Finite Group Databasebeta

← D₅×2₋¹⁺⁶ #13013198

Subgroups — D₅×2₋¹⁺⁶ #13013198

Order 1280.

59186 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.

normal subgroups only

This group has 59186 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×#585	C₂	1	36	yes	maximal, complemented
3	#10619440	C₂	1	36	yes	maximal, complemented
4	#10619439	C₂	1	27	yes	maximal, complemented
5	C₂×D₅×2₋¹⁺⁴	C₂	1	27	yes	maximal, complemented
6	C₅×#585	C₂²	1	36	yes	complemented
7	C₂×C₅×2₋¹⁺⁴	C₂²	1	27	yes	complemented
8	#66287	C₂²	1	216	yes	complemented
9	D₅×2₋¹⁺⁴	C₂²	1	216	yes	complemented
10	D₅×2₋¹⁺⁴	C₂²	1	432	yes	complemented
11	#66285	C₂²	1	360	yes	complemented
12	D₅×H(5,𝔽₂)	C₂²	1	120	yes	complemented
13	#66283	C₂²	1	540	yes	complemented
14	C₂×#4150	C₂²	1	270	yes	complemented
15	C₂×D₅×#41	C₂²	1	270	yes	complemented
16	C₂×#4147	C₂²	1	135	yes	complemented
17	C₂²×Q₈×D₅	C₂²	1	45	yes	complemented
18	C₂×2₋¹⁺⁶	—	5	1	no	maximal, Sylow
19	C₅×2₋¹⁺⁴	C₂³	1	216	yes	complemented
20	C₅×H(5,𝔽₂)	C₂³	1	120	yes	complemented
21	C₂×C₅×#41	C₂³	1	270	yes	complemented
22	C₂²×C₅×Q₈	C₂³	1	45	yes	complemented
23	#4150	C₂³	1	2160	yes	complemented
24	#4149	C₂³	1	720	yes	complemented
25	D₅×#41	C₂³	1	720	yes	—
26	D₅×#41	C₂³	1	2160	yes	complemented
27	#4147	C₂³	1	1080	yes	complemented
28	C₂×#713	C₂³	1	270	yes	complemented
29	C₂×Q₈×D₅	C₂³	1	270	yes	—
30	C₂×Q₈×D₅	C₂³	1	540	yes	complemented
31	#4144	C₂³	1	1080	yes	complemented
32	C₂×#711	C₂³	1	810	yes	complemented
33	C₂×D₄×D₅	C₂³	1	270	yes	—
34	C₂×#709	C₂³	1	810	yes	complemented
35	C₂²×C₄×D₅	C₂³	1	135	yes	—
36	C₂²×Dic₁₀	C₂³	1	135	yes	complemented
37	2₋¹⁺⁶	—	5	36	no	—
38	2₋¹⁺⁶	—	5	27	no	—
39	2₋¹⁺⁶	D₅	1	1	yes	characteristic, direct product factor
40	C₂×#585	—	5	36	no	—
41	C₂²×2₋¹⁺⁴	—	5	27	no	—
42	C₂³×D₅	C₂⁴	1	45	yes	—
43	C₅×#41	C₂⁴	1	720	yes	—
44	C₂×C₅×Q₈	C₂⁴	1	270	yes	—
45	C₂×C₅×D₄	C₂⁴	1	270	yes	—
46	C₂²×C₄×C₅	C₂⁴	1	135	yes	—
47	C₂×C₂²⋊D₅	C₂⁴	1	540	yes	—
48	C₂²×Dic₅	C₂⁴	1	135	yes	—
49	#713	C₂⁴	1	720	yes	—
50	Q₈×D₅	C₂⁴	1	120	yes	—
51	Q₈×D₅	C₂⁴	1	1080	yes	—
52	#711	C₂⁴	1	2160	yes	—
53	D₄×D₅	C₂⁴	1	216	yes	—
54	D₄×D₅	C₂⁴	1	1080	yes	—
55	#709	C₂⁴	1	2160	yes	—
56	C₂×D₂₀	C₂⁴	1	270	yes	—
57	C₂×C₄×D₅	C₂⁴	1	270	yes	—
58	C₂×C₄×D₅	C₂⁴	1	810	yes	—
59	C₂×Dic₁₀	C₂⁴	1	810	yes	—
60	#585	—	5	216	no	—
61	#585	—	5	360	no	—
62	#585	—	5	540	no	—
63	#585	C₂×D₅	1	36	yes	complemented
64	C₂×2₋¹⁺⁴	—	5	216	no	—
65	C₂×2₋¹⁺⁴	—	5	432	no	—
66	C₂×2₋¹⁺⁴	—	5	270	no	—
67	C₂×2₋¹⁺⁴	—	5	135	no	—
68	C₂×2₋¹⁺⁴	C₂×D₅	1	27	yes	complemented
69	C₂×H(5,𝔽₂)	—	5	120	no	—
70	C₂²×#41	—	5	270	no	—
71	C₂³×Q₈	—	5	45	no	—
72	C₂³×C₅	C₂⁵	1	45	yes	—
73	C₂²×D₅	—	4	90	no	—
74	C₂²×D₅	C₂⁵	1	27	yes	—
75	C₂²×D₅	C₂⁵	1	135	yes	—
76	C₅×Q₈	C₂⁵	1	120	yes	—
77	C₅×D₄	C₂⁵	1	216	yes	—
78	C₂×C₄×C₅	C₂⁵	1	270	yes	—
79	C₂²⋊D₅	C₂⁵	1	432	yes	—
80	C₂×Dic₅	C₂⁵	1	270	yes	—
81	D₂₀	C₂⁵	1	216	yes	—
82	C₄×D₅	C₂⁵	1	36	yes	—
83	C₄×D₅	C₂⁵	1	540	yes	—
84	Dic₁₀	C₂⁵	1	360	yes	—
85	2₋¹⁺⁴	—	5	2160	no	—
86	2₋¹⁺⁴	—	5	1080	no	—
87	2₋¹⁺⁴	C₂²×D₅	1	216	yes	complemented
88	H(5,𝔽₂)	—	5	720	no	—
89	H(5,𝔽₂)	—	5	1080	no	—
90	H(5,𝔽₂)	C₂²×D₅	1	120	yes	complemented
91	C₂×#41	—	5	720	no	—
92	C₂×#41	—	5	2160	no	—
93	C₂×#41	—	5	270	no	—
94	C₂×#41	—	5	810	no	—
95	C₂×#41	—	5	810	no	—
96	C₂×#41	C₂²×D₅	1	270	yes	complemented
97	C₂²×Q₈	—	5	270	no	—
98	C₂²×Q₈	—	5	540	no	—
99	C₂²×Q₈	—	5	135	no	—
100	C₂²×Q₈	C₂²×D₅	1	45	yes	complemented
101	C₂²×D₄	—	5	270	no	—
102	C₂³×C₄	—	5	135	no	—
103	C₂²×C₅	—	4	45	no	—
104	C₂²×C₅	C₂⁶	1	27	yes	—
105	C₂×D₅	—	2	54	no	—
106	C₂×D₅	—	4	135	no	—
107	C₂×D₅	C₂⁶	1	27	yes	—
108	C₂×D₅	C₂⁶	1	1	yes	characteristic
109	C₄×C₅	C₂⁶	1	36	yes	—
110	Dic₅	C₂⁶	1	36	yes	—
111	C₂⁴	—	5	45	no	—
112	#41	—	5	720	no	—
113	#41	—	5	2160	no	—
114	#41	—	5	2160	no	—
115	#41	C₂³×D₅	1	720	yes	—
116	C₂×Q₈	—	5	120	no	—
117	C₂×Q₈	—	5	1080	no	—
118	C₂×Q₈	—	5	810	no	—
119	C₂×Q₈	C₂³×D₅	1	270	yes	—
120	C₂×D₄	—	5	540	no	—
121	C₂×D₄	—	5	216	no	—
122	C₂×D₄	—	5	1080	no	—
123	C₂×D₄	—	5	270	no	—
124	C₂×D₄	C₂³×D₅	1	270	yes	—
125	C₂²×C₄	—	5	135	no	—
126	C₂²×C₄	—	5	270	no	—
127	C₂²×C₄	—	5	810	no	—
128	C₂²×C₄	C₂³×D₅	1	135	yes	—
129	C₂×C₅	—	2	27	no	—
130	C₂×C₅	C₂⁷	1	1	yes	characteristic
131	D₅	—	2	27	no	—
132	D₅	2₋¹⁺⁶	1	2	yes	direct product factor
133	C₂³	—	5	27	no	—
134	C₂³	—	5	135	no	—
135	C₂³	—	20	90	no	—
136	C₂³	C₂⁴×D₅	1	45	yes	—
137	Q₈	—	5	360	no	—
138	Q₈	C₂⁴×D₅	1	120	yes	—
139	D₄	—	5	432	no	—
140	D₄	—	5	216	no	—
141	D₄	C₂⁴×D₅	1	216	yes	—
142	C₂×C₄	—	5	270	no	—
143	C₂×C₄	—	5	36	no	—
144	C₂×C₄	—	5	540	no	—
145	C₂×C₄	C₂⁴×D₅	1	270	yes	—
146	C₅	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	4	45	no	—
148	C₂²	—	5	1	no	—
149	C₂²	—	5	27	no	—
150	C₂²	—	10	54	no	—
151	C₂²	—	20	135	no	—
152	C₂²	C₂⁵×D₅	1	27	yes	—
153	C₄	—	5	36	no	—
154	C₄	C₂⁵×D₅	1	36	yes	—
155	C₂	—	2	27	no	—
156	C₂	—	5	2	no	—
157	C₂	—	10	27	no	—
158	C₂	C₂⁶×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₅
commutator, socle, cosocle	C₂×C₅	C₂⁷
Fitting	C₅×2₋¹⁺⁶	C₂
hypercenter	2₋¹⁺⁶	D₅
hypocenter	C₅	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁶

Complements

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

N	H ≅ G/N	type
1. C₅×2₋¹⁺⁶	C₂	semidirect
2. D₅×#585	C₂	semidirect
3. #10619440	C₂	semidirect
4. #10619439	C₂	semidirect
5. C₂×D₅×2₋¹⁺⁴	C₂	semidirect
6. C₅×#585	C₂²	semidirect
7. C₂×C₅×2₋¹⁺⁴	C₂²	semidirect
8. #66287	C₂²	semidirect
9. D₅×2₋¹⁺⁴	C₂²	semidirect
10. D₅×2₋¹⁺⁴	C₂²	semidirect
11. #66285	C₂²	semidirect
12. D₅×H(5,𝔽₂)	C₂²	semidirect
13. #66283	C₂²	semidirect
14. C₂×#4150	C₂²	semidirect
15. C₂×D₅×#41	C₂²	semidirect
16. C₂×#4147	C₂²	semidirect
17. C₂²×Q₈×D₅	C₂²	semidirect
19. C₅×2₋¹⁺⁴	C₂³	semidirect
20. C₅×H(5,𝔽₂)	C₂³	semidirect
21. C₂×C₅×#41	C₂³	semidirect
22. C₂²×C₅×Q₈	C₂³	semidirect
23. #4150	C₂³	semidirect
24. #4149	C₂³	semidirect
26. D₅×#41	C₂³	semidirect
27. #4147	C₂³	semidirect
28. C₂×#713	C₂³	semidirect
30. C₂×Q₈×D₅	C₂³	semidirect
31. #4144	C₂³	semidirect
32. C₂×#711	C₂³	semidirect
34. C₂×#709	C₂³	semidirect
36. C₂²×Dic₁₀	C₂³	semidirect
39. 2₋¹⁺⁶	D₅	direct (N × H)
63. #585	C₂×D₅	semidirect
68. C₂×2₋¹⁺⁴	C₂×D₅	semidirect
87. 2₋¹⁺⁴	C₂²×D₅	semidirect
90. H(5,𝔽₂)	C₂²×D₅	semidirect
96. C₂×#41	C₂²×D₅	semidirect
100. C₂²×Q₈	C₂²×D₅	semidirect
132. D₅	2₋¹⁺⁶	direct (N × H)
146. C₅	C₂×2₋¹⁺⁶	semidirect

Maximal subgroups

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

subgroup	index
C₅×2₋¹⁺⁶	2
D₅×#585	2
#10619440	2
#10619439	2
C₂×D₅×2₋¹⁺⁴	2
C₂×2₋¹⁺⁶	5