Subgroups — C₂³×C₃×C₈×C₁₂₇ #72201682

Order 24384.

676 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈×C₁₂₇	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₃×C₁₂₇	C₄	1	1	yes	characteristic
4	C₂²×C₃×C₄×C₁₂₇	C₄	1	7	yes	—
5	C₂²×C₃×C₄×C₁₂₇	C₂²	1	7	yes	—
6	C₂×C₃×C₈×C₁₂₇	C₂²	1	28	yes	direct product factor
7	C₂³×C₃×C₁₂₇	C₈	1	8	yes	direct product factor
8	C₂³×C₃×C₁₂₇	C₂×C₄	1	7	yes	—
9	C₂×C₃×C₄×C₁₂₇	C₂×C₄	1	21	yes	—
10	C₂×C₃×C₄×C₁₂₇	C₂³	1	7	yes	—
11	C₃×C₈×C₁₂₇	C₂³	1	8	yes	direct product factor
12	C₂³×C₈×C₁₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₄×C₁₂₇	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	C₂×C₃	1	14	yes	direct product factor
15	C₂⁴×C₁₂₇	C₃×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	C₃×C₄	1	7	yes	—
17	C₂²×C₄×C₁₂₇	C₂²×C₃	1	7	yes	—
18	C₂×C₈×C₁₂₇	C₂²×C₃	1	28	yes	direct product factor
19	C₂²×C₃×C₁₂₇	C₂×C₈	1	28	yes	direct product factor
20	C₂²×C₃×C₁₂₇	C₂²×C₄	1	7	yes	—
21	C₃×C₄×C₁₂₇	C₂²×C₄	1	7	yes	—
22	C₃×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
23	C₂³×C₁₂₇	C₃×C₈	1	8	yes	direct product factor
24	C₂³×C₁₂₇	C₂×C₃×C₄	1	7	yes	—
25	C₂×C₄×C₁₂₇	C₂×C₃×C₄	1	21	yes	—
26	C₂×C₄×C₁₂₇	C₂³×C₃	1	7	yes	—
27	C₈×C₁₂₇	C₂³×C₃	1	8	yes	direct product factor
28	C₂×C₃×C₁₂₇	C₂²×C₈	1	14	yes	direct product factor
29	C₂×C₃×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
30	C₂²×C₁₂₇	C₂×C₃×C₈	1	28	yes	direct product factor
31	C₂²×C₁₂₇	C₂²×C₃×C₄	1	7	yes	—
32	C₄×C₁₂₇	C₂²×C₃×C₄	1	7	yes	—
33	C₄×C₁₂₇	C₂⁴×C₃	1	1	yes	characteristic
34	C₃×C₁₂₇	C₂³×C₈	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₁₂₇	C₂²×C₃×C₈	1	14	yes	direct product factor
36	C₂×C₁₂₇	C₂³×C₃×C₄	1	1	yes	characteristic
37	C₂³×C₃×C₈	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₁₂₇	C₂³×C₃×C₈	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³×C₃×C₄	C₂×C₁₂₇	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂×C₁₂₇	1	14	yes	direct product factor
41	C₂³×C₈	C₃×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂⁴×C₃	C₄×C₁₂₇	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₄×C₁₂₇	1	7	yes	—
44	C₂²×C₃×C₄	C₂²×C₁₂₇	1	7	yes	—
45	C₂×C₃×C₈	C₂²×C₁₂₇	1	28	yes	direct product factor
46	C₂³×C₄	C₂×C₃×C₁₂₇	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₃×C₁₂₇	1	14	yes	direct product factor
48	C₂³×C₃	C₈×C₁₂₇	1	8	yes	direct product factor
49	C₂³×C₃	C₂×C₄×C₁₂₇	1	7	yes	—
50	C₂×C₃×C₄	C₂×C₄×C₁₂₇	1	21	yes	—
51	C₂×C₃×C₄	C₂³×C₁₂₇	1	7	yes	—
52	C₃×C₈	C₂³×C₁₂₇	1	8	yes	direct product factor
53	C₂⁴	C₃×C₄×C₁₂₇	1	1	yes	characteristic
54	C₂²×C₄	C₃×C₄×C₁₂₇	1	7	yes	—
55	C₂²×C₄	C₂²×C₃×C₁₂₇	1	7	yes	—
56	C₂×C₈	C₂²×C₃×C₁₂₇	1	28	yes	direct product factor
57	C₂²×C₃	C₂×C₈×C₁₂₇	1	28	yes	direct product factor
58	C₂²×C₃	C₂²×C₄×C₁₂₇	1	7	yes	—
59	C₃×C₄	C₂²×C₄×C₁₂₇	1	7	yes	—
60	C₃×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
61	C₂³	C₃×C₈×C₁₂₇	1	8	yes	direct product factor
62	C₂³	C₂×C₃×C₄×C₁₂₇	1	7	yes	—
63	C₂×C₄	C₂×C₃×C₄×C₁₂₇	1	21	yes	—
64	C₂×C₄	C₂³×C₃×C₁₂₇	1	7	yes	—
65	C₈	C₂³×C₃×C₁₂₇	1	8	yes	direct product factor
66	C₂×C₃	C₂²×C₈×C₁₂₇	1	14	yes	direct product factor
67	C₂×C₃	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
68	C₂²	C₂×C₃×C₈×C₁₂₇	1	28	yes	direct product factor
69	C₂²	C₂²×C₃×C₄×C₁₂₇	1	7	yes	—
70	C₄	C₂²×C₃×C₄×C₁₂₇	1	7	yes	—
71	C₄	C₂⁴×C₃×C₁₂₇	1	1	yes	characteristic
72	C₃	C₂³×C₈×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₂²×C₃×C₈×C₁₂₇	1	14	yes	direct product factor
74	C₂	C₂³×C₃×C₄×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁴×C₃×C₁₂₇
socle, prime radical	C₂⁴×C₃×C₁₂₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈×C₁₂₇	C₂	direct (N × H)
6. C₂×C₃×C₈×C₁₂₇	C₂²	direct (N × H)
7. C₂³×C₃×C₁₂₇	C₈	direct (N × H)
11. C₃×C₈×C₁₂₇	C₂³	direct (N × H)
12. C₂³×C₈×C₁₂₇	C₃	direct (N × H)
14. C₂²×C₈×C₁₂₇	C₂×C₃	direct (N × H)
18. C₂×C₈×C₁₂₇	C₂²×C₃	direct (N × H)
19. C₂²×C₃×C₁₂₇	C₂×C₈	direct (N × H)
23. C₂³×C₁₂₇	C₃×C₈	direct (N × H)
27. C₈×C₁₂₇	C₂³×C₃	direct (N × H)
28. C₂×C₃×C₁₂₇	C₂²×C₈	direct (N × H)
30. C₂²×C₁₂₇	C₂×C₃×C₈	direct (N × H)
34. C₃×C₁₂₇	C₂³×C₈	direct (N × H)
35. C₂×C₁₂₇	C₂²×C₃×C₈	direct (N × H)
37. C₂³×C₃×C₈	C₁₂₇	direct (N × H)
38. C₁₂₇	C₂³×C₃×C₈	direct (N × H)
40. C₂²×C₃×C₈	C₂×C₁₂₇	direct (N × H)
41. C₂³×C₈	C₃×C₁₂₇	direct (N × H)
45. C₂×C₃×C₈	C₂²×C₁₂₇	direct (N × H)
47. C₂²×C₈	C₂×C₃×C₁₂₇	direct (N × H)
48. C₂³×C₃	C₈×C₁₂₇	direct (N × H)
52. C₃×C₈	C₂³×C₁₂₇	direct (N × H)
56. C₂×C₈	C₂²×C₃×C₁₂₇	direct (N × H)
57. C₂²×C₃	C₂×C₈×C₁₂₇	direct (N × H)
61. C₂³	C₃×C₈×C₁₂₇	direct (N × H)
65. C₈	C₂³×C₃×C₁₂₇	direct (N × H)
66. C₂×C₃	C₂²×C₈×C₁₂₇	direct (N × H)
68. C₂²	C₂×C₃×C₈×C₁₂₇	direct (N × H)
72. C₃	C₂³×C₈×C₁₂₇	direct (N × H)
73. C₂	C₂²×C₃×C₈×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₁₂₇	2
C₂²×C₃×C₈×C₁₂₇	2
C₂³×C₈×C₁₂₇	3
C₂³×C₃×C₈	127

Subgroups — C23×C3×C8×C127 #72201682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₈×C₁₂₇ #72201682