Subgroups — C₂²×#16861206 #30652002

Order 16256.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₀₈	C₂	1	6	yes	maximal, complemented
2	C₂×#16861206	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₂₇	C₂²	1	4	yes	—
5	C₂²×D₅₀₈	—	2	48	no	—
6	C₂²×D₅₀₈	C₂²	1	72	yes	complemented
7	#16861206	C₂²	1	64	yes	direct product factor
8	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	12	yes	complemented
10	C₂³×D₁₂₇	—	2	48	no	—
11	C₂³×D₁₂₇	C₂³	1	24	yes	—
12	C₂×D₅₀₈	—	2	288	no	—
13	C₂×D₅₀₈	C₂³	1	96	yes	complemented
14	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	D₄	1	24	yes	complemented
16	C₂²×C₄×C₁₂₇	C₂³	1	18	yes	—
17	C₄²×C₁₂₇	C₂³	1	16	yes	complemented
18	C₂²×D₁₂₇	—	2	144	no	—
19	C₂²×D₁₂₇	—	4	64	no	—
20	C₂²×D₁₂₇	C₂⁴	1	16	yes	—
21	D₅₀₈	—	2	192	no	—
22	C₂³×C₁₂₇	C₂×D₄	1	12	yes	—
23	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
24	C₂×C₄×C₁₂₇	C₂×D₄	1	72	yes	complemented
25	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
26	C₂²×C₁₂₇	#127	1	16	yes	complemented
27	C₂²×C₁₂₇	C₂²×D₄	1	18	yes	—
28	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
29	C₂×D₁₂₇	—	2	48	no	—
30	C₂×D₁₂₇	—	4	96	no	—
31	C₄×C₁₂₇	C₂²×D₄	1	24	yes	complemented
32	C₂×C₁₂₇	C₂×#127	1	12	yes	complemented
33	C₂×C₁₂₇	C₂³×D₄	1	3	yes	—
34	D₁₂₇	—	4	16	no	—
35	C₂²×#127	—	127	1	no	maximal, Sylow
36	C₁₂₇	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
37	C₂³×D₄	—	127	6	no	—
38	C₂×#127	—	127	24	no	—
39	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic, complemented
40	C₂⁵	—	127	4	no	—
41	C₂²×D₄	—	127	72	no	—
42	C₂²×D₄	—	254	48	no	—
43	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
44	#127	—	127	64	no	—
45	C₂×C₄²	C₂×D₁₂₇	1	12	yes	complemented
46	C₂⁴	—	127	24	no	—
47	C₂⁴	—	254	48	no	—
48	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂×D₄	—	127	96	no	—
50	C₂×D₄	—	254	288	no	—
51	C₂²×C₄	D₅₀₈	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₁₂₇	1	18	yes	—
53	C₄²	C₂²×D₁₂₇	1	16	yes	complemented
54	C₂³	—	127	16	no	—
55	C₂³	—	508	64	no	—
56	C₂³	—	254	144	no	—
57	C₂³	C₂×D₅₀₈	1	12	yes	—
58	C₂³	C₂³×D₁₂₇	1	3	yes	—
59	D₄	—	254	192	no	—
60	C₂×C₄	C₂×D₅₀₈	1	72	yes	complemented
61	C₂×C₄	C₂³×D₁₂₇	1	12	yes	—
62	C₂²	—	508	96	no	—
63	C₂²	—	254	48	no	—
64	C₂²	#16861206	1	16	yes	direct product factor
65	C₂²	C₂²×D₅₀₈	1	18	yes	—
66	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
67	C₄	C₂²×D₅₀₈	1	24	yes	complemented
68	C₂	—	508	16	no	—
69	C₂	C₂×#16861206	1	12	yes	direct product factor
70	C₂	C₂³×D₅₀₈	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂²×#127
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅₀₈	C₂	semidirect
2. C₂×#16861206	C₂	direct (N × H)
3. C₂²×C₄²×C₁₂₇	C₂	semidirect
6. C₂²×D₅₀₈	C₂²	semidirect
7. #16861206	C₂²	direct (N × H)
9. C₂×C₄²×C₁₂₇	C₂²	semidirect
13. C₂×D₅₀₈	C₂³	semidirect
15. C₂²×C₄×C₁₂₇	D₄	semidirect
17. C₄²×C₁₂₇	C₂³	semidirect
24. C₂×C₄×C₁₂₇	C₂×D₄	semidirect
26. C₂²×C₁₂₇	#127	semidirect
31. C₄×C₁₂₇	C₂²×D₄	semidirect
32. C₂×C₁₂₇	C₂×#127	semidirect
36. C₁₂₇	C₂²×#127	semidirect
39. C₂²×C₄²	D₁₂₇	semidirect
45. C₂×C₄²	C₂×D₁₂₇	semidirect
51. C₂²×C₄	D₅₀₈	semidirect
53. C₄²	C₂²×D₁₂₇	semidirect
60. C₂×C₄	C₂×D₅₀₈	semidirect
64. C₂²	#16861206	direct (N × H)
67. C₄	C₂²×D₅₀₈	semidirect
69. C₂	C₂×#16861206	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×D₅₀₈	2
C₂×#16861206	2
C₂²×C₄²×C₁₂₇	2
C₂²×#127	127

Subgroups — C22×#16861206 #30652002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861206 #30652002