Subgroups — C₅×#68278 #14283021

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5596	C₂	1	1	yes	maximal, characteristic
2	C₅×#4719	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4648	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4537	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1505	—	3	1	no	maximal, Hall
9	C₂×C₅×#237	—	2	1	no	—
10	C₃×C₅×#121	C₂²	1	1	yes	characteristic
11	C₅×#940	C₄	1	1	yes	characteristic, complemented
12	C₅×#939	—	2	1	no	—
13	C₅×D₄×Dic₃	C₄	1	1	yes	characteristic, complemented
14	C₅×#930	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×Dic₃	—	2	1	no	—
16	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₅×#893	C₄	1	1	yes	characteristic, complemented
18	C₅×#888	C₄	1	1	yes	characteristic, complemented
19	C₅×#885	—	2	1	no	—
20	C₅×#67⋊C₄	—	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	#68278	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₅×#532	—	3	1	no	—
30	C₅×#432	—	3	1	no	—
31	C₅×#376	—	3	1	no	—
32	C₅×#327	—	3	1	no	—
33	C₅×#327	—	3	1	no	—
34	C₅×#327	—	3	1	no	—
35	C₅×#327	D₃	1	1	yes	characteristic, complemented
36	C₂×C₃×C₅×D₄	—	2	1	no	—
37	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	—	2	1	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×#67	—	2	1	no	—
41	C₂×C₅×#67	—	2	1	no	—
42	C₂×C₅×#67	—	2	1	no	—
43	C₂²×C₅×Dic₃	—	2	1	no	—
44	C₂²×C₅×Dic₃	—	2	1	no	—
45	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
46	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
47	C₅×#237	—	4	2	no	—
48	C₅×#237	—	4	2	no	—
49	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅×D₃	—	2	1	no	—
51	C₂×C₅×Dic₆	—	2	1	no	—
52	C₂×C₃×C₅×C₈	—	2	1	no	—
53	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
55	C₅×#217	—	2	1	no	—
56	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
57	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
58	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
59	C₅×Dic₃⋊C₄	—	2	1	no	—
60	C₄×C₅×Dic₃	—	2	4	no	—
61	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
64	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
65	C₂×C₅×C₃⋊C₈	—	2	1	no	—
66	#5596	C₂×C₅	1	1	yes	characteristic
67	#4719	C₂×C₅	1	1	yes	characteristic, complemented
68	#4648	C₂×C₅	1	1	yes	characteristic, complemented
69	C₃×#327	C₂×C₅	1	1	yes	characteristic, complemented
70	#4537	C₂×C₅	1	1	yes	characteristic, complemented
71	#4470	C₂×C₅	1	1	yes	characteristic, complemented
72	#4451	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₅×#41	—	6	1	no	—
74	C₅×C₄⋊Q₈	—	3	1	no	—
75	C₅×#127	—	3	1	no	—
76	C₅×#124	—	6	1	no	—
77	C₅×#121	—	3	1	no	—
78	C₅×#121	—	6	1	no	—
79	C₅×#121	C₂×D₃	1	1	yes	characteristic
80	C₄×C₅×D₄	—	3	1	no	—
81	C₄×C₅×D₄	—	3	1	no	—
82	C₄×C₅×D₄	—	6	1	no	—
83	C₂×C₄²×C₅	—	3	1	no	—
84	C₂×C₄²×C₅	—	6	1	no	—
85	C₅×C₄⋊C₈	—	6	1	no	—
86	C₅×C₂²⋊C₈	—	3	1	no	—
87	C₅×C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
88	C₅×#97	—	6	1	no	—
89	C₅×#95	—	3	1	no	—
90	C₅×#95	—	6	1	no	—
91	C₅×#95	C₂×D₃	1	1	yes	characteristic
92	#1505	—	3	1	no	Sylow
93	C₂³×C₃×C₅	—	2	1	no	—
94	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₅×D₃	—	2	1	no	—
96	C₃×C₅×D₄	—	4	1	no	—
97	C₃×C₅×D₄	—	4	1	no	—
98	C₃×C₅×D₄	—	4	1	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	—	2	1	no	—
101	C₂×C₃×C₄×C₅	—	4	1	no	—
102	C₂×C₃×C₄×C₅	—	4	1	no	—
103	C₂×C₃×C₄×C₅	—	4	1	no	—
104	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₅×#67	—	4	2	no	—
107	C₅×#67	—	4	1	no	—
108	C₅×#67	—	4	1	no	—
109	C₅×#67	—	4	2	no	—
110	C₂×C₅×Dic₃	—	2	1	no	—
111	C₂×C₅×Dic₃	—	2	1	no	—
112	C₂×C₅×Dic₃	—	2	2	no	—
113	C₂×C₅×Dic₃	—	2	2	no	—
114	C₂×C₅×Dic₃	—	2	1	no	—
115	C₂×C₅×Dic₃	—	4	2	no	—
116	C₂×C₅×Dic₃	—	4	1	no	—
117	C₂×C₅×Dic₃	—	4	1	no	—
118	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₅×D₁₂	—	4	1	no	—
123	C₄×C₅×D₃	—	4	2	no	—
124	C₅×Dic₆	—	4	2	no	—
125	C₃×C₅×C₈	—	4	1	no	—
126	C₅×C₃⋊C₈	—	4	1	no	—
127	C₂×#237	—	2	1	no	—
128	C₃×#121	C₂²×C₅	1	1	yes	characteristic
129	#940	C₄×C₅	1	1	yes	characteristic, complemented
130	#939	—	2	1	no	—
131	D₄×Dic₃	C₄×C₅	1	1	yes	characteristic, complemented
132	#930	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₄×Dic₃	—	2	1	no	—
134	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
135	#893	C₄×C₅	1	1	yes	characteristic, complemented
136	#888	C₄×C₅	1	1	yes	characteristic, complemented
137	#885	—	2	1	no	—
138	#67⋊C₄	—	2	1	no	—
139	C₃×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
140	C₃×#95	C₂²×C₅	1	1	yes	characteristic
141	C₃⋊#95	—	2	1	no	—
142	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
143	#830	C₂²×C₅	1	1	yes	characteristic
144	#815	—	2	1	no	—
145	Dic₃⋊C₈	—	2	1	no	—
146	C₅×#41	—	12	2	no	—
147	C₅×#41	—	12	2	no	—
148	C₂×C₅×Q₈	—	6	1	no	—
149	C₂×C₅×D₄	—	2	1	no	—
150	C₂×C₅×D₄	—	3	1	no	—
151	C₂×C₅×D₄	—	6	1	no	—
152	C₂×C₅×D₄	—	6	1	no	—
153	C₂×C₅×D₄	—	6	1	no	—
154	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
155	C₂²×C₄×C₅	—	2	1	no	—
156	C₂²×C₄×C₅	—	3	1	no	—
157	C₂²×C₄×C₅	—	3	1	no	—
158	C₂²×C₄×C₅	—	6	1	no	—
159	C₂²×C₄×C₅	—	6	1	no	—
160	C₂²×C₄×C₅	—	6	1	no	—
161	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₅×C₈	—	2	1	no	—
163	C₂×C₅×C₈	—	6	1	no	—
164	C₅×C₄⋊C₄	—	3	1	no	—
165	C₅×C₄⋊C₄	—	6	1	no	—
166	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
167	C₅×C₂²⋊C₄	—	2	1	no	—
168	C₅×C₂²⋊C₄	—	6	1	no	—
169	C₅×C₂²⋊C₄	—	6	1	no	—
170	C₅×C₂²⋊C₄	—	6	1	no	—
171	C₄²×C₅	—	3	1	no	—
172	C₄²×C₅	—	3	1	no	—
173	C₄²×C₅	—	3	1	no	—
174	C₄²×C₅	—	3	1	no	—
175	C₄²×C₅	—	6	4	no	—
176	#532	—	3	1	no	—
177	#432	—	3	1	no	—
178	#376	—	3	1	no	—
179	#327	—	3	1	no	—
180	#327	—	3	1	no	—
181	#327	—	3	1	no	—
182	#327	C₅×D₃	1	1	yes	characteristic, complemented
183	C₂²×C₃×C₅	—	2	1	no	—
184	C₂²×C₃×C₅	—	2	1	no	—
185	C₂²×C₃×C₅	—	4	1	no	—
186	C₂²×C₃×C₅	—	4	1	no	—
187	C₂²×C₃×C₅	—	4	1	no	—
188	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂×C₅×D₃	—	4	1	no	—
192	C₂×C₅×D₃	—	4	1	no	—
193	C₂×C₅×D₃	—	4	1	no	—
194	C₃×C₄×C₅	—	2	1	no	—
195	C₃×C₄×C₅	—	2	1	no	—
196	C₃×C₄×C₅	—	4	2	no	—
197	C₃×C₄×C₅	—	4	1	no	—
198	C₅×Dic₃	—	2	1	no	—
199	C₅×Dic₃	—	2	1	no	—
200	C₅×Dic₃	—	2	1	no	—
201	C₅×Dic₃	—	4	2	no	—
202	C₅×Dic₃	—	4	1	no	—
203	C₅×Dic₃	#104	1	2	yes	—
204	C₂×C₃×D₄	—	2	1	no	—
205	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₃×C₄	—	2	1	no	—
207	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
208	C₂×#67	—	2	1	no	—
209	C₂×#67	—	2	1	no	—
210	C₂×#67	—	2	1	no	—
211	C₂²×Dic₃	—	2	1	no	—
212	C₂²×Dic₃	—	2	1	no	—
213	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
214	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
215	#237	—	4	2	no	—
216	#237	—	4	2	no	—
217	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
218	C₂×C₄×D₃	—	2	1	no	—
219	C₂×Dic₆	—	2	1	no	—
220	C₂×C₃×C₈	—	2	1	no	—
221	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
222	C₃×C₂²⋊C₄	—	2	1	no	—
223	#217	—	2	1	no	—
224	(C₂×D₃)⋊C₄	—	2	1	no	—
225	(C₂×D₃)⋊C₄	—	2	1	no	—
226	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
227	Dic₃⋊C₄	—	2	1	no	—
228	C₄×Dic₃	—	2	4	no	—
229	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
230	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
231	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
232	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
233	C₂×C₃⋊C₈	—	2	1	no	—
234	C₂³×C₅	—	2	1	no	—
235	C₂³×C₅	—	6	1	no	—
236	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
237	C₅×Q₈	—	12	2	no	—
238	C₅×D₄	—	4	1	no	—
239	C₅×D₄	—	4	1	no	—
240	C₅×D₄	—	4	1	no	—
241	C₅×D₄	—	12	1	no	—
242	C₅×D₄	—	12	2	no	—
243	C₅×D₄	—	12	2	no	—
244	C₅×D₄	—	12	1	no	—
245	C₅×D₄	—	12	1	no	—
246	C₂×C₄×C₅	—	2	2	no	—
247	C₂×C₄×C₅	—	2	1	no	—
248	C₂×C₄×C₅	—	3	1	no	—
249	C₂×C₄×C₅	—	3	1	no	—
250	C₂×C₄×C₅	—	3	1	no	—
251	C₂×C₄×C₅	—	3	1	no	—
252	C₂×C₄×C₅	—	4	1	no	—
253	C₂×C₄×C₅	—	4	1	no	—
254	C₂×C₄×C₅	—	4	1	no	—
255	C₂×C₄×C₅	—	6	2	no	—
256	C₂×C₄×C₅	—	6	1	no	—
257	C₂×C₄×C₅	—	6	1	no	—
258	C₂×C₄×C₅	—	6	2	no	—
259	C₂×C₄×C₅	—	6	1	no	—
260	C₂×C₄×C₅	—	12	2	no	—
261	C₂×C₄×C₅	—	12	2	no	—
262	C₂×C₄×C₅	—	12	1	no	—
263	C₂×C₄×C₅	—	12	1	no	—
264	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
265	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
266	C₅×C₈	—	4	1	no	—
267	C₅×C₈	—	12	1	no	—
268	C₂×#41	—	6	1	no	—
269	C₄⋊Q₈	—	3	1	no	—
270	#127	—	3	1	no	—
271	#124	—	6	1	no	—
272	#121	—	3	1	no	—
273	#121	—	6	1	no	—
274	#121	C₂×C₅×D₃	1	1	yes	characteristic
275	C₄×D₄	—	3	1	no	—
276	C₄×D₄	—	3	1	no	—
277	C₄×D₄	—	6	1	no	—
278	C₂×C₄²	—	3	1	no	—
279	C₂×C₄²	—	6	1	no	—
280	C₄⋊C₈	—	6	1	no	—
281	C₂²⋊C₈	—	3	1	no	—
282	C₂²⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
283	#97	—	6	1	no	—
284	#95	—	3	1	no	—
285	#95	—	6	1	no	—
286	#95	C₂×C₅×D₃	1	1	yes	characteristic
287	C₂×C₃×C₅	—	2	1	no	—
288	C₂×C₃×C₅	—	2	1	no	—
289	C₂×C₃×C₅	—	8	1	no	—
290	C₂×C₃×C₅	#410	1	1	yes	characteristic
291	C₂×C₃×C₅	#418	1	1	yes	characteristic
292	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
293	C₅×D₃	—	8	1	no	—
294	C₂³×C₃	—	2	1	no	—
295	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
296	C₂²×D₃	—	2	1	no	—
297	C₃×D₄	—	4	1	no	—
298	C₃×D₄	—	4	1	no	—
299	C₃×D₄	—	4	1	no	—
300	C₂×C₃×C₄	—	2	2	no	—
301	C₂×C₃×C₄	—	2	1	no	—
302	C₂×C₃×C₄	—	4	1	no	—
303	C₂×C₃×C₄	—	4	1	no	—
304	C₂×C₃×C₄	—	4	1	no	—
305	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
306	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
307	#67	—	4	2	no	—
308	#67	—	4	1	no	—
309	#67	—	4	1	no	—
310	#67	—	4	2	no	—
311	C₂×Dic₃	—	2	1	no	—
312	C₂×Dic₃	—	2	1	no	—
313	C₂×Dic₃	—	2	2	no	—
314	C₂×Dic₃	—	2	2	no	—
315	C₂×Dic₃	—	2	1	no	—
316	C₂×Dic₃	—	4	2	no	—
317	C₂×Dic₃	—	4	1	no	—
318	C₂×Dic₃	—	4	1	no	—
319	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
321	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
322	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
323	D₁₂	—	4	1	no	—
324	C₄×D₃	—	4	2	no	—
325	Dic₆	—	4	2	no	—
326	C₃×C₈	—	4	1	no	—
327	C₃⋊C₈	—	4	1	no	—
328	C₂²×C₅	—	2	1	no	—
329	C₂²×C₅	—	2	1	no	—
330	C₂²×C₅	—	4	1	no	—
331	C₂²×C₅	—	4	1	no	—
332	C₂²×C₅	—	4	1	no	—
333	C₂²×C₅	—	12	1	no	—
334	C₂²×C₅	—	12	1	no	—
335	C₂²×C₅	—	12	1	no	—
336	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
337	C₂²×C₅	#911	1	1	yes	characteristic
338	C₂²×C₅	#914	1	1	yes	characteristic
339	C₄×C₅	—	2	1	no	—
340	C₄×C₅	—	2	1	no	—
341	C₄×C₅	—	3	2	no	—
342	C₄×C₅	—	4	2	no	—
343	C₄×C₅	—	4	1	no	—
344	C₄×C₅	—	6	1	no	—
345	C₄×C₅	—	6	1	no	—
346	C₄×C₅	—	6	1	no	—
347	C₄×C₅	—	12	2	no	—
348	C₄×C₅	—	12	1	no	—
349	#41	—	12	2	no	—
350	#41	—	12	2	no	—
351	C₂×Q₈	—	6	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	3	1	no	—
354	C₂×D₄	—	6	1	no	—
355	C₂×D₄	—	6	1	no	—
356	C₂×D₄	—	6	1	no	—
357	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	3	1	no	—
360	C₂²×C₄	—	3	1	no	—
361	C₂²×C₄	—	6	1	no	—
362	C₂²×C₄	—	6	1	no	—
363	C₂²×C₄	—	6	1	no	—
364	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	6	1	no	—
367	C₄⋊C₄	—	3	1	no	—
368	C₄⋊C₄	—	6	1	no	—
369	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	6	1	no	—
372	C₂²⋊C₄	—	6	1	no	—
373	C₂²⋊C₄	—	6	1	no	—
374	C₄²	—	3	1	no	—
375	C₄²	—	3	1	no	—
376	C₄²	—	3	1	no	—
377	C₄²	—	3	1	no	—
378	C₄²	—	6	4	no	—
379	C₃×C₅	#1505	1	1	yes	characteristic, complemented, Hall
380	C₂²×C₃	—	2	1	no	—
381	C₂²×C₃	—	2	1	no	—
382	C₂²×C₃	—	4	1	no	—
383	C₂²×C₃	—	4	1	no	—
384	C₂²×C₃	—	4	1	no	—
385	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
386	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
387	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
388	C₂×D₃	—	4	1	no	—
389	C₂×D₃	—	4	1	no	—
390	C₂×D₃	—	4	1	no	—
391	C₃×C₄	—	2	1	no	—
392	C₃×C₄	—	2	1	no	—
393	C₃×C₄	—	4	2	no	—
394	C₃×C₄	—	4	1	no	—
395	Dic₃	—	2	1	no	—
396	Dic₃	—	2	1	no	—
397	Dic₃	—	2	1	no	—
398	Dic₃	—	4	2	no	—
399	Dic₃	—	4	1	no	—
400	Dic₃	C₅×#104	1	2	yes	—
401	C₂×C₅	—	2	1	no	—
402	C₂×C₅	—	2	1	no	—
403	C₂×C₅	—	8	1	no	—
404	C₂×C₅	—	24	1	no	—
405	C₂×C₅	#4740	1	1	yes	characteristic
406	C₂×C₅	#4770	1	1	yes	characteristic
407	C₂×C₅	D₃×#104	1	1	yes	characteristic
408	C₂³	—	2	1	no	—
409	C₂³	—	6	1	no	—
410	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
411	Q₈	—	12	2	no	—
412	D₄	—	4	1	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	1	no	—
415	D₄	—	12	1	no	—
416	D₄	—	12	2	no	—
417	D₄	—	12	2	no	—
418	D₄	—	12	1	no	—
419	D₄	—	12	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	3	1	no	—
423	C₂×C₄	—	3	1	no	—
424	C₂×C₄	—	3	1	no	—
425	C₂×C₄	—	3	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	6	2	no	—
430	C₂×C₄	—	6	1	no	—
431	C₂×C₄	—	6	1	no	—
432	C₂×C₄	—	6	2	no	—
433	C₂×C₄	—	6	1	no	—
434	C₂×C₄	—	12	2	no	—
435	C₂×C₄	—	12	2	no	—
436	C₂×C₄	—	12	1	no	—
437	C₂×C₄	—	12	1	no	—
438	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
439	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
440	C₈	—	4	1	no	—
441	C₈	—	12	1	no	—
442	C₂×C₃	—	2	1	no	—
443	C₂×C₃	—	2	1	no	—
444	C₂×C₃	—	8	1	no	—
445	C₂×C₃	C₅×#410	1	1	yes	characteristic
446	C₂×C₃	C₅×#418	1	1	yes	characteristic
447	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
448	D₃	—	8	1	no	—
449	C₅	#68278	1	1	yes	characteristic, direct product factor, Sylow
450	C₂²	—	2	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	12	1	no	—
456	C₂²	—	12	1	no	—
457	C₂²	—	12	1	no	—
458	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
459	C₂²	C₅×#911	1	1	yes	characteristic
460	C₂²	C₅×#914	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	3	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	6	1	no	—
467	C₄	—	6	1	no	—
468	C₄	—	6	1	no	—
469	C₄	—	12	2	no	—
470	C₄	—	12	1	no	—
471	C₃	C₅×#1505	1	1	yes	characteristic, complemented, Sylow
472	C₂	—	2	1	no	—
473	C₂	—	2	1	no	—
474	C₂	—	8	1	no	—
475	C₂	—	24	1	no	—
476	C₂	C₅×#4740	1	1	yes	characteristic
477	C₂	C₅×#4770	1	1	yes	characteristic
478	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×#327	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5596	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#327	D₃
hypocenter	C₃	C₅×#1505
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#327

Complements

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

N	H ≅ G/N	type
2. C₅×#4719	C₂	semidirect
3. C₅×#4648	C₂	semidirect
4. C₃×C₅×#327	C₂	semidirect
5. C₅×#4537	C₂	semidirect
6. C₅×#4470	C₂	semidirect
7. C₅×#4451	C₂	semidirect
11. C₅×#940	C₄	semidirect
13. C₅×D₄×Dic₃	C₄	semidirect
17. C₅×#893	C₄	semidirect
18. C₅×#888	C₄	semidirect
28. #68278	C₅	direct (N × H)
35. C₅×#327	D₃	semidirect
67. #4719	C₂×C₅	semidirect
68. #4648	C₂×C₅	semidirect
69. C₃×#327	C₂×C₅	semidirect
70. #4537	C₂×C₅	semidirect
71. #4470	C₂×C₅	semidirect
72. #4451	C₂×C₅	semidirect
129. #940	C₄×C₅	semidirect
131. D₄×Dic₃	C₄×C₅	semidirect
135. #893	C₄×C₅	semidirect
136. #888	C₄×C₅	semidirect
182. #327	C₅×D₃	semidirect
379. C₃×C₅	#1505	semidirect
449. C₅	#68278	direct (N × H)
471. C₃	C₅×#1505	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5596	2
C₅×#4719	2
C₅×#4648	2
C₃×C₅×#327	2
C₅×#4537	2
C₅×#4470	2
C₅×#4451	2
C₅×#1505	3
#68278	5

Subgroups — C5×#68278 #14283021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68278 #14283021