Subgroups — C₂²×2₋¹⁺⁴×C₁₃₁ #30663775

Order 16768.

5872 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₁₃₁	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#41×C₁₃₁	C₂	1	10	yes	maximal, complemented
3	C₂³×Q₈×C₁₃₁	C₂	1	5	yes	maximal, complemented
4	2₋¹⁺⁴×C₁₃₁	C₂²	1	256	yes	direct product factor
5	C₂×#41×C₁₃₁	C₂²	1	240	yes	complemented
6	C₂²×Q₈×C₁₃₁	C₂²	1	10	yes	—
7	C₂²×Q₈×C₁₃₁	C₂²	1	120	yes	complemented
8	C₂²×D₄×C₁₃₁	C₂²	1	10	yes	—
9	C₂³×C₄×C₁₃₁	C₂²	1	15	yes	—
10	C₂⁴×C₁₃₁	C₂³	1	5	yes	—
11	#41×C₁₃₁	C₂³	1	640	yes	complemented
12	C₂×Q₈×C₁₃₁	C₂³	1	120	yes	—
13	C₂×Q₈×C₁₃₁	C₂³	1	320	yes	complemented
14	C₂×D₄×C₁₃₁	C₂³	1	120	yes	—
15	C₂²×C₄×C₁₃₁	C₂³	1	180	yes	—
16	C₂²×C₄×C₁₃₁	C₂³	1	10	yes	—
17	C₂³×C₁₃₁	—	2	20	no	—
18	C₂³×C₁₃₁	C₂⁴	1	30	yes	—
19	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
20	Q₈×C₁₃₁	C₂⁴	1	160	yes	—
21	D₄×C₁₃₁	C₂⁴	1	160	yes	—
22	C₂×C₄×C₁₃₁	C₂⁴	1	60	yes	—
23	C₂×C₄×C₁₃₁	C₂⁴	1	240	yes	—
24	C₂²×C₁₃₁	—	2	60	no	—
25	C₂²×C₁₃₁	2₋¹⁺⁴	1	4	yes	direct product factor
26	C₂²×C₁₃₁	C₂⁵	1	20	yes	—
27	C₂²×C₁₃₁	C₂⁵	1	3	yes	—
28	C₄×C₁₃₁	C₂⁵	1	40	yes	—
29	C₂×C₁₃₁	—	2	20	no	—
30	C₂×C₁₃₁	C₂×2₋¹⁺⁴	1	6	yes	direct product factor
31	C₂×C₁₃₁	C₂⁶	1	1	yes	characteristic
32	C₁₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²×2₋¹⁺⁴	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×2₋¹⁺⁴	C₂×C₁₃₁	1	48	yes	direct product factor
35	C₂²×#41	C₂×C₁₃₁	1	10	yes	complemented
36	C₂³×Q₈	C₂×C₁₃₁	1	5	yes	complemented
37	2₋¹⁺⁴	C₂²×C₁₃₁	1	256	yes	direct product factor
38	C₂×#41	C₂²×C₁₃₁	1	240	yes	complemented
39	C₂²×Q₈	C₂²×C₁₃₁	1	10	yes	—
40	C₂²×Q₈	C₂²×C₁₃₁	1	120	yes	complemented
41	C₂²×D₄	C₂²×C₁₃₁	1	10	yes	—
42	C₂³×C₄	C₂²×C₁₃₁	1	15	yes	—
43	C₂⁴	C₂³×C₁₃₁	1	5	yes	—
44	#41	C₂³×C₁₃₁	1	640	yes	complemented
45	C₂×Q₈	C₂³×C₁₃₁	1	120	yes	—
46	C₂×Q₈	C₂³×C₁₃₁	1	320	yes	complemented
47	C₂×D₄	C₂³×C₁₃₁	1	120	yes	—
48	C₂²×C₄	C₂³×C₁₃₁	1	10	yes	—
49	C₂²×C₄	C₂³×C₁₃₁	1	180	yes	—
50	C₂³	—	2	20	no	—
51	C₂³	C₂⁴×C₁₃₁	1	30	yes	—
52	C₂³	C₂⁴×C₁₃₁	1	1	yes	characteristic
53	Q₈	C₂⁴×C₁₃₁	1	160	yes	—
54	D₄	C₂⁴×C₁₃₁	1	160	yes	—
55	C₂×C₄	C₂⁴×C₁₃₁	1	60	yes	—
56	C₂×C₄	C₂⁴×C₁₃₁	1	240	yes	—
57	C₂²	—	2	60	no	—
58	C₂²	2₋¹⁺⁴×C₁₃₁	1	4	yes	direct product factor
59	C₂²	C₂⁵×C₁₃₁	1	20	yes	—
60	C₂²	C₂⁵×C₁₃₁	1	3	yes	—
61	C₄	C₂⁵×C₁₃₁	1	40	yes	—
62	C₂	—	2	20	no	—
63	C₂	C₂×2₋¹⁺⁴×C₁₃₁	1	6	yes	direct product factor
64	C₂	C₂⁶×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₁₃₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₁₃₁	C₂	direct (N × H)
2. C₂²×#41×C₁₃₁	C₂	semidirect
3. C₂³×Q₈×C₁₃₁	C₂	semidirect
4. 2₋¹⁺⁴×C₁₃₁	C₂²	direct (N × H)
5. C₂×#41×C₁₃₁	C₂²	semidirect
7. C₂²×Q₈×C₁₃₁	C₂²	semidirect
11. #41×C₁₃₁	C₂³	semidirect
13. C₂×Q₈×C₁₃₁	C₂³	semidirect
25. C₂²×C₁₃₁	2₋¹⁺⁴	direct (N × H)
30. C₂×C₁₃₁	C₂×2₋¹⁺⁴	direct (N × H)
32. C₁₃₁	C₂²×2₋¹⁺⁴	direct (N × H)
33. C₂²×2₋¹⁺⁴	C₁₃₁	direct (N × H)
34. C₂×2₋¹⁺⁴	C₂×C₁₃₁	direct (N × H)
35. C₂²×#41	C₂×C₁₃₁	semidirect
36. C₂³×Q₈	C₂×C₁₃₁	semidirect
37. 2₋¹⁺⁴	C₂²×C₁₃₁	direct (N × H)
38. C₂×#41	C₂²×C₁₃₁	semidirect
40. C₂²×Q₈	C₂²×C₁₃₁	semidirect
44. #41	C₂³×C₁₃₁	semidirect
46. C₂×Q₈	C₂³×C₁₃₁	semidirect
58. C₂²	2₋¹⁺⁴×C₁₃₁	direct (N × H)
63. C₂	C₂×2₋¹⁺⁴×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₁₃₁	2
C₂²×#41×C₁₃₁	2
C₂³×Q₈×C₁₃₁	2
C₂²×2₋¹⁺⁴	131

Subgroups — C22×2−1+4×C131 #30663775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×2₋¹⁺⁴×C₁₃₁ #30663775