Group #19745 — subgroups

Subgroups — #19745

Order 256.

497 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3184	C₂	1	3	yes	maximal, complemented
2	#3151	C₂	1	3	yes	maximal, complemented
3	#1721	C₂	1	3	yes	maximal, complemented
4	#1715	C₂	1	3	yes	maximal, complemented
5	#1712	C₂	1	1	yes	maximal, characteristic, complemented
6	#1705	C₂	1	1	yes	maximal, characteristic, complemented
7	#1455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)	—	2	6	no	—
9	#532	C₂²	1	3	yes	—
10	C₂×#127	C₂²	1	1	yes	characteristic
11	#498	C₂²	1	3	yes	complemented
12	#496	—	2	6	no	—
13	#493	C₂²	1	3	yes	—
14	#493	C₂²	1	3	yes	—
15	#493	C₂²	1	3	yes	complemented
16	#492	C₂²	1	3	yes	—
17	#492	C₂²	1	3	yes	—
18	#486	C₂²	1	6	yes	complemented
19	#469	—	2	6	no	—
20	#442	—	2	6	no	—
21	C₄×M₁₆	C₂²	1	3	yes	—
22	#322	C₂²	1	3	yes	complemented
23	C₈²	C₂²	1	1	yes	characteristic, complemented
24	C₂×#41	—	2	3	no	—
25	C₂²×D₄	—	2	3	no	—
26	AGL(1,ℤ/8)	—	4	24	no	—
27	C₂×SD₁₆	—	2	6	no	—
28	C₂×SD₁₆	—	2	6	no	—
29	C₂×D₈	—	2	6	no	—
30	C₂×D₈	—	2	6	no	—
31	C₂×D₈	—	2	6	no	—
32	C₂×M₁₆	D₄	1	6	yes	—
33	C₄⋊Q₈	C₂³	1	3	yes	—
34	#127	—	2	3	no	—
35	#127	C₂³	1	1	yes	characteristic
36	#127	C₂³	1	1	yes	characteristic
37	#127	C₂³	1	3	yes	—
38	#124	—	2	3	no	—
39	#121	—	2	6	no	—
40	C₄×D₄	—	2	6	no	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	#106	—	2	6	no	—
43	#102	—	2	12	no	—
44	#97	D₄	1	6	yes	—
45	C₄×C₈	C₂³	1	3	yes	—
46	C₄×C₈	C₂³	1	3	yes	—
47	C₂⁴	—	4	1	no	—
48	#41	—	4	12	no	—
49	C₂×Q₈	—	2	3	no	—
50	C₂×D₄	—	2	6	no	—
51	C₂×D₄	—	2	3	no	—
52	C₂×D₄	—	2	3	no	—
53	C₂×D₄	—	2	3	no	—
54	C₂×D₄	—	4	12	no	—
55	C₂×D₄	—	4	3	no	—
56	C₂×D₄	—	4	3	no	—
57	C₂²×C₄	—	4	3	no	—
58	C₂²×C₄	C₂×D₄	1	3	yes	—
59	SD₁₆	—	4	12	no	—
60	SD₁₆	—	4	12	no	—
61	D₈	—	4	12	no	—
62	D₈	—	4	12	no	—
63	D₈	—	4	12	no	—
64	M₁₆	—	2	12	no	—
65	C₂×C₈	C₂×D₄	1	6	yes	—
66	C₂×C₈	C₂×D₄	1	6	yes	—
67	C₄⋊C₄	—	2	6	no	—
68	C₂²⋊C₄	—	4	6	no	—
69	C₄²	C₂×D₄	1	3	yes	—
70	C₄²	C₂⁴	1	1	yes	characteristic
71	C₂³	—	4	3	no	—
72	C₂³	—	4	1	no	—
73	C₂³	—	4	1	no	—
74	C₂³	—	8	6	no	—
75	C₂³	#127	1	1	yes	characteristic
76	Q₈	—	4	6	no	—
77	D₄	—	4	6	no	—
78	D₄	—	4	6	no	—
79	D₄	—	4	6	no	—
80	D₄	—	4	12	no	—
81	D₄	—	8	6	no	—
82	D₄	—	8	6	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	4	3	no	—
85	C₂×C₄	—	8	6	no	—
86	C₂×C₄	#127	1	3	yes	—
87	C₂×C₄	C₂²×D₄	1	3	yes	—
88	C₈	—	2	6	no	—
89	C₈	AGL(1,ℤ/8)	1	12	yes	complemented
90	C₂²	—	2	3	no	—
91	C₂²	—	8	3	no	—
92	C₂²	—	8	3	no	—
93	C₂²	—	8	3	no	—
94	C₂²	—	8	6	no	—
95	C₂²	—	16	4	no	—
96	C₂²	C₂×#127	1	1	yes	characteristic
97	C₄	—	2	3	no	—
98	C₄	—	8	3	no	—
99	C₄	C₂×AGL(1,ℤ/8)	1	6	yes	—
100	C₂	—	4	1	no	—
101	C₂	—	16	3	no	—
102	C₂	—	16	1	no	—
103	C₂	—	16	1	no	—
104	C₂	#3151	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3184

C₂

yes

maximal, complemented

#3151

C₂

yes

maximal, complemented

#1721

C₂

yes

maximal, complemented

#1715

C₂

yes

maximal, complemented

#1712

C₂

yes

maximal, characteristic, complemented

#1705

C₂

yes

maximal, characteristic, complemented

#1455

C₂

yes

maximal, characteristic, complemented

C₂×AGL(1,ℤ/8)

—

#532

C₂²

yes

—

C₂×#127

C₂²

yes

characteristic

#498

C₂²

yes

complemented

#496

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#493

C₂²

yes

complemented

#492

C₂²

yes

—

#492

C₂²

yes

—

#486

C₂²

yes

complemented

#469

—

#442

—

C₄×M₁₆

C₂²

yes

—

#322

C₂²

yes

complemented

C₈²

C₂²

yes

characteristic, complemented

C₂×#41

—

C₂²×D₄

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

C₂³

yes

characteristic

#127

C₂³

yes

characteristic

#127

C₂³

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

—

—

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

C₂²×D₄

yes

—

C₈

—

C₈

AGL(1,ℤ/8)

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#127

yes

characteristic

C₄

—

C₄

—

C₄

C₂×AGL(1,ℤ/8)

yes

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

104

C₂

#3151

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3184	C₂	semidirect
2. #3151	C₂	semidirect
3. #1721	C₂	semidirect
4. #1715	C₂	semidirect
5. #1712	C₂	semidirect
6. #1705	C₂	semidirect
7. #1455	C₂	semidirect
11. #498	C₂²	semidirect
15. #493	C₂²	semidirect
18. #486	C₂²	semidirect
22. #322	C₂²	semidirect
23. C₈²	C₂²	semidirect
89. C₈	AGL(1,ℤ/8)	semidirect

H ≅ G/N

type

1. #3184

C₂

semidirect

2. #3151

C₂

semidirect

3. #1721

C₂

semidirect

4. #1715

C₂

semidirect

5. #1712

C₂

semidirect

6. #1705

C₂

semidirect

7. #1455

C₂

semidirect

11. #498

C₂²

semidirect

15. #493

C₂²

semidirect

18. #486

C₂²

semidirect

22. #322

C₂²

semidirect

23. C₈²

C₂²

semidirect

89. C₈

AGL(1,ℤ/8)

semidirect

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

subgroup	index
#3184	2
#3151	2
#1721	2
#1715	2
#1712	2
#1705	2
#1455	2

subgroup

index

Subgroups — #19745

Conjugacy classes of subgroups

Complements

Maximal subgroups