Group #44529861 — subgroups

Subgroups — #44529861

Order 31168.

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	#29510262	C₂	1	1	yes	maximal, characteristic, complemented
2	#29510245	C₂	1	3	yes	maximal, complemented
3	#29510241	C₂	1	3	yes	maximal, complemented
4	#29510235	C₂	1	3	yes	maximal, complemented
5	C₂×#21441249	C₂	1	3	yes	maximal, complemented
6	#29510222	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#21441261	C₂²	1	3	yes	complemented
9	#21441260	—	2	3	no	—
10	C₂×Dic₉₇₄	C₂²	1	3	yes	complemented
11	Q₈×D₄₈₇	—	2	1	no	—
12	#21441251	C₂²	1	3	yes	complemented
13	#21441249	C₂²	1	3	yes	complemented
14	#21441249	C₂²	1	6	yes	complemented
15	#21441248	C₂²	1	1	yes	characteristic, complemented
16	#21441246	C₂²	1	3	yes	complemented
17	#21441245	C₂²	1	3	yes	complemented
18	C₂×C₄₈₇⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₄₈₇	—	2	1	no	—
20	#41×C₄₈₇	C₂²	1	3	yes	—
21	#41×C₄₈₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₄₈₇	—	2	3	no	—
23	C₂×D₄×C₄₈₇	C₂²	1	3	yes	—
24	Dic₉₇₄	—	2	3	no	—
25	Dic₉₇₄	C₂³	1	3	yes	—
26	D₁₉₄₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₄₈₇	—	2	3	no	—
28	C₂×Dic₄₈₇	—	2	3	no	—
29	C₄×D₄₈₇	—	2	3	no	—
30	C₄₈₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₄₈₇⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₄₈₇	—	2	3	no	—
33	Q₈×C₄₈₇	D₄	1	1	yes	characteristic
34	Q₈×C₄₈₇	C₂³	1	1	yes	characteristic
35	D₄×C₄₈₇	—	2	3	no	—
36	D₄×C₄₈₇	—	2	3	no	—
37	D₄×C₄₈₇	D₄	1	3	yes	—
38	D₄×C₄₈₇	C₂³	1	3	yes	—
39	C₂×C₄×C₄₈₇	—	2	3	no	—
40	C₂×C₄×C₄₈₇	C₂³	1	3	yes	—
41	C₂²×C₄₈₇	—	2	3	no	—
42	C₂²×C₄₈₇	—	8	3	no	—
43	C₂²×C₄₈₇	C₂×D₄	1	3	yes	—
44	C₂×D₄₈₇	—	2	1	no	—
45	C₄×C₄₈₇	—	2	1	no	—
46	C₄×C₄₈₇	C₂×D₄	1	3	yes	—
47	C₄×C₄₈₇	C₂⁴	1	1	yes	characteristic
48	Dic₄₈₇	—	2	3	no	—
49	C₂×C₄₈₇	—	2	3	no	—
50	C₂×C₄₈₇	—	4	3	no	—
51	C₂×C₄₈₇	C₂²×D₄	1	1	yes	characteristic
52	D₄₈₇	—	4	1	no	—
53	C₄₈₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	487	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	487	1	no	—
56	H(5,𝔽₂)	D₄₈₇	1	1	yes	characteristic, complemented
57	#137	—	487	3	no	—
58	AGL(1,ℤ/8)	—	487	3	no	—
59	#135	—	487	3	no	—
60	C₂×SD₁₆	—	487	3	no	—
61	#131	—	487	1	no	—
62	#41	—	2	1	no	—
63	#41	—	974	3	no	—
64	#41	—	487	3	no	—
65	#41	C₂×D₄₈₇	1	3	yes	—
66	#41	C₂×D₄₈₇	1	1	yes	characteristic
67	C₂×Q₈	—	974	1	no	—
68	C₂×Q₈	—	487	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₄₈₇	1	3	yes	—
71	Q₁₆	—	487	3	no	—
72	SD₁₆	—	487	1	no	—
73	SD₁₆	—	487	3	no	—
74	SD₁₆	—	487	6	no	—
75	D₈	—	487	3	no	—
76	M₁₆	—	487	3	no	—
77	C₂×C₈	—	487	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	974	3	no	—
80	Q₈	—	487	3	no	—
81	Q₈	C₂²⋊D₄₈₇	1	1	yes	characteristic
82	Q₈	C₂²×D₄₈₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	974	3	no	—
86	D₄	—	487	1	no	—
87	D₄	C₂²⋊D₄₈₇	1	3	yes	—
88	D₄	C₂²×D₄₈₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	974	3	no	—
91	C₂×C₄	—	974	3	no	—
92	C₂×C₄	C₂²×D₄₈₇	1	3	yes	—
93	C₈	—	487	1	no	—
94	C₈	—	487	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	974	1	no	—
98	C₂²	C₂×C₂²⋊D₄₈₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	974	3	no	—
101	C₄	C₂×C₂²⋊D₄₈₇	1	3	yes	—
102	C₄	C₂³×D₄₈₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1948	1	no	—
106	C₂	C₂²×C₂²⋊D₄₈₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29510262

C₂

yes

maximal, characteristic, complemented

#29510245

C₂

yes

maximal, complemented

#29510241

C₂

yes

maximal, complemented

#29510235

C₂

yes

maximal, complemented

C₂×#21441249

C₂

yes

maximal, complemented

#29510222

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₄₈₇

C₂

yes

maximal, characteristic, complemented

#21441261

C₂²

yes

complemented

#21441260

—

C₂×Dic₉₇₄

C₂²

yes

complemented

Q₈×D₄₈₇

—

#21441251

C₂²

yes

complemented

#21441249

C₂²

yes

complemented

#21441249

C₂²

yes

complemented

#21441248

C₂²

yes

characteristic, complemented

#21441246

C₂²

yes

complemented

#21441245

C₂²

yes

complemented

C₂×C₄₈₇⋊C₈

C₂²

yes

complemented

#41×C₄₈₇

—

#41×C₄₈₇

C₂²

yes

—

#41×C₄₈₇

C₂²

yes

characteristic

C₂×D₄×C₄₈₇

—

C₂×D₄×C₄₈₇

C₂²

yes

—

Dic₉₇₄

—

Dic₉₇₄

C₂³

yes

—

D₁₉₄₈

C₂³

yes

characteristic

C₂²⋊D₄₈₇

—

C₂×Dic₄₈₇

—

C₄×D₄₈₇

—

C₄₈₇⋊C₈

C₂³

yes

characteristic

C₄₈₇⋊C₈

C₂³

yes

—

C₂³×C₄₈₇

—

Q₈×C₄₈₇

D₄

yes

characteristic

Q₈×C₄₈₇

C₂³

yes

characteristic

D₄×C₄₈₇

—

D₄×C₄₈₇

—

D₄×C₄₈₇

D₄

yes

—

D₄×C₄₈₇

C₂³

yes

—

C₂×C₄×C₄₈₇

—

C₂×C₄×C₄₈₇

C₂³

yes

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

C₂×D₄

yes

—

C₂×D₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

C₂×D₄

yes

—

C₄×C₄₈₇

C₂⁴

yes

characteristic

Dic₄₈₇

—

C₂×C₄₈₇

—

C₂×C₄₈₇

—

C₂×C₄₈₇

C₂²×D₄

yes

characteristic

D₄₈₇

—

C₄₈₇

#577

yes

characteristic, complemented, Sylow

#577

—

487

maximal, Sylow

2₋¹⁺⁴

—

487

—

H(5,𝔽₂)

D₄₈₇

yes

characteristic, complemented

#137

—

487

—

AGL(1,ℤ/8)

—

487

—

#135

—

487

—

C₂×SD₁₆

—

487

—

#131

—

487

—

#41

—

#41

—

974

—

#41

—

487

—

#41

C₂×D₄₈₇

yes

—

#41

C₂×D₄₈₇

yes

characteristic

C₂×Q₈

—

974

—

C₂×Q₈

—

487

—

C₂×D₄

—

C₂×D₄

C₂×D₄₈₇

yes

—

Q₁₆

—

487

—

SD₁₆

—

487

—

SD₁₆

—

487

—

SD₁₆

—

487

—

D₈

—

487

—

M₁₆

—

487

—

C₂×C₈

—

487

—

C₂³

—

Q₈

—

974

—

Q₈

—

487

—

Q₈

C₂²⋊D₄₈₇

yes

characteristic

Q₈

C₂²×D₄₈₇

yes

characteristic

D₄

—

D₄

—

D₄

—

974

—

D₄

—

487

—

D₄

C₂²⋊D₄₈₇

yes

—

D₄

C₂²×D₄₈₇

yes

—

C₂×C₄

—

C₂×C₄

—

974

—

C₂×C₄

—

974

—

C₂×C₄

C₂²×D₄₈₇

yes

—

C₈

—

487

—

C₈

—

487

—

C₂²

—

C₂²

—

C₂²

—

974

—

C₂²

C₂×C₂²⋊D₄₈₇

yes

—

C₄

—

100

C₄

—

974

—

101

C₄

C₂×C₂²⋊D₄₈₇

yes

—

102

C₄

C₂³×D₄₈₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1948

—

106

C₂

C₂²×C₂²⋊D₄₈₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₄₈₇
commutator, cosocle	C₄×C₄₈₇	C₂⁴
Frattini	C₄	C₂³×D₄₈₇
Fitting	H(5,𝔽₂)×C₄₈₇	C₂
socle	C₂×C₄₈₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₄₈₇
hypocenter	C₄₈₇	#577
LCS	G ▷ C₄×C₄₈₇ ▷ C₂×C₄₈₇ ▷ C₄₈₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₄₈₇

commutator, cosocle

C₄×C₄₈₇

C₂⁴

Frattini

C₄

C₂³×D₄₈₇

Fitting

H(5,𝔽₂)×C₄₈₇

C₂

socle

C₂×C₄₈₇

C₂²×D₄

D₄₈₇

C₄₈₇

G ▷ C₄×C₄₈₇ ▷ C₂×C₄₈₇ ▷ C₄₈₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #29510262	C₂	semidirect
2. #29510245	C₂	semidirect
3. #29510241	C₂	semidirect
4. #29510235	C₂	semidirect
5. C₂×#21441249	C₂	semidirect
6. #29510222	C₂	semidirect
7. H(5,𝔽₂)×C₄₈₇	C₂	semidirect
8. #21441261	C₂²	semidirect
10. C₂×Dic₉₇₄	C₂²	semidirect
12. #21441251	C₂²	semidirect
13. #21441249	C₂²	semidirect
14. #21441249	C₂²	semidirect
15. #21441248	C₂²	semidirect
16. #21441246	C₂²	semidirect
17. #21441245	C₂²	semidirect
18. C₂×C₄₈₇⋊C₈	C₂²	semidirect
53. C₄₈₇	#577	semidirect
56. H(5,𝔽₂)	D₄₈₇	semidirect

H ≅ G/N

type

1. #29510262

C₂

semidirect

2. #29510245

C₂

semidirect

3. #29510241

C₂

semidirect

4. #29510235

C₂

semidirect

5. C₂×#21441249

C₂

semidirect

6. #29510222

C₂

semidirect

7. H(5,𝔽₂)×C₄₈₇

C₂

semidirect

8. #21441261

C₂²

semidirect

10. C₂×Dic₉₇₄

C₂²

semidirect

12. #21441251

C₂²

semidirect

13. #21441249

C₂²

semidirect

14. #21441249

C₂²

semidirect

15. #21441248

C₂²

semidirect

16. #21441246

C₂²

semidirect

17. #21441245

C₂²

semidirect

18. C₂×C₄₈₇⋊C₈

C₂²

semidirect

53. C₄₈₇

#577

semidirect

56. H(5,𝔽₂)

D₄₈₇

semidirect

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

subgroup	index
#29510262	2
#29510245	2
#29510241	2
#29510235	2
C₂×#21441249	2
#29510222	2
H(5,𝔽₂)×C₄₈₇	2
#577	487

subgroup

index

H(5,𝔽₂)×C₄₈₇

487

Subgroups — #44529861

Conjugacy classes of subgroups

Complements

Maximal subgroups