Subgroups — C₂×C₇²×C₉×C₂₇ #72201124

Order 23814.

720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₉×C₂₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇²×C₉²	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃×C₇²×C₂₇	C₃	1	3	yes	maximal
4	C₇²×C₉²	C₂×C₃	1	1	yes	characteristic
5	C₃×C₇²×C₂₇	C₂×C₃	1	3	yes	—
6	C₂×C₇×C₉×C₂₇	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₃×C₇²×C₉	C₉	1	3	yes	—
8	C₂×C₃×C₇²×C₉	C₃²	1	1	yes	characteristic
9	C₂×C₇²×C₂₇	C₉	1	9	yes	direct product factor
10	C₇×C₉×C₂₇	C₂×C₇	1	8	yes	direct product factor
11	C₃×C₇²×C₉	C₂×C₉	1	3	yes	—
12	C₃×C₇²×C₉	C₂×C₃²	1	1	yes	characteristic
13	C₇²×C₂₇	C₂×C₉	1	9	yes	direct product factor
14	C₂×C₃×C₇×C₂₇	C₃×C₇	1	24	yes	—
15	C₂×C₇×C₉²	C₃×C₇	1	8	yes	—
16	C₂×C₃²×C₇²	C₃×C₉	1	1	yes	characteristic
17	C₂×C₇²×C₉	C₂₇	1	9	yes	direct product factor
18	C₂×C₇²×C₉	C₃×C₉	1	3	yes	—
19	C₃×C₇×C₂₇	C₂×C₃×C₇	1	24	yes	—
20	C₇×C₉²	C₂×C₃×C₇	1	8	yes	—
21	C₂×C₉×C₂₇	C₇²	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₇²	C₂×C₃×C₉	1	1	yes	characteristic
23	C₇²×C₉	C₂×C₂₇	1	9	yes	direct product factor
24	C₇²×C₉	C₂×C₃×C₉	1	3	yes	—
25	C₂×C₃×C₇×C₉	C₇×C₉	1	24	yes	—
26	C₂×C₃×C₇×C₉	C₃²×C₇	1	8	yes	—
27	C₂×C₇×C₂₇	C₇×C₉	1	72	yes	direct product factor
28	C₂×C₃×C₇²	C₉²	1	1	yes	characteristic
29	C₂×C₃×C₇²	C₃×C₂₇	1	3	yes	—
30	C₉×C₂₇	C₂×C₇²	1	1	yes	characteristic, direct product factor, Sylow
31	C₃×C₇×C₉	C₂×C₇×C₉	1	24	yes	—
32	C₃×C₇×C₉	C₂×C₃²×C₇	1	8	yes	—
33	C₇×C₂₇	C₂×C₇×C₉	1	72	yes	direct product factor
34	C₂×C₃×C₂₇	C₃×C₇²	1	3	yes	—
35	C₂×C₉²	C₃×C₇²	1	1	yes	characteristic
36	C₃×C₇²	C₂×C₉²	1	1	yes	characteristic
37	C₃×C₇²	C₂×C₃×C₂₇	1	3	yes	—
38	C₂×C₃²×C₇	C₃×C₇×C₉	1	8	yes	—
39	C₂×C₇×C₉	C₇×C₂₇	1	72	yes	direct product factor
40	C₂×C₇×C₉	C₃×C₇×C₉	1	24	yes	—
41	C₂×C₇²	C₉×C₂₇	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₂₇	C₂×C₃×C₇²	1	3	yes	—
43	C₉²	C₂×C₃×C₇²	1	1	yes	characteristic
44	C₃²×C₇	C₂×C₃×C₇×C₉	1	8	yes	—
45	C₇×C₉	C₂×C₇×C₂₇	1	72	yes	direct product factor
46	C₇×C₉	C₂×C₃×C₇×C₉	1	24	yes	—
47	C₂×C₃×C₉	C₇²×C₉	1	3	yes	—
48	C₂×C₃×C₉	C₃²×C₇²	1	1	yes	characteristic
49	C₂×C₂₇	C₇²×C₉	1	9	yes	direct product factor
50	C₇²	C₂×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×C₇	C₇×C₉²	1	8	yes	—
52	C₂×C₃×C₇	C₃×C₇×C₂₇	1	24	yes	—
53	C₃×C₉	C₂×C₇²×C₉	1	3	yes	—
54	C₃×C₉	C₂×C₃²×C₇²	1	1	yes	characteristic
55	C₂₇	C₂×C₇²×C₉	1	9	yes	direct product factor
56	C₃×C₇	C₂×C₇×C₉²	1	8	yes	—
57	C₃×C₇	C₂×C₃×C₇×C₂₇	1	24	yes	—
58	C₂×C₃²	C₃×C₇²×C₉	1	1	yes	characteristic
59	C₂×C₉	C₇²×C₂₇	1	9	yes	direct product factor
60	C₂×C₉	C₃×C₇²×C₉	1	3	yes	—
61	C₂×C₇	C₇×C₉×C₂₇	1	8	yes	direct product factor
62	C₃²	C₂×C₃×C₇²×C₉	1	1	yes	characteristic
63	C₉	C₂×C₇²×C₂₇	1	9	yes	direct product factor
64	C₉	C₂×C₃×C₇²×C₉	1	3	yes	—
65	C₇	C₂×C₇×C₉×C₂₇	1	8	yes	direct product factor
66	C₂×C₃	C₃×C₇²×C₂₇	1	3	yes	—
67	C₂×C₃	C₇²×C₉²	1	1	yes	characteristic
68	C₃	C₂×C₃×C₇²×C₂₇	1	3	yes	—
69	C₃	C₂×C₇²×C₉²	1	1	yes	characteristic
70	C₂	C₇²×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₉	C₂×C₃²×C₇²
socle, prime radical	C₂×C₃²×C₇²	C₃×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇²×C₉×C₂₇	C₂	direct (N × H)
6. C₂×C₇×C₉×C₂₇	C₇	direct (N × H)
9. C₂×C₇²×C₂₇	C₉	direct (N × H)
10. C₇×C₉×C₂₇	C₂×C₇	direct (N × H)
13. C₇²×C₂₇	C₂×C₉	direct (N × H)
17. C₂×C₇²×C₉	C₂₇	direct (N × H)
21. C₂×C₉×C₂₇	C₇²	direct (N × H)
23. C₇²×C₉	C₂×C₂₇	direct (N × H)
27. C₂×C₇×C₂₇	C₇×C₉	direct (N × H)
30. C₉×C₂₇	C₂×C₇²	direct (N × H)
33. C₇×C₂₇	C₂×C₇×C₉	direct (N × H)
39. C₂×C₇×C₉	C₇×C₂₇	direct (N × H)
41. C₂×C₇²	C₉×C₂₇	direct (N × H)
45. C₇×C₉	C₂×C₇×C₂₇	direct (N × H)
49. C₂×C₂₇	C₇²×C₉	direct (N × H)
50. C₇²	C₂×C₉×C₂₇	direct (N × H)
55. C₂₇	C₂×C₇²×C₉	direct (N × H)
59. C₂×C₉	C₇²×C₂₇	direct (N × H)
61. C₂×C₇	C₇×C₉×C₂₇	direct (N × H)
63. C₉	C₂×C₇²×C₂₇	direct (N × H)
65. C₇	C₂×C₇×C₉×C₂₇	direct (N × H)
70. C₂	C₇²×C₉×C₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₉×C₂₇	2
C₂×C₇²×C₉²	3
C₂×C₃×C₇²×C₂₇	3
C₂×C₇×C₉×C₂₇	7

Subgroups — C2×C72×C9×C27 #72201124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×C₉×C₂₇ #72201124