Group #22581756 — subgroups

Subgroups — #22581756

Order 8896.

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

normal subgroups only

This group has 268 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	#17984888	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984868	C₂	1	3	yes	maximal, complemented
3	#17984867	C₂	1	3	yes	maximal, complemented
4	#17984862	C₂	1	3	yes	maximal, complemented
5	C₂×#14555770	C₂	1	3	yes	maximal, complemented
6	#17984849	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555786	—	2	1	no	—
9	#14555785	C₂²	1	3	yes	complemented
10	C₂×D₅₅₆	C₂²	1	3	yes	complemented
11	D₄×D₁₃₉	—	2	3	no	—
12	#14555775	C₂²	1	1	yes	characteristic, complemented
13	#14555773	C₂²	1	3	yes	complemented
14	#14555772	C₂²	1	3	yes	complemented
15	#14555770	C₂²	1	3	yes	complemented
16	#14555770	C₂²	1	6	yes	complemented
17	#14555769	C₂²	1	3	yes	complemented
18	C₂×C₁₃₉⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₃₉	—	2	1	no	—
20	#41×C₁₃₉	C₂²	1	1	yes	characteristic
21	#41×C₁₃₉	C₂²	1	3	yes	complemented
22	C₂×D₄×C₁₃₉	—	2	3	no	—
23	C₂×D₄×C₁₃₉	C₂²	1	3	yes	complemented
24	C₂²×D₁₃₉	—	2	3	no	—
25	Dic₂₇₈	C₂³	1	1	yes	characteristic
26	D₅₅₆	—	2	3	no	—
27	D₅₅₆	C₂³	1	3	yes	—
28	C₂²⋊D₁₃₉	—	2	3	no	—
29	C₄×D₁₃₉	—	2	3	no	—
30	C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₃₉⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₃₉	—	2	3	no	—
33	Q₈×C₁₃₉	D₄	1	1	yes	characteristic
34	Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
35	D₄×C₁₃₉	—	2	3	no	—
36	D₄×C₁₃₉	—	2	3	no	—
37	D₄×C₁₃₉	D₄	1	3	yes	—
38	D₄×C₁₃₉	C₂³	1	3	yes	—
39	C₂×C₄×C₁₃₉	—	2	3	no	—
40	C₂×C₄×C₁₃₉	C₂³	1	3	yes	—
41	C₂²×C₁₃₉	—	2	3	no	—
42	C₂²×C₁₃₉	—	8	3	no	—
43	C₂²×C₁₃₉	C₂×D₄	1	3	yes	—
44	C₂×D₁₃₉	—	2	3	no	—
45	C₂×D₁₃₉	—	8	3	no	—
46	C₄×C₁₃₉	—	2	1	no	—
47	C₄×C₁₃₉	C₂×D₄	1	3	yes	—
48	C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
49	Dic₁₃₉	—	2	1	no	—
50	C₂×C₁₃₉	—	2	3	no	—
51	C₂×C₁₃₉	—	4	3	no	—
52	C₂×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
53	D₁₃₉	—	4	3	no	—
54	C₁₃₉	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	139	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	139	1	no	—
57	H(5,𝔽₂)	D₁₃₉	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	139	3	no	—
59	AGL(1,ℤ/8)	—	139	3	no	—
60	#135	—	139	3	no	—
61	C₂×D₈	—	139	3	no	—
62	#131	—	139	1	no	—
63	#41	—	2	1	no	—
64	#41	—	278	1	no	—
65	#41	—	139	3	no	—
66	#41	C₂×D₁₃₉	1	1	yes	characteristic
67	#41	C₂×D₁₃₉	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	278	3	no	—
70	C₂×D₄	—	139	3	no	—
71	C₂×D₄	C₂×D₁₃₉	1	3	yes	complemented
72	Q₁₆	—	139	1	no	—
73	SD₁₆	—	139	3	no	—
74	SD₁₆	—	139	3	no	—
75	D₈	—	139	3	no	—
76	D₈	—	139	6	no	—
77	M₁₆	—	139	3	no	—
78	C₂×C₈	—	139	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	278	3	no	—
81	Q₈	—	139	1	no	—
82	Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
83	Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	278	3	no	—
87	D₄	—	278	3	no	—
88	D₄	—	139	3	no	—
89	D₄	C₂²⋊D₁₃₉	1	3	yes	—
90	D₄	C₂²×D₁₃₉	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	278	3	no	—
93	C₂×C₄	C₂²×D₁₃₉	1	3	yes	—
94	C₈	—	139	1	no	—
95	C₈	—	139	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	278	3	no	—
99	C₂²	—	1112	3	no	—
100	C₂²	C₂×C₂²⋊D₁₃₉	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	278	1	no	—
103	C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
104	C₄	C₂³×D₁₃₉	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	556	3	no	—
108	C₂	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984888

C₂

yes

maximal, characteristic, complemented

#17984868

C₂

yes

maximal, complemented

#17984867

C₂

yes

maximal, complemented

#17984862

C₂

yes

maximal, complemented

C₂×#14555770

C₂

yes

maximal, complemented

#17984849

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555786

—

#14555785

C₂²

yes

complemented

C₂×D₅₅₆

C₂²

yes

complemented

D₄×D₁₃₉

—

#14555775

C₂²

yes

characteristic, complemented

#14555773

C₂²

yes

complemented

#14555772

C₂²

yes

complemented

#14555770

C₂²

yes

complemented

#14555770

C₂²

yes

complemented

#14555769

C₂²

yes

complemented

C₂×C₁₃₉⋊C₈

C₂²

yes

complemented

#41×C₁₃₉

—

#41×C₁₃₉

C₂²

yes

characteristic

#41×C₁₃₉

C₂²

yes

complemented

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂²

yes

complemented

C₂²×D₁₃₉

—

Dic₂₇₈

C₂³

yes

characteristic

D₅₅₆

—

D₅₅₆

C₂³

yes

—

C₂²⋊D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

C₂³

yes

characteristic

C₁₃₉⋊C₈

C₂³

yes

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

D₄

yes

characteristic

Q₈×C₁₃₉

C₂³

yes

characteristic

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

—

D₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

—

C₄×C₁₃₉

C₂⁴

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂²×D₄

yes

characteristic

D₁₃₉

—

C₁₃₉

#576

yes

characteristic, complemented, Sylow

#576

—

139

maximal, Sylow

H(5,𝔽₂)

—

139

—

H(5,𝔽₂)

D₁₃₉

yes

characteristic, complemented

AGL(1,ℤ/8)

—

139

—

AGL(1,ℤ/8)

—

139

—

#135

—

139

—

C₂×D₈

—

139

—

#131

—

139

—

#41

—

#41

—

278

—

#41

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

#41

C₂×D₁₃₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

complemented

Q₁₆

—

139

—

SD₁₆

—

139

—

SD₁₆

—

139

—

D₈

—

139

—

D₈

—

139

—

M₁₆

—

139

—

C₂×C₈

—

139

—

C₂³

—

C₂³

—

278

—

Q₈

—

139

—

Q₈

C₂²⋊D₁₃₉

yes

characteristic

Q₈

C₂²×D₁₃₉

yes

characteristic

D₄

—

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

139

—

D₄

C₂²⋊D₁₃₉

yes

—

D₄

C₂²×D₁₃₉

yes

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₈

—

139

—

C₈

—

139

—

C₂²

—

C₂²

—

C₂²

—

278

—

C₂²

—

1112

—

100

C₂²

C₂×C₂²⋊D₁₃₉

yes

—

101

C₄

—

102

C₄

—

278

—

103

C₄

C₂×C₂²⋊D₁₃₉

yes

—

104

C₄

C₂³×D₁₃₉

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

556

—

108

C₂

C₂²×C₂²⋊D₁₃₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂⁴
Frattini	C₄	C₂³×D₁₃₉
Fitting	H(5,𝔽₂)×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₃₉
hypocenter	C₁₃₉	#576
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₃₉

commutator

C₄×C₁₃₉

C₂⁴

Frattini

C₄

C₂³×D₁₃₉

Fitting

H(5,𝔽₂)×C₁₃₉

C₂

socle

C₂×C₁₃₉

C₂²×D₄

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #17984888	C₂	semidirect
2. #17984868	C₂	semidirect
3. #17984867	C₂	semidirect
4. #17984862	C₂	semidirect
5. C₂×#14555770	C₂	semidirect
6. #17984849	C₂	semidirect
7. H(5,𝔽₂)×C₁₃₉	C₂	semidirect
9. #14555785	C₂²	semidirect
10. C₂×D₅₅₆	C₂²	semidirect
12. #14555775	C₂²	semidirect
13. #14555773	C₂²	semidirect
14. #14555772	C₂²	semidirect
15. #14555770	C₂²	semidirect
16. #14555770	C₂²	semidirect
17. #14555769	C₂²	semidirect
18. C₂×C₁₃₉⋊C₈	C₂²	semidirect
21. #41×C₁₃₉	C₂²	semidirect
23. C₂×D₄×C₁₃₉	C₂²	semidirect
54. C₁₃₉	#576	semidirect
57. H(5,𝔽₂)	D₁₃₉	semidirect
67. #41	C₂×D₁₃₉	semidirect
71. C₂×D₄	C₂×D₁₃₉	semidirect

H ≅ G/N

type

1. #17984888

C₂

semidirect

2. #17984868

C₂

semidirect

3. #17984867

C₂

semidirect

4. #17984862

C₂

semidirect

5. C₂×#14555770

C₂

semidirect

6. #17984849

C₂

semidirect

7. H(5,𝔽₂)×C₁₃₉

C₂

semidirect

9. #14555785

C₂²

semidirect

10. C₂×D₅₅₆

C₂²

semidirect

12. #14555775

C₂²

semidirect

13. #14555773

C₂²

semidirect

14. #14555772

C₂²

semidirect

15. #14555770

C₂²

semidirect

16. #14555770

C₂²

semidirect

17. #14555769

C₂²

semidirect

18. C₂×C₁₃₉⋊C₈

C₂²

semidirect

21. #41×C₁₃₉

C₂²

semidirect

23. C₂×D₄×C₁₃₉

C₂²

semidirect

54. C₁₃₉

#576

semidirect

57. H(5,𝔽₂)

D₁₃₉

semidirect

67. #41

C₂×D₁₃₉

semidirect

71. C₂×D₄

C₂×D₁₃₉

semidirect

Subgroups — #22581756

Conjugacy classes of subgroups

Complements