Subgroups — C₂×C₃×C₇×C₁₁×C₂₇ #72190815

Order 12474.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₁₁×C₂₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃×C₇×C₉×C₁₁	C₃	1	1	yes	maximal, characteristic
3	C₂×C₇×C₁₁×C₂₇	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₇×C₉×C₁₁	C₂×C₃	1	1	yes	characteristic
5	C₇×C₁₁×C₂₇	C₂×C₃	1	3	yes	direct product factor
6	C₂×C₃×C₁₁×C₂₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃²×C₇×C₁₁	C₉	1	1	yes	characteristic
8	C₂×C₇×C₉×C₁₁	C₉	1	2	yes	—
9	C₂×C₇×C₉×C₁₁	C₃²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₂₇	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₁₁×C₂₇	C₂×C₇	1	1	yes	characteristic, direct product factor, Hall
12	C₃²×C₇×C₁₁	C₂×C₉	1	1	yes	characteristic
13	C₇×C₉×C₁₁	C₂×C₉	1	2	yes	—
14	C₇×C₉×C₁₁	C₂×C₃²	1	1	yes	characteristic
15	C₂×C₃×C₉×C₁₁	C₃×C₇	1	1	yes	characteristic
16	C₂×C₁₁×C₂₇	C₃×C₇	1	3	yes	direct product factor
17	C₃×C₇×C₂₇	C₂×C₁₁	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇×C₁₁	C₂₇	1	3	yes	direct product factor
19	C₂×C₃×C₇×C₁₁	C₃×C₉	1	1	yes	characteristic
20	C₂×C₃×C₇×C₉	C₃×C₁₁	1	1	yes	characteristic
21	C₂×C₇×C₂₇	C₃×C₁₁	1	3	yes	direct product factor
22	C₃×C₉×C₁₁	C₂×C₃×C₇	1	1	yes	characteristic
23	C₁₁×C₂₇	C₂×C₃×C₇	1	3	yes	direct product factor
24	C₃×C₇×C₁₁	C₂×C₂₇	1	3	yes	direct product factor
25	C₃×C₇×C₁₁	C₂×C₃×C₉	1	1	yes	characteristic
26	C₂×C₃²×C₁₁	C₇×C₉	1	1	yes	characteristic
27	C₂×C₉×C₁₁	C₇×C₉	1	2	yes	—
28	C₂×C₉×C₁₁	C₃²×C₇	1	1	yes	characteristic
29	C₃×C₇×C₉	C₂×C₃×C₁₁	1	1	yes	characteristic
30	C₇×C₂₇	C₂×C₃×C₁₁	1	3	yes	direct product factor
31	C₂×C₃×C₂₇	C₇×C₁₁	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₇×C₁₁	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₃²×C₇	C₉×C₁₁	1	1	yes	characteristic
34	C₂×C₇×C₉	C₉×C₁₁	1	2	yes	—
35	C₂×C₇×C₉	C₃²×C₁₁	1	1	yes	characteristic
36	C₃²×C₁₁	C₂×C₇×C₉	1	1	yes	characteristic
37	C₉×C₁₁	C₂×C₇×C₉	1	2	yes	—
38	C₉×C₁₁	C₂×C₃²×C₇	1	1	yes	characteristic
39	C₃×C₂₇	C₂×C₇×C₁₁	1	1	yes	characteristic, direct product factor, Sylow
40	C₇×C₁₁	C₂×C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
41	C₂×C₃×C₁₁	C₇×C₂₇	1	3	yes	direct product factor
42	C₂×C₃×C₁₁	C₃×C₇×C₉	1	1	yes	characteristic
43	C₃²×C₇	C₂×C₉×C₁₁	1	1	yes	characteristic
44	C₇×C₉	C₂×C₉×C₁₁	1	2	yes	—
45	C₇×C₉	C₂×C₃²×C₁₁	1	1	yes	characteristic
46	C₂×C₃×C₉	C₃×C₇×C₁₁	1	1	yes	characteristic
47	C₂×C₂₇	C₃×C₇×C₁₁	1	3	yes	direct product factor
48	C₂×C₃×C₇	C₁₁×C₂₇	1	3	yes	direct product factor
49	C₂×C₃×C₇	C₃×C₉×C₁₁	1	1	yes	characteristic
50	C₃×C₁₁	C₂×C₇×C₂₇	1	3	yes	direct product factor
51	C₃×C₁₁	C₂×C₃×C₇×C₉	1	1	yes	characteristic
52	C₃×C₉	C₂×C₃×C₇×C₁₁	1	1	yes	characteristic
53	C₂₇	C₂×C₃×C₇×C₁₁	1	3	yes	direct product factor
54	C₂×C₁₁	C₃×C₇×C₂₇	1	1	yes	characteristic, direct product factor, Hall
55	C₃×C₇	C₂×C₁₁×C₂₇	1	3	yes	direct product factor
56	C₃×C₇	C₂×C₃×C₉×C₁₁	1	1	yes	characteristic
57	C₂×C₃²	C₇×C₉×C₁₁	1	1	yes	characteristic
58	C₂×C₉	C₇×C₉×C₁₁	1	2	yes	—
59	C₂×C₉	C₃²×C₇×C₁₁	1	1	yes	characteristic
60	C₂×C₇	C₃×C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Hall
61	C₁₁	C₂×C₃×C₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
62	C₃²	C₂×C₇×C₉×C₁₁	1	1	yes	characteristic
63	C₉	C₂×C₇×C₉×C₁₁	1	2	yes	—
64	C₉	C₂×C₃²×C₇×C₁₁	1	1	yes	characteristic
65	C₇	C₂×C₃×C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	C₇×C₁₁×C₂₇	1	3	yes	direct product factor
67	C₂×C₃	C₃×C₇×C₉×C₁₁	1	1	yes	characteristic
68	C₃	C₂×C₇×C₁₁×C₂₇	1	3	yes	direct product factor
69	C₃	C₂×C₃×C₇×C₉×C₁₁	1	1	yes	characteristic
70	C₂	C₃×C₇×C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂×C₃²×C₇×C₁₁
socle, prime radical	C₂×C₃²×C₇×C₁₁	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₁₁×C₂₇	C₂	direct (N × H)
3. C₂×C₇×C₁₁×C₂₇	C₃	direct (N × H)
5. C₇×C₁₁×C₂₇	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₁×C₂₇	C₇	direct (N × H)
10. C₂×C₃×C₇×C₂₇	C₁₁	direct (N × H)
11. C₃×C₁₁×C₂₇	C₂×C₇	direct (N × H)
16. C₂×C₁₁×C₂₇	C₃×C₇	direct (N × H)
17. C₃×C₇×C₂₇	C₂×C₁₁	direct (N × H)
18. C₂×C₃×C₇×C₁₁	C₂₇	direct (N × H)
21. C₂×C₇×C₂₇	C₃×C₁₁	direct (N × H)
23. C₁₁×C₂₇	C₂×C₃×C₇	direct (N × H)
24. C₃×C₇×C₁₁	C₂×C₂₇	direct (N × H)
30. C₇×C₂₇	C₂×C₃×C₁₁	direct (N × H)
31. C₂×C₃×C₂₇	C₇×C₁₁	direct (N × H)
32. C₂×C₇×C₁₁	C₃×C₂₇	direct (N × H)
39. C₃×C₂₇	C₂×C₇×C₁₁	direct (N × H)
40. C₇×C₁₁	C₂×C₃×C₂₇	direct (N × H)
41. C₂×C₃×C₁₁	C₇×C₂₇	direct (N × H)
47. C₂×C₂₇	C₃×C₇×C₁₁	direct (N × H)
48. C₂×C₃×C₇	C₁₁×C₂₇	direct (N × H)
50. C₃×C₁₁	C₂×C₇×C₂₇	direct (N × H)
53. C₂₇	C₂×C₃×C₇×C₁₁	direct (N × H)
54. C₂×C₁₁	C₃×C₇×C₂₇	direct (N × H)
55. C₃×C₇	C₂×C₁₁×C₂₇	direct (N × H)
60. C₂×C₇	C₃×C₁₁×C₂₇	direct (N × H)
61. C₁₁	C₂×C₃×C₇×C₂₇	direct (N × H)
65. C₇	C₂×C₃×C₁₁×C₂₇	direct (N × H)
66. C₂×C₃	C₇×C₁₁×C₂₇	direct (N × H)
68. C₃	C₂×C₇×C₁₁×C₂₇	direct (N × H)
70. C₂	C₃×C₇×C₁₁×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
C₃×C₇×C₁₁×C₂₇	2
C₂×C₃×C₇×C₉×C₁₁	3
C₂×C₇×C₁₁×C₂₇	3
C₂×C₃×C₁₁×C₂₇	7
C₂×C₃×C₇×C₂₇	11

Subgroups — C2×C3×C7×C11×C27 #72190815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×C₁₁×C₂₇ #72190815