	index
#19133343	2
C₂×C₄×C₁₆×C₄₃	2
#7155	43

subgroup

index

#19133343

C₂×C₄×C₁₆×C₄₃

#7155

Subgroups — #24983469

Conjugacy classes of subgroups

Complements

Maximal subgroups