Subgroups — D₁₅₅⋊D₅₅ #46802196

Order 34100.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁×D₁₅₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₁₇₀₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₅⋊D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₁₁×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅₅⋊D₃₁	—	5	1	no	maximal
6	D₃₁×D₅₅	—	5	1	no	maximal
7	D₁₇₀₅	D₅	1	1	yes	characteristic, complemented
8	C₃₁×D₅₅	—	5	1	no	—
9	C₃₁×D₅₅	—	5	1	no	—
10	C₃₁×D₅₅	—	10	2	no	—
11	C₁₁×D₁₅₅	D₅	1	1	yes	characteristic, complemented
12	C₅×D₃₄₁	—	5	1	no	—
13	C₅×C₁₁×D₃₁	—	5	1	no	—
14	(C₅⋊D₅)⋊D₃₁	—	11	1	no	maximal, Hall
15	C₅×C₁₁×C₃₁	—	2	2	no	—
16	C₅×C₁₁×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented
17	C₅×C₁₁×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented
18	C₃₁×C₅⋊D₅	—	11	1	no	—
19	C₅×D₁₅₅	—	11	1	no	—
20	C₅×D₁₅₅	D₁₁	1	1	yes	characteristic, complemented
21	D₁₁×D₃₁	—	25	1	no	Hall
22	D₅×D₅₅	—	31	1	no	maximal, Hall
23	C₅²×C₃₁	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
24	D₃₄₁	—	5	1	no	—
25	C₁₁×D₃₁	—	5	1	no	—
26	D₁₁×C₃₁	—	25	1	no	—
27	D₅×D₃₁	—	55	1	no	—
28	D₅×D₃₁	—	55	1	no	—
29	C₅⋊D₅₅	D₃₁	1	1	yes	characteristic, complemented
30	C₅×D₅₅	—	31	1	no	—
31	C₅×D₅×C₁₁	—	31	1	no	—
32	C₁₁×C₃₁	D₅²	1	1	yes	characteristic, complemented, Hall
33	D₁₅₅	—	11	1	no	—
34	D₁₅₅	D₅₅	1	1	yes	characteristic, complemented
35	C₅×D₃₁	—	5	1	no	—
36	C₅×D₃₁	—	55	1	no	—
37	D₅×C₃₁	—	55	1	no	—
38	D₅×C₃₁	—	55	1	no	—
39	D₅×C₃₁	—	110	2	no	—
40	C₅²×C₁₁	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
41	C₂×D₅₅	—	155	1	no	—
42	D₅×D₁₁	—	155	1	no	—
43	C₅×C₃₁	—	2	2	no	—
44	C₅×C₃₁	D₅×D₁₁	1	1	yes	characteristic, complemented
45	C₅×C₃₁	C₂×D₅₅	1	1	yes	characteristic, complemented
46	C₂×D₃₁	—	275	1	no	Hall
47	C₂×C₅×C₁₁	—	155	1	no	—
48	D₅₅	—	5	1	no	—
49	D₅₅	—	5	1	no	—
50	D₅₅	—	10	2	no	—
51	D₅₅	—	31	1	no	—
52	C₅×D₁₁	—	155	1	no	—
53	D₅×C₁₁	—	31	1	no	—
54	D₅²	—	341	1	no	Hall
55	C₂×C₃₁	—	275	1	no	—
56	D₃₁	—	5	1	no	—
57	D₃₁	—	55	1	no	—
58	C₅×C₁₁	—	2	2	no	—
59	C₅×C₁₁	D₅×D₃₁	1	1	yes	characteristic, complemented
60	C₅×C₁₁	D₅×D₃₁	1	1	yes	characteristic, complemented
61	C₅⋊D₅	—	11	1	no	—
62	C₅×D₅	—	31	1	no	—
63	C₅×D₅	—	341	1	no	—
64	C₂×D₁₁	—	775	1	no	Hall
65	C₃₁	D₅×D₅₅	1	1	yes	characteristic, complemented, Sylow
66	C₅²	D₁₁×D₃₁	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₁₁	—	155	1	no	—
68	D₁₁	—	25	1	no	—
69	D₁₁	—	155	1	no	—
70	C₂×D₅	—	1705	1	no	—
71	C₂×D₅	—	1705	1	no	—
72	C₁₁	(C₅⋊D₅)⋊D₃₁	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₅	—	155	1	no	—
74	C₂×C₅	—	1705	1	no	—
75	D₅	—	31	1	no	—
76	D₅	—	55	1	no	—
77	D₅	—	55	1	no	—
78	D₅	—	341	1	no	—
79	D₅	—	110	2	no	—
80	C₅	—	2	2	no	—
81	C₅	D₃₁×D₅₅	1	1	yes	characteristic, complemented
82	C₅	D₅₅⋊D₃₁	1	1	yes	characteristic, complemented
83	C₂²	—	8525	1	no	Sylow
84	C₂	—	275	1	no	—
85	C₂	—	155	1	no	—
86	C₂	—	1705	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. C₅×C₁₁×D₁₅₅	C₂	semidirect
2. C₅×D₁₇₀₅	C₂	semidirect
3. C₃₁×C₅⋊D₅₅	C₂	semidirect
4. C₅²×C₁₁×C₃₁	C₂²	semidirect
7. D₁₇₀₅	D₅	semidirect
11. C₁₁×D₁₅₅	D₅	semidirect
16. C₅×C₁₁×C₃₁	C₂×D₅	semidirect
17. C₅×C₁₁×C₃₁	C₂×D₅	semidirect
20. C₅×D₁₅₅	D₁₁	semidirect
23. C₅²×C₃₁	C₂×D₁₁	semidirect
29. C₅⋊D₅₅	D₃₁	semidirect
32. C₁₁×C₃₁	D₅²	semidirect
34. D₁₅₅	D₅₅	semidirect
40. C₅²×C₁₁	C₂×D₃₁	semidirect
44. C₅×C₃₁	D₅×D₁₁	semidirect
45. C₅×C₃₁	C₂×D₅₅	semidirect
59. C₅×C₁₁	D₅×D₃₁	semidirect
60. C₅×C₁₁	D₅×D₃₁	semidirect
65. C₃₁	D₅×D₅₅	semidirect
66. C₅²	D₁₁×D₃₁	semidirect
72. C₁₁	(C₅⋊D₅)⋊D₃₁	semidirect
81. C₅	D₃₁×D₅₅	semidirect
82. C₅	D₅₅⋊D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₁₁×D₁₅₅	2
C₅×D₁₇₀₅	2
C₃₁×C₅⋊D₅₅	2
D₅₅⋊D₃₁	5
D₃₁×D₅₅	5
(C₅⋊D₅)⋊D₃₁	11
D₅×D₅₅	31

Subgroups — D155⋊D55 #46802196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₅₅⋊D₅₅ #46802196