Subgroups — C₂²×C₄×C₈×C₁₂₇ #30635821

Order 16256.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₁₂₇	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₂₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
5	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₁₂₇	C₄	1	16	yes	direct product factor
7	C₂²×C₈×C₁₂₇	C₂²	1	12	yes	—
8	C₂×C₄²×C₁₂₇	C₄	1	6	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	6	yes	—
10	C₄×C₈×C₁₂₇	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	C₈	1	16	yes	direct product factor
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
15	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
17	C₂×C₈×C₁₂₇	C₂×C₄	1	48	yes	direct product factor
18	C₂×C₈×C₁₂₇	C₂³	1	8	yes	—
19	C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
20	C₄²×C₁₂₇	C₂³	1	4	yes	—
21	C₂³×C₁₂₇	C₄²	1	4	yes	—
22	C₂³×C₁₂₇	C₂×C₈	1	8	yes	—
23	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₁₂₇	C₄²	1	12	yes	—
25	C₂×C₄×C₁₂₇	C₂×C₈	1	48	yes	direct product factor
26	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
27	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
29	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂²×C₄	1	16	yes	direct product factor
31	C₂²×C₁₂₇	C₄×C₈	1	16	yes	direct product factor
32	C₂²×C₁₂₇	C₂×C₄²	1	6	yes	—
33	C₂²×C₁₂₇	C₂²×C₈	1	12	yes	—
34	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₁₂₇	C₂×C₄²	1	6	yes	—
36	C₄×C₁₂₇	C₂²×C₈	1	16	yes	direct product factor
37	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
38	C₂×C₁₂₇	C₂×C₄×C₈	1	12	yes	direct product factor
39	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂³×C₈	1	2	yes	—
41	C₂²×C₄×C₈	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₁₂₇	C₂²×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³×C₈	C₂×C₁₂₇	1	2	yes	—
44	C₂²×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₁₂₇	1	12	yes	direct product factor
46	C₂³×C₄	C₄×C₁₂₇	1	2	yes	—
47	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
48	C₂²×C₈	C₄×C₁₂₇	1	16	yes	direct product factor
49	C₂²×C₈	C₂²×C₁₂₇	1	12	yes	—
50	C₂×C₄²	C₄×C₁₂₇	1	6	yes	—
51	C₂×C₄²	C₂²×C₁₂₇	1	6	yes	—
52	C₄×C₈	C₂²×C₁₂₇	1	16	yes	direct product factor
53	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
54	C₂²×C₄	C₈×C₁₂₇	1	16	yes	direct product factor
55	C₂²×C₄	C₂×C₄×C₁₂₇	1	3	yes	—
56	C₂²×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
57	C₂²×C₄	C₂×C₄×C₁₂₇	1	12	yes	—
58	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
59	C₂×C₈	C₂×C₄×C₁₂₇	1	48	yes	direct product factor
60	C₂×C₈	C₂³×C₁₂₇	1	8	yes	—
61	C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
62	C₄²	C₂³×C₁₂₇	1	4	yes	—
63	C₂³	C₄²×C₁₂₇	1	4	yes	—
64	C₂³	C₂×C₈×C₁₂₇	1	8	yes	—
65	C₂³	C₂²×C₄×C₁₂₇	1	3	yes	—
66	C₂×C₄	C₄²×C₁₂₇	1	12	yes	—
67	C₂×C₄	C₂×C₈×C₁₂₇	1	48	yes	direct product factor
68	C₂×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
69	C₂×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
70	C₂×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
71	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
72	C₈	C₂²×C₄×C₁₂₇	1	16	yes	direct product factor
73	C₂²	C₄×C₈×C₁₂₇	1	16	yes	direct product factor
74	C₂²	C₂×C₄²×C₁₂₇	1	6	yes	—
75	C₂²	C₂²×C₈×C₁₂₇	1	12	yes	—
76	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
77	C₄	C₂×C₄²×C₁₂₇	1	6	yes	—
78	C₄	C₂²×C₈×C₁₂₇	1	16	yes	direct product factor
79	C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
80	C₂	C₂×C₄×C₈×C₁₂₇	1	12	yes	direct product factor
81	C₂	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
82	C₂	C₂³×C₈×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₈×C₁₂₇	C₂	direct (N × H)
6. C₂²×C₈×C₁₂₇	C₄	direct (N × H)
10. C₄×C₈×C₁₂₇	C₂²	direct (N × H)
12. C₂²×C₄×C₁₂₇	C₈	direct (N × H)
17. C₂×C₈×C₁₂₇	C₂×C₄	direct (N × H)
25. C₂×C₄×C₁₂₇	C₂×C₈	direct (N × H)
30. C₈×C₁₂₇	C₂²×C₄	direct (N × H)
31. C₂²×C₁₂₇	C₄×C₈	direct (N × H)
36. C₄×C₁₂₇	C₂²×C₈	direct (N × H)
38. C₂×C₁₂₇	C₂×C₄×C₈	direct (N × H)
41. C₂²×C₄×C₈	C₁₂₇	direct (N × H)
42. C₁₂₇	C₂²×C₄×C₈	direct (N × H)
45. C₂×C₄×C₈	C₂×C₁₂₇	direct (N × H)
48. C₂²×C₈	C₄×C₁₂₇	direct (N × H)
52. C₄×C₈	C₂²×C₁₂₇	direct (N × H)
54. C₂²×C₄	C₈×C₁₂₇	direct (N × H)
59. C₂×C₈	C₂×C₄×C₁₂₇	direct (N × H)
67. C₂×C₄	C₂×C₈×C₁₂₇	direct (N × H)
72. C₈	C₂²×C₄×C₁₂₇	direct (N × H)
73. C₂²	C₄×C₈×C₁₂₇	direct (N × H)
78. C₄	C₂²×C₈×C₁₂₇	direct (N × H)
80. C₂	C₂×C₄×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×C₈×C₁₂₇	2
C₂²×C₄²×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂²×C₄×C₈	127

Subgroups — C22×C4×C8×C127 #30635821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₈×C₁₂₇ #30635821