Group #72261698 — subgroups

Subgroups — #72261698

Order 12288.

At least 16840 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has at least 16840 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72261592	C₂	1	1	yes	maximal, characteristic, complemented
2	#72260766	C₂	1	1	yes	maximal, characteristic
3	#72240363	C₂	1	1	yes	maximal, characteristic, complemented
4	#72240348	C₂²	1	1	yes	characteristic
5	#72240347	D₄	1	1	yes	characteristic
6	#72233929	D₃	1	1	yes	characteristic
7	#72233925	C₂×D₃	1	1	yes	characteristic
8	C₂³×#392	S₄	1	1	yes	characteristic
9	C₂²×#2380	S₄	1	1	yes	characteristic
10	#158342	S₄	1	1	yes	characteristic
11	#154074	#67	1	1	yes	characteristic
12	C₂³×#127	C₂×S₄	1	1	yes	characteristic
13	C₂³×#95	C₂×S₄	1	1	yes	characteristic
14	#7555	C₂×S₄	1	1	yes	characteristic
15	C₂⁵×C₄	#1020	1	1	yes	characteristic
16	C₂²×#127	#984	1	1	yes	characteristic
17	C₂²×#127	#985	1	1	yes	characteristic
18	C₂³×C₄²	GL(2,ℤ/4)	1	1	yes	characteristic
19	C₂²×#95	GL(2,ℤ/4)	1	1	yes	characteristic
20	C₂²×#95	GL(2,ℤ/4)	1	1	yes	characteristic
21	C₂⁶	C₂×#1020	1	1	yes	characteristic
22	C₂×#127	#857⋊C₂	1	2	yes	—
23	C₂²×C₄²	#5425	1	1	yes	characteristic
24	C₂⁵	#87508	1	1	yes	characteristic
25	C₂⁵	#87509	1	1	yes	characteristic
26	C₂⁵	#87524	1	1	yes	characteristic
27	#127	#72995	1	2	yes	—
28	#127	#72996	1	2	yes	—
29	C₂×C₄²	#73084	1	2	yes	—
30	C₂⁴	#11715690	1	1	yes	characteristic
31	C₂⁴	#11715699	1	1	yes	characteristic
32	C₄²	#11711240	1	2	yes	complemented
33	C₂³	AGL(2,ℤ/4)	1	1	yes	characteristic
34	C₂³	#72224760	1	1	yes	characteristic
35	C₂³	#72226631	1	1	yes	characteristic
36	C₂³	#72231237	1	1	yes	characteristic
37	C₂³	#72231238	1	1	yes	characteristic
38	C₂²	#72231239	1	1	yes	characteristic
39	C₂²	#72260821	1	1	yes	characteristic
40	C₂²	#72261269	1	1	yes	characteristic
41	C₂	#72260915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72261592

C₂

yes

maximal, characteristic, complemented

#72260766

C₂

yes

maximal, characteristic

#72240363

C₂

yes

maximal, characteristic, complemented

#72240348

C₂²

yes

characteristic

#72240347

D₄

yes

characteristic

#72233929

D₃

yes

characteristic

#72233925

C₂×D₃

yes

characteristic

C₂³×#392

S₄

yes

characteristic

C₂²×#2380

S₄

yes

characteristic

#158342

S₄

yes

yes

C₂³×#127

C₂×S₄

yes

characteristic

C₂³×#95

C₂×S₄

yes

characteristic

#7555

C₂×S₄

yes

C₂⁵×C₄

yes

C₂²×#127

yes

C₂²×#127

yes

C₂³×C₄²

yes

C₂²×#95

yes

C₂²×#95

yes

C₂⁶

yes

yes

—

C₂²×C₄²

yes

C₂⁵

yes

C₂⁵

yes

C₂⁵

yes

yes

—

#127

#72996

yes

—

C₂×C₄²

#73084

yes

—

C₂⁴

yes

C₂⁴

yes

C₄²

yes

C₂³

yes

C₂³

yes

C₂³

yes

C₂³

yes

C₂³

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂

yes

	subgroup	G/H
center, hypercenter	C₂	#72260915
commutator, cosocle	#72240348	C₂²
Frattini	C₂⁶	C₂×#1020
Fitting	#72233929	D₃
socle	C₂⁵	#87524
prime radical	#72260766	C₂
hypocenter	#72240347	D₄
LCS	G ▷ #72240348 ▷ #72240347
UCS	C₁ ◁ C₂

subgroup

G/H

C₂

C₂²

C₂⁶

D₃

C₂⁵

C₂

D₄

G ▷ #72240348 ▷ #72240347

UCS

C₁ ◁ C₂

Complements

N	H ≅ G/N	type
1. #72261592	C₂	semidirect
3. #72240363	C₂	semidirect
32. C₄²	#11711240	semidirect

H ≅ G/N

type

1. #72261592

C₂

semidirect

3. #72240363

C₂

semidirect

32. C₄²

#11711240

semidirect

Maximal subgroups

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

subgroup	index
#72261592	2
#72261452	3
#72261431	4
#72261420	4
#72260766	2
#72240363	2
#72231239	4

subgroup

index