Group #72261866 — subgroups

Subgroups — #72261866

Order 6144.

Only the 89 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 89 normal subgroups are shown; non-normal subgroups are omitted.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72261348	C₂	1	2	yes	maximal, complemented
2	#72260994	C₂	1	1	yes	maximal, characteristic
3	#72260975	C₂²	1	1	yes	characteristic
4	#72237305	D₃	1	1	yes	characteristic
5	#72235027	C₄	1	2	yes	complemented
6	C₂²⋊#5464	C₂×C₄	1	1	yes	characteristic, complemented
7	#10586796	Dic₃	1	2	yes	complemented
8	C₂²×#3549	C₂×D₃	1	1	yes	characteristic
9	#63040	S₄	1	2	yes	—
10	#62832	S₄	1	2	yes	—
11	C₂×#3549	S₄	1	2	yes	—
12	C₂²×H(3,𝔽₄)	C₂×Dic₃	1	1	yes	characteristic, complemented
13	C₂×#3444	S₄	1	1	yes	characteristic
14	#3549	SL(2,ℤ/4)	1	2	yes	complemented
15	#3549	SL(2,ℤ/4)	1	4	yes	complemented
16	#3549	C₂×S₄	1	2	yes	—
17	#3549	C₂×S₄	1	2	yes	—
18	C₂×H(3,𝔽₄)	SL(2,ℤ/4)	1	2	yes	complemented
19	#3444	SL(2,ℤ/4)	1	4	yes	—
20	#3444	SL(2,ℤ/4)	1	2	yes	—
21	C₂²×#127	C₂×S₄	1	2	yes	—
22	C₂³×C₄²	C₂×S₄	1	1	yes	characteristic
23	H(3,𝔽₄)	C₂×SL(2,ℤ/4)	1	2	yes	—
24	H(3,𝔽₄)	C₂×SL(2,ℤ/4)	1	2	yes	—
25	C₂×#127	#1020	1	4	yes	—
26	C₂×#127	#1020	1	1	yes	characteristic
27	C₂×#120	#1020	1	1	yes	characteristic
28	C₂²×C₄²	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
29	C₂²×C₄²	C₂×SL(2,ℤ/4)	1	2	yes	—
30	C₂²×C₄²	#1020	1	1	yes	characteristic
31	C₂⁵	C₂×#1020	1	1	yes	characteristic
32	#127	#5936	1	4	yes	—
33	#127	#5936	1	2	yes	—
34	#127	C₂×#1020	1	4	yes	—
35	#120	#5936	1	2	yes	—
36	C₂×C₄²	#5936	1	4	yes	—
37	C₂×C₄²	#5936	1	2	yes	—
38	C₂×C₄²	C₂×#1020	1	1	yes	characteristic
39	C₂×C₄²	C₂×#1020	1	1	yes	characteristic
40	C₂⁴	C₂×#5936	1	1	yes	characteristic
41	C₂⁴	#87588	1	1	yes	characteristic
42	C₄²	C₂×#5936	1	1	yes	characteristic
43	C₄²	C₂×#5936	1	4	yes	—
44	C₄²	C₂×#5936	1	1	yes	characteristic
45	C₂³	#11715843	1	2	yes	—
46	C₂³	C₂×#87588	1	1	yes	characteristic
47	C₂²	C₂×#11715843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72261348

C₂

yes

maximal, complemented

#72260994

C₂

yes

maximal, characteristic

#72260975

C₂²

yes

characteristic

#72237305

D₃

yes

characteristic

#72235027

C₄

yes

complemented

C₂²⋊#5464

C₂×C₄

yes

characteristic, complemented

#10586796

Dic₃

yes

complemented

C₂²×#3549

C₂×D₃

yes

characteristic

#63040

S₄

yes

—

#62832

S₄

yes

—

C₂×#3549

S₄

yes

—

C₂²×H(3,𝔽₄)

C₂×Dic₃

yes

characteristic, complemented

C₂×#3444

S₄

yes

yes

yes

C₂×S₄

yes

—

#3549

C₂×S₄

yes

—

C₂×H(3,𝔽₄)

yes

yes

—

#3444

SL(2,ℤ/4)

yes

—

C₂²×#127

C₂×S₄

yes

—

C₂³×C₄²

C₂×S₄

yes

characteristic

H(3,𝔽₄)

C₂×SL(2,ℤ/4)

yes

—

H(3,𝔽₄)

C₂×SL(2,ℤ/4)

yes

—

C₂×#127

#1020

yes

—

yes

yes

C₂²×C₄²

yes

C₂²×C₄²

yes

—

C₂²×C₄²

yes

C₂⁵

yes

yes

—

#127

#5936

yes

—

#127

C₂×#1020

yes

—

#120

#5936

yes

—

C₂×C₄²

#5936

yes

—

C₂×C₄²

#5936

yes

—

C₂×C₄²

yes

C₂×C₄²

yes

C₂⁴

yes

C₂⁴

yes

C₄²

yes

C₄²

yes

—

C₄²

C₂×#5936

yes

characteristic

C₂³

#11715843

yes

—

C₂³

yes

C₂²

yes

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₂²⋊#5464	C₂×C₄
Frattini	C₂³	C₂×#87588
Fitting	#72237305	D₃
socle	C₂²	C₂×#11715843
prime radical	#72260994	C₂
cosocle	#72260975	C₂²
LCS	G ▷ C₂²⋊#5464
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, hypocenter

C₂²⋊#5464

C₂×C₄

C₂³

D₃

C₂²

C₂

C₂²

G ▷ C₂²⋊#5464

C₁

N	H ≅ G/N	type
1. #72261348	C₂	semidirect
5. #72235027	C₄	semidirect
6. C₂²⋊#5464	C₂×C₄	semidirect
7. #10586796	Dic₃	semidirect
12. C₂²×H(3,𝔽₄)	C₂×Dic₃	semidirect
14. #3549	SL(2,ℤ/4)	semidirect
15. #3549	SL(2,ℤ/4)	semidirect
18. C₂×H(3,𝔽₄)	SL(2,ℤ/4)	semidirect

H ≅ G/N

type

1. #72261348

C₂

semidirect

5. #72235027

C₄

semidirect

6. C₂²⋊#5464

C₂×C₄

semidirect

7. #10586796

Dic₃

semidirect

12. C₂²×H(3,𝔽₄)

C₂×Dic₃

semidirect

14. #3549

SL(2,ℤ/4)

semidirect

15. #3549

SL(2,ℤ/4)

semidirect

18. C₂×H(3,𝔽₄)

SL(2,ℤ/4)

semidirect

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

subgroup	index
#72261840	3
#72261664	4
#72261348	2
#72260994	2
#72235439	4
#72235204	4
C₂×#11714278	4

subgroup

index

Subgroups — #72261866

Conjugacy classes of subgroups

Complements

Maximal subgroups