Subgroups — C₂²×#16862243 #30679233

Order 16768.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₂₄	C₂	1	6	yes	maximal, complemented
2	C₂×#16862243	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₃₁	C₂²	1	4	yes	—
5	C₂²×D₅₂₄	—	2	48	no	—
6	C₂²×D₅₂₄	C₂²	1	72	yes	complemented
7	#16862243	C₂²	1	64	yes	direct product factor
8	C₂³×C₄×C₁₃₁	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₃₁	C₂²	1	12	yes	complemented
10	C₂³×D₁₃₁	—	2	48	no	—
11	C₂³×D₁₃₁	C₂³	1	24	yes	—
12	C₂×D₅₂₄	—	2	288	no	—
13	C₂×D₅₂₄	C₂³	1	96	yes	complemented
14	C₂⁴×C₁₃₁	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₁₃₁	D₄	1	24	yes	complemented
16	C₂²×C₄×C₁₃₁	C₂³	1	18	yes	—
17	C₄²×C₁₃₁	C₂³	1	16	yes	complemented
18	C₂²×D₁₃₁	—	2	144	no	—
19	C₂²×D₁₃₁	—	4	64	no	—
20	C₂²×D₁₃₁	C₂⁴	1	16	yes	—
21	D₅₂₄	—	2	192	no	—
22	C₂³×C₁₃₁	C₂×D₄	1	12	yes	—
23	C₂³×C₁₃₁	C₂⁴	1	3	yes	—
24	C₂×C₄×C₁₃₁	C₂×D₄	1	72	yes	complemented
25	C₂×C₄×C₁₃₁	C₂⁴	1	12	yes	—
26	C₂²×C₁₃₁	#127	1	16	yes	complemented
27	C₂²×C₁₃₁	C₂²×D₄	1	18	yes	—
28	C₂²×C₁₃₁	C₂⁵	1	1	yes	characteristic
29	C₂×D₁₃₁	—	2	48	no	—
30	C₂×D₁₃₁	—	4	96	no	—
31	C₄×C₁₃₁	C₂²×D₄	1	24	yes	complemented
32	C₂×C₁₃₁	C₂×#127	1	12	yes	complemented
33	C₂×C₁₃₁	C₂³×D₄	1	3	yes	—
34	D₁₃₁	—	4	16	no	—
35	C₁₃₁	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
36	C₂²×#127	—	131	1	no	maximal, Sylow
37	C₂³×D₄	—	131	6	no	—
38	C₂×#127	—	131	24	no	—
39	C₂²×C₄²	D₁₃₁	1	1	yes	characteristic, complemented
40	C₂⁵	—	131	4	no	—
41	C₂²×D₄	—	262	48	no	—
42	C₂²×D₄	—	131	72	no	—
43	C₂³×C₄	C₂×D₁₃₁	1	3	yes	—
44	#127	—	131	64	no	—
45	C₂×C₄²	C₂×D₁₃₁	1	12	yes	complemented
46	C₂⁴	—	262	48	no	—
47	C₂⁴	—	131	24	no	—
48	C₂⁴	C₂²×D₁₃₁	1	1	yes	characteristic
49	C₂×D₄	—	262	288	no	—
50	C₂×D₄	—	131	96	no	—
51	C₂²×C₄	D₅₂₄	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₁₃₁	1	18	yes	—
53	C₄²	C₂²×D₁₃₁	1	16	yes	complemented
54	C₂³	—	262	144	no	—
55	C₂³	—	524	64	no	—
56	C₂³	—	131	16	no	—
57	C₂³	C₂×D₅₂₄	1	12	yes	—
58	C₂³	C₂³×D₁₃₁	1	3	yes	—
59	D₄	—	262	192	no	—
60	C₂×C₄	C₂×D₅₂₄	1	72	yes	complemented
61	C₂×C₄	C₂³×D₁₃₁	1	12	yes	—
62	C₂²	—	262	48	no	—
63	C₂²	—	524	96	no	—
64	C₂²	#16862243	1	16	yes	direct product factor
65	C₂²	C₂²×D₅₂₄	1	18	yes	—
66	C₂²	C₂⁴×D₁₃₁	1	1	yes	characteristic
67	C₄	C₂²×D₅₂₄	1	24	yes	complemented
68	C₂	—	524	16	no	—
69	C₂	C₂×#16862243	1	12	yes	direct product factor
70	C₂	C₂³×D₅₂₄	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₃₁
commutator, cosocle	C₂²×C₁₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₁
Fitting	C₂²×C₄²×C₁₃₁	C₂
socle	C₂⁴×C₁₃₁	C₂³
hypercenter	C₂²×C₄²	D₁₃₁
hypocenter	C₁₃₁	C₂²×#127
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅₂₄	C₂	semidirect
2. C₂×#16862243	C₂	direct (N × H)
3. C₂²×C₄²×C₁₃₁	C₂	semidirect
6. C₂²×D₅₂₄	C₂²	semidirect
7. #16862243	C₂²	direct (N × H)
9. C₂×C₄²×C₁₃₁	C₂²	semidirect
13. C₂×D₅₂₄	C₂³	semidirect
15. C₂²×C₄×C₁₃₁	D₄	semidirect
17. C₄²×C₁₃₁	C₂³	semidirect
24. C₂×C₄×C₁₃₁	C₂×D₄	semidirect
26. C₂²×C₁₃₁	#127	semidirect
31. C₄×C₁₃₁	C₂²×D₄	semidirect
32. C₂×C₁₃₁	C₂×#127	semidirect
35. C₁₃₁	C₂²×#127	semidirect
39. C₂²×C₄²	D₁₃₁	semidirect
45. C₂×C₄²	C₂×D₁₃₁	semidirect
51. C₂²×C₄	D₅₂₄	semidirect
53. C₄²	C₂²×D₁₃₁	semidirect
60. C₂×C₄	C₂×D₅₂₄	semidirect
64. C₂²	#16862243	direct (N × H)
67. C₄	C₂²×D₅₂₄	semidirect
69. C₂	C₂×#16862243	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×D₅₂₄	2
C₂×#16862243	2
C₂²×C₄²×C₁₃₁	2
C₂²×#127	131

Subgroups — C22×#16862243 #30679233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16862243 #30679233