Subgroups — D₃₁×C₅⋊D₅₅ #46802201

Order 34100.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₁×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×C₅⋊D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅⋊D₁₇₀₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₁₁×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₃₁×D₅₅	—	5	6	no	maximal
6	D₁₇₀₅	—	5	6	no	—
7	C₃₁×D₅₅	—	5	6	no	—
8	C₅×C₁₁×D₃₁	D₅	1	6	yes	complemented
9	C₅⋊D₅×D₃₁	—	11	1	no	maximal, Hall
10	C₅×C₁₁×C₃₁	C₂×D₅	1	6	yes	complemented
11	C₅⋊D₁₅₅	—	11	1	no	—
12	C₃₁×C₅⋊D₅	—	11	1	no	—
13	C₅²×D₃₁	D₁₁	1	1	yes	characteristic, complemented
14	D₁₁×D₃₁	—	25	1	no	Hall
15	C₂×C₅⋊D₅₅	—	31	1	no	maximal, Hall
16	C₅²×C₃₁	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
17	D₃₄₁	—	25	1	no	—
18	C₁₁×D₃₁	C₅⋊D₅	1	1	yes	characteristic, complemented
19	D₁₁×C₃₁	—	25	1	no	—
20	D₅×D₃₁	—	55	6	no	—
21	C₂×C₅²×C₁₁	—	31	1	no	—
22	C₅⋊D₅₅	—	31	1	no	—
23	C₅⋊D₅₅	D₃₁	1	1	yes	characteristic, direct product factor
24	C₁₁×C₃₁	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
25	D₁₅₅	—	55	6	no	—
26	C₅×D₃₁	D₅₅	1	6	yes	complemented
27	D₅×C₃₁	—	55	6	no	—
28	C₅²×C₁₁	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
29	C₂×D₅₅	—	155	6	no	—
30	C₅×C₃₁	C₂×D₅₅	1	6	yes	complemented
31	C₂×D₃₁	—	275	1	no	Hall
32	C₂×C₅×C₁₁	—	31	6	no	—
33	D₅₅	—	5	6	no	—
34	D₅₅	—	155	6	no	—
35	C₂×C₅⋊D₅	—	341	1	no	Hall
36	C₂×C₃₁	—	275	1	no	—
37	D₃₁	—	275	1	no	—
38	D₃₁	C₅⋊D₅₅	1	1	yes	characteristic, direct product factor
39	C₅×C₁₁	D₅×D₃₁	1	6	yes	complemented
40	C₂×C₅²	—	31	1	no	—
41	C₅⋊D₅	—	11	1	no	—
42	C₅⋊D₅	—	341	1	no	—
43	C₂×D₁₁	—	775	1	no	Hall
44	C₃₁	C₂×C₅⋊D₅₅	1	1	yes	characteristic, complemented, Sylow
45	C₅²	D₁₁×D₃₁	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₁₁	—	31	1	no	—
47	D₁₁	—	775	1	no	—
48	D₁₁	—	25	1	no	—
49	C₂×D₅	—	1705	6	no	—
50	C₁₁	C₅⋊D₅×D₃₁	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₅	—	31	6	no	—
52	D₅	—	55	6	no	—
53	D₅	—	1705	6	no	—
54	C₅	D₃₁×D₅₅	1	6	yes	complemented
55	C₂²	—	8525	1	no	Sylow
56	C₂	—	275	1	no	—
57	C₂	—	31	1	no	—
58	C₂	—	8525	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₁₁×C₃₁	C₂²
LCS	G ▷ C₅²×C₁₁×C₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₁×D₃₁	C₂	semidirect
2. C₃₁×C₅⋊D₅₅	C₂	semidirect
3. C₅⋊D₁₇₀₅	C₂	semidirect
4. C₅²×C₁₁×C₃₁	C₂²	semidirect
8. C₅×C₁₁×D₃₁	D₅	semidirect
10. C₅×C₁₁×C₃₁	C₂×D₅	semidirect
13. C₅²×D₃₁	D₁₁	semidirect
16. C₅²×C₃₁	C₂×D₁₁	semidirect
18. C₁₁×D₃₁	C₅⋊D₅	semidirect
23. C₅⋊D₅₅	D₃₁	direct (N × H)
24. C₁₁×C₃₁	C₂×C₅⋊D₅	semidirect
26. C₅×D₃₁	D₅₅	semidirect
28. C₅²×C₁₁	C₂×D₃₁	semidirect
30. C₅×C₃₁	C₂×D₅₅	semidirect
38. D₃₁	C₅⋊D₅₅	direct (N × H)
39. C₅×C₁₁	D₅×D₃₁	semidirect
44. C₃₁	C₂×C₅⋊D₅₅	semidirect
45. C₅²	D₁₁×D₃₁	semidirect
50. C₁₁	C₅⋊D₅×D₃₁	semidirect
54. C₅	D₃₁×D₅₅	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₁₁×D₃₁	2
C₃₁×C₅⋊D₅₅	2
C₅⋊D₁₇₀₅	2
D₃₁×D₅₅	5
C₅⋊D₅×D₃₁	11
C₂×C₅⋊D₅₅	31

Subgroups — D31×C5⋊D55 #46802201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×C₅⋊D₅₅ #46802201