Finite Group Databasebeta

Subgroups — #13013201

Order 1280.

59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.

normal subgroups only

This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 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×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×#585	C₂	1	28	yes	maximal, complemented
3	#10619440	C₂	1	28	yes	maximal, complemented
4	#10619436	C₂	1	35	yes	maximal, complemented
5	C₂×#66285	C₂	1	35	yes	maximal, complemented
6	C₅×#585	C₂²	1	28	yes	complemented
7	C₂×C₅×H(5,𝔽₂)	C₂²	1	35	yes	complemented
8	#66287	C₂²	1	56	yes	complemented
9	D₅×2₋¹⁺⁴	C₂²	1	168	yes	complemented
10	#66285	C₂²	1	280	yes	complemented
11	#66285	C₂²	1	560	yes	complemented
12	D₅×H(5,𝔽₂)	C₂²	1	280	yes	complemented
13	#66283	C₂²	1	420	yes	complemented
14	C₂×#4150	C₂²	1	210	yes	complemented
15	C₂×D₅×#41	C₂²	1	210	yes	complemented
16	C₂×#4144	C₂²	1	315	yes	complemented
17	C₂²×#711	C₂²	1	105	yes	complemented
18	#63103	—	5	1	no	maximal, Sylow
19	C₅×2₋¹⁺⁴	C₂³	1	56	yes	complemented
20	C₅×H(5,𝔽₂)	C₂³	1	280	yes	complemented
21	C₂×C₅×#41	C₂³	1	210	yes	complemented
22	C₂²×C₅×D₄	C₂³	1	105	yes	complemented
23	C₂²×C₂²⋊D₅	C₂³	1	210	yes	complemented
24	C₂³×Dic₅	C₂³	1	30	yes	complemented
25	#4150	C₂³	1	1680	yes	complemented
26	#4149	C₂³	1	560	yes	complemented
27	D₅×#41	C₂³	1	560	yes	complemented
28	D₅×#41	C₂³	1	1680	yes	complemented
29	#4147	C₂³	1	280	yes	complemented
30	C₂×#713	C₂³	1	70	yes	complemented
31	C₂×Q₈×D₅	C₂³	1	210	yes	complemented
32	#4144	C₂³	1	2520	yes	complemented
33	C₂×#711	C₂³	1	630	yes	complemented
34	C₂×#711	C₂³	1	1260	yes	complemented
35	C₂×D₄×D₅	C₂³	1	630	yes	complemented
36	C₂×#709	C₂³	1	630	yes	complemented
37	C₂²×C₄×D₅	C₂³	1	105	yes	complemented
38	C₂²×Dic₁₀	C₂³	1	105	yes	complemented
39	2₋¹⁺⁶	—	5	28	no	—
40	H(7,𝔽₂)	—	5	35	no	—
41	H(7,𝔽₂)	D₅	1	1	yes	characteristic, complemented
42	C₂×#585	—	5	28	no	—
43	C₂×#585	—	5	35	no	—
44	C₂⁴×C₅	C₂⁴	1	30	yes	—
45	C₂³×D₅	C₂⁴	1	105	yes	—
46	C₅×#41	C₂⁴	1	560	yes	—
47	C₂×C₅×Q₈	C₂⁴	1	70	yes	—
48	C₂×C₅×D₄	C₂⁴	1	630	yes	—
49	C₂²×C₄×C₅	C₂⁴	1	105	yes	—
50	C₂×C₂²⋊D₅	C₂⁴	1	1260	yes	—
51	C₂²×Dic₅	C₂⁴	1	210	yes	—
52	C₂²×Dic₅	C₂⁴	1	105	yes	—
53	#713	C₂⁴	1	56	yes	—
54	#713	C₂⁴	1	280	yes	—
55	Q₈×D₅	C₂⁴	1	560	yes	—
56	#711	C₂⁴	1	280	yes	—
57	#711	C₂⁴	1	2520	yes	—
58	D₄×D₅	C₂⁴	1	1680	yes	—
59	#709	C₂⁴	1	1680	yes	—
60	C₂×D₂₀	C₂⁴	1	210	yes	—
61	C₂×C₄×D₅	C₂⁴	1	210	yes	—
62	C₂×C₄×D₅	C₂⁴	1	630	yes	—
63	C₂×Dic₁₀	C₂⁴	1	630	yes	—
64	#585	—	5	56	no	—
65	#585	—	5	280	no	—
66	#585	—	5	560	no	—
67	#585	—	5	420	no	—
68	#585	C₂×D₅	1	28	yes	complemented
69	C₂×2₋¹⁺⁴	—	5	168	no	—
70	C₂×2₋¹⁺⁴	—	5	210	no	—
71	C₂×H(5,𝔽₂)	—	5	280	no	—
72	C₂×H(5,𝔽₂)	—	5	315	no	—
73	C₂×H(5,𝔽₂)	C₂×D₅	1	35	yes	complemented
74	C₂²×#41	—	5	210	no	—
75	C₂²×#41	—	5	105	no	—
76	C₂³×C₅	—	8	30	no	—
77	C₂³×C₅	C₂⁵	1	105	yes	—
78	C₂²×D₅	—	8	105	no	—
79	C₂²×D₅	C₂⁵	1	210	yes	—
80	C₅×Q₈	C₂⁵	1	56	yes	—
81	C₅×D₄	C₂⁵	1	280	yes	—
82	C₂×C₄×C₅	C₂⁵	1	210	yes	—
83	C₂²⋊D₅	C₂⁵	1	560	yes	—
84	C₂×Dic₅	C₂⁵	1	315	yes	—
85	C₂×Dic₅	C₂⁵	1	35	yes	—
86	D₂₀	C₂⁵	1	168	yes	—
87	C₄×D₅	C₂⁵	1	28	yes	—
88	C₄×D₅	C₂⁵	1	420	yes	—
89	Dic₁₀	C₂⁵	1	280	yes	—
90	2₋¹⁺⁴	—	5	1680	no	—
91	2₋¹⁺⁴	—	5	280	no	—
92	2₋¹⁺⁴	C₂²×D₅	1	56	yes	complemented
93	H(5,𝔽₂)	—	5	560	no	—
94	H(5,𝔽₂)	—	5	2520	no	—
95	H(5,𝔽₂)	C₂²×D₅	1	280	yes	complemented
96	C₂×#41	—	5	560	no	—
97	C₂×#41	—	5	1680	no	—
98	C₂×#41	—	5	70	no	—
99	C₂×#41	—	5	630	no	—
100	C₂×#41	—	5	1260	no	—
101	C₂×#41	—	5	630	no	—
102	C₂×#41	C₂²×D₅	1	210	yes	complemented
103	C₂²×Q₈	—	5	210	no	—
104	C₂²×Q₈	—	5	105	no	—
105	C₂²×D₄	—	5	210	no	—
106	C₂²×D₄	—	5	630	no	—
107	C₂²×D₄	C₂²×D₅	1	105	yes	complemented
108	C₂³×C₄	—	5	30	no	—
109	C₂³×C₄	—	5	105	no	—
110	C₂²×C₅	—	4	105	no	—
111	C₂²×C₅	C₂⁶	1	35	yes	—
112	C₂×D₅	—	4	210	no	—
113	C₂×D₅	C₂⁶	1	28	yes	—
114	C₄×C₅	C₂⁶	1	28	yes	—
115	Dic₅	C₂⁶	1	35	yes	—
116	Dic₅	C₂⁶	1	1	yes	characteristic
117	C₂⁴	—	5	105	no	—
118	C₂⁴	C₂³×D₅	1	30	yes	—
119	#41	—	5	56	no	—
120	#41	—	5	280	no	—
121	#41	—	5	280	no	—
122	#41	—	5	2520	no	—
123	#41	—	5	1680	no	—
124	#41	C₂³×D₅	1	560	yes	—
125	C₂×Q₈	—	5	560	no	—
126	C₂×Q₈	—	5	630	no	—
127	C₂×Q₈	C₂³×D₅	1	70	yes	—
128	C₂×D₄	—	5	1260	no	—
129	C₂×D₄	—	5	1680	no	—
130	C₂×D₄	—	5	210	no	—
131	C₂×D₄	C₂³×D₅	1	630	yes	—
132	C₂²×C₄	—	5	105	no	—
133	C₂²×C₄	—	5	210	no	—
134	C₂²×C₄	—	5	210	no	—
135	C₂²×C₄	—	5	630	no	—
136	C₂²×C₄	C₂³×D₅	1	105	yes	—
137	C₂×C₅	—	2	35	no	—
138	C₂×C₅	C₂⁷	1	1	yes	characteristic
139	D₅	—	2	28	no	—
140	C₂³	—	5	210	no	—
141	C₂³	—	8	30	no	—
142	C₂³	—	40	105	no	—
143	C₂³	C₂⁴×D₅	1	105	yes	—
144	Q₈	—	5	280	no	—
145	Q₈	C₂⁴×D₅	1	56	yes	—
146	D₄	—	5	560	no	—
147	D₄	—	5	168	no	—
148	D₄	C₂⁴×D₅	1	280	yes	—
149	C₂×C₄	—	5	35	no	—
150	C₂×C₄	—	5	315	no	—
151	C₂×C₄	—	5	28	no	—
152	C₂×C₄	—	5	420	no	—
153	C₂×C₄	C₂⁴×D₅	1	210	yes	—
154	C₅	#63103	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	4	105	no	—
156	C₂²	—	5	28	no	—
157	C₂²	—	20	210	no	—
158	C₂²	C₂⁵×D₅	1	35	yes	—
159	C₄	—	5	1	no	—
160	C₄	—	5	35	no	—
161	C₄	C₂⁵×D₅	1	28	yes	—
162	C₂	—	2	35	no	—
163	C₂	—	10	28	no	—
164	C₂	C₂⁶×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₅
commutator, socle, cosocle	C₂×C₅	C₂⁷
Fitting	C₅×H(7,𝔽₂)	C₂
hypercenter	H(7,𝔽₂)	D₅
hypocenter	C₅	#63103
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₅×H(7,𝔽₂)	C₂	semidirect
2. D₅×#585	C₂	semidirect
3. #10619440	C₂	semidirect
4. #10619436	C₂	semidirect
5. C₂×#66285	C₂	semidirect
6. C₅×#585	C₂²	semidirect
7. C₂×C₅×H(5,𝔽₂)	C₂²	semidirect
8. #66287	C₂²	semidirect
9. D₅×2₋¹⁺⁴	C₂²	semidirect
10. #66285	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×#4144	C₂²	semidirect
17. C₂²×#711	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×D₄	C₂³	semidirect
23. C₂²×C₂²⋊D₅	C₂³	semidirect
24. C₂³×Dic₅	C₂³	semidirect
25. #4150	C₂³	semidirect
26. #4149	C₂³	semidirect
27. D₅×#41	C₂³	semidirect
28. D₅×#41	C₂³	semidirect
29. #4147	C₂³	semidirect
30. C₂×#713	C₂³	semidirect
31. C₂×Q₈×D₅	C₂³	semidirect
32. #4144	C₂³	semidirect
33. C₂×#711	C₂³	semidirect
34. C₂×#711	C₂³	semidirect
35. C₂×D₄×D₅	C₂³	semidirect
36. C₂×#709	C₂³	semidirect
37. C₂²×C₄×D₅	C₂³	semidirect
38. C₂²×Dic₁₀	C₂³	semidirect
41. H(7,𝔽₂)	D₅	semidirect
68. #585	C₂×D₅	semidirect
73. C₂×H(5,𝔽₂)	C₂×D₅	semidirect
92. 2₋¹⁺⁴	C₂²×D₅	semidirect
95. H(5,𝔽₂)	C₂²×D₅	semidirect
102. C₂×#41	C₂²×D₅	semidirect
107. C₂²×D₄	C₂²×D₅	semidirect
154. C₅	#63103	semidirect

Maximal subgroups

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

subgroup	index
C₅×H(7,𝔽₂)	2
D₅×#585	2
#10619440	2
#10619436	2
C₂×#66285	2
#63103	5