Subgroups — C₂²×#67355 #13094608

Order 1512.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4428	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67355	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×AGL(1,𝔽₇)	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₉×D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₂×#4428	C₂²	1	3	yes	complemented
7	#67355	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×AGL(1,𝔽₇)	C₂×C₃	1	6	yes	—
10	C₂²×C₇×C₉	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₉×D₇	C₂×C₃	1	6	yes	complemented
12	C₂²×C₇⋊C₉	C₂×C₃	1	2	yes	complemented
13	C₂×D₇⋊C₉	C₂×C₃	1	12	yes	complemented
14	C₂³×3₋¹⁺²	—	7	1	no	maximal, Hall
15	#4428	C₂³	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₃×D₇	C₃²	1	1	yes	characteristic
17	C₂²×AGL(1,𝔽₇)	—	3	1	no	—
18	C₂×C₃×ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	—
19	C₃×AGL(1,𝔽₇)	C₂²×C₃	1	4	yes	—
20	C₂×C₇×C₉	C₂²×C₃	1	3	yes	complemented
21	C₉×D₇	C₂²×C₃	1	4	yes	complemented
22	C₂×C₇⋊C₉	C₂²×C₃	1	6	yes	complemented
23	D₇⋊C₉	C₂²×C₃	1	8	yes	complemented
24	C₂²×3₋¹⁺²	—	7	6	no	—
25	C₂²×3₋¹⁺²	—	7	1	no	—
26	C₂²×C₃×C₇	C₂×C₃²	1	1	yes	characteristic
27	C₂×C₃×D₇	C₂×C₃²	1	6	yes	—
28	C₂²×ΓL(1,𝔽₈)	—	3	1	no	—
29	C₂×AGL(1,𝔽₇)	—	3	6	no	—
30	C₂³×C₃²	—	7	1	no	—
31	C₂³×C₉	—	7	2	no	—
32	C₂³×C₉	—	7	1	no	—
33	C₃×ΓL(1,𝔽₈)	C₂³×C₃	1	1	yes	characteristic
34	C₇×C₉	C₂³×C₃	1	1	yes	characteristic, complemented
35	C₇⋊C₉	C₂³×C₃	1	2	yes	complemented
36	C₂²×D₇	3₋¹⁺²	1	1	yes	characteristic, complemented, Hall
37	C₂×3₋¹⁺²	—	7	4	no	—
38	C₂×3₋¹⁺²	—	7	3	no	—
39	C₂×C₃×C₇	C₂²×C₃²	1	3	yes	—
40	C₃×D₇	C₂²×C₃²	1	4	yes	—
41	C₂×ΓL(1,𝔽₈)	—	3	3	no	—
42	AGL(1,𝔽₇)	—	3	4	no	—
43	C₂²×C₃²	—	7	6	no	—
44	C₂²×C₃²	—	7	1	no	—
45	C₂²×C₉	—	7	12	no	—
46	C₂²×C₉	—	7	6	no	—
47	C₂²×C₉	—	7	2	no	—
48	C₂²×C₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
49	C₂²×C₇	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
50	C₂×D₇	C₂×3₋¹⁺²	1	6	yes	complemented
51	3₋¹⁺²	—	7	1	no	Sylow
52	C₂³×C₃	—	7	1	no	—
53	C₂³×C₃	—	21	1	no	—
54	C₃×C₇	C₂³×C₃²	1	1	yes	characteristic
55	ΓL(1,𝔽₈)	—	3	1	no	—
56	C₂×C₃²	—	7	4	no	—
57	C₂×C₃²	—	7	3	no	—
58	C₂×C₉	—	7	8	no	—
59	C₂×C₉	—	7	4	no	—
60	C₂×C₉	—	7	6	no	—
61	C₂×C₉	C₂×AGL(1,𝔽₇)	1	3	yes	complemented
62	C₂×C₇	C₂²×3₋¹⁺²	1	3	yes	complemented
63	D₇	C₂²×3₋¹⁺²	1	4	yes	complemented
64	C₂²×C₃	—	7	6	no	—
65	C₂²×C₃	—	21	6	no	—
66	C₂²×C₃	—	21	1	no	—
67	C₂²×C₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
68	C₃²	—	7	1	no	—
69	C₉	—	7	2	no	—
70	C₉	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
71	C₂³	—	7	1	no	Sylow
72	C₇	C₂³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	7	4	no	—
74	C₂×C₃	—	21	4	no	—
75	C₂×C₃	—	21	3	no	—
76	C₂×C₃	C₂×C₃×AGL(1,𝔽₇)	1	3	yes	—
77	C₂²	—	7	6	no	—
78	C₂²	#67355	1	1	yes	characteristic, direct product factor
79	C₃	—	21	1	no	—
80	C₃	C₂²×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
81	C₂	—	7	4	no	—
82	C₂	C₂×#67355	1	3	yes	direct product factor

	subgroup	G/H
center	C₂²×C₃	C₃×AGL(1,𝔽₇)
commutator, cosocle	C₃×C₇	C₂³×C₃²
Frattini	C₃	C₂²×C₃×AGL(1,𝔽₇)
Fitting	C₂²×C₇×C₉	C₂×C₃
socle	C₂²×C₃×C₇	C₂×C₃²
prime radical	C₂²×C₃×AGL(1,𝔽₇)	C₃
hypercenter	C₂²×C₉	AGL(1,𝔽₇)
hypocenter	C₇	C₂³×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#4428	C₂	semidirect
2. C₂×#67355	C₂	direct (N × H)
4. C₂²×C₉×D₇	C₃	semidirect
5. C₂²×D₇⋊C₉	C₃	semidirect
6. C₂×#4428	C₂²	semidirect
7. #67355	C₂²	direct (N × H)
10. C₂²×C₇×C₉	C₂×C₃	semidirect
11. C₂×C₉×D₇	C₂×C₃	semidirect
12. C₂²×C₇⋊C₉	C₂×C₃	semidirect
13. C₂×D₇⋊C₉	C₂×C₃	semidirect
15. #4428	C₂³	semidirect
20. C₂×C₇×C₉	C₂²×C₃	semidirect
21. C₉×D₇	C₂²×C₃	semidirect
22. C₂×C₇⋊C₉	C₂²×C₃	semidirect
23. D₇⋊C₉	C₂²×C₃	semidirect
34. C₇×C₉	C₂³×C₃	semidirect
35. C₇⋊C₉	C₂³×C₃	semidirect
36. C₂²×D₇	3₋¹⁺²	semidirect
48. C₂²×C₉	AGL(1,𝔽₇)	semidirect
49. C₂²×C₇	C₂×3₋¹⁺²	semidirect
50. C₂×D₇	C₂×3₋¹⁺²	semidirect
61. C₂×C₉	C₂×AGL(1,𝔽₇)	semidirect
62. C₂×C₇	C₂²×3₋¹⁺²	semidirect
63. D₇	C₂²×3₋¹⁺²	semidirect
70. C₉	C₂²×AGL(1,𝔽₇)	semidirect
72. C₇	C₂³×3₋¹⁺²	semidirect
78. C₂²	#67355	direct (N × H)
82. C₂	C₂×#67355	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#4428	2
C₂×#67355	2
C₂²×C₃×AGL(1,𝔽₇)	3
C₂²×C₉×D₇	3
C₂²×D₇⋊C₉	3
C₂³×3₋¹⁺²	7

Subgroups — C22×#67355 #13094608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#67355 #13094608