Subgroups — 2₋¹⁺⁶×C₁₃₁ #30663778

Order 16768.

5796 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×C₁₃₁	C₂	1	36	yes	maximal, complemented
2	C₂×2₋¹⁺⁴×C₁₃₁	C₂	1	27	yes	maximal, complemented
3	2₋¹⁺⁴×C₁₃₁	C₂²	1	216	yes	complemented
4	H(5,𝔽₂)×C₁₃₁	C₂²	1	120	yes	complemented
5	C₂×#41×C₁₃₁	C₂²	1	270	yes	complemented
6	C₂²×Q₈×C₁₃₁	C₂²	1	45	yes	complemented
7	#41×C₁₃₁	C₂³	1	720	yes	—
8	C₂×Q₈×C₁₃₁	C₂³	1	270	yes	—
9	C₂×D₄×C₁₃₁	C₂³	1	270	yes	—
10	C₂²×C₄×C₁₃₁	C₂³	1	135	yes	—
11	C₂³×C₁₃₁	C₂⁴	1	45	yes	—
12	Q₈×C₁₃₁	C₂⁴	1	120	yes	—
13	D₄×C₁₃₁	C₂⁴	1	216	yes	—
14	C₂×C₄×C₁₃₁	C₂⁴	1	270	yes	—
15	C₂²×C₁₃₁	—	4	45	no	—
16	C₂²×C₁₃₁	C₂⁵	1	27	yes	—
17	C₄×C₁₃₁	C₂⁵	1	36	yes	—
18	C₂×C₁₃₁	—	2	27	no	—
19	C₂×C₁₃₁	C₂⁶	1	1	yes	characteristic
20	C₁₃₁	2₋¹⁺⁶	1	1	yes	characteristic, direct product factor, Sylow
21	2₋¹⁺⁶	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	#585	C₂×C₁₃₁	1	36	yes	complemented
23	C₂×2₋¹⁺⁴	C₂×C₁₃₁	1	27	yes	complemented
24	2₋¹⁺⁴	C₂²×C₁₃₁	1	216	yes	complemented
25	H(5,𝔽₂)	C₂²×C₁₃₁	1	120	yes	complemented
26	C₂×#41	C₂²×C₁₃₁	1	270	yes	complemented
27	C₂²×Q₈	C₂²×C₁₃₁	1	45	yes	complemented
28	#41	C₂³×C₁₃₁	1	720	yes	—
29	C₂×Q₈	C₂³×C₁₃₁	1	270	yes	—
30	C₂×D₄	C₂³×C₁₃₁	1	270	yes	—
31	C₂²×C₄	C₂³×C₁₃₁	1	135	yes	—
32	C₂³	C₂⁴×C₁₃₁	1	45	yes	—
33	Q₈	C₂⁴×C₁₃₁	1	120	yes	—
34	D₄	C₂⁴×C₁₃₁	1	216	yes	—
35	C₂×C₄	C₂⁴×C₁₃₁	1	270	yes	—
36	C₂²	—	4	45	no	—
37	C₂²	C₂⁵×C₁₃₁	1	27	yes	—
38	C₄	C₂⁵×C₁₃₁	1	36	yes	—
39	C₂	—	2	27	no	—
40	C₂	C₂⁶×C₁₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #585×C₁₃₁	C₂	semidirect
2. C₂×2₋¹⁺⁴×C₁₃₁	C₂	semidirect
3. 2₋¹⁺⁴×C₁₃₁	C₂²	semidirect
4. H(5,𝔽₂)×C₁₃₁	C₂²	semidirect
5. C₂×#41×C₁₃₁	C₂²	semidirect
6. C₂²×Q₈×C₁₃₁	C₂²	semidirect
20. C₁₃₁	2₋¹⁺⁶	direct (N × H)
21. 2₋¹⁺⁶	C₁₃₁	direct (N × H)
22. #585	C₂×C₁₃₁	semidirect
23. C₂×2₋¹⁺⁴	C₂×C₁₃₁	semidirect
24. 2₋¹⁺⁴	C₂²×C₁₃₁	semidirect
25. H(5,𝔽₂)	C₂²×C₁₃₁	semidirect
26. C₂×#41	C₂²×C₁₃₁	semidirect
27. C₂²×Q₈	C₂²×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#585×C₁₃₁	2
C₂×2₋¹⁺⁴×C₁₃₁	2
2₋¹⁺⁶	131

Subgroups — 2−1+6×C131 #30663778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁶×C₁₃₁ #30663778