Group #72337699 — subgroups

Subgroups — #72337699

Order 4096.

At least 8610 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has at least 8610 conjugacy classes of subgroups — too many to draw. Only its 71 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	#72337698	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337695	C₂	1	1	yes	maximal, characteristic, complemented
3	#72337686	C₂	1	1	yes	maximal, characteristic, complemented
4	#72337683	C₂²	1	1	yes	characteristic
5	#72252412	C₂	1	1	yes	maximal, characteristic, complemented
6	#72241312	C₂	1	1	yes	maximal, characteristic, complemented
7	#72241085	C₂	1	1	yes	maximal, characteristic
8	#72236935	C₂	1	1	yes	maximal, characteristic, complemented
9	#72227588	C₂²	1	1	yes	characteristic
10	#72213758	C₂²	1	1	yes	characteristic
11	#72213500	C₂²	1	1	yes	characteristic
12	#72213462	C₂²	1	1	yes	characteristic
13	#72210529	C₂²	1	1	yes	characteristic
14	#72210439	C₂²	1	1	yes	characteristic
15	H(9,𝔽₂)	D₄	1	1	yes	characteristic
16	#8934366	D₄	1	1	yes	characteristic
17	#8933848	D₄	1	1	yes	characteristic, complemented
18	#6628901	C₂³	1	1	yes	characteristic
19	#392136	D₄	1	1	yes	characteristic
20	#213861	D₄	1	1	yes	characteristic, complemented
21	#213860	D₄	1	1	yes	characteristic
22	#133835	D₄	1	1	yes	characteristic, complemented
23	#128064	D₄	1	2	yes	complemented
24	#128062	D₄	1	2	yes	—
25	#97460	D₄	1	1	yes	characteristic, complemented
26	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
27	#35527	C₂×D₄	1	1	yes	characteristic
28	#35527	C₂×D₄	1	1	yes	characteristic
29	#35527	C₂×D₄	1	1	yes	characteristic
30	#20586	C₂×D₄	1	1	yes	characteristic
31	#20586	C₂×D₄	1	1	yes	characteristic
32	#20586	#41	1	1	yes	characteristic
33	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
34	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
35	C₂²×H(5,𝔽₂)	#121	1	1	yes	characteristic
36	C₂²×#127	#120	1	1	yes	characteristic
37	#2309	#121	1	1	yes	characteristic
38	#2309	#121	1	1	yes	characteristic
39	#2309	#127	1	1	yes	characteristic
40	C₂³×D₄	#392	1	1	yes	characteristic
41	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
42	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
43	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
44	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
45	C₂⁵	#2021	1	1	yes	characteristic
46	C₂⁵	#2021	1	1	yes	characteristic
47	C₂⁵	#2196	1	2	yes	—
48	C₂⁵	#2199	1	2	yes	—
49	C₂³×C₄	#2021	1	1	yes	characteristic
50	C₂³×C₄	#2196	1	2	yes	—
51	C₂³×C₄	#2199	1	2	yes	—
52	C₂⁴	#10336	1	1	yes	characteristic
53	C₂⁴	#13343	1	2	yes	—
54	C₂⁴	#13343	1	2	yes	—
55	C₂³	#144366	1	1	yes	characteristic
56	C₂³	#144366	1	1	yes	characteristic
57	C₂³	#144378	1	1	yes	characteristic
58	C₂²	#72210565	1	1	yes	characteristic
59	C₂	#72210568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72337698

C₂

yes

maximal, characteristic, complemented

#72337695

C₂

yes

maximal, characteristic, complemented

#72337686

C₂

yes

maximal, characteristic, complemented

#72337683

C₂²

yes

characteristic

#72252412

C₂

yes

maximal, characteristic, complemented

#72241312

C₂

yes

maximal, characteristic, complemented

#72241085

C₂

yes

maximal, characteristic

#72236935

C₂

yes

maximal, characteristic, complemented

#72227588

C₂²

yes

characteristic

#72213758

C₂²

yes

characteristic

#72213500

C₂²

yes

characteristic

#72213462

C₂²

yes

characteristic

#72210529

C₂²

yes

characteristic

#72210439

C₂²

yes

characteristic

H(9,𝔽₂)

D₄

yes

characteristic

#8934366

D₄

yes

characteristic

#8933848

D₄

yes

characteristic, complemented

#6628901

C₂³

yes

characteristic

#392136

D₄

yes

characteristic

#213861

D₄

yes

characteristic, complemented

#213860

D₄

yes

characteristic

#133835

D₄

yes

characteristic, complemented

#128064

D₄

yes

complemented

#128062

D₄

yes

—

#97460

D₄

yes

characteristic, complemented

C₂×H(7,𝔽₂)

C₂×D₄

yes

characteristic

#35527

C₂×D₄

yes

characteristic

#35527

C₂×D₄

yes

characteristic

#35527

C₂×D₄

yes

characteristic

#20586

C₂×D₄

yes

characteristic

#20586

C₂×D₄

yes

yes

C₂²×H(5,𝔽₂)

yes

C₂²×H(5,𝔽₂)

yes

C₂²×H(5,𝔽₂)

yes

C₂²×#127

yes

yes

yes

yes

C₂³×D₄

yes

C₂³×D₄

yes

—

C₂³×D₄

B(4,𝔽₂)

yes

—

C₂³×D₄

yes

C₂²×C₄²

yes

C₂⁵

yes

C₂⁵

yes

C₂⁵

yes

—

C₂⁵

#2199

yes

—

C₂³×C₄

#2021

yes

characteristic

C₂³×C₄

#2196

yes

—

C₂³×C₄

#2199

yes

—

C₂⁴

#10336

yes

characteristic

C₂⁴

#13343

yes

—

C₂⁴

#13343

yes

—

C₂³

yes

C₂³

yes

C₂³

yes

C₂²

yes

C₂

yes

	subgroup	G/H
center, socle	C₂	#72210568
commutator, Frattini, cosocle	#6628901	C₂³
prime radical	#72241085	C₂
LCS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G
LECS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#72210568

commutator, Frattini, cosocle

C₂³

C₂

G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G

LECS

G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

N	H ≅ G/N	type
1. #72337698	C₂	semidirect
2. #72337695	C₂	semidirect
3. #72337686	C₂	semidirect
5. #72252412	C₂	semidirect
6. #72241312	C₂	semidirect
8. #72236935	C₂	semidirect
17. #8933848	D₄	semidirect
20. #213861	D₄	semidirect
22. #133835	D₄	semidirect
23. #128064	D₄	semidirect
25. #97460	D₄	semidirect

H ≅ G/N

type

1. #72337698

C₂

semidirect

2. #72337695

C₂

semidirect

3. #72337686

C₂

semidirect

5. #72252412

C₂

semidirect

6. #72241312

C₂

semidirect

8. #72236935

C₂

semidirect

17. #8933848

D₄

semidirect

20. #213861

D₄

semidirect

22. #133835

D₄

semidirect

23. #128064

D₄

semidirect

25. #97460

D₄

semidirect

Maximal subgroups

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

subgroup	index
#72337698	2
#72337695	2
#72337686	2
#72252412	2
#72241312	2
#72241085	2
#72236935	2

subgroup

index