Group #72271449 — subgroups

Subgroups — #72271449

Order 6144.

Only the 114 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 114 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	#72271429	C₂	1	1	yes	maximal, characteristic, complemented
2	#72271405	C₂	1	2	yes	maximal, complemented
3	C₂×#11715584	C₂²	1	1	yes	characteristic, complemented
4	#72268140	C₂	1	1	yes	maximal, characteristic, complemented
5	#72268131	C₂	1	2	yes	maximal, complemented
6	#72268130	C₂²	1	2	yes	complemented
7	#72268129	C₂²	1	1	yes	characteristic, complemented
8	#72268063	C₂	1	1	yes	maximal, characteristic, complemented
9	#72268062	D₃	1	1	yes	characteristic, complemented
10	#72268058	C₂²	1	2	yes	—
11	C₂×#62844⋊C₃	C₂²	1	1	yes	characteristic
12	#62904⋊C₃	D₄	1	2	yes	complemented
13	#62894⋊C₃	D₄	1	2	yes	complemented
14	C₂×#3459⋊C₃	D₄	1	1	yes	characteristic
15	C₂×#3459⋊C₃	C₂³	1	1	yes	characteristic
16	C₂²×#543⋊C₃	D₄	1	1	yes	characteristic
17	#10586407	C₂×D₃	1	2	yes	—
18	C₂×#62844	C₂×D₃	1	1	yes	characteristic
19	#3459⋊C₃	C₂×D₄	1	2	yes	—
20	C₂×#543⋊C₃	C₂×D₄	1	1	yes	characteristic
21	#62904	#67	1	2	yes	complemented
22	#62894	#67	1	2	yes	complemented
23	#62893	S₄	1	2	yes	complemented
24	C₂×#3459	#67	1	1	yes	characteristic
25	C₂×#3459	C₂²×D₃	1	1	yes	characteristic
26	C₂²×#543	#67	1	1	yes	characteristic
27	C₂²×#534	S₄	1	1	yes	characteristic
28	#543⋊C₃	#120	1	1	yes	characteristic
29	#3489	C₂×S₄	1	4	yes	—
30	C₂×#548	C₂×S₄	1	2	yes	—
31	#3459	C₂×#67	1	2	yes	—
32	C₂×#543	C₂×#67	1	1	yes	characteristic
33	C₂²×#121	C₂×S₄	1	2	yes	—
34	C₂³×C₂²⋊C₄	C₂×S₄	1	1	yes	characteristic
35	C₂⁶	GL(2,ℤ/4)	1	1	yes	characteristic
36	C₂×2₋¹⁺⁴	GL(2,ℤ/4)	1	4	yes	complemented
37	C₂²×#41	C₂²×S₄	1	2	yes	—
38	C₂³×Q₈	GL(2,ℤ/4)	1	2	yes	—
39	C₂³×D₄	GL(2,ℤ/4)	1	2	yes	—
40	C₂⁴×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
41	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
42	#543	#953	1	1	yes	characteristic
43	#534	#1020	1	1	yes	characteristic
44	#520	GL(2,ℤ/4)	1	4	yes	complemented
45	C₂×C₄×Q₈	GL(2,ℤ/4)	1	2	yes	—
46	C₂²×C₄⋊C₄	GL(2,ℤ/4)	1	2	yes	—
47	C₂⁵	#5431	1	2	yes	complemented
48	C₂⁵	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
49	C₂×#41	C₂×GL(2,ℤ/4)	1	4	yes	—
50	C₂²×Q₈	C₂×GL(2,ℤ/4)	1	2	yes	—
51	C₂³×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
52	#121	C₂×#1020	1	2	yes	—
53	C₂×C₂²⋊C₄	C₂×#1020	1	1	yes	characteristic
54	C₂⁴	C₂×#5431	1	1	yes	characteristic
55	C₂⁴	#85533	1	1	yes	characteristic
56	C₂⁴	#85546	1	1	yes	characteristic
57	C₂⁴	#87524	1	1	yes	characteristic
58	C₂×Q₈	#85546	1	2	yes	—
59	C₂×D₄	#87524	1	2	yes	—
60	C₂²×C₄	#87524	1	1	yes	characteristic
61	C₂²×C₄	C₂²×#1020	1	1	yes	characteristic
62	C₄⋊C₄	#87524	1	2	yes	—
63	C₂³	#11713951	1	1	yes	characteristic
64	C₂³	#11715702	1	2	yes	—
65	C₂³	C₂×#87524	1	1	yes	characteristic
66	C₂×C₄	C₂×#87524	1	2	yes	—
67	C₂²	#72209718	1	1	yes	characteristic
68	C₂²	#72227573	1	1	yes	characteristic
69	C₂²	C₂×#11715702	1	1	yes	characteristic
70	C₂	#72227580	1	1	yes	characteristic
71	C₂	#72227648	1	1	yes	characteristic
72	C₂	#72227958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72271429

C₂

yes

maximal, characteristic, complemented

#72271405

C₂

yes

maximal, complemented

C₂×#11715584

C₂²

yes

characteristic, complemented

#72268140

C₂

yes

maximal, characteristic, complemented

#72268131

C₂

yes

maximal, complemented

#72268130

C₂²

yes

complemented

#72268129

C₂²

yes

characteristic, complemented

#72268063

C₂

yes

maximal, characteristic, complemented

#72268062

D₃

yes

characteristic, complemented

#72268058

C₂²

yes

—

C₂×#62844⋊C₃

C₂²

yes

characteristic

#62904⋊C₃

D₄

yes

complemented

#62894⋊C₃

D₄

yes

complemented

C₂×#3459⋊C₃

D₄

yes

characteristic

C₂×#3459⋊C₃

C₂³

yes

characteristic

C₂²×#543⋊C₃

D₄

yes

characteristic

#10586407

C₂×D₃

yes

—

C₂×#62844

C₂×D₃

yes

characteristic

#3459⋊C₃

C₂×D₄

yes

—

C₂×#543⋊C₃

C₂×D₄

yes

yes

yes

S₄

yes

yes

C₂²×D₃

yes

characteristic

C₂²×#543

#67

yes

characteristic

C₂²×#534

S₄

yes

yes

C₂×S₄

yes

—

C₂×#548

C₂×S₄

yes

—

#3459

C₂×#67

yes

—

C₂×#543

C₂×#67

yes

characteristic

C₂²×#121

C₂×S₄

yes

—

C₂³×C₂²⋊C₄

C₂×S₄

yes

C₂⁶

yes

C₂×2₋¹⁺⁴

yes

C₂²×#41

C₂²×S₄

yes

—

C₂³×Q₈

GL(2,ℤ/4)

yes

—

C₂³×D₄

GL(2,ℤ/4)

yes

—

C₂⁴×C₄

GL(2,ℤ/4)

yes

characteristic

C₂⁴×C₄

C₂²×S₄

yes

yes

yes

yes

C₂×C₄×Q₈

yes

—

C₂²×C₄⋊C₄

GL(2,ℤ/4)

yes

—

C₂⁵

yes

C₂⁵

yes

C₂×#41

yes

—

C₂²×Q₈

C₂×GL(2,ℤ/4)

yes

—

C₂³×C₄

C₂×GL(2,ℤ/4)

yes

—

#121

C₂×#1020

yes

—

C₂×C₂²⋊C₄

yes

C₂⁴

yes

C₂⁴

yes

C₂⁴

yes

C₂⁴

yes

C₂×Q₈

yes

—

C₂×D₄

#87524

yes

—

C₂²×C₄

#87524

yes

characteristic

C₂²×C₄

C₂²×#1020

yes

characteristic

C₄⋊C₄

#87524

yes

—

C₂³

#11713951

yes

characteristic

C₂³

#11715702

yes

—

C₂³

C₂×#87524

yes

characteristic

C₂×C₄

C₂×#87524

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²	#72209718
commutator, cosocle	C₂×#3459⋊C₃	C₂³
Frattini	C₂²×C₄	C₂²×#1020
Fitting	#72268062	D₃
hypercenter	#534	#1020
hypocenter	#543⋊C₃	#120
LCS	G ▷ C₂×#3459⋊C₃ ▷ #543⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #534

subgroup

G/H

center, socle

C₂²

#72209718

commutator, cosocle

C₂×#3459⋊C₃

C₂³

Frattini

C₂²×C₄

C₂²×#1020

D₃

G ▷ C₂×#3459⋊C₃ ▷ #543⋊C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #534

N	H ≅ G/N	type
1. #72271429	C₂	semidirect
2. #72271405	C₂	semidirect
3. C₂×#11715584	C₂²	semidirect
4. #72268140	C₂	semidirect
5. #72268131	C₂	semidirect
6. #72268130	C₂²	semidirect
7. #72268129	C₂²	semidirect
8. #72268063	C₂	semidirect
9. #72268062	D₃	semidirect
12. #62904⋊C₃	D₄	semidirect
13. #62894⋊C₃	D₄	semidirect
21. #62904	#67	semidirect
22. #62894	#67	semidirect
23. #62893	S₄	semidirect
36. C₂×2₋¹⁺⁴	GL(2,ℤ/4)	semidirect
44. #520	GL(2,ℤ/4)	semidirect
47. C₂⁵	#5431	semidirect

H ≅ G/N

type

1. #72271429

C₂

semidirect

2. #72271405

C₂

semidirect

3. C₂×#11715584

C₂²

semidirect

4. #72268140

C₂

semidirect

5. #72268131

C₂

semidirect

6. #72268130

C₂²

semidirect

7. #72268129

C₂²

semidirect

8. #72268063

C₂

semidirect

9. #72268062

D₃

semidirect

12. #62904⋊C₃

D₄

semidirect

13. #62894⋊C₃

D₄

semidirect

21. #62904

#67

semidirect

22. #62894

#67

semidirect

23. #62893

S₄

semidirect

36. C₂×2₋¹⁺⁴

GL(2,ℤ/4)

semidirect

44. #520

GL(2,ℤ/4)

semidirect

47. C₂⁵

#5431

semidirect

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

subgroup	index
#72271438	4
#72271435	4
#72271429	2
#72271405	2
#72268216	3
#72268140	2
#72268131	2
#72268063	2

subgroup

index

Subgroups — #72271449

Conjugacy classes of subgroups

Complements

Maximal subgroups