Subgroups — C₅×#3510 #10619295

Order 640.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#559	C₂	1	4	yes	maximal, complemented
2	C₅×#556	C₂	1	4	yes	maximal, complemented
3	C₅×#551	C₂	1	8	yes	maximal, complemented
4	C₅×#542	C₂	1	4	yes	maximal, complemented
5	C₅×#536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#535	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#529	C₂	1	4	yes	maximal, complemented
9	C₂×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₅×#127	C₂²	1	2	yes	—
19	C₅×#126	C₂²	1	8	yes	complemented
20	C₅×#126	C₂²	1	8	yes	complemented
21	C₅×#125	C₂²	1	2	yes	complemented
22	C₅×#124	C₂²	1	2	yes	—
23	C₅×#124	C₂²	1	2	yes	—
24	C₅×#124	C₂²	1	4	yes	—
25	C₅×#124	C₂²	1	2	yes	—
26	C₅×#124	C₂²	1	2	yes	complemented
27	C₅×#124	C₂²	1	8	yes	complemented
28	C₅×#123	C₂²	1	4	yes	—
29	C₅×#123	C₂²	1	8	yes	complemented
30	C₅×#123	C₂²	1	8	yes	complemented
31	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
32	C₅×C₂²⋊Q₈	C₂²	1	4	yes	complemented
33	C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
34	C₅×#121	C₂²	1	4	yes	—
35	C₅×#120	C₂²	1	4	yes	—
36	C₄×C₅×Q₈	C₂²	1	2	yes	complemented
37	C₄×C₅×Q₈	C₂²	1	4	yes	complemented
38	C₄×C₅×Q₈	C₂²	1	4	yes	complemented
39	C₄×C₅×Q₈	C₂²	1	4	yes	complemented
40	C₄×C₅×D₄	C₂²	1	4	yes	—
41	C₄×C₅×D₄	C₂²	1	4	yes	—
42	C₄×C₅×D₄	C₂²	1	4	yes	—
43	C₄×C₅×D₄	C₂²	1	2	yes	complemented
44	C₄×C₅×D₄	C₂²	1	8	yes	complemented
45	C₅×#117	C₂²	1	1	yes	characteristic
46	C₅×#117	C₂²	1	1	yes	characteristic
47	C₅×#117	C₂²	1	1	yes	characteristic
48	C₅×#117	C₂²	1	4	yes	—
49	C₅×#117	C₂²	1	2	yes	complemented
50	C₅×#117	C₂²	1	8	yes	complemented
51	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
52	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
53	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂²	1	2	yes	complemented
55	#3510	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₅	C₂³	1	2	yes	—
57	C₅×#41	—	2	4	no	—
58	C₂×C₅×Q₈	—	2	4	no	—
59	C₂×C₅×Q₈	C₂³	1	2	yes	—
60	C₂×C₅×Q₈	C₂³	1	2	yes	—
61	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×Q₈	C₂³	1	4	yes	—
63	C₂×C₅×D₄	—	2	4	no	—
64	C₂×C₅×D₄	C₂³	1	2	yes	—
65	C₂×C₅×D₄	C₂³	1	2	yes	—
66	C₂×C₅×D₄	C₂³	1	2	yes	—
67	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
68	C₂×C₅×D₄	C₂³	1	8	yes	—
69	C₂²×C₄×C₅	C₂³	1	2	yes	—
70	C₂²×C₄×C₅	C₂³	1	2	yes	—
71	C₂²×C₄×C₅	C₂³	1	2	yes	—
72	C₂²×C₄×C₅	C₂³	1	4	yes	—
73	C₂²×C₄×C₅	C₂³	1	8	yes	—
74	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₅×C₄⋊C₄	C₂³	1	2	yes	—
76	C₅×C₄⋊C₄	C₂³	1	2	yes	—
77	C₅×C₄⋊C₄	C₂³	1	2	yes	—
78	C₅×C₄⋊C₄	C₂³	1	8	yes	—
79	C₅×C₄⋊C₄	C₂³	1	4	yes	—
80	C₅×C₄⋊C₄	C₂³	1	4	yes	—
81	C₅×C₄⋊C₄	C₂³	1	8	yes	—
82	C₅×C₄⋊C₄	C₂³	1	4	yes	—
83	C₅×C₄⋊C₄	C₂³	1	4	yes	—
84	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
85	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
86	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
87	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
88	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
89	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
90	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
91	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₄²×C₅	C₂³	1	1	yes	characteristic
94	C₄²×C₅	C₂³	1	1	yes	characteristic
95	C₄²×C₅	C₂³	1	2	yes	—
96	C₄²×C₅	C₂³	1	2	yes	—
97	C₄²×C₅	C₂³	1	4	yes	—
98	C₄²×C₅	C₂³	1	8	yes	—
99	C₄²×C₅	C₂³	1	2	yes	—
100	C₄²×C₅	C₂³	1	2	yes	—
101	#559	C₂×C₅	1	4	yes	complemented
102	#556	C₂×C₅	1	4	yes	complemented
103	#551	C₂×C₅	1	8	yes	complemented
104	#542	C₂×C₅	1	4	yes	complemented
105	#536	C₂×C₅	1	1	yes	characteristic, complemented
106	#535	C₂×C₅	1	1	yes	characteristic
107	#533	C₂×C₅	1	1	yes	characteristic, complemented
108	#529	C₂×C₅	1	4	yes	complemented
109	C₂×#124	C₂×C₅	1	1	yes	characteristic, complemented
110	#519	C₂×C₅	1	1	yes	characteristic, complemented
111	#518	C₂×C₅	1	1	yes	characteristic, complemented
112	C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	—	4	4	no	—
115	C₂³×C₅	C₂⁴	1	4	yes	—
116	C₂³×C₅	C₂⁴	1	2	yes	—
117	C₂³×C₅	C₂⁴	1	1	yes	characteristic
118	C₅×Q₈	—	2	4	no	—
119	C₅×Q₈	—	2	4	no	—
120	C₅×Q₈	—	2	2	no	—
121	C₅×Q₈	—	4	4	no	—
122	C₅×D₄	—	2	4	no	—
123	C₅×D₄	—	2	4	no	—
124	C₅×D₄	—	2	4	no	—
125	C₅×D₄	—	2	2	no	—
126	C₅×D₄	—	4	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	8	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	#41	1	4	yes	—
133	C₂×C₄×C₅	C₂⁴	1	2	yes	—
134	C₂×C₄×C₅	C₂⁴	1	4	yes	—
135	C₂×C₄×C₅	C₂⁴	1	4	yes	—
136	C₂×C₄×C₅	C₂⁴	1	4	yes	—
137	C₂×C₄×C₅	C₂⁴	1	2	yes	—
138	C₂×C₄×C₅	C₂⁴	1	2	yes	—
139	C₂×C₄×C₅	C₂⁴	1	4	yes	—
140	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
142	C₂×#41	C₂²×C₅	1	1	yes	characteristic
143	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
144	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
145	C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
146	C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
147	#127	C₂²×C₅	1	2	yes	—
148	#126	C₂²×C₅	1	8	yes	complemented
149	#126	C₂²×C₅	1	8	yes	complemented
150	#125	C₂²×C₅	1	2	yes	complemented
151	#124	C₂²×C₅	1	2	yes	—
152	#124	C₂²×C₅	1	2	yes	—
153	#124	C₂²×C₅	1	4	yes	—
154	#124	C₂²×C₅	1	2	yes	—
155	#124	C₂²×C₅	1	2	yes	complemented
156	#124	C₂²×C₅	1	8	yes	complemented
157	#123	C₂²×C₅	1	4	yes	—
158	#123	C₂²×C₅	1	8	yes	complemented
159	#123	C₂²×C₅	1	8	yes	complemented
160	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
161	C₂²⋊Q₈	C₂²×C₅	1	4	yes	complemented
162	C₂²⋊Q₈	C₂²×C₅	1	8	yes	complemented
163	#121	C₂²×C₅	1	4	yes	—
164	#120	C₂²×C₅	1	4	yes	—
165	C₄×Q₈	C₂²×C₅	1	2	yes	complemented
166	C₄×Q₈	C₂²×C₅	1	4	yes	complemented
167	C₄×Q₈	C₂²×C₅	1	4	yes	complemented
168	C₄×Q₈	C₂²×C₅	1	4	yes	complemented
169	C₄×D₄	C₂²×C₅	1	4	yes	—
170	C₄×D₄	C₂²×C₅	1	4	yes	—
171	C₄×D₄	C₂²×C₅	1	4	yes	—
172	C₄×D₄	C₂²×C₅	1	2	yes	complemented
173	C₄×D₄	C₂²×C₅	1	8	yes	complemented
174	#117	C₂²×C₅	1	1	yes	characteristic
175	#117	C₂²×C₅	1	1	yes	characteristic
176	#117	C₂²×C₅	1	1	yes	characteristic
177	#117	C₂²×C₅	1	4	yes	—
178	#117	C₂²×C₅	1	2	yes	complemented
179	#117	C₂²×C₅	1	8	yes	complemented
180	C₂×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
182	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₄²	C₂²×C₅	1	2	yes	complemented
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	8	no	—
189	C₂²×C₅	—	8	4	no	—
190	C₂²×C₅	C₂×#41	1	2	yes	—
191	C₂²×C₅	C₂⁵	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	4	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	4	no	—
196	C₄×C₅	—	2	4	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	C₂×#41	1	2	yes	—
200	C₄×C₅	C₂×#41	1	2	yes	—
201	C₂⁴	C₂³×C₅	1	2	yes	—
202	#41	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₂³×C₅	1	2	yes	—
205	C₂×Q₈	C₂³×C₅	1	2	yes	—
206	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
207	C₂×Q₈	C₂³×C₅	1	4	yes	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₂³×C₅	1	2	yes	—
210	C₂×D₄	C₂³×C₅	1	2	yes	—
211	C₂×D₄	C₂³×C₅	1	2	yes	—
212	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
213	C₂×D₄	C₂³×C₅	1	8	yes	—
214	C₂²×C₄	C₂³×C₅	1	2	yes	—
215	C₂²×C₄	C₂³×C₅	1	2	yes	—
216	C₂²×C₄	C₂³×C₅	1	2	yes	—
217	C₂²×C₄	C₂³×C₅	1	4	yes	—
218	C₂²×C₄	C₂³×C₅	1	8	yes	—
219	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
220	C₄⋊C₄	C₂³×C₅	1	2	yes	—
221	C₄⋊C₄	C₂³×C₅	1	2	yes	—
222	C₄⋊C₄	C₂³×C₅	1	2	yes	—
223	C₄⋊C₄	C₂³×C₅	1	8	yes	—
224	C₄⋊C₄	C₂³×C₅	1	4	yes	—
225	C₄⋊C₄	C₂³×C₅	1	4	yes	—
226	C₄⋊C₄	C₂³×C₅	1	8	yes	—
227	C₄⋊C₄	C₂³×C₅	1	4	yes	—
228	C₄⋊C₄	C₂³×C₅	1	4	yes	—
229	C₂²⋊C₄	C₂³×C₅	1	2	yes	—
230	C₂²⋊C₄	C₂³×C₅	1	2	yes	—
231	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
232	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
233	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
234	C₂²⋊C₄	C₂³×C₅	1	2	yes	—
235	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
236	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
237	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
238	C₄²	C₂³×C₅	1	1	yes	characteristic
239	C₄²	C₂³×C₅	1	1	yes	characteristic
240	C₄²	C₂³×C₅	1	2	yes	—
241	C₄²	C₂³×C₅	1	2	yes	—
242	C₄²	C₂³×C₅	1	4	yes	—
243	C₄²	C₂³×C₅	1	8	yes	—
244	C₄²	C₂³×C₅	1	2	yes	—
245	C₄²	C₂³×C₅	1	2	yes	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	4	2	no	—
248	C₂×C₅	—	4	4	no	—
249	C₂×C₅	C₂²×#41	1	1	yes	characteristic
250	C₂×C₅	#585	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	—	4	4	no	—
253	C₂³	C₂⁴×C₅	1	4	yes	—
254	C₂³	C₂⁴×C₅	1	2	yes	—
255	C₂³	C₂⁴×C₅	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	2	no	—
259	Q₈	—	4	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	2	2	no	—
264	D₄	—	4	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	C₅×#41	1	4	yes	—
271	C₂×C₄	C₂⁴×C₅	1	2	yes	—
272	C₂×C₄	C₂⁴×C₅	1	4	yes	—
273	C₂×C₄	C₂⁴×C₅	1	4	yes	—
274	C₂×C₄	C₂⁴×C₅	1	4	yes	—
275	C₂×C₄	C₂⁴×C₅	1	2	yes	—
276	C₂×C₄	C₂⁴×C₅	1	2	yes	—
277	C₂×C₄	C₂⁴×C₅	1	4	yes	—
278	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
279	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
280	C₅	#3510	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₅×#41	1	2	yes	—
288	C₂²	C₂⁵×C₅	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	C₂×C₅×#41	1	2	yes	—
297	C₄	C₂×C₅×#41	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	4	2	no	—
301	C₂	C₂²×C₅×#41	1	1	yes	characteristic
302	C₂	C₅×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
prime radical	C₅×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#559	C₂	semidirect
2. C₅×#556	C₂	semidirect
3. C₅×#551	C₂	semidirect
4. C₅×#542	C₂	semidirect
5. C₅×#536	C₂	semidirect
7. C₅×#533	C₂	semidirect
8. C₅×#529	C₂	semidirect
9. C₂×C₅×#124	C₂	semidirect
10. C₅×#519	C₂	semidirect
11. C₅×#518	C₂	semidirect
12. C₄×C₅×#41	C₂	semidirect
16. C₅×C₄⋊Q₈	C₂²	semidirect
17. C₅×C₄⋊Q₈	C₂²	semidirect
19. C₅×#126	C₂²	semidirect
20. C₅×#126	C₂²	semidirect
21. C₅×#125	C₂²	semidirect
26. C₅×#124	C₂²	semidirect
27. C₅×#124	C₂²	semidirect
29. C₅×#123	C₂²	semidirect
30. C₅×#123	C₂²	semidirect
32. C₅×C₂²⋊Q₈	C₂²	semidirect
33. C₅×C₂²⋊Q₈	C₂²	semidirect
36. C₄×C₅×Q₈	C₂²	semidirect
37. C₄×C₅×Q₈	C₂²	semidirect
38. C₄×C₅×Q₈	C₂²	semidirect
39. C₄×C₅×Q₈	C₂²	semidirect
43. C₄×C₅×D₄	C₂²	semidirect
44. C₄×C₅×D₄	C₂²	semidirect
49. C₅×#117	C₂²	semidirect
50. C₅×#117	C₂²	semidirect
54. C₂×C₄²×C₅	C₂²	semidirect
55. #3510	C₅	direct (N × H)
101. #559	C₂×C₅	semidirect
102. #556	C₂×C₅	semidirect
103. #551	C₂×C₅	semidirect
104. #542	C₂×C₅	semidirect
105. #536	C₂×C₅	semidirect
107. #533	C₂×C₅	semidirect
108. #529	C₂×C₅	semidirect
109. C₂×#124	C₂×C₅	semidirect
110. #519	C₂×C₅	semidirect
111. #518	C₂×C₅	semidirect
112. C₄×#41	C₂×C₅	semidirect
145. C₄⋊Q₈	C₂²×C₅	semidirect
146. C₄⋊Q₈	C₂²×C₅	semidirect
148. #126	C₂²×C₅	semidirect
149. #126	C₂²×C₅	semidirect
150. #125	C₂²×C₅	semidirect
155. #124	C₂²×C₅	semidirect
156. #124	C₂²×C₅	semidirect
158. #123	C₂²×C₅	semidirect
159. #123	C₂²×C₅	semidirect
161. C₂²⋊Q₈	C₂²×C₅	semidirect
162. C₂²⋊Q₈	C₂²×C₅	semidirect
165. C₄×Q₈	C₂²×C₅	semidirect
166. C₄×Q₈	C₂²×C₅	semidirect
167. C₄×Q₈	C₂²×C₅	semidirect
168. C₄×Q₈	C₂²×C₅	semidirect
172. C₄×D₄	C₂²×C₅	semidirect
173. C₄×D₄	C₂²×C₅	semidirect
178. #117	C₂²×C₅	semidirect
179. #117	C₂²×C₅	semidirect
183. C₂×C₄²	C₂²×C₅	semidirect
280. C₅	#3510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#559	2
C₅×#556	2
C₅×#551	2
C₅×#542	2
C₅×#536	2
C₅×#535	2
C₅×#533	2
C₅×#529	2
C₂×C₅×#124	2
C₅×#519	2
C₅×#518	2
C₄×C₅×#41	2
#3510	5

Subgroups — C5×#3510 #10619295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3510 #10619295