Subgroups — C₅×#10602610 #72226425

Order 3200.

974 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#532	C₂	1	1	yes	maximal, characteristic
2	C₅×#66153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#65525	C₂	1	2	yes	maximal, complemented
4	C₅×#65529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#64743	C₂	1	2	yes	maximal, complemented
6	C₂×C₅²×#41	C₂²	1	1	yes	characteristic
7	C₂×C₅²×#41	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	C₅²×#127	C₄	1	1	yes	characteristic, complemented
10	C₅²×#124	C₄	1	1	yes	characteristic, complemented
11	C₅²×#124	C₄	1	1	yes	characteristic, complemented
12	C₅²×#121	—	2	2	no	—
13	C₄×C₅²×D₄	—	2	2	no	—
14	C₂×C₄²×C₅²	C₂²	1	1	yes	characteristic
15	C₅×#4094	—	2	4	no	—
16	C₅×#4072	C₂²	1	2	yes	—
17	C₂×C₅×#681	—	2	2	no	—
18	C₂×C₅×#681	C₂²	1	2	yes	—
19	C₅×#3969	—	2	4	no	—
20	C₅×#3966	—	2	2	no	—
21	C₅×#1884	—	5	1	no	maximal
22	#10602610	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₅²×#41	—	2	2	no	—
24	C₅²×#41	—	2	2	no	—
25	C₅²×#41	—	2	2	no	—
26	C₅²×#41	—	2	2	no	—
27	C₂×C₅²×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅²×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅²×D₄	—	2	2	no	—
30	C₂×C₅²×D₄	—	2	1	no	—
31	C₂×C₅²×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅²×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅²×D₄	D₄	1	2	yes	—
34	C₂²×C₄×C₅²	—	2	1	no	—
35	C₂²×C₄×C₅²	—	2	1	no	—
36	C₂²×C₄×C₅²	D₄	1	2	yes	—
37	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
38	C₅²×C₄⋊C₄	—	2	2	no	—
39	C₅²×C₂²⋊C₄	—	2	2	no	—
40	C₅²×C₂²⋊C₄	—	2	2	no	—
41	C₄²×C₅²	—	2	1	no	—
42	C₄²×C₅²	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅²	C₂×C₄	1	1	yes	characteristic
44	C₅×C₂²⋊Dic₅	—	2	2	no	—
45	C₅×C₄⋊Dic₅	—	2	2	no	—
46	C₄×C₅×Dic₅	—	2	2	no	—
47	C₅×#681	—	2	4	no	—
48	C₅×#681	—	2	2	no	—
49	C₅×#681	D₄	1	4	yes	complemented
50	C₂×C₅×#168	—	2	4	no	—
51	C₂×C₅×#168	—	2	2	no	—
52	C₅×#568	—	5	1	no	—
53	C₅×#532	—	2	2	no	—
54	C₅×#532	D₅	1	1	yes	characteristic
55	C₅×#532	C₂×C₅	1	1	yes	characteristic
56	#66153	C₂×C₅	1	1	yes	characteristic, complemented
57	C₅×#421	—	5	2	no	—
58	C₅×#410	—	5	1	no	—
59	#65529	C₂×C₅	1	1	yes	characteristic, complemented
60	#65525	C₂×C₅	1	2	yes	complemented
61	C₅×#339	—	5	2	no	—
62	#64743	C₂×C₅	1	2	yes	complemented
63	C₂³×C₅²	—	2	1	no	—
64	C₂³×C₅²	C₂²⋊C₄	1	2	yes	—
65	C₂³×C₅²	C₂×D₄	1	1	yes	characteristic
66	C₅²×Q₈	—	2	2	no	—
67	C₅²×Q₈	—	2	2	no	—
68	C₅²×D₄	—	2	4	no	—
69	C₅²×D₄	—	2	2	no	—
70	C₅²×D₄	—	2	2	no	—
71	C₅²×D₄	—	4	4	no	—
72	C₅²×D₄	—	4	2	no	—
73	C₂×C₄×C₅²	—	2	1	no	—
74	C₂×C₄×C₅²	—	2	2	no	—
75	C₂×C₄×C₅²	—	2	2	no	—
76	C₂×C₄×C₅²	—	2	1	no	—
77	C₂×C₄×C₅²	—	2	1	no	—
78	C₂×C₄×C₅²	—	4	2	no	—
79	C₂×C₄×C₅²	—	4	1	no	—
80	C₂×C₄×C₅²	—	4	1	no	—
81	C₂×C₄×C₅²	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅²	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅²	#41	1	2	yes	—
86	C₂×C₅×Dic₅	—	4	2	no	—
87	C₅×#168	—	2	4	no	—
88	C₂×C₅×#41	—	2	2	no	—
89	C₂×C₅×#41	—	2	2	no	—
90	C₂×C₅×#41	C₂×D₅	1	1	yes	characteristic
91	C₂×C₅×#41	C₂×D₅	1	1	yes	characteristic
92	C₂×C₅×#41	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₅×#41	C₂²×C₅	1	1	yes	characteristic
94	C₅×#131	—	10	4	no	—
95	C₂×C₅×M₁₆	—	5	2	no	—
96	C₂×C₅×M₁₆	—	10	2	no	—
97	C₅×C₄⋊Q₈	—	2	2	no	—
98	C₅×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
99	C₅×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic, complemented
100	C₅×#127	—	2	2	no	—
101	C₅×#127	Dic₅	1	1	yes	characteristic, complemented
102	C₅×#127	C₄×C₅	1	1	yes	characteristic, complemented
103	C₅×#124	—	2	2	no	—
104	C₅×#124	—	2	2	no	—
105	C₅×#124	Dic₅	1	1	yes	characteristic, complemented
106	C₅×#124	Dic₅	1	1	yes	characteristic, complemented
107	C₅×#124	C₄×C₅	1	1	yes	characteristic, complemented
108	C₅×#124	C₄×C₅	1	1	yes	characteristic, complemented
109	C₅×#121	—	2	8	no	—
110	C₅×#121	—	2	2	no	—
111	C₅×#121	—	2	2	no	—
112	C₄×C₅×D₄	—	2	8	no	—
113	C₄×C₅×D₄	—	2	2	no	—
114	C₄×C₅×D₄	—	2	2	no	—
115	C₅×#117	—	5	2	no	—
116	C₂×C₄²×C₅	—	2	2	no	—
117	C₂×C₄²×C₅	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
119	#4094	—	2	4	no	—
120	#4072	C₂²×C₅	1	2	yes	—
121	C₂×#681	—	2	2	no	—
122	C₂×#681	C₂²×C₅	1	2	yes	—
123	C₅×#104	—	10	4	no	—
124	C₅×#99	—	10	2	no	—
125	#3969	—	2	4	no	—
126	#3966	—	2	2	no	—
127	#1884	—	5	1	no	Sylow
128	C₂²×C₅²	—	2	2	no	—
129	C₂²×C₅²	—	4	1	no	—
130	C₂²×C₅²	—	4	2	no	—
131	C₂²×C₅²	—	4	2	no	—
132	C₂²×C₅²	—	4	1	no	—
133	C₂²×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₅²	#121	1	2	yes	—
135	C₄×C₅²	—	2	2	no	—
136	C₄×C₅²	—	2	2	no	—
137	C₄×C₅²	—	4	1	no	—
138	C₄×C₅²	—	4	1	no	—
139	C₄×C₅²	C₄×D₄	1	2	yes	—
140	C₄×C₅²	#120	1	1	yes	characteristic
141	C₄×C₅²	#123	1	1	yes	characteristic
142	C₅×Dic₅	—	8	2	no	—
143	C₅×#41	—	2	2	no	—
144	C₅×#41	—	2	2	no	—
145	C₅×#41	—	2	2	no	—
146	C₅×#41	—	2	2	no	—
147	C₅×#41	—	2	2	no	—
148	C₅×#41	—	2	2	no	—
149	C₅×#41	—	2	2	no	—
150	C₅×#41	—	2	2	no	—
151	C₅×#41	—	4	4	no	—
152	C₅×#41	—	4	4	no	—
153	C₅×#41	—	4	4	no	—
154	C₅×#41	—	4	4	no	—
155	C₂×C₅×Q₈	—	2	2	no	—
156	C₂×C₅×Q₈	—	2	2	no	—
157	C₂×C₅×Q₈	C₂×Dic₅	1	1	yes	characteristic
158	C₂×C₅×Q₈	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₅×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₅×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×C₅×D₄	—	2	8	no	—
162	C₂×C₅×D₄	—	2	4	no	—
163	C₂×C₅×D₄	—	2	2	no	—
164	C₂×C₅×D₄	—	2	2	no	—
165	C₂×C₅×D₄	—	2	1	no	—
166	C₂×C₅×D₄	—	2	1	no	—
167	C₂×C₅×D₄	—	2	2	no	—
168	C₂×C₅×D₄	—	2	2	no	—
169	C₂×C₅×D₄	—	2	4	no	—
170	C₂×C₅×D₄	C₂×Dic₅	1	1	yes	characteristic
171	C₂×C₅×D₄	C₂×Dic₅	1	1	yes	characteristic
172	C₂×C₅×D₄	C₂²⋊D₅	1	2	yes	—
173	C₂×C₅×D₄	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₅×D₄	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₅×D₄	C₅×D₄	1	2	yes	—
176	C₂²×C₄×C₅	—	2	4	no	—
177	C₂²×C₄×C₅	—	2	4	no	—
178	C₂²×C₄×C₅	—	2	1	no	—
179	C₂²×C₄×C₅	—	2	1	no	—
180	C₂²×C₄×C₅	—	2	1	no	—
181	C₂²×C₄×C₅	—	2	1	no	—
182	C₂²×C₄×C₅	—	2	4	no	—
183	C₂²×C₄×C₅	—	2	2	no	—
184	C₂²×C₄×C₅	C₂²⋊D₅	1	2	yes	—
185	C₂²×C₄×C₅	C₅×D₄	1	2	yes	—
186	C₂²×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
188	C₅×M₁₆	—	5	4	no	—
189	C₅×M₁₆	—	10	4	no	—
190	C₅×M₁₆	—	10	2	no	—
191	C₂×C₅×C₈	—	10	4	no	—
192	C₂×C₅×C₈	—	10	2	no	—
193	C₅×C₄⋊C₄	—	2	8	no	—
194	C₅×C₄⋊C₄	—	2	2	no	—
195	C₅×C₄⋊C₄	—	2	2	no	—
196	C₅×C₄⋊C₄	—	10	2	no	—
197	C₅×C₂²⋊C₄	—	2	8	no	—
198	C₅×C₂²⋊C₄	—	2	8	no	—
199	C₅×C₂²⋊C₄	—	2	2	no	—
200	C₅×C₂²⋊C₄	—	2	2	no	—
201	C₅×C₂²⋊C₄	—	2	2	no	—
202	C₅×C₂²⋊C₄	—	2	2	no	—
203	C₅×C₂²⋊C₄	—	10	2	no	—
204	C₄²×C₅	—	2	4	no	—
205	C₄²×C₅	—	2	1	no	—
206	C₄²×C₅	—	2	1	no	—
207	C₄²×C₅	—	2	2	no	—
208	C₄²×C₅	—	2	2	no	—
209	C₄²×C₅	—	10	2	no	—
210	C₄²×C₅	C₂×Dic₅	1	1	yes	characteristic
211	C₄²×C₅	C₂×Dic₅	1	1	yes	characteristic
212	C₄²×C₅	C₂×C₄×C₅	1	1	yes	characteristic
213	C₄²×C₅	C₂×C₄×C₅	1	1	yes	characteristic
214	C₂²⋊Dic₅	—	2	2	no	—
215	C₄⋊Dic₅	—	2	2	no	—
216	C₄×Dic₅	—	2	2	no	—
217	#681	—	2	4	no	—
218	#681	—	2	2	no	—
219	#681	C₅×D₄	1	4	yes	complemented
220	C₂×#168	—	2	4	no	—
221	C₂×#168	—	2	2	no	—
222	#568	—	5	1	no	—
223	#532	C₅×D₅	1	1	yes	characteristic
224	#421	—	5	2	no	—
225	#410	—	5	1	no	—
226	#339	—	5	2	no	—
227	C₂×C₅²	—	2	1	no	—
228	C₂×C₅²	—	2	2	no	—
229	C₂×C₅²	—	4	2	no	—
230	C₂×C₅²	—	8	1	no	—
231	C₂×C₅²	#386	1	1	yes	characteristic
232	C₂³×C₅	—	2	4	no	—
233	C₂³×C₅	—	2	1	no	—
234	C₂³×C₅	—	2	1	no	—
235	C₂³×C₅	—	2	2	no	—
236	C₂³×C₅	—	2	4	no	—
237	C₂³×C₅	C₂²⋊Dic₅	1	2	yes	—
238	C₂³×C₅	C₅×C₂²⋊C₄	1	2	yes	—
239	C₂³×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₂³×C₅	C₂×C₅×D₄	1	1	yes	characteristic
241	C₅×Q₈	—	2	2	no	—
242	C₅×Q₈	—	2	2	no	—
243	C₅×Q₈	—	2	2	no	—
244	C₅×Q₈	—	2	2	no	—
245	C₅×Q₈	—	4	4	no	—
246	C₅×Q₈	—	4	4	no	—
247	C₅×D₄	—	2	4	no	—
248	C₅×D₄	—	2	4	no	—
249	C₅×D₄	—	2	2	no	—
250	C₅×D₄	—	2	2	no	—
251	C₅×D₄	—	2	2	no	—
252	C₅×D₄	—	2	2	no	—
253	C₅×D₄	—	4	4	no	—
254	C₅×D₄	—	4	4	no	—
255	C₅×D₄	—	4	8	no	—
256	C₅×D₄	—	4	4	no	—
257	C₅×D₄	—	4	4	no	—
258	C₅×D₄	—	4	16	no	—
259	C₅×D₄	—	4	8	no	—
260	C₅×D₄	—	4	2	no	—
261	C₅×D₄	—	4	2	no	—
262	C₂×C₄×C₅	—	2	4	no	—
263	C₂×C₄×C₅	—	2	1	no	—
264	C₂×C₄×C₅	—	2	1	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	2	no	—
268	C₂×C₄×C₅	—	2	2	no	—
269	C₂×C₄×C₅	—	2	4	no	—
270	C₂×C₄×C₅	—	2	4	no	—
271	C₂×C₄×C₅	—	2	1	no	—
272	C₂×C₄×C₅	—	2	1	no	—
273	C₂×C₄×C₅	—	2	1	no	—
274	C₂×C₄×C₅	—	2	1	no	—
275	C₂×C₄×C₅	—	2	4	no	—
276	C₂×C₄×C₅	—	2	4	no	—
277	C₂×C₄×C₅	—	2	2	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	4	no	—
280	C₂×C₄×C₅	—	4	4	no	—
281	C₂×C₄×C₅	—	4	4	no	—
282	C₂×C₄×C₅	—	4	2	no	—
283	C₂×C₄×C₅	—	4	2	no	—
284	C₂×C₄×C₅	—	4	8	no	—
285	C₂×C₄×C₅	—	4	4	no	—
286	C₂×C₄×C₅	—	4	4	no	—
287	C₂×C₄×C₅	—	4	1	no	—
288	C₂×C₄×C₅	—	4	1	no	—
289	C₂×C₄×C₅	—	4	1	no	—
290	C₂×C₄×C₅	—	4	1	no	—
291	C₂×C₄×C₅	—	20	2	no	—
292	C₂×C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
293	C₂×C₄×C₅	C₅×C₂²⋊C₄	1	2	yes	—
294	C₂×C₄×C₅	D₄×D₅	1	2	yes	—
295	C₂×C₄×C₅	#711	1	2	yes	—
296	C₂×C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
297	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
298	C₂×C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
299	C₂×C₄×C₅	C₂×C₅×D₄	1	2	yes	—
300	C₂×C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄×C₅	C₅×#41	1	2	yes	—
302	C₂×Dic₅	—	4	2	no	—
303	C₅×C₈	—	10	4	no	—
304	#168	—	2	4	no	—
305	C₂×#41	C₂×C₅×D₅	1	1	yes	characteristic
306	C₂×#41	C₂×C₅×D₅	1	1	yes	characteristic
307	#131	—	10	4	no	—
308	C₂×M₁₆	—	5	2	no	—
309	C₂×M₁₆	—	10	2	no	—
310	C₄⋊Q₈	C₅×Dic₅	1	1	yes	characteristic, complemented
311	#127	C₅×Dic₅	1	1	yes	characteristic, complemented
312	#124	C₅×Dic₅	1	1	yes	characteristic, complemented
313	#124	C₅×Dic₅	1	1	yes	characteristic, complemented
314	#121	—	2	2	no	—
315	C₄×D₄	—	2	2	no	—
316	#117	—	5	2	no	—
317	C₂×C₄²	C₂×C₅×D₅	1	1	yes	characteristic
318	#104	—	10	4	no	—
319	#99	—	10	2	no	—
320	C₅²	#1884	1	1	yes	characteristic, complemented, Sylow
321	C₂²×C₅	—	2	2	no	—
322	C₂²×C₅	—	2	2	no	—
323	C₂²×C₅	—	2	4	no	—
324	C₂²×C₅	—	2	2	no	—
325	C₂²×C₅	—	4	4	no	—
326	C₂²×C₅	—	4	1	no	—
327	C₂²×C₅	—	4	1	no	—
328	C₂²×C₅	—	4	2	no	—
329	C₂²×C₅	—	4	2	no	—
330	C₂²×C₅	—	4	4	no	—
331	C₂²×C₅	—	4	4	no	—
332	C₂²×C₅	—	4	8	no	—
333	C₂²×C₅	—	4	8	no	—
334	C₂²×C₅	—	4	2	no	—
335	C₂²×C₅	—	4	2	no	—
336	C₂²×C₅	—	4	1	no	—
337	C₂²×C₅	—	4	1	no	—
338	C₂²×C₅	#4085	1	2	yes	—
339	C₂²×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
340	C₂²×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
341	C₂²×C₅	C₅×#121	1	2	yes	—
342	C₄×C₅	—	2	8	no	—
343	C₄×C₅	—	2	2	no	—
344	C₄×C₅	—	2	2	no	—
345	C₄×C₅	—	2	2	no	—
346	C₄×C₅	—	2	2	no	—
347	C₄×C₅	—	2	4	no	—
348	C₄×C₅	—	2	2	no	—
349	C₄×C₅	—	2	2	no	—
350	C₄×C₅	—	4	4	no	—
351	C₄×C₅	—	4	4	no	—
352	C₄×C₅	—	4	1	no	—
353	C₄×C₅	—	4	1	no	—
354	C₄×C₅	—	4	4	no	—
355	C₄×C₅	—	4	1	no	—
356	C₄×C₅	—	4	1	no	—
357	C₄×C₅	—	40	2	no	—
358	C₄×C₅	D₄×Dic₅	1	2	yes	—
359	C₄×C₅	#4081	1	1	yes	characteristic
360	C₄×C₅	#4083	1	1	yes	characteristic
361	C₄×C₅	C₄×C₅×D₄	1	2	yes	—
362	C₄×C₅	C₅×#120	1	1	yes	characteristic
363	C₄×C₅	C₅×#123	1	1	yes	characteristic
364	Dic₅	—	8	2	no	—
365	#41	—	2	2	no	—
366	#41	—	2	2	no	—
367	#41	—	2	2	no	—
368	#41	—	2	2	no	—
369	C₂×Q₈	C₂×C₅×Dic₅	1	1	yes	characteristic
370	C₂×Q₈	C₂×C₅×Dic₅	1	1	yes	characteristic
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	C₂×C₅×Dic₅	1	1	yes	characteristic
374	C₂×D₄	C₂×C₅×Dic₅	1	1	yes	characteristic
375	C₂×D₄	C₅×C₂²⋊D₅	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	C₅×C₂²⋊D₅	1	2	yes	—
379	C₂²×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
380	M₁₆	—	5	4	no	—
381	M₁₆	—	10	4	no	—
382	M₁₆	—	10	2	no	—
383	C₂×C₈	—	10	4	no	—
384	C₂×C₈	—	10	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	10	2	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	10	2	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	10	2	no	—
392	C₄²	C₂×C₅×Dic₅	1	1	yes	characteristic
393	C₄²	C₂×C₅×Dic₅	1	1	yes	characteristic
394	C₂×C₅	—	2	4	no	—
395	C₂×C₅	—	2	1	no	—
396	C₂×C₅	—	2	2	no	—
397	C₂×C₅	—	2	2	no	—
398	C₂×C₅	—	2	1	no	—
399	C₂×C₅	—	2	2	no	—
400	C₂×C₅	—	4	4	no	—
401	C₂×C₅	—	4	2	no	—
402	C₂×C₅	—	4	2	no	—
403	C₂×C₅	—	8	4	no	—
404	C₂×C₅	—	8	4	no	—
405	C₂×C₅	—	8	1	no	—
406	C₂×C₅	—	8	1	no	—
407	C₂×C₅	#65509	1	1	yes	characteristic
408	C₂×C₅	C₅×#386	1	1	yes	characteristic
409	C₂³	—	2	1	no	—
410	C₂³	C₅×C₂²⋊Dic₅	1	2	yes	—
411	C₂³	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
412	Q₈	—	2	2	no	—
413	Q₈	—	2	2	no	—
414	D₄	—	2	4	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	20	2	no	—
428	C₂×C₄	C₅×C₂²⋊Dic₅	1	2	yes	—
429	C₂×C₄	C₅×D₄×D₅	1	2	yes	—
430	C₂×C₄	C₅×#711	1	2	yes	—
431	C₂×C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
432	C₂×C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
433	C₈	—	10	4	no	—
434	C₅	—	2	2	no	—
435	C₅	#10602610	1	1	yes	characteristic, direct product factor
436	C₅	C₅×#1884	1	1	yes	characteristic, complemented
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	1	no	—
442	C₂²	C₅×#4085	1	2	yes	—
443	C₂²	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	1	no	—
447	C₄	—	4	1	no	—
448	C₄	—	40	2	no	—
449	C₄	C₅×D₄×Dic₅	1	2	yes	—
450	C₄	C₅×#4081	1	1	yes	characteristic
451	C₄	C₅×#4083	1	1	yes	characteristic
452	C₂	—	2	1	no	—
453	C₂	—	2	2	no	—
454	C₂	—	4	2	no	—
455	C₂	—	8	1	no	—
456	C₂	C₅×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#4083
commutator	C₂×C₄×C₅	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₅
Fitting, prime radical	C₅²×#532	C₂
socle	C₂×C₅²	#386
cosocle	C₂²×C₄×C₅	C₂³×C₅
hypercenter	C₅×#532	D₅
hypocenter	C₅	C₅×#1884
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532

Complements

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

N	H ≅ G/N	type
2. C₅×#66153	C₂	semidirect
3. C₅×#65525	C₂	semidirect
4. C₅×#65529	C₂	semidirect
5. C₅×#64743	C₂	semidirect
8. C₅²×C₄⋊Q₈	C₄	semidirect
9. C₅²×#127	C₄	semidirect
10. C₅²×#124	C₄	semidirect
11. C₅²×#124	C₄	semidirect
22. #10602610	C₅	direct (N × H)
49. C₅×#681	D₄	semidirect
56. #66153	C₂×C₅	semidirect
59. #65529	C₂×C₅	semidirect
60. #65525	C₂×C₅	semidirect
62. #64743	C₂×C₅	semidirect
98. C₅×C₄⋊Q₈	Dic₅	semidirect
99. C₅×C₄⋊Q₈	C₄×C₅	semidirect
101. C₅×#127	Dic₅	semidirect
102. C₅×#127	C₄×C₅	semidirect
105. C₅×#124	Dic₅	semidirect
106. C₅×#124	Dic₅	semidirect
107. C₅×#124	C₄×C₅	semidirect
108. C₅×#124	C₄×C₅	semidirect
219. #681	C₅×D₄	semidirect
310. C₄⋊Q₈	C₅×Dic₅	semidirect
311. #127	C₅×Dic₅	semidirect
312. #124	C₅×Dic₅	semidirect
313. #124	C₅×Dic₅	semidirect
320. C₅²	#1884	semidirect
435. C₅	#10602610	direct (N × H)
436. C₅	C₅×#1884	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#532	2
C₅×#66153	2
C₅×#65525	2
C₅×#65529	2
C₅×#64743	2
C₅×#1884	5
#10602610	5

Subgroups — C5×#10602610 #72226425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10602610 #72226425