Group #72340467 — subgroups

Subgroups — #72340467

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	#72340456	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337639	C₂	1	1	yes	maximal, characteristic, complemented
3	#72337631	C₂	1	1	yes	maximal, characteristic, complemented
4	#72337626	C₂²	1	1	yes	characteristic
5	#72337582	C₂	1	4	yes	maximal, complemented
6	#72337576	C₂	1	4	yes	maximal, complemented
7	#72311150	C₂²	1	2	yes	complemented
8	#72241085	C₂	1	2	yes	maximal, complemented
9	#72241006	C₂²	1	4	yes	—
10	#72236992	C₂	1	2	yes	maximal, complemented
11	#72236925	C₂²	1	4	yes	complemented
12	#72227588	C₂²	1	1	yes	characteristic
13	#72227300	C₂²	1	4	yes	complemented
14	#72226369	C₂²	1	8	yes	complemented
15	#72213462	C₂²	1	2	yes	—
16	#72213462	C₂²	1	1	yes	characteristic
17	#72210439	C₂²	1	2	yes	—
18	#72210035	C₂²	1	2	yes	complemented
19	#72210025	C₂²	1	4	yes	complemented
20	H(9,𝔽₂)	C₂³	1	1	yes	characteristic
21	#8934366	D₄	1	2	yes	complemented
22	#8933802	D₄	1	1	yes	characteristic, complemented
23	#8926668	D₄	1	4	yes	complemented
24	#6628901	C₂³	1	1	yes	characteristic
25	#6628897	C₂³	1	2	yes	—
26	#6628896	C₂³	1	4	yes	—
27	#392136	C₂³	1	1	yes	characteristic
28	#392135	C₂³	1	2	yes	—
29	#392135	C₂³	1	4	yes	—
30	#213860	D₄	1	2	yes	complemented
31	#213860	D₄	1	1	yes	characteristic, complemented
32	#213855	D₄	1	2	yes	complemented
33	#213854	D₄	1	4	yes	complemented
34	#213830	D₄	1	4	yes	complemented
35	#213828	D₄	1	8	yes	complemented
36	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
37	#35527	C₂×D₄	1	1	yes	characteristic
38	#35527	C₂×D₄	1	2	yes	—
39	#35446	C₂×D₄	1	4	yes	—
40	#35446	C₂×D₄	1	2	yes	—
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	#20585	C₂×D₄	1	8	yes	—
46	#20585	C₂×D₄	1	4	yes	—
47	#20585	C₂×D₄	1	4	yes	—
48	#20585	C₂×D₄	1	2	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×#123	#120	1	2	yes	—
54	C₂²×#120	#120	1	4	yes	—
55	#2309	#120	1	1	yes	characteristic
56	#2309	#120	1	2	yes	—
57	#2308	#120	1	8	yes	—
58	#2308	#120	1	4	yes	—
59	#2308	#120	1	2	yes	—
60	#2308	#120	1	4	yes	—
61	C₂⁶	B(4,𝔽₂)	1	4	yes	—
62	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
63	C₂³×D₄	C₂×#120	1	1	yes	characteristic
64	C₂³×D₄	C₂×#120	1	4	yes	—
65	C₂³×D₄	C₂×#120	1	2	yes	—
66	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
67	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
68	C₂²×C₂²⋊C₄	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	2	yes	—
72	C₂⁵	C₂×B(4,𝔽₂)	1	4	yes	—
73	C₂⁵	#3026	1	1	yes	characteristic
74	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
75	C₂³×C₄	#3026	1	4	yes	—
76	C₂³×C₄	#3026	1	2	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

#72340456

C₂

yes

maximal, characteristic, complemented

#72337639

C₂

yes

maximal, characteristic, complemented

#72337631

C₂

yes

maximal, characteristic, complemented

#72337626

C₂²

yes

characteristic

#72337582

C₂

yes

maximal, complemented

#72337576

C₂

yes

maximal, complemented

#72311150

C₂²

yes

complemented

#72241085

C₂

yes

maximal, complemented

#72241006

C₂²

yes

—

#72236992

C₂

yes

maximal, complemented

#72236925

C₂²

yes

complemented

#72227588

C₂²

yes

characteristic

#72227300

C₂²

yes

complemented

#72226369

C₂²

yes

complemented

#72213462

C₂²

yes

—

#72213462

C₂²

yes

characteristic

#72210439

C₂²

yes

—

#72210035

C₂²

yes

complemented

#72210025

C₂²

yes

complemented

H(9,𝔽₂)

C₂³

yes

characteristic

#8934366

D₄

yes

complemented

#8933802

D₄

yes

characteristic, complemented

#8926668

D₄

yes

complemented

#6628901

C₂³

yes

characteristic

#6628897

C₂³

yes

—

#6628896

C₂³

yes

—

#392136

C₂³

yes

characteristic

#392135

C₂³

yes

—

#392135

C₂³

yes

—

#213860

D₄

yes

complemented

#213860

D₄

yes

characteristic, complemented

#213855

D₄

yes

complemented

#213854

D₄

yes

complemented

#213830

D₄

yes

complemented

#213828

D₄

yes

complemented

C₂×H(7,𝔽₂)

C₂⁴

yes

characteristic

#35527

C₂×D₄

yes

characteristic

#35527

C₂×D₄

yes

—

#35446

C₂×D₄

yes

—

#35446

C₂×D₄

yes

—

#35346

C₂×D₄

yes

—

#35346

C₂×D₄

yes

—

#20586

C₂×D₄

yes

—

#20586

C₂×D₄

yes

characteristic

#20585

C₂×D₄

yes

—

#20585

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×#123

yes

—

C₂²×#120

#120

yes

—

yes

yes

—

#2308

#120

yes

—

#2308

#120

yes

—

#2308

#120

yes

—

#2308

#120

yes

—

C₂⁶

B(4,𝔽₂)

yes

—

C₂³×D₄

yes

C₂³×D₄

yes

C₂³×D₄

yes

—

C₂³×D₄

C₂×#120

yes

—

C₂⁴×C₄

B(4,𝔽₂)

yes

—

C₂²×C₂²⋊C₄

B(4,𝔽₂)

yes

—

C₂²×C₂²⋊C₄

B(4,𝔽₂)

yes

—

C₂²×C₄²

yes

C₂⁵

yes

C₂⁵

C₂×B(4,𝔽₂)

yes

—

C₂⁵

C₂×B(4,𝔽₂)

yes

—

C₂⁵

#3026

yes

characteristic

C₂³×C₄

C₂×B(4,𝔽₂)

yes

characteristic

C₂³×C₄

#3026

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. #72340456	C₂	semidirect
2. #72337639	C₂	semidirect
3. #72337631	C₂	semidirect
5. #72337582	C₂	semidirect
6. #72337576	C₂	semidirect
7. #72311150	C₂²	semidirect
8. #72241085	C₂	semidirect
10. #72236992	C₂	semidirect
11. #72236925	C₂²	semidirect
13. #72227300	C₂²	semidirect
14. #72226369	C₂²	semidirect
18. #72210035	C₂²	semidirect
19. #72210025	C₂²	semidirect
21. #8934366	D₄	semidirect
22. #8933802	D₄	semidirect
23. #8926668	D₄	semidirect
30. #213860	D₄	semidirect
31. #213860	D₄	semidirect
32. #213855	D₄	semidirect
33. #213854	D₄	semidirect
34. #213830	D₄	semidirect
35. #213828	D₄	semidirect

H ≅ G/N

type

1. #72340456

C₂

semidirect

2. #72337639

C₂

semidirect

3. #72337631

C₂

semidirect

5. #72337582

C₂

semidirect

6. #72337576

C₂

semidirect

7. #72311150

C₂²

semidirect

8. #72241085

C₂

semidirect

10. #72236992

C₂

semidirect

11. #72236925

C₂²

semidirect

13. #72227300

C₂²

semidirect

14. #72226369

C₂²

semidirect

18. #72210035

C₂²

semidirect

19. #72210025

C₂²

semidirect

21. #8934366

D₄

semidirect

22. #8933802

D₄

semidirect

23. #8926668

D₄

semidirect

30. #213860

D₄

semidirect

31. #213860

D₄

semidirect

32. #213855

D₄

semidirect

33. #213854

D₄

semidirect

34. #213830

D₄

semidirect

35. #213828

D₄

semidirect

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

subgroup	index
#72340456	2
#72337639	2
#72337631	2
#72337582	2
#72337576	2
#72241085	2
#72236992	2

subgroup

index

Subgroups — #72340467

Conjugacy classes of subgroups

Complements

Maximal subgroups