Subgroups — C₂²×D₁₆×C₁₂₇ #30636360

Order 16256.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₆×C₁₂₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₁₂₇	—	2	2	no	—
5	C₂×D₈×C₁₂₇	—	2	6	no	—
6	C₂×D₈×C₁₂₇	C₂²	1	12	yes	complemented
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	D₁₆×C₁₂₇	C₂²	1	16	yes	direct product factor
9	C₂×C₁₆×C₁₂₇	C₂²	1	6	yes	complemented
10	C₂⁴×C₁₂₇	—	4	2	no	—
11	C₂×D₄×C₁₂₇	—	2	12	no	—
12	C₂×D₄×C₁₂₇	—	4	6	no	—
13	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
14	D₈×C₁₂₇	—	2	12	no	—
15	D₈×C₁₂₇	C₂³	1	8	yes	complemented
16	C₂×C₈×C₁₂₇	D₄	1	3	yes	—
17	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
18	C₁₆×C₁₂₇	C₂³	1	4	yes	complemented
19	C₂³×C₁₂₇	—	4	12	no	—
20	C₂³×C₁₂₇	—	8	8	no	—
21	C₂³×C₁₂₇	D₈	1	1	yes	characteristic
22	D₄×C₁₂₇	—	2	8	no	—
23	D₄×C₁₂₇	—	4	12	no	—
24	C₂×C₄×C₁₂₇	D₈	1	3	yes	—
25	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
26	C₈×C₁₂₇	C₂×D₄	1	3	yes	—
27	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
28	C₂²×C₁₂₇	—	4	8	no	—
29	C₂²×C₁₂₇	—	8	24	no	—
30	C₂²×C₁₂₇	D₁₆	1	4	yes	direct product factor
31	C₂²×C₁₂₇	C₂×D₈	1	3	yes	—
32	C₄×C₁₂₇	C₂×D₈	1	3	yes	—
33	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₂₇	—	8	8	no	—
35	C₂×C₁₂₇	C₂×D₁₆	1	6	yes	direct product factor
36	C₂×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
37	C₂²×D₁₆	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₁₂₇	C₂²×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×D₈	C₂×C₁₂₇	1	2	yes	complemented
40	C₂×D₁₆	C₂×C₁₂₇	1	12	yes	direct product factor
41	C₂²×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	2	2	no	—
43	C₂×D₈	—	2	6	no	—
44	C₂×D₈	C₂²×C₁₂₇	1	12	yes	complemented
45	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
46	D₁₆	C₂²×C₁₂₇	1	16	yes	direct product factor
47	C₂×C₁₆	C₂²×C₁₂₇	1	6	yes	complemented
48	C₂⁴	—	4	2	no	—
49	C₂×D₄	—	2	12	no	—
50	C₂×D₄	—	4	6	no	—
51	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
52	D₈	—	2	12	no	—
53	D₈	C₂³×C₁₂₇	1	8	yes	complemented
54	C₂×C₈	D₄×C₁₂₇	1	3	yes	—
55	C₂×C₈	C₂³×C₁₂₇	1	3	yes	—
56	C₁₆	C₂³×C₁₂₇	1	4	yes	complemented
57	C₂³	—	4	12	no	—
58	C₂³	—	8	8	no	—
59	C₂³	D₈×C₁₂₇	1	1	yes	characteristic
60	D₄	—	2	8	no	—
61	D₄	—	4	12	no	—
62	C₂×C₄	D₈×C₁₂₇	1	3	yes	—
63	C₂×C₄	C₂×D₄×C₁₂₇	1	3	yes	—
64	C₈	C₂×D₄×C₁₂₇	1	3	yes	—
65	C₈	C₂⁴×C₁₂₇	1	1	yes	characteristic
66	C₂²	—	4	8	no	—
67	C₂²	—	8	24	no	—
68	C₂²	D₁₆×C₁₂₇	1	4	yes	direct product factor
69	C₂²	C₂×D₈×C₁₂₇	1	3	yes	—
70	C₄	C₂×D₈×C₁₂₇	1	3	yes	—
71	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
72	C₂	—	8	8	no	—
73	C₂	C₂×D₁₆×C₁₂₇	1	6	yes	direct product factor
74	C₂	C₂²×D₈×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₂₇	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₂₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈×C₁₂₇	C₂	semidirect
2. C₂×D₁₆×C₁₂₇	C₂	direct (N × H)
3. C₂²×C₁₆×C₁₂₇	C₂	semidirect
6. C₂×D₈×C₁₂₇	C₂²	semidirect
8. D₁₆×C₁₂₇	C₂²	direct (N × H)
9. C₂×C₁₆×C₁₂₇	C₂²	semidirect
15. D₈×C₁₂₇	C₂³	semidirect
18. C₁₆×C₁₂₇	C₂³	semidirect
30. C₂²×C₁₂₇	D₁₆	direct (N × H)
35. C₂×C₁₂₇	C₂×D₁₆	direct (N × H)
37. C₂²×D₁₆	C₁₂₇	direct (N × H)
38. C₁₂₇	C₂²×D₁₆	direct (N × H)
39. C₂²×D₈	C₂×C₁₂₇	semidirect
40. C₂×D₁₆	C₂×C₁₂₇	direct (N × H)
41. C₂²×C₁₆	C₂×C₁₂₇	semidirect
44. C₂×D₈	C₂²×C₁₂₇	semidirect
46. D₁₆	C₂²×C₁₂₇	direct (N × H)
47. C₂×C₁₆	C₂²×C₁₂₇	semidirect
53. D₈	C₂³×C₁₂₇	semidirect
56. C₁₆	C₂³×C₁₂₇	semidirect
68. C₂²	D₁₆×C₁₂₇	direct (N × H)
73. C₂	C₂×D₁₆×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
C₂²×D₈×C₁₂₇	2
C₂×D₁₆×C₁₂₇	2
C₂²×C₁₆×C₁₂₇	2
C₂²×D₁₆	127

Subgroups — C22×D16×C127 #30636360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₆×C₁₂₇ #30636360