Subgroups — C₃²×C₇×#127 #72246346

Order 2016.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×D₄	C₂	1	6	yes	maximal, complemented
2	C₃²×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#127	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²×C₇	C₂²	1	4	yes	—
5	C₃²×C₇×D₄	—	2	12	no	—
6	C₂×C₃²×C₄×C₇	C₂²	1	3	yes	—
7	C₂×C₃×C₇×D₄	C₂×C₃	1	24	yes	complemented
8	C₃×C₄²×C₇	C₂×C₃	1	4	yes	complemented
9	C₃²×#127	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃²×C₇	—	2	12	no	—
11	C₂²×C₃²×C₇	C₂³	1	1	yes	characteristic
12	C₃²×C₄×C₇	D₄	1	6	yes	complemented
13	C₇×#127	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂³×C₃×C₇	C₂²×C₃	1	16	yes	—
15	C₃×C₇×D₄	—	2	48	no	—
16	C₂×C₃×C₄×C₇	C₂²×C₃	1	12	yes	—
17	C₂×C₃²×D₄	C₂×C₇	1	6	yes	complemented
18	C₃²×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×C₃²×C₇	—	4	4	no	—
20	C₂×C₃²×C₇	C₂×D₄	1	3	yes	—
21	C₂×C₇×D₄	C₂×C₃²	1	6	yes	complemented
22	C₄²×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
23	C₃×#127	C₃×C₇	1	4	yes	direct product factor
24	C₂²×C₃×C₇	—	2	48	no	—
25	C₂²×C₃×C₇	C₂³×C₃	1	4	yes	—
26	C₃×C₄×C₇	C₃×D₄	1	24	yes	complemented
27	C₂³×C₃²	C₂²×C₇	1	4	yes	—
28	C₃²×D₄	—	2	12	no	—
29	C₂×C₃²×C₄	C₂²×C₇	1	3	yes	—
30	C₃²×C₇	#127	1	1	yes	characteristic, direct product factor, Hall
31	C₂³×C₇	C₂²×C₃²	1	4	yes	—
32	C₇×D₄	—	2	12	no	—
33	C₂×C₄×C₇	C₂²×C₃²	1	3	yes	—
34	C₂×C₃×D₄	C₂×C₃×C₇	1	24	yes	complemented
35	C₃×C₄²	C₂×C₃×C₇	1	4	yes	complemented
36	C₂×C₃×C₇	—	4	16	no	—
37	C₂×C₃×C₇	C₂×C₃×D₄	1	12	yes	—
38	C₂²×C₃²	—	2	12	no	—
39	C₂²×C₃²	C₂³×C₇	1	1	yes	characteristic
40	C₃²×C₄	C₇×D₄	1	6	yes	complemented
41	#127	C₃²×C₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₇	—	2	12	no	—
43	C₂²×C₇	C₂³×C₃²	1	1	yes	characteristic
44	C₄×C₇	C₃²×D₄	1	6	yes	complemented
45	C₂³×C₃	C₂²×C₃×C₇	1	16	yes	—
46	C₃×D₄	—	2	48	no	—
47	C₂×C₃×C₄	C₂²×C₃×C₇	1	12	yes	—
48	C₃×C₇	C₃×#127	1	4	yes	direct product factor
49	C₂×C₃²	—	4	4	no	—
50	C₂×C₃²	C₂×C₇×D₄	1	3	yes	—
51	C₂×D₄	C₂×C₃²×C₇	1	6	yes	complemented
52	C₄²	C₂×C₃²×C₇	1	1	yes	characteristic, complemented
53	C₂×C₇	—	4	4	no	—
54	C₂×C₇	C₂×C₃²×D₄	1	3	yes	—
55	C₂²×C₃	—	2	48	no	—
56	C₂²×C₃	C₂³×C₃×C₇	1	4	yes	—
57	C₃×C₄	C₃×C₇×D₄	1	24	yes	complemented
58	C₃²	C₇×#127	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³	C₂²×C₃²×C₇	1	4	yes	—
60	D₄	—	2	12	no	—
61	C₂×C₄	C₂²×C₃²×C₇	1	3	yes	—
62	C₇	C₃²×#127	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	—	4	16	no	—
64	C₂×C₃	C₂×C₃×C₇×D₄	1	12	yes	—
65	C₂²	—	2	12	no	—
66	C₂²	C₂³×C₃²×C₇	1	1	yes	characteristic
67	C₄	C₃²×C₇×D₄	1	6	yes	complemented
68	C₃	C₃×C₇×#127	1	4	yes	direct product factor
69	C₂	—	4	4	no	—
70	C₂	C₂×C₃²×C₇×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₇×D₄	C₂	semidirect
2. C₃²×C₄²×C₇	C₂	semidirect
3. C₃×C₇×#127	C₃	direct (N × H)
7. C₂×C₃×C₇×D₄	C₂×C₃	semidirect
8. C₃×C₄²×C₇	C₂×C₃	semidirect
9. C₃²×#127	C₇	direct (N × H)
12. C₃²×C₄×C₇	D₄	semidirect
13. C₇×#127	C₃²	direct (N × H)
17. C₂×C₃²×D₄	C₂×C₇	semidirect
18. C₃²×C₄²	C₂×C₇	semidirect
21. C₂×C₇×D₄	C₂×C₃²	semidirect
22. C₄²×C₇	C₂×C₃²	semidirect
23. C₃×#127	C₃×C₇	direct (N × H)
26. C₃×C₄×C₇	C₃×D₄	semidirect
30. C₃²×C₇	#127	direct (N × H)
34. C₂×C₃×D₄	C₂×C₃×C₇	semidirect
35. C₃×C₄²	C₂×C₃×C₇	semidirect
40. C₃²×C₄	C₇×D₄	semidirect
41. #127	C₃²×C₇	direct (N × H)
44. C₄×C₇	C₃²×D₄	semidirect
48. C₃×C₇	C₃×#127	direct (N × H)
51. C₂×D₄	C₂×C₃²×C₇	semidirect
52. C₄²	C₂×C₃²×C₇	semidirect
57. C₃×C₄	C₃×C₇×D₄	semidirect
58. C₃²	C₇×#127	direct (N × H)
62. C₇	C₃²×#127	direct (N × H)
67. C₄	C₃²×C₇×D₄	semidirect
68. C₃	C₃×C₇×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₇×D₄	2
C₃²×C₄²×C₇	2
C₃×C₇×#127	3
C₃²×#127	7

Subgroups — C32×C7×#127 #72246346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₇×#127 #72246346