Group #72340527 — subgroups

Subgroups — #72340527

Order 4096.

Only the 225 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 225 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	#72340524	C₂	1	4	yes	maximal, complemented
2	#72340518	C₂	1	1	yes	maximal, characteristic, complemented
3	#72337696	C₂	1	1	yes	maximal, characteristic, complemented
4	#72337686	C₂	1	1	yes	maximal, characteristic, complemented
5	#72337683	C₂²	1	1	yes	characteristic
6	#72337574	C₂	1	4	yes	maximal, complemented
7	#72310930	C₂²	1	4	yes	complemented
8	#72284724	C₂²	1	4	yes	—
9	#72241085	C₂	1	2	yes	maximal, complemented
10	#72236999	C₂	1	2	yes	maximal, complemented
11	#72236925	C₂²	1	4	yes	complemented
12	#72236925	C₂²	1	8	yes	complemented
13	#72227734	C₂²	1	2	yes	complemented
14	#72227731	C₂²	1	2	yes	complemented
15	#72227588	C₂²	1	1	yes	characteristic
16	#72213500	C₂²	1	1	yes	characteristic
17	#72213462	C₂²	1	2	yes	—
18	#72210439	C₂²	1	2	yes	—
19	#72210025	C₂²	1	4	yes	complemented
20	H(9,𝔽₂)	C₂³	1	1	yes	characteristic
21	#8934366	D₄	1	2	yes	complemented
22	#8933848	D₄	1	1	yes	characteristic, complemented
23	#8928621	D₄	1	4	yes	complemented
24	#6628902	C₂³	1	2	yes	—
25	#6628901	C₂³	1	1	yes	characteristic
26	#6628896	C₂³	1	4	yes	—
27	#392136	C₂³	1	1	yes	characteristic
28	#392136	C₂³	1	2	yes	—
29	#392135	C₂³	1	4	yes	—
30	#213863	D₄	1	2	yes	complemented
31	#213862	D₄	1	4	yes	complemented
32	#213861	D₄	1	1	yes	characteristic, complemented
33	#213860	D₄	1	2	yes	complemented
34	#213854	D₄	1	4	yes	complemented
35	#213854	D₄	1	8	yes	complemented
36	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
37	#35616	C₂×D₄	1	4	yes	—
38	#35616	C₂×D₄	1	2	yes	—
39	#35527	C₂×D₄	1	2	yes	—
40	#35527	C₂×D₄	1	1	yes	characteristic
41	#35346	C₂×D₄	1	8	yes	—
42	#35346	C₂×D₄	1	4	yes	—
43	#20586	C₂×D₄	1	2	yes	—
44	#20586	C₂×D₄	1	1	yes	characteristic
45	#20586	C₂×D₄	1	4	yes	—
46	#20586	C₂×D₄	1	2	yes	—
47	#20585	C₂×D₄	1	4	yes	—
48	#20585	C₂×D₄	1	8	yes	—
49	C₂²×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
50	C₂²×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
51	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
52	C₂²×#127	#120	1	1	yes	characteristic
53	C₂²×#124	#120	1	2	yes	—
54	C₂²×#120	#120	1	4	yes	—
55	#2309	#120	1	4	yes	—
56	#2309	#120	1	2	yes	—
57	#2309	#120	1	1	yes	characteristic
58	#2309	#120	1	2	yes	—
59	#2308	#120	1	8	yes	—
60	#2308	#120	1	4	yes	—
61	C₂⁶	B(4,𝔽₂)	1	4	yes	—
62	C₂³×Q₈	B(4,𝔽₂)	1	2	yes	—
63	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
64	C₂³×D₄	C₂×#120	1	2	yes	—
65	C₂³×D₄	C₂×#120	1	1	yes	characteristic
66	C₂³×D₄	C₂×#120	1	4	yes	—
67	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
68	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
69	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
70	C₂⁵	#2846	1	1	yes	characteristic
71	C₂⁵	C₂×B(4,𝔽₂)	1	4	yes	—
72	C₂⁵	#3026	1	2	yes	—
73	C₂⁵	#3026	1	1	yes	characteristic
74	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
75	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
76	C₂³×C₄	#3026	1	4	yes	—
77	C₂⁴	#32898	1	1	yes	characteristic
78	C₂⁴	#32898	1	2	yes	—
79	C₂⁴	#32898	1	4	yes	—
80	C₂³	#347042	1	1	yes	characteristic
81	C₂³	#347042	1	2	yes	—
82	C₂³	#347042	1	4	yes	—
83	C₂²	#72216432	1	1	yes	characteristic
84	C₂	#72216433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72340524

C₂

yes

maximal, complemented

#72340518

C₂

yes

maximal, characteristic, complemented

#72337696

C₂

yes

maximal, characteristic, complemented

#72337686

C₂

yes

maximal, characteristic, complemented

#72337683

C₂²

yes

characteristic

#72337574

C₂

yes

maximal, complemented

#72310930

C₂²

yes

complemented

#72284724

C₂²

yes

—

#72241085

C₂

yes

maximal, complemented

#72236999

C₂

yes

maximal, complemented

#72236925

C₂²

yes

complemented

#72236925

C₂²

yes

complemented

#72227734

C₂²

yes

complemented

#72227731

C₂²

yes

complemented

#72227588

C₂²

yes

characteristic

#72213500

C₂²

yes

characteristic

#72213462

C₂²

yes

—

#72210439

C₂²

yes

—

#72210025

C₂²

yes

complemented

H(9,𝔽₂)

C₂³

yes

characteristic

#8934366

D₄

yes

complemented

#8933848

D₄

yes

characteristic, complemented

#8928621

D₄

yes

complemented

#6628902

C₂³

yes

—

#6628901

C₂³

yes

characteristic

#6628896

C₂³

yes

—

#392136

C₂³

yes

characteristic

#392136

C₂³

yes

—

#392135

C₂³

yes

—

#213863

D₄

yes

complemented

#213862

D₄

yes

complemented

#213861

D₄

yes

characteristic, complemented

#213860

D₄

yes

complemented

#213854

D₄

yes

complemented

#213854

D₄

yes

complemented

C₂×H(7,𝔽₂)

C₂⁴

yes

characteristic

#35616

C₂×D₄

yes

—

#35616

C₂×D₄

yes

—

#35527

C₂×D₄

yes

—

#35527

C₂×D₄

yes

characteristic

#35346

C₂×D₄

yes

—

#35346

C₂×D₄

yes

—

#20586

C₂×D₄

yes

—

#20586

C₂×D₄

yes

characteristic

#20586

C₂×D₄

yes

—

#20586

C₂×D₄

yes

—

#20585

C₂×D₄

yes

—

#20585

C₂×D₄

yes

—

C₂²×H(5,𝔽₂)

C₂²×D₄

yes

—

C₂²×H(5,𝔽₂)

C₂²×D₄

yes

—

C₂²×H(5,𝔽₂)

C₂²×D₄

yes

C₂²×#127

yes

C₂²×#124

yes

—

C₂²×#120

#120

yes

—

#2309

#120

yes

—

#2309

#120

yes

—

yes

yes

—

#2308

#120

yes

—

#2308

#120

yes

—

C₂⁶

B(4,𝔽₂)

yes

—

C₂³×Q₈

B(4,𝔽₂)

yes

—

C₂³×D₄

B(4,𝔽₂)

yes

characteristic

C₂³×D₄

C₂×#120

yes

—

C₂³×D₄

C₂×#120

yes

characteristic

C₂³×D₄

C₂×#120

yes

—

C₂²×C₂²⋊C₄

B(4,𝔽₂)

yes

—

C₂²×C₄²

B(4,𝔽₂)

yes

—

C₂²×C₄²

yes

C₂⁵

yes

C₂⁵

C₂×B(4,𝔽₂)

yes

—

C₂⁵

#3026

yes

—

C₂⁵

#3026

yes

characteristic

C₂³×C₄

C₂×B(4,𝔽₂)

yes

characteristic

C₂³×C₄

C₂×B(4,𝔽₂)

yes

—

C₂³×C₄

#3026

yes

—

C₂⁴

#32898

yes

characteristic

C₂⁴

#32898

yes

—

C₂⁴

#32898

yes

—

C₂³

#347042

yes

characteristic

C₂³

#347042

yes

—

C₂³

#347042

yes

—

C₂²

yes

C₂

yes

	subgroup	G/H
center, socle	C₂	#72216433
commutator, Frattini, cosocle	C₂×H(7,𝔽₂)	C₂⁴
LCS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁵ ◁ C₂×H(7,𝔽₂) ◁ G
LECS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#72216433

commutator, Frattini, cosocle

C₂×H(7,𝔽₂)

C₂⁴

LCS

G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂⁵ ◁ C₂×H(7,𝔽₂) ◁ G

LECS

G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #72340524	C₂	semidirect
2. #72340518	C₂	semidirect
3. #72337696	C₂	semidirect
4. #72337686	C₂	semidirect
6. #72337574	C₂	semidirect
7. #72310930	C₂²	semidirect
9. #72241085	C₂	semidirect
10. #72236999	C₂	semidirect
11. #72236925	C₂²	semidirect
12. #72236925	C₂²	semidirect
13. #72227734	C₂²	semidirect
14. #72227731	C₂²	semidirect
19. #72210025	C₂²	semidirect
21. #8934366	D₄	semidirect
22. #8933848	D₄	semidirect
23. #8928621	D₄	semidirect
30. #213863	D₄	semidirect
31. #213862	D₄	semidirect
32. #213861	D₄	semidirect
33. #213860	D₄	semidirect
34. #213854	D₄	semidirect
35. #213854	D₄	semidirect

H ≅ G/N

type

1. #72340524

C₂

semidirect

2. #72340518

C₂

semidirect

3. #72337696

C₂

semidirect

4. #72337686

C₂

semidirect

6. #72337574

C₂

semidirect

7. #72310930

C₂²

semidirect

9. #72241085

C₂

semidirect

10. #72236999

C₂

semidirect

11. #72236925

C₂²

semidirect

12. #72236925

C₂²

semidirect

13. #72227734

C₂²

semidirect

14. #72227731

C₂²

semidirect

19. #72210025

C₂²

semidirect

21. #8934366

D₄

semidirect

22. #8933848

D₄

semidirect

23. #8928621

D₄

semidirect

30. #213863

D₄

semidirect

31. #213862

D₄

semidirect

32. #213861

D₄

semidirect

33. #213860

D₄

semidirect

34. #213854

D₄

semidirect

35. #213854

D₄

semidirect

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

subgroup	index
#72340524	2
#72340518	2
#72337696	2
#72337686	2
#72337574	2
#72241085	2
#72236999	2

subgroup

index

Subgroups — #72340527

Conjugacy classes of subgroups

Complements

Maximal subgroups