Subgroups — C₅×#65820 #13106641

Order 1600.

808 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
5	C₅×#4027	C₂	1	4	yes	maximal, complemented
6	C₅×D₅×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₅×#4023	C₂	1	2	yes	maximal, complemented
8	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
9	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
11	C₂×C₅×C₂²⋊D₅	C₄	1	8	yes	complemented
12	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	2	yes	complemented
16	C₂×C₄×C₅×D₅	C₂²	1	4	yes	complemented
17	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
19	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
21	C₅×#685	C₂²	1	4	yes	complemented
22	C₅×#683	C₂²	1	4	yes	complemented
23	C₄×C₅×Dic₅	C₂²	1	4	yes	complemented
24	C₅×#518	—	5	1	no	maximal
25	#65820	C₅	1	1	yes	maximal, characteristic, direct product factor
26	C₂³×C₅²	—	4	2	no	—
27	C₂³×C₅²	C₂×C₄	1	4	yes	complemented
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂³×C₅²	C₂³	1	1	yes	characteristic
30	C₂³×C₅²	C₂³	1	1	yes	characteristic
31	C₂²×C₅×D₅	—	4	2	no	—
32	C₂²×C₅×D₅	C₂×C₄	1	4	yes	complemented
33	C₂²×C₅×D₅	C₂³	1	2	yes	—
34	C₂×C₄×C₅²	—	4	2	no	—
35	C₂×C₄×C₅²	C₂³	1	4	yes	—
36	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
37	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
38	C₂×C₅×Dic₅	—	4	2	no	—
39	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
40	C₂×C₅×Dic₅	C₂³	1	4	yes	—
41	C₂×C₅×Dic₅	C₂³	1	2	yes	—
42	C₄×C₅×D₅	—	4	4	no	—
43	C₂²×C₅×D₄	—	5	1	no	—
44	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic
45	C₄×C₅×D₄	—	5	4	no	—
46	C₄×C₅×D₄	—	5	4	no	—
47	C₅×#117	—	5	2	no	—
48	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
50	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
51	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
52	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
54	C₄×C₂²⋊D₅	C₂×C₅	1	4	yes	complemented
55	#4027	C₂×C₅	1	4	yes	complemented
56	D₅×C₂²⋊C₄	C₂×C₅	1	2	yes	complemented
57	#4023	C₂×C₅	1	2	yes	complemented
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	2	2	no	—
60	C₂²×C₅²	—	2	2	no	—
61	C₂²×C₅²	—	4	4	no	—
62	C₂²×C₅²	C₂²×C₄	1	2	yes	—
63	C₂²×C₅²	C₂²×C₄	1	2	yes	—
64	C₂²×C₅²	C₂²×C₄	1	2	yes	—
65	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
66	C₂×C₅×D₅	—	2	4	no	—
67	C₂×C₅×D₅	—	4	4	no	—
68	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
69	C₄×C₅²	—	2	4	no	—
70	C₅×Dic₅	—	2	4	no	—
71	C₅×Dic₅	C₂²×C₄	1	4	yes	—
72	C₂⁴×C₅	—	2	2	no	—
73	C₂⁴×C₅	—	5	1	no	—
74	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
75	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₅×D₄	—	5	8	no	—
78	C₂×C₅×D₄	—	5	2	no	—
79	C₂×C₅×D₄	—	5	2	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	5	1	no	—
82	C₂²×C₄×C₅	—	5	2	no	—
83	C₂²×C₄×C₅	—	5	4	no	—
84	C₂²×C₄×C₅	C₂×D₅	1	2	yes	complemented
85	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	complemented
86	C₂×C₂²⋊D₅	C₄×C₅	1	8	yes	complemented
87	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
88	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
89	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
90	C₂²×Dic₅	C₂²×C₅	1	2	yes	complemented
91	C₂×C₄×D₅	C₂²×C₅	1	4	yes	complemented
92	C₅×C₄⋊C₄	—	5	4	no	—
93	C₅×C₂²⋊C₄	—	2	4	no	—
94	C₅×C₂²⋊C₄	—	2	4	no	—
95	C₅×C₂²⋊C₄	—	5	2	no	—
96	C₅×C₂²⋊C₄	—	5	2	no	—
97	C₅×C₂²⋊C₄	—	5	4	no	—
98	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
99	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
100	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
101	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
102	C₄²×C₅	—	5	4	no	—
103	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
104	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	complemented
105	#685	C₂²×C₅	1	4	yes	complemented
106	#683	C₂²×C₅	1	4	yes	complemented
107	C₄×Dic₅	C₂²×C₅	1	4	yes	complemented
108	#518	—	5	1	no	Sylow
109	C₂×C₅²	—	2	2	no	—
110	C₂×C₅²	—	2	2	no	—
111	C₂×C₅²	—	2	2	no	—
112	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
114	C₅×D₅	—	2	4	no	—
115	C₂³×C₅	—	2	8	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	—	4	2	no	—
121	C₂³×C₅	—	4	8	no	—
122	C₂³×C₅	—	5	2	no	—
123	C₂³×C₅	—	5	4	no	—
124	C₂³×C₅	—	20	2	no	—
125	C₂³×C₅	C₄×D₅	1	4	yes	complemented
126	C₂³×C₅	C₂×C₄×C₅	1	4	yes	complemented
127	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
128	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
131	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
132	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
133	C₂²×D₅	—	4	2	no	—
134	C₂²×D₅	C₂×C₄×C₅	1	4	yes	complemented
135	C₂²×D₅	C₂³×C₅	1	2	yes	—
136	C₅×D₄	—	5	8	no	—
137	C₅×D₄	—	5	8	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	8	no	—
142	C₂×C₄×C₅	—	5	4	no	—
143	C₂×C₄×C₅	—	5	2	no	—
144	C₂×C₄×C₅	—	5	4	no	—
145	C₂×C₄×C₅	—	20	4	no	—
146	C₂×C₄×C₅	—	20	2	no	—
147	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
148	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
149	C₂²⋊D₅	C₂×C₄×C₅	1	8	yes	complemented
150	C₂²⋊D₅	C₂×C₄×C₅	1	8	yes	complemented
151	C₂×Dic₅	—	4	2	no	—
152	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	complemented
153	C₂×Dic₅	C₂³×C₅	1	4	yes	—
154	C₂×Dic₅	C₂³×C₅	1	2	yes	—
155	C₄×D₅	—	4	4	no	—
156	C₂²×D₄	—	5	1	no	—
157	C₄×D₄	—	5	4	no	—
158	C₄×D₄	—	5	4	no	—
159	#117	—	5	2	no	—
160	C₂×C₂²⋊C₄	—	5	1	no	—
161	C₂×C₂²⋊C₄	—	5	2	no	—
162	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
163	C₅²	#518	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₅	—	2	8	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	4	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	4	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	4	4	no	—
176	C₂²×C₅	—	4	4	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	—	4	16	no	—
180	C₂²×C₅	—	5	4	no	—
181	C₂²×C₅	—	10	4	no	—
182	C₂²×C₅	—	20	4	no	—
183	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
184	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
185	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
186	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
187	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
188	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
189	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
190	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
191	C₂×D₅	—	2	4	no	—
192	C₂×D₅	—	4	4	no	—
193	C₂×D₅	C₂²×C₄×C₅	1	4	yes	—
194	C₄×C₅	—	2	4	no	—
195	C₄×C₅	—	2	4	no	—
196	C₄×C₅	—	4	8	no	—
197	C₄×C₅	—	5	4	no	—
198	C₄×C₅	—	10	4	no	—
199	Dic₅	—	2	4	no	—
200	Dic₅	C₂²×C₄×C₅	1	4	yes	—
201	C₂⁴	—	5	1	no	—
202	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
203	C₂×D₄	—	5	8	no	—
204	C₂×D₄	—	5	2	no	—
205	C₂×D₄	—	5	2	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂²×C₄	—	5	4	no	—
209	C₂²×C₄	C₂×C₅×D₅	1	2	yes	complemented
210	C₄⋊C₄	—	5	4	no	—
211	C₂²⋊C₄	—	5	2	no	—
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	5	4	no	—
214	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	—
215	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	complemented
216	C₄²	—	5	4	no	—
217	C₂×C₅	—	2	8	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	2	2	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	2	2	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	4	no	—
226	C₂×C₅	—	4	4	no	—
227	C₂×C₅	—	4	4	no	—
228	C₂×C₅	—	10	4	no	—
229	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
230	C₂×C₅	#4144	1	2	yes	—
231	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
232	C₂×C₅	C₅×H(5,𝔽₂)	1	2	yes	—
233	D₅	—	2	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	5	2	no	—
236	C₂³	—	5	4	no	—
237	C₂³	—	20	2	no	—
238	C₂³	C₄×C₅×D₅	1	4	yes	complemented
239	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
240	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
241	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
242	D₄	—	5	8	no	—
243	D₄	—	5	8	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	5	4	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	5	4	no	—
248	C₂×C₄	—	20	4	no	—
249	C₂×C₄	—	20	2	no	—
250	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
251	C₅	—	2	2	no	—
252	C₅	#65820	1	1	yes	characteristic, direct product factor
253	C₅	C₅×#518	1	1	yes	characteristic, complemented
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	5	4	no	—
259	C₂²	—	10	4	no	—
260	C₂²	—	20	4	no	—
261	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
262	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
263	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
264	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	5	4	no	—
267	C₄	—	10	4	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	—	10	4	no	—
272	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
273	C₂	C₅×#4144	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂²×C₅×C₂²⋊D₅	C₂
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₅×#518
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
3. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
4. C₄×C₅×C₂²⋊D₅	C₂	semidirect
5. C₅×#4027	C₂	semidirect
6. C₅×D₅×C₂²⋊C₄	C₂	semidirect
7. C₅×#4023	C₂	semidirect
10. C₂²×C₄×C₅²	C₂²	semidirect
11. C₂×C₅×C₂²⋊D₅	C₄	semidirect
15. C₂²×C₅×Dic₅	C₂²	semidirect
16. C₂×C₄×C₅×D₅	C₂²	semidirect
18. C₅²×C₂²⋊C₄	C₂²	semidirect
20. C₅×C₂²⋊Dic₅	C₂²	semidirect
21. C₅×#685	C₂²	semidirect
22. C₅×#683	C₂²	semidirect
23. C₄×C₅×Dic₅	C₂²	semidirect
25. #65820	C₅	direct (N × H)
27. C₂³×C₅²	C₂×C₄	semidirect
32. C₂²×C₅×D₅	C₂×C₄	semidirect
36. C₅×C₂²⋊D₅	C₂×C₄	semidirect
37. C₅×C₂²⋊D₅	C₂×C₄	semidirect
39. C₂×C₅×Dic₅	C₂×C₄	semidirect
51. C₂×C₅×C₂²⋊C₄	D₅	semidirect
52. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
53. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
54. C₄×C₂²⋊D₅	C₂×C₅	semidirect
55. #4027	C₂×C₅	semidirect
56. D₅×C₂²⋊C₄	C₂×C₅	semidirect
57. #4023	C₂×C₅	semidirect
84. C₂²×C₄×C₅	C₂×D₅	semidirect
85. C₂²×C₄×C₅	C₂²×C₅	semidirect
86. C₂×C₂²⋊D₅	C₄×C₅	semidirect
90. C₂²×Dic₅	C₂²×C₅	semidirect
91. C₂×C₄×D₅	C₂²×C₅	semidirect
99. C₅×C₂²⋊C₄	C₂×D₅	semidirect
101. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
104. C₂²⋊Dic₅	C₂²×C₅	semidirect
105. #685	C₂²×C₅	semidirect
106. #683	C₂²×C₅	semidirect
107. C₄×Dic₅	C₂²×C₅	semidirect
125. C₂³×C₅	C₄×D₅	semidirect
126. C₂³×C₅	C₂×C₄×C₅	semidirect
134. C₂²×D₅	C₂×C₄×C₅	semidirect
149. C₂²⋊D₅	C₂×C₄×C₅	semidirect
150. C₂²⋊D₅	C₂×C₄×C₅	semidirect
152. C₂×Dic₅	C₂×C₄×C₅	semidirect
162. C₂×C₂²⋊C₄	C₅×D₅	semidirect
163. C₅²	#518	semidirect
209. C₂²×C₄	C₂×C₅×D₅	semidirect
215. C₂²⋊C₄	C₂×C₅×D₅	semidirect
238. C₂³	C₄×C₅×D₅	semidirect
252. C₅	#65820	direct (N × H)
253. C₅	C₅×#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₂²⋊D₅	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4027	2
C₅×D₅×C₂²⋊C₄	2
C₅×#4023	2
C₅×#518	5
#65820	5

Subgroups — C5×#65820 #13106641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65820 #13106641