Subgroups — C₅×#68310 #14283064

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5622	C₂	1	1	yes	maximal, characteristic
2	C₅×#4719	C₂	1	1	yes	maximal, characteristic
3	C₅×#4648	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#328	C₂	1	1	yes	maximal, characteristic
5	C₅×#4538	C₂	1	1	yes	maximal, characteristic
6	C₅×#4467	C₂	1	1	yes	maximal, characteristic
7	C₅×#4452	C₂	1	1	yes	maximal, characteristic
8	C₅×#1506	—	3	1	no	maximal, Hall
9	C₃×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×Q₈×Dic₃	C₄	1	1	yes	characteristic, complemented
11	C₅×#944	C₄	1	1	yes	characteristic, complemented
12	C₅×#924	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Dic₃	—	2	1	no	—
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₅×#889	—	2	1	no	—
16	C₅×#888	C₄	1	1	yes	characteristic, complemented
17	C₅×#887	—	2	1	no	—
18	C₅×#887	C₄	1	1	yes	characteristic, complemented
19	C₅×#878	—	2	1	no	—
20	C₅×#877	—	2	1	no	—
21	C₃×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₅×#95	C₂²	1	1	yes	characteristic
23	C₅×C₃⋊#95	—	2	1	no	—
24	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
25	C₅×#830	C₂²	1	1	yes	characteristic
26	C₅×#815	—	2	1	no	—
27	C₅×Dic₃⋊C₈	—	2	1	no	—
28	#68310	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₅×#533	—	3	1	no	—
30	C₅×#432	—	3	1	no	—
31	C₅×#376	—	3	1	no	—
32	C₅×#328	—	3	1	no	—
33	C₅×#328	—	3	1	no	—
34	C₅×#328	—	3	1	no	—
35	C₅×#328	D₃	1	1	yes	characteristic
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	—	2	1	no	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×Dic₃	—	2	1	no	—
40	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
41	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
42	C₂×C₅×Dic₆	—	2	1	no	—
43	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
48	C₅×#217	—	2	1	no	—
49	C₅×C₄⋊Dic₃	—	2	1	no	—
50	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₅×Dic₃⋊C₄	—	2	1	no	—
54	C₅×Dic₃⋊C₄	—	2	1	no	—
55	C₅×Dic₃⋊C₄	—	2	1	no	—
56	C₄×C₅×Dic₃	—	2	4	no	—
57	C₄×C₅×Dic₃	—	2	1	no	—
58	C₄×C₅×Dic₃	—	2	1	no	—
59	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
62	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
63	C₂×C₅×C₃⋊C₈	—	2	1	no	—
64	#5622	C₂×C₅	1	1	yes	characteristic
65	#4719	C₂×C₅	1	1	yes	characteristic
66	#4648	C₂×C₅	1	1	yes	characteristic
67	C₃×#328	C₂×C₅	1	1	yes	characteristic
68	#4538	C₂×C₅	1	1	yes	characteristic
69	#4467	C₂×C₅	1	1	yes	characteristic
70	#4452	C₂×C₅	1	1	yes	characteristic
71	C₅×C₄⋊Q₈	—	3	1	no	—
72	C₅×C₄⋊Q₈	—	3	1	no	—
73	C₅×#125	—	6	1	no	—
74	C₅×C₂²⋊Q₈	—	3	1	no	—
75	C₅×C₂²⋊Q₈	—	6	1	no	—
76	C₅×C₂²⋊Q₈	C₂×D₃	1	1	yes	characteristic
77	C₄×C₅×Q₈	—	3	1	no	—
78	C₄×C₅×Q₈	—	3	1	no	—
79	C₄×C₅×Q₈	—	6	1	no	—
80	C₅×#117	—	6	1	no	—
81	C₂×C₄²×C₅	—	3	1	no	—
82	C₂×C₄²×C₅	—	6	1	no	—
83	C₅×C₄⋊C₈	—	6	1	no	—
84	C₅×C₂²⋊C₈	—	3	1	no	—
85	C₅×C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
86	C₅×#97	—	6	1	no	—
87	C₅×#95	—	3	1	no	—
88	C₅×#95	—	6	1	no	—
89	C₅×#95	C₂×D₃	1	1	yes	characteristic
90	#1506	—	3	1	no	Sylow
91	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
92	C₃×C₅×Q₈	—	4	1	no	—
93	C₂×C₃×C₄×C₅	—	2	2	no	—
94	C₂×C₃×C₄×C₅	—	2	1	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	—	4	1	no	—
97	C₂×C₃×C₄×C₅	—	4	1	no	—
98	C₂×C₃×C₄×C₅	—	4	1	no	—
99	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×Dic₃	—	2	1	no	—
102	C₂×C₅×Dic₃	—	2	1	no	—
103	C₂×C₅×Dic₃	—	2	2	no	—
104	C₂×C₅×Dic₃	—	2	2	no	—
105	C₂×C₅×Dic₃	—	2	1	no	—
106	C₂×C₅×Dic₃	—	2	1	no	—
107	C₂×C₅×Dic₃	—	4	1	no	—
108	C₂×C₅×Dic₃	—	4	1	no	—
109	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
113	C₅×Dic₆	—	4	2	no	—
114	C₅×Dic₆	—	4	1	no	—
115	C₃×C₅×C₈	—	4	1	no	—
116	C₅×C₃⋊C₈	—	4	1	no	—
117	C₃×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
118	Q₈×Dic₃	C₄×C₅	1	1	yes	characteristic, complemented
119	#944	C₄×C₅	1	1	yes	characteristic, complemented
120	#924	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×Dic₃	—	2	1	no	—
122	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
123	#889	—	2	1	no	—
124	#888	C₄×C₅	1	1	yes	characteristic, complemented
125	#887	—	2	1	no	—
126	#887	C₄×C₅	1	1	yes	characteristic, complemented
127	#878	—	2	1	no	—
128	#877	—	2	1	no	—
129	C₃×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
130	C₃×#95	C₂²×C₅	1	1	yes	characteristic
131	C₃⋊#95	—	2	1	no	—
132	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
133	#830	C₂²×C₅	1	1	yes	characteristic
134	#815	—	2	1	no	—
135	Dic₃⋊C₈	—	2	1	no	—
136	C₂×C₅×Q₈	—	3	1	no	—
137	C₂×C₅×Q₈	—	6	1	no	—
138	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
139	C₂²×C₄×C₅	—	2	1	no	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	1	no	—
142	C₂²×C₄×C₅	—	6	1	no	—
143	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	6	1	no	—
146	C₅×C₄⋊C₄	—	2	1	no	—
147	C₅×C₄⋊C₄	—	3	1	no	—
148	C₅×C₄⋊C₄	—	6	1	no	—
149	C₅×C₄⋊C₄	—	6	1	no	—
150	C₅×C₄⋊C₄	—	6	1	no	—
151	C₅×C₄⋊C₄	—	6	1	no	—
152	C₅×C₄⋊C₄	—	6	1	no	—
153	C₅×C₄⋊C₄	—	6	1	no	—
154	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
155	C₅×C₂²⋊C₄	—	2	1	no	—
156	C₅×C₂²⋊C₄	—	6	1	no	—
157	C₄²×C₅	—	3	1	no	—
158	C₄²×C₅	—	3	1	no	—
159	C₄²×C₅	—	3	1	no	—
160	C₄²×C₅	—	3	1	no	—
161	C₄²×C₅	—	6	1	no	—
162	C₄²×C₅	—	6	1	no	—
163	C₄²×C₅	—	6	4	no	—
164	#533	—	3	1	no	—
165	#432	—	3	1	no	—
166	#376	—	3	1	no	—
167	#328	—	3	1	no	—
168	#328	—	3	1	no	—
169	#328	—	3	1	no	—
170	#328	C₅×D₃	1	1	yes	characteristic
171	C₂²×C₃×C₅	—	2	1	no	—
172	C₂²×C₃×C₅	—	2	1	no	—
173	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₃×C₅	#137	1	1	yes	characteristic
175	C₂²×C₃×C₅	#137	1	1	yes	characteristic
176	C₃×C₄×C₅	—	2	1	no	—
177	C₃×C₄×C₅	—	2	1	no	—
178	C₃×C₄×C₅	—	4	2	no	—
179	C₃×C₄×C₅	—	4	1	no	—
180	C₃×C₄×C₅	—	4	1	no	—
181	C₅×Dic₃	—	2	1	no	—
182	C₅×Dic₃	—	2	1	no	—
183	C₅×Dic₃	—	2	1	no	—
184	C₅×Dic₃	—	4	2	no	—
185	C₅×Dic₃	—	4	1	no	—
186	C₅×Dic₃	—	4	1	no	—
187	C₅×Dic₃	#104	1	2	yes	—
188	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₃×C₄	—	2	1	no	—
190	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
191	C₂²×Dic₃	—	2	1	no	—
192	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
193	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
194	C₂×Dic₆	—	2	1	no	—
195	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₃×C₈	—	2	1	no	—
197	C₃×C₄⋊C₄	—	2	1	no	—
198	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
199	C₃×C₂²⋊C₄	—	2	1	no	—
200	#217	—	2	1	no	—
201	C₄⋊Dic₃	—	2	1	no	—
202	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
203	Dic₃⋊C₄	—	2	1	no	—
204	Dic₃⋊C₄	—	2	1	no	—
205	Dic₃⋊C₄	—	2	1	no	—
206	Dic₃⋊C₄	—	2	1	no	—
207	Dic₃⋊C₄	—	2	1	no	—
208	C₄×Dic₃	—	2	4	no	—
209	C₄×Dic₃	—	2	1	no	—
210	C₄×Dic₃	—	2	1	no	—
211	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
212	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
213	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
214	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
215	C₂×C₃⋊C₈	—	2	1	no	—
216	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
217	C₅×Q₈	—	4	1	no	—
218	C₅×Q₈	—	12	1	no	—
219	C₅×Q₈	—	12	2	no	—
220	C₂×C₄×C₅	—	2	2	no	—
221	C₂×C₄×C₅	—	2	1	no	—
222	C₂×C₄×C₅	—	2	1	no	—
223	C₂×C₄×C₅	—	3	1	no	—
224	C₂×C₄×C₅	—	3	1	no	—
225	C₂×C₄×C₅	—	3	1	no	—
226	C₂×C₄×C₅	—	3	1	no	—
227	C₂×C₄×C₅	—	4	1	no	—
228	C₂×C₄×C₅	—	4	1	no	—
229	C₂×C₄×C₅	—	4	1	no	—
230	C₂×C₄×C₅	—	6	2	no	—
231	C₂×C₄×C₅	—	6	1	no	—
232	C₂×C₄×C₅	—	6	1	no	—
233	C₂×C₄×C₅	—	6	2	no	—
234	C₂×C₄×C₅	—	6	1	no	—
235	C₂×C₄×C₅	—	6	1	no	—
236	C₂×C₄×C₅	—	12	1	no	—
237	C₂×C₄×C₅	—	12	1	no	—
238	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
239	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
240	C₅×C₈	—	4	1	no	—
241	C₅×C₈	—	12	1	no	—
242	C₄⋊Q₈	—	3	1	no	—
243	C₄⋊Q₈	—	3	1	no	—
244	#125	—	6	1	no	—
245	C₂²⋊Q₈	—	3	1	no	—
246	C₂²⋊Q₈	—	6	1	no	—
247	C₂²⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
248	C₄×Q₈	—	3	1	no	—
249	C₄×Q₈	—	3	1	no	—
250	C₄×Q₈	—	6	1	no	—
251	#117	—	6	1	no	—
252	C₂×C₄²	—	3	1	no	—
253	C₂×C₄²	—	6	1	no	—
254	C₄⋊C₈	—	6	1	no	—
255	C₂²⋊C₈	—	3	1	no	—
256	C₂²⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
257	#97	—	6	1	no	—
258	#95	—	3	1	no	—
259	#95	—	6	1	no	—
260	#95	C₂×C₅×D₃	1	1	yes	characteristic
261	C₂×C₃×C₅	—	2	1	no	—
262	C₂×C₃×C₅	—	2	1	no	—
263	C₂×C₃×C₅	#410	1	1	yes	characteristic
264	C₂×C₃×C₅	#419	1	1	yes	characteristic
265	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
266	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
267	C₃×Q₈	—	4	1	no	—
268	C₂×C₃×C₄	—	2	2	no	—
269	C₂×C₃×C₄	—	2	1	no	—
270	C₂×C₃×C₄	—	2	1	no	—
271	C₂×C₃×C₄	—	4	1	no	—
272	C₂×C₃×C₄	—	4	1	no	—
273	C₂×C₃×C₄	—	4	1	no	—
274	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
275	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂×Dic₃	—	2	1	no	—
277	C₂×Dic₃	—	2	1	no	—
278	C₂×Dic₃	—	2	2	no	—
279	C₂×Dic₃	—	2	2	no	—
280	C₂×Dic₃	—	2	1	no	—
281	C₂×Dic₃	—	2	1	no	—
282	C₂×Dic₃	—	4	1	no	—
283	C₂×Dic₃	—	4	1	no	—
284	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
288	Dic₆	—	4	2	no	—
289	Dic₆	—	4	1	no	—
290	C₃×C₈	—	4	1	no	—
291	C₃⋊C₈	—	4	1	no	—
292	C₂²×C₅	—	2	1	no	—
293	C₂²×C₅	—	2	1	no	—
294	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
295	C₂²×C₅	#915	1	1	yes	characteristic
296	C₂²×C₅	#918	1	1	yes	characteristic
297	C₄×C₅	—	2	1	no	—
298	C₄×C₅	—	2	1	no	—
299	C₄×C₅	—	3	2	no	—
300	C₄×C₅	—	4	2	no	—
301	C₄×C₅	—	4	1	no	—
302	C₄×C₅	—	4	1	no	—
303	C₄×C₅	—	6	1	no	—
304	C₄×C₅	—	6	1	no	—
305	C₄×C₅	—	6	1	no	—
306	C₄×C₅	—	12	2	no	—
307	C₄×C₅	—	12	1	no	—
308	C₄×C₅	—	12	1	no	—
309	C₂×Q₈	—	3	1	no	—
310	C₂×Q₈	—	6	1	no	—
311	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	3	1	no	—
314	C₂²×C₄	—	3	1	no	—
315	C₂²×C₄	—	6	1	no	—
316	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	6	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	3	1	no	—
321	C₄⋊C₄	—	6	1	no	—
322	C₄⋊C₄	—	6	1	no	—
323	C₄⋊C₄	—	6	1	no	—
324	C₄⋊C₄	—	6	1	no	—
325	C₄⋊C₄	—	6	1	no	—
326	C₄⋊C₄	—	6	1	no	—
327	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
328	C₂²⋊C₄	—	2	1	no	—
329	C₂²⋊C₄	—	6	1	no	—
330	C₄²	—	3	1	no	—
331	C₄²	—	3	1	no	—
332	C₄²	—	3	1	no	—
333	C₄²	—	3	1	no	—
334	C₄²	—	6	1	no	—
335	C₄²	—	6	1	no	—
336	C₄²	—	6	4	no	—
337	C₃×C₅	#1506	1	1	yes	characteristic, complemented, Hall
338	C₂²×C₃	—	2	1	no	—
339	C₂²×C₃	—	2	1	no	—
340	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
341	C₂²×C₃	C₅×#137	1	1	yes	characteristic
342	C₂²×C₃	C₅×#137	1	1	yes	characteristic
343	C₃×C₄	—	2	1	no	—
344	C₃×C₄	—	2	1	no	—
345	C₃×C₄	—	4	2	no	—
346	C₃×C₄	—	4	1	no	—
347	C₃×C₄	—	4	1	no	—
348	Dic₃	—	2	1	no	—
349	Dic₃	—	2	1	no	—
350	Dic₃	—	2	1	no	—
351	Dic₃	—	4	2	no	—
352	Dic₃	—	4	1	no	—
353	Dic₃	—	4	1	no	—
354	Dic₃	C₅×#104	1	2	yes	—
355	C₂×C₅	—	2	1	no	—
356	C₂×C₅	—	2	1	no	—
357	C₂×C₅	#4740	1	1	yes	characteristic
358	C₂×C₅	#4802	1	1	yes	characteristic
359	C₂×C₅	D₃×#104	1	1	yes	characteristic
360	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
361	Q₈	—	4	1	no	—
362	Q₈	—	12	1	no	—
363	Q₈	—	12	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	3	1	no	—
368	C₂×C₄	—	3	1	no	—
369	C₂×C₄	—	3	1	no	—
370	C₂×C₄	—	3	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	6	2	no	—
375	C₂×C₄	—	6	1	no	—
376	C₂×C₄	—	6	1	no	—
377	C₂×C₄	—	6	2	no	—
378	C₂×C₄	—	6	1	no	—
379	C₂×C₄	—	6	1	no	—
380	C₂×C₄	—	12	1	no	—
381	C₂×C₄	—	12	1	no	—
382	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
383	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	12	1	no	—
386	C₂×C₃	—	2	1	no	—
387	C₂×C₃	—	2	1	no	—
388	C₂×C₃	C₅×#410	1	1	yes	characteristic
389	C₂×C₃	C₅×#419	1	1	yes	characteristic
390	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
391	C₅	#68310	1	1	yes	characteristic, direct product factor, Sylow
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
395	C₂²	C₅×#915	1	1	yes	characteristic
396	C₂²	C₅×#918	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	3	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	6	1	no	—
404	C₄	—	6	1	no	—
405	C₄	—	6	1	no	—
406	C₄	—	12	2	no	—
407	C₄	—	12	1	no	—
408	C₄	—	12	1	no	—
409	C₃	C₅×#1506	1	1	yes	characteristic, complemented, Sylow
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	C₅×#4740	1	1	yes	characteristic
413	C₂	C₅×#4802	1	1	yes	characteristic
414	C₂	C₅×D₃×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#328	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#328	D₃
hypocenter	C₃	C₅×#1506
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#328

Complements

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

N	H ≅ G/N	type
10. C₅×Q₈×Dic₃	C₄	semidirect
11. C₅×#944	C₄	semidirect
16. C₅×#888	C₄	semidirect
18. C₅×#887	C₄	semidirect
28. #68310	C₅	direct (N × H)
118. Q₈×Dic₃	C₄×C₅	semidirect
119. #944	C₄×C₅	semidirect
124. #888	C₄×C₅	semidirect
126. #887	C₄×C₅	semidirect
337. C₃×C₅	#1506	semidirect
391. C₅	#68310	direct (N × H)
409. C₃	C₅×#1506	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5622	2
C₅×#4719	2
C₅×#4648	2
C₃×C₅×#328	2
C₅×#4538	2
C₅×#4467	2
C₅×#4452	2
C₅×#1506	3
#68310	5

Subgroups — C5×#68310 #14283064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68310 #14283064