Group #72268229 — subgroups

Subgroups — #72268229

Order 4096.

Only the 323 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 323 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	#72268228	C₂	1	6	yes	maximal, complemented
2	#72268227	C₂	1	2	yes	maximal, complemented
3	#72268226	C₂	1	3	yes	maximal, complemented
4	#72268225	C₂²	1	6	yes	complemented
5	#72268224	C₂	1	1	yes	maximal, characteristic, complemented
6	#72268222	C₂²	1	3	yes	complemented
7	#72268220	C₂²	1	1	yes	characteristic, complemented
8	#72268216	C₂	1	3	yes	maximal, complemented
9	#72268215	C₂²	1	6	yes	—
10	#72268214	C₂²	1	3	yes	—
11	#72268085	C₂²	1	6	yes	complemented
12	#72268072	C₂²	1	6	yes	complemented
13	#72268070	C₂²	1	3	yes	complemented
14	#72268062	C₂²	1	1	yes	characteristic, complemented
15	#10586405	C₂³	1	2	yes	—
16	C₂×#62843	C₂³	1	1	yes	characteristic
17	#6499839	D₄	1	12	yes	complemented
18	#4928940	D₄	1	3	yes	—
19	#764240	D₄	1	3	yes	—
20	#764240	C₂³	1	3	yes	—
21	#362714	D₄	1	3	yes	—
22	#362714	C₂³	1	3	yes	—
23	#354795	D₄	1	3	yes	—
24	#188123	C₂³	1	6	yes	—
25	#186831	D₄	1	12	yes	complemented
26	#62819	C₂×D₄	1	12	yes	—
27	C₂²×#543	C₂×D₄	1	3	yes	—
28	C₂²×#543	C₂⁴	1	1	yes	characteristic
29	C₂²×#518	C₂×D₄	1	3	yes	—
30	C₂×#2435	C₂×D₄	1	6	yes	—
31	C₂²×#392	C₂×D₄	1	3	yes	—
32	#21753	C₂×D₄	1	12	yes	—
33	#14338	C₂×D₄	1	3	yes	—
34	#14159	C₂×D₄	1	6	yes	—
35	#13775	C₂×D₄	1	3	yes	—
36	#13775	C₂×D₄	1	3	yes	—
37	C₂⁴×D₄	#120	1	3	yes	—
38	C₂⁴×D₄	#120	1	1	yes	characteristic
39	#3486	#120	1	8	yes	—
40	C₂²×C₂²⋊Q₈	C₂²×D₄	1	6	yes	—
41	C₂²×C₂²⋊Q₈	H(5,𝔽₂)	1	6	yes	—
42	C₂³×C₂²⋊C₄	#120	1	3	yes	—
43	C₂³×C₂²⋊C₄	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₂²⋊C₄	#120	1	6	yes	—
45	C₂²×#95	#120	1	3	yes	—
46	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
47	C₂⁶	C₂×#120	1	1	yes	characteristic
48	C₂×2₋¹⁺⁴	B(4,𝔽₂)	1	8	yes	complemented
49	C₂³×Q₈	C₂×#120	1	2	yes	—
50	C₂³×Q₈	H(3,𝔽₄)	1	2	yes	—
51	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
52	C₂⁴×C₄	B(4,𝔽₂)	1	6	yes	—
53	C₂⁴×C₄	C₂×#120	1	3	yes	—
54	C₂⁴×C₄	#534	1	3	yes	—
55	#519	B(4,𝔽₂)	1	8	yes	complemented
56	C₂²×C₄⋊C₄	#534	1	6	yes	—
57	C₂²×C₄⋊C₄	D₄²	1	6	yes	—
58	C₂²×C₂²⋊C₄	D₄²	1	12	yes	—
59	C₂⁵	#2202	1	2	yes	complemented
60	C₂⁵	#2403	1	3	yes	—
61	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
62	C₂⁵	C₂×B(4,𝔽₂)	1	3	yes	—
63	C₂²×Q₈	C₂×B(4,𝔽₂)	1	8	yes	—
64	C₂³×C₄	#2403	1	6	yes	—
65	C₂³×C₄	#2679	1	6	yes	—
66	C₂⁴	#15947	1	1	yes	characteristic
67	C₂⁴	#23900	1	3	yes	—
68	C₂⁴	#23900	1	3	yes	—
69	C₂⁴	C₂×#2202	1	1	yes	characteristic
70	C₂⁴	#33568	1	3	yes	—
71	C₂²×C₄	#23900	1	12	yes	—
72	C₂³	#449475	1	2	yes	—
73	C₂³	#449475	1	1	yes	characteristic
74	C₂³	#499856	1	3	yes	—
75	C₂³	#499867	1	3	yes	—
76	C₂²	#72209716	1	1	yes	characteristic
77	C₂²	#72210362	1	1	yes	characteristic
78	C₂²	#72227409	1	1	yes	characteristic
79	C₂	#72227297	1	1	yes	characteristic
80	C₂	#72227420	1	1	yes	characteristic
81	C₂	#72227590	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72268228

C₂

yes

maximal, complemented

#72268227

C₂

yes

maximal, complemented

#72268226

C₂

yes

maximal, complemented

#72268225

C₂²

yes

complemented

#72268224

C₂

yes

maximal, characteristic, complemented

#72268222

C₂²

yes

complemented

#72268220

C₂²

yes

characteristic, complemented

#72268216

C₂

yes

maximal, complemented

#72268215

C₂²

yes

—

#72268214

C₂²

yes

—

#72268085

C₂²

yes

complemented

#72268072

C₂²

yes

complemented

#72268070

C₂²

yes

complemented

#72268062

C₂²

yes

characteristic, complemented

#10586405

C₂³

yes

—

C₂×#62843

C₂³

yes

characteristic

#6499839

D₄

yes

complemented

#4928940

D₄

yes

—

#764240

D₄

yes

—

#764240

C₂³

yes

—

#362714

D₄

yes

—

#362714

C₂³

yes

—

#354795

D₄

yes

—

#188123

C₂³

yes

—

#186831

D₄

yes

complemented

#62819

C₂×D₄

yes

—

C₂²×#543

C₂×D₄

yes

—

C₂²×#543

C₂⁴

yes

characteristic

C₂²×#518

C₂×D₄

yes

—

C₂×#2435

C₂×D₄

yes

—

C₂²×#392

C₂×D₄

yes

—

#21753

C₂×D₄

yes

—

#14338

C₂×D₄

yes

—

#14159

C₂×D₄

yes

—

#13775

C₂×D₄

yes

—

#13775

C₂×D₄

yes

—

C₂⁴×D₄

#120

yes

—

C₂⁴×D₄

yes

yes

—

C₂²×C₂²⋊Q₈

C₂²×D₄

yes

—

C₂²×C₂²⋊Q₈

H(5,𝔽₂)

yes

—

C₂³×C₂²⋊C₄

#120

yes

—

C₂³×C₂²⋊C₄

C₂²×D₄

yes

—

C₂×C₄×C₂²⋊C₄

#120

yes

—

C₂²×#95

#120

yes

—

C₂⁶

yes

C₂⁶

yes

C₂×2₋¹⁺⁴

yes

C₂³×Q₈

yes

—

C₂³×Q₈

H(3,𝔽₄)

yes

—

C₂⁴×C₄

B(4,𝔽₂)

yes

characteristic

C₂⁴×C₄

B(4,𝔽₂)

yes

—

C₂⁴×C₄

C₂×#120

yes

—

C₂⁴×C₄

#534

yes

—

yes

C₂²×C₄⋊C₄

yes

—

C₂²×C₄⋊C₄

D₄²

yes

—

C₂²×C₂²⋊C₄

D₄²

yes

—

C₂⁵

#2202

yes

complemented

C₂⁵

#2403

yes

—

C₂⁵

C₂×B(4,𝔽₂)

yes

characteristic

C₂⁵

C₂×B(4,𝔽₂)

yes

—

C₂²×Q₈

C₂×B(4,𝔽₂)

yes

—

C₂³×C₄

#2403

yes

—

C₂³×C₄

#2679

yes

—

C₂⁴

#15947

yes

characteristic

C₂⁴

#23900

yes

—

C₂⁴

#23900

yes

—

C₂⁴

C₂×#2202

yes

characteristic

C₂⁴

#33568

yes

—

C₂²×C₄

#23900

yes

—

C₂³

#449475

yes

—

C₂³

#449475

yes

characteristic

C₂³

#499856

yes

—

C₂³

#499867

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²	#72209716
commutator, Frattini, cosocle	C₂²×#543	C₂⁴
LCS	G ▷ C₂²×#543 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#543 ◁ G
LECS	G ▷ C₂²×#543 ▷ C₂⁴ ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#72209716

commutator, Frattini, cosocle

C₂²×#543

C₂⁴

LCS

G ▷ C₂²×#543 ▷ C₂⁴ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#543 ◁ G

LECS

G ▷ C₂²×#543 ▷ C₂⁴ ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #72268228	C₂	semidirect
2. #72268227	C₂	semidirect
3. #72268226	C₂	semidirect
4. #72268225	C₂²	semidirect
5. #72268224	C₂	semidirect
6. #72268222	C₂²	semidirect
7. #72268220	C₂²	semidirect
8. #72268216	C₂	semidirect
11. #72268085	C₂²	semidirect
12. #72268072	C₂²	semidirect
13. #72268070	C₂²	semidirect
14. #72268062	C₂²	semidirect
17. #6499839	D₄	semidirect
25. #186831	D₄	semidirect
48. C₂×2₋¹⁺⁴	B(4,𝔽₂)	semidirect
55. #519	B(4,𝔽₂)	semidirect
59. C₂⁵	#2202	semidirect

H ≅ G/N

type

1. #72268228

C₂

semidirect

2. #72268227

C₂

semidirect

3. #72268226

C₂

semidirect

4. #72268225

C₂²

semidirect

5. #72268224

C₂

semidirect

6. #72268222

C₂²

semidirect

7. #72268220

C₂²

semidirect

8. #72268216

C₂

semidirect

11. #72268085

C₂²

semidirect

12. #72268072

C₂²

semidirect

13. #72268070

C₂²

semidirect

14. #72268062

C₂²

semidirect

17. #6499839

D₄

semidirect

25. #186831

D₄

semidirect

48. C₂×2₋¹⁺⁴

B(4,𝔽₂)

semidirect

55. #519

B(4,𝔽₂)

semidirect

59. C₂⁵

#2202

semidirect

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

subgroup	index
#72268228	2
#72268227	2
#72268226	2
#72268224	2
#72268216	2

subgroup

index

Subgroups — #72268229

Conjugacy classes of subgroups

Complements

Maximal subgroups