Group #24990371 — subgroups

Subgroups — #24990371

Order 11008.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#19134752	C₂	1	2	yes	maximal
2	#19134723	C₂	1	4	yes	maximal
3	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063257	—	2	4	no	—
5	#14582769	C₂²	1	4	yes	—
6	#14582662	—	2	8	no	—
7	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₈²×C₄₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₃	—	2	8	no	—
11	C₂×C₄⋊Dic₄₃	—	2	4	no	—
12	#13063257	—	4	8	no	—
13	C₂²×C₈×C₄₃	D₄	1	4	yes	—
14	C₂²×C₈×C₄₃	D₄	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₄₃	Q₈	1	4	yes	—
19	C₄×C₈×C₄₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
22	C₂²×Dic₄₃	—	4	4	no	—
23	C₂×C₈×C₄₃	—	2	8	no	—
24	C₂×C₈×C₄₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₄₃	SD₁₆	1	8	yes	—
27	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
29	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
30	C₄⋊Dic₄₃	—	4	8	no	—
31	C₄⋊Dic₄₃	—	4	16	no	—
32	C₂³×C₄₃	#127	1	1	yes	characteristic
33	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₄₃	#135	1	8	yes	—
39	C₂×Dic₄₃	—	8	4	no	—
40	C₂×Dic₄₃	—	8	8	no	—
41	C₈×C₄₃	—	2	8	no	—
42	C₈×C₄₃	#106	1	8	yes	—
43	#8815	—	43	1	no	maximal, Sylow
44	C₂²×C₄₃	#384	1	1	yes	characteristic
45	C₂²×C₄₃	#492	1	2	yes	—
46	C₂²×C₄₃	#495	1	4	yes	—
47	C₄×C₄₃	C₂×#106	1	2	yes	—
48	C₄×C₄₃	#427	1	4	yes	—
49	C₄×C₄₃	#498	1	2	yes	—
50	C₄×C₄₃	#499	1	4	yes	—
51	Dic₄₃	—	16	4	no	—
52	#1844	—	43	2	no	—
53	#1834	—	43	4	no	—
54	C₂×C₈²	D₄₃	1	1	yes	characteristic
55	C₂×C₄₃	#1708	1	4	yes	—
56	C₂×C₄₃	#1844	1	1	yes	characteristic
57	C₂×C₄₃	#1846	1	2	yes	—
58	C₂×#106	—	86	4	no	—
59	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
61	#384	—	43	4	no	—
62	#340	—	86	8	no	—
63	C₈²	Dic₄₃	1	4	yes	complemented
64	C₄₃	#8815	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	D₁₇₂	1	4	yes	—
66	C₂²×C₈	D₁₇₂	1	2	yes	—
67	C₂×C₄⋊C₄	—	86	8	no	—
68	C₂×C₄⋊C₄	—	86	4	no	—
69	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
70	#106	—	172	8	no	—
71	C₄×C₈	Dic₈₆	1	4	yes	—
72	C₄×C₈	Dic₈₆	1	2	yes	—
73	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
74	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
75	C₂²×C₄	—	172	4	no	—
76	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	#10621873	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₄₃	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
82	C₄⋊C₄	—	172	8	no	—
83	C₄⋊C₄	—	172	16	no	—
84	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
85	C₄²	C₂×Dic₈₆	1	2	yes	—
86	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
87	C₂³	#13063322	1	1	yes	characteristic
88	C₂×C₄	—	344	4	no	—
89	C₂×C₄	—	344	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
92	C₂×C₄	#13063325	1	1	yes	characteristic
93	C₂×C₄	#13063325	1	2	yes	—
94	C₂×C₄	C₂×#10621873	1	4	yes	—
95	C₂×C₄	#13063347	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#13063257	1	8	yes	—
98	C₂²	#14582769	1	1	yes	characteristic
99	C₂²	#14583272	1	2	yes	—
100	C₂²	#14583283	1	4	yes	—
101	C₄	—	688	4	no	—
102	C₄	C₂×#13063257	1	2	yes	—
103	C₄	#14582937	1	4	yes	—
104	C₄	#14583297	1	2	yes	—
105	C₄	#14583301	1	4	yes	—
106	C₂	#19134196	1	4	yes	—
107	C₂	#19134752	1	1	yes	characteristic
108	C₂	#19134759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134752

C₂

yes

maximal

#19134723

C₂

yes

maximal

C₂×C₈²×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063257

—

#14582769

C₂²

yes

—

#14582662

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063257

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#106

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#495

yes

—

C₄×C₄₃

C₂×#106

yes

—

C₄×C₄₃

#427

yes

—

C₄×C₄₃

#498

yes

—

C₄×C₄₃

#499

yes

—

Dic₄₃

—

#1844

—

#1834

—

C₂×C₈²

D₄₃

yes

characteristic

C₂×C₄₃

#1708

yes

—

C₂×C₄₃

#1844

yes

characteristic

C₂×C₄₃

#1846

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#384

—

#340

—

C₈²

Dic₄₃

yes

complemented

C₄₃

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₇₂

yes

—

C₂²×C₈

D₁₇₂

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#106

—

172

—

C₄×C₈

Dic₈₆

yes

—

C₄×C₈

Dic₈₆

yes

—

C₄×C₈

C₂×Dic₄₃

yes

—

C₄×C₈

C₂×Dic₄₃

yes

—

C₂²×C₄

—

172

—

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

C₂²×C₄

C₂×D₁₇₂

yes

—

C₂×C₈

—

C₂×C₈

#10621873

yes

—

C₂×C₈

C₄⋊Dic₄₃

yes

—

C₂×C₈

C₄⋊Dic₄₃

yes

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄²

C₂×Dic₈₆

yes

characteristic

C₄²

C₂×Dic₈₆

yes

—

C₄²

C₂²×Dic₄₃

yes

characteristic

C₂³

#13063322

yes

characteristic

C₂×C₄

—

344

—

C₂×C₄

—

344

—

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

—

C₂×C₄

#13063325

yes

characteristic

C₂×C₄

#13063325

yes

—

C₂×C₄

C₂×#10621873

yes

—

C₂×C₄

#13063347

yes

—

C₈

—

C₈

#13063257

yes

—

C₂²

#14582769

yes

characteristic

C₂²

#14583272

yes

—

100

C₂²

#14583283

yes

—

101

C₄

—

688

—

102

C₄

C₂×#13063257

yes

—

103

C₄

#14582937

yes

—

104

C₄

#14583297

yes

—

105

C₄

#14583301

yes

—

106

C₂

#19134196

yes

—

107

C₂

#19134752

yes

characteristic

108

C₂

#19134759

yes

—

	subgroup	G/H
center	C₂³	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#8815
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13063322

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₈²×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

#127

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₂×C₈²

D₄₃

hypocenter

C₄₃

#8815

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₄₃	C₄	semidirect
63. C₈²	Dic₄₃	semidirect
64. C₄₃	#8815	semidirect

H ≅ G/N

type

9. C₈²×C₄₃

C₄

semidirect

63. C₈²

Dic₄₃

semidirect

64. C₄₃

#8815

semidirect

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

subgroup	index
#19134752	2
#19134723	2
C₂×C₈²×C₄₃	2
#8815	43

subgroup

index

#19134752

#19134723

C₂×C₈²×C₄₃

#8815

Subgroups — #24990371

Conjugacy classes of subgroups

Complements

Maximal subgroups