Group #15689934 — subgroups

Subgroups — #15689934

Order 3008.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#13094071	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094054	C₂	1	3	yes	maximal, complemented
3	#13094050	C₂	1	3	yes	maximal, complemented
4	#13094044	C₂	1	3	yes	maximal, complemented
5	C₂×#10625335	C₂	1	3	yes	maximal, complemented
6	#13094031	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×C₄₇	—	2	1	no	—
9	#41×C₄₇	C₂²	1	3	yes	—
10	#41×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₇	—	2	3	no	—
12	C₂×D₄×C₄₇	C₂²	1	3	yes	—
13	Q₈×D₄₇	—	2	1	no	—
14	#10625352	—	2	3	no	—
15	#10625350	C₂²	1	3	yes	complemented
16	C₂×Dic₉₄	C₂²	1	3	yes	complemented
17	#10625337	C₂²	1	3	yes	complemented
18	#10625336	C₂²	1	1	yes	characteristic, complemented
19	#10625335	C₂²	1	3	yes	complemented
20	#10625335	C₂²	1	6	yes	complemented
21	#10625334	C₂²	1	3	yes	complemented
22	#10625329	C₂²	1	3	yes	complemented
23	C₂×C₄₇⋊C₈	C₂²	1	3	yes	complemented
24	C₂³×C₄₇	—	2	3	no	—
25	Q₈×C₄₇	D₄	1	1	yes	characteristic
26	Q₈×C₄₇	C₂³	1	1	yes	characteristic
27	D₄×C₄₇	—	2	3	no	—
28	D₄×C₄₇	—	2	3	no	—
29	D₄×C₄₇	D₄	1	3	yes	—
30	D₄×C₄₇	C₂³	1	3	yes	—
31	C₂×C₄×C₄₇	—	2	3	no	—
32	C₂×C₄×C₄₇	C₂³	1	3	yes	—
33	C₂²⋊D₄₇	—	2	3	no	—
34	C₂×Dic₄₇	—	2	3	no	—
35	D₁₈₈	C₂³	1	1	yes	characteristic
36	C₄×D₄₇	—	2	3	no	—
37	Dic₉₄	—	2	3	no	—
38	Dic₉₄	C₂³	1	3	yes	—
39	C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₄₇⋊C₈	C₂³	1	3	yes	—
41	C₂²×C₄₇	—	2	3	no	—
42	C₂²×C₄₇	—	8	3	no	—
43	C₂²×C₄₇	C₂×D₄	1	3	yes	—
44	C₂×D₄₇	—	2	1	no	—
45	C₄×C₄₇	—	2	1	no	—
46	C₄×C₄₇	C₂×D₄	1	3	yes	—
47	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	Dic₄₇	—	2	3	no	—
49	C₂×C₄₇	—	2	3	no	—
50	C₂×C₄₇	—	4	3	no	—
51	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
52	D₄₇	—	4	1	no	—
53	#577	—	47	1	no	maximal, Sylow
54	C₄₇	#577	1	1	yes	characteristic, complemented, Sylow
55	2₋¹⁺⁴	—	47	1	no	—
56	H(5,𝔽₂)	D₄₇	1	1	yes	characteristic, complemented
57	#137	—	47	3	no	—
58	AGL(1,ℤ/8)	—	47	3	no	—
59	#135	—	47	3	no	—
60	C₂×SD₁₆	—	47	3	no	—
61	#131	—	47	1	no	—
62	#41	—	2	1	no	—
63	#41	—	47	3	no	—
64	#41	—	94	3	no	—
65	#41	C₂×D₄₇	1	3	yes	—
66	#41	C₂×D₄₇	1	1	yes	characteristic
67	C₂×Q₈	—	47	3	no	—
68	C₂×Q₈	—	94	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₄₇	1	3	yes	—
71	Q₁₆	—	47	3	no	—
72	SD₁₆	—	47	1	no	—
73	SD₁₆	—	47	3	no	—
74	SD₁₆	—	47	6	no	—
75	D₈	—	47	3	no	—
76	M₁₆	—	47	3	no	—
77	C₂×C₈	—	47	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	47	3	no	—
80	Q₈	—	94	3	no	—
81	Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
82	Q₈	C₂²×D₄₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	47	1	no	—
86	D₄	—	94	3	no	—
87	D₄	C₂²⋊D₄₇	1	3	yes	—
88	D₄	C₂²×D₄₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	94	3	no	—
91	C₂×C₄	—	94	3	no	—
92	C₂×C₄	C₂²×D₄₇	1	3	yes	—
93	C₈	—	47	1	no	—
94	C₈	—	47	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	94	1	no	—
98	C₂²	C₂×C₂²⋊D₄₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	94	3	no	—
101	C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
102	C₄	C₂³×D₄₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	188	1	no	—
106	C₂	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094071

C₂

yes

maximal, characteristic, complemented

#13094054

C₂

yes

maximal, complemented

#13094050

C₂

yes

maximal, complemented

#13094044

C₂

yes

maximal, complemented

C₂×#10625335

C₂

yes

maximal, complemented

#13094031

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₄₇

C₂

yes

maximal, characteristic, complemented

#41×C₄₇

—

#41×C₄₇

C₂²

yes

—

#41×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²

yes

—

Q₈×D₄₇

—

#10625352

—

#10625350

C₂²

yes

complemented

C₂×Dic₉₄

C₂²

yes

complemented

#10625337

C₂²

yes

complemented

#10625336

C₂²

yes

characteristic, complemented

#10625335

C₂²

yes

complemented

#10625335

C₂²

yes

complemented

#10625334

C₂²

yes

complemented

#10625329

C₂²

yes

complemented

C₂×C₄₇⋊C₈

C₂²

yes

complemented

C₂³×C₄₇

—

Q₈×C₄₇

D₄

yes

characteristic

Q₈×C₄₇

C₂³

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

D₄

yes

—

D₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

C₂³

yes

characteristic

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂³

yes

—

C₄₇⋊C₈

C₂³

yes

characteristic

C₄₇⋊C₈

C₂³

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₄

yes

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

—

C₄×C₄₇

C₂⁴

yes

characteristic

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

D₄₇

—

#577

—

maximal, Sylow

C₄₇

#577

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

D₄₇

yes

characteristic, complemented

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×SD₁₆

—

—

—

—

—

C₂×D₄₇

yes

—

#41

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

Q₈

—

Q₈

—

Q₈

C₂²⋊D₄₇

yes

characteristic

Q₈

C₂²×D₄₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²⋊D₄₇

yes

—

D₄

C₂²×D₄₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₄₇

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₄₇

yes

—

C₄

—

100

C₄

—

101

C₄

C₂×C₂²⋊D₄₇

yes

—

102

C₄

C₂³×D₄₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

188

—

106

C₂

C₂²×C₂²⋊D₄₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	H(5,𝔽₂)×C₄₇	C₂
socle	C₂×C₄₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₄₇
hypocenter	C₄₇	#577
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₄₇

commutator, cosocle

C₄×C₄₇

C₂⁴

Frattini

C₄

C₂³×D₄₇

Fitting

H(5,𝔽₂)×C₄₇

C₂

socle

C₂×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #13094071	C₂	semidirect
2. #13094054	C₂	semidirect
3. #13094050	C₂	semidirect
4. #13094044	C₂	semidirect
5. C₂×#10625335	C₂	semidirect
6. #13094031	C₂	semidirect
7. H(5,𝔽₂)×C₄₇	C₂	semidirect
15. #10625350	C₂²	semidirect
16. C₂×Dic₉₄	C₂²	semidirect
17. #10625337	C₂²	semidirect
18. #10625336	C₂²	semidirect
19. #10625335	C₂²	semidirect
20. #10625335	C₂²	semidirect
21. #10625334	C₂²	semidirect
22. #10625329	C₂²	semidirect
23. C₂×C₄₇⋊C₈	C₂²	semidirect
54. C₄₇	#577	semidirect
56. H(5,𝔽₂)	D₄₇	semidirect

H ≅ G/N

type

1. #13094071

C₂

semidirect

2. #13094054

C₂

semidirect

3. #13094050

C₂

semidirect

4. #13094044

C₂

semidirect

5. C₂×#10625335

C₂

semidirect

6. #13094031

C₂

semidirect

7. H(5,𝔽₂)×C₄₇

C₂

semidirect

15. #10625350

C₂²

semidirect

16. C₂×Dic₉₄

C₂²

semidirect

17. #10625337

C₂²

semidirect

18. #10625336

C₂²

semidirect

19. #10625335

C₂²

semidirect

20. #10625335

C₂²

semidirect

21. #10625334

C₂²

semidirect

22. #10625329

C₂²

semidirect

23. C₂×C₄₇⋊C₈

C₂²

semidirect

54. C₄₇

#577

semidirect

56. H(5,𝔽₂)

D₄₇

semidirect

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

subgroup	index
#13094071	2
#13094054	2
#13094050	2
#13094044	2
C₂×#10625335	2
#13094031	2
H(5,𝔽₂)×C₄₇	2
#577	47

subgroup

index

H(5,𝔽₂)×C₄₇

Subgroups — #15689934

Conjugacy classes of subgroups

Complements

Maximal subgroups