Subgroups — C₅×#83570 #14460267

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#550	C₂	1	1	yes	maximal, characteristic
2	C₅×#5542	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5239	C₂	1	2	yes	maximal, complemented
4	C₅×#5233	C₂	1	2	yes	maximal, complemented
5	C₅×#5075	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5070	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5041	C₂	1	2	yes	maximal, complemented
8	C₅×#5035	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×#216	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#5020	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3304	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
14	C₃×C₅×#127	—	2	1	no	—
15	C₃×C₅×#127	C₂²	1	1	yes	characteristic
16	C₃×C₅×#121	—	2	2	no	—
17	C₃×C₅×#121	C₂²	1	2	yes	—
18	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₅×D₄	—	2	1	no	—
20	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₅×#216	C₂²	1	1	yes	characteristic
22	C₅×#936	—	2	1	no	—
23	C₂×C₅×#214	C₂²	1	2	yes	—
24	C₅×#926	C₂²	1	2	yes	—
25	C₅×#924	C₂²	1	2	yes	—
26	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
27	C₅×#887	—	2	1	no	—
28	C₅×#879	—	2	2	no	—
29	C₅×#869	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
31	C₅×#835	C₂²	1	1	yes	characteristic
32	C₅×#835	C₂²	1	2	yes	—
33	C₅×#832	C₂²	1	2	yes	—
34	C₅×#832	C₂²	1	2	yes	—
35	C₅×#832	C₂²	1	2	yes	—
36	C₅×#830	C₂²	1	2	yes	—
37	C₅×#810	C₂²	1	1	yes	characteristic
38	C₅×#808	C₂²	1	2	yes	—
39	C₅×#807	C₂²	1	1	yes	characteristic
40	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	#83570	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#556	—	3	1	no	—
44	C₅×#550	D₃	1	1	yes	characteristic
45	C₅×#486	—	3	1	no	—
46	C₅×#482	—	3	2	no	—
47	C₅×#475	—	3	1	no	—
48	C₅×#465	—	3	2	no	—
49	C₅×#460	—	3	1	no	—
50	C₅×#452	—	3	2	no	—
51	C₄×C₅×Q₁₆	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₅×#416	—	3	2	no	—
54	C₃×C₅×#41	—	2	4	no	—
55	C₃×C₅×#41	—	4	2	no	—
56	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₅×D₄	—	2	2	no	—
58	C₂×C₃×C₅×D₄	—	2	2	no	—
59	C₂×C₃×C₅×D₄	—	2	1	no	—
60	C₂×C₃×C₅×D₄	—	2	1	no	—
61	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
62	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	—	2	2	no	—
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂×C₅×Dic₆	—	2	1	no	—
66	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
67	C₃×C₅×C₄⋊C₄	—	2	1	no	—
68	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
72	C₃×C₄²×C₅	—	2	1	no	—
73	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
74	C₅×#217	—	2	2	no	—
75	C₅×#217	—	2	2	no	—
76	C₅×#216	—	2	2	no	—
77	C₅×#214	—	4	2	no	—
78	C₅×C₄⋊Dic₃	—	2	1	no	—
79	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
81	C₅×Dic₃⋊C₄	—	2	1	no	—
82	C₅×Dic₃⋊C₄	—	2	2	no	—
83	C₄×C₅×Dic₃	—	2	1	no	—
84	C₄×C₅×Dic₃	—	2	2	no	—
85	C₂×C₅×C₃⋊C₈	—	4	2	no	—
86	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
87	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
88	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₃×#550	C₂×C₅	1	1	yes	characteristic
90	#5542	C₂×C₅	1	1	yes	characteristic, complemented
91	#5239	C₂×C₅	1	2	yes	complemented
92	#5233	C₂×C₅	1	2	yes	complemented
93	#5075	C₂×C₅	1	1	yes	characteristic, complemented
94	#5070	C₂×C₅	1	1	yes	characteristic, complemented
95	#5041	C₂×C₅	1	2	yes	complemented
96	#5035	C₂×C₅	1	2	yes	complemented
97	C₄×#216	C₂×C₅	1	1	yes	characteristic, complemented
98	#5020	C₂×C₅	1	1	yes	characteristic, complemented
99	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×C₅×#41	C₂×D₃	1	2	yes	—
101	C₂×C₅×Q₁₆	—	3	1	no	—
102	C₂×C₅×SD₁₆	—	3	2	no	—
103	C₂²×C₅×C₈	—	3	2	no	—
104	C₅×C₄⋊Q₈	—	3	1	no	—
105	C₅×#127	—	2	1	no	—
106	C₅×#127	C₂×D₃	1	1	yes	characteristic
107	C₅×#126	—	6	2	no	—
108	C₅×#124	—	6	1	no	—
109	C₅×C₂²⋊Q₈	—	3	2	no	—
110	C₅×#121	—	2	2	no	—
111	C₅×#121	C₂×D₃	1	2	yes	—
112	C₄×C₅×Q₈	—	3	1	no	—
113	C₄×C₅×Q₈	—	6	1	no	—
114	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
115	C₄×C₅×D₄	—	2	1	no	—
116	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
117	C₅×#117	—	3	2	no	—
118	C₅×#107	—	3	1	no	—
119	C₅×#106	—	3	2	no	—
120	C₅×C₄⋊C₈	—	3	1	no	—
121	C₅×#103	—	3	1	no	—
122	C₅×#103	—	3	2	no	—
123	C₅×#103	—	3	1	no	—
124	C₅×#102	—	3	2	no	—
125	C₅×#102	—	3	2	no	—
126	C₅×#102	—	3	2	no	—
127	C₅×C₂²⋊C₈	—	3	2	no	—
128	C₄×C₅×C₈	—	3	1	no	—
129	#3304	—	3	1	no	Sylow
130	C₂³×C₃×C₅	—	2	2	no	—
131	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
132	C₃×C₅×Q₈	—	2	1	no	—
133	C₃×C₅×Q₈	#41	1	2	yes	—
134	C₃×C₅×D₄	—	2	2	no	—
135	C₃×C₅×D₄	—	2	2	no	—
136	C₃×C₅×D₄	—	2	2	no	—
137	C₃×C₅×D₄	—	4	4	no	—
138	C₃×C₅×D₄	—	4	2	no	—
139	C₃×C₅×D₄	—	8	1	no	—
140	C₃×C₅×D₄	—	8	2	no	—
141	C₂×C₃×C₄×C₅	—	2	1	no	—
142	C₂×C₃×C₄×C₅	—	2	2	no	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	—	2	1	no	—
145	C₂×C₃×C₄×C₅	—	4	4	no	—
146	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
147	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
150	C₂×C₅×Dic₃	—	2	1	no	—
151	C₂×C₅×Dic₃	—	2	1	no	—
152	C₂×C₅×Dic₃	—	2	2	no	—
153	C₅×Dic₆	—	2	1	no	—
154	C₅×Dic₆	—	2	1	no	—
155	C₅×Dic₆	—	4	2	no	—
156	C₅×C₃⋊C₈	—	2	1	no	—
157	C₅×C₃⋊C₈	—	2	1	no	—
158	C₅×C₃⋊C₈	—	2	2	no	—
159	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
160	C₃×#127	—	2	1	no	—
161	C₃×#127	C₂²×C₅	1	1	yes	characteristic
162	C₃×#121	—	2	2	no	—
163	C₃×#121	C₂²×C₅	1	2	yes	—
164	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
165	C₃×C₄×D₄	—	2	1	no	—
166	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
167	C₂×#216	C₂²×C₅	1	1	yes	characteristic
168	#936	—	2	1	no	—
169	C₂×#214	C₂²×C₅	1	2	yes	—
170	#926	C₂²×C₅	1	2	yes	—
171	#924	C₂²×C₅	1	2	yes	—
172	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
173	#887	—	2	1	no	—
174	#879	—	2	2	no	—
175	#869	C₂²×C₅	1	1	yes	characteristic
176	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
177	#835	C₂²×C₅	1	1	yes	characteristic
178	#835	C₂²×C₅	1	2	yes	—
179	#832	C₂²×C₅	1	2	yes	—
180	#832	C₂²×C₅	1	2	yes	—
181	#832	C₂²×C₅	1	2	yes	—
182	#830	C₂²×C₅	1	2	yes	—
183	#810	C₂²×C₅	1	1	yes	characteristic
184	#808	C₂²×C₅	1	2	yes	—
185	#807	C₂²×C₅	1	1	yes	characteristic
186	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
187	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
188	C₅×#41	—	2	4	no	—
189	C₅×#41	—	4	2	no	—
190	C₂×C₅×Q₈	—	3	1	no	—
191	C₂×C₅×Q₈	—	6	1	no	—
192	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
193	C₂×C₅×D₄	—	2	2	no	—
194	C₂×C₅×D₄	—	2	2	no	—
195	C₂×C₅×D₄	—	2	1	no	—
196	C₂×C₅×D₄	—	2	1	no	—
197	C₂×C₅×D₄	#67	1	1	yes	characteristic
198	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
199	C₂²×C₄×C₅	—	2	2	no	—
200	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
201	C₅×Q₁₆	—	6	2	no	—
202	C₅×SD₁₆	—	12	2	no	—
203	C₂×C₅×C₈	—	3	1	no	—
204	C₂×C₅×C₈	—	3	2	no	—
205	C₂×C₅×C₈	—	3	1	no	—
206	C₂×C₅×C₈	—	12	2	no	—
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	3	1	no	—
209	C₅×C₄⋊C₄	—	3	2	no	—
210	C₅×C₄⋊C₄	—	6	1	no	—
211	C₅×C₄⋊C₄	—	6	2	no	—
212	C₅×C₄⋊C₄	—	6	1	no	—
213	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
214	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
215	C₅×C₂²⋊C₄	—	2	2	no	—
216	C₅×C₂²⋊C₄	—	6	2	no	—
217	C₅×C₂²⋊C₄	—	6	2	no	—
218	C₅×C₂²⋊C₄	#67	1	2	yes	—
219	C₄²×C₅	—	2	1	no	—
220	C₄²×C₅	—	6	2	no	—
221	C₄²×C₅	—	6	1	no	—
222	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
223	#556	—	3	1	no	—
224	#550	C₅×D₃	1	1	yes	characteristic
225	#486	—	3	1	no	—
226	#482	—	3	2	no	—
227	#475	—	3	1	no	—
228	#465	—	3	2	no	—
229	#460	—	3	1	no	—
230	#452	—	3	2	no	—
231	C₄×Q₁₆	—	3	1	no	—
232	C₈×D₄	—	3	1	no	—
233	#416	—	3	2	no	—
234	C₂²×C₃×C₅	—	2	2	no	—
235	C₂²×C₃×C₅	—	2	2	no	—
236	C₂²×C₃×C₅	—	2	2	no	—
237	C₂²×C₃×C₅	—	4	2	no	—
238	C₂²×C₃×C₅	—	4	2	no	—
239	C₂²×C₃×C₅	—	4	2	no	—
240	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
241	C₃×C₄×C₅	—	2	2	no	—
242	C₃×C₄×C₅	—	2	2	no	—
243	C₃×C₄×C₅	—	2	1	no	—
244	C₃×C₄×C₅	—	4	1	no	—
245	C₃×C₄×C₅	#135	1	2	yes	—
246	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
247	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
248	C₅×Dic₃	—	4	1	no	—
249	C₅×Dic₃	—	4	2	no	—
250	C₅×Dic₃	—	4	1	no	—
251	C₃×#41	—	2	4	no	—
252	C₃×#41	—	4	2	no	—
253	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
254	C₂×C₃×D₄	—	2	2	no	—
255	C₂×C₃×D₄	—	2	2	no	—
256	C₂×C₃×D₄	—	2	1	no	—
257	C₂×C₃×D₄	—	2	1	no	—
258	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
259	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
260	C₂²×C₃×C₄	—	2	2	no	—
261	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
262	C₂×Dic₆	—	2	1	no	—
263	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
264	C₃×C₄⋊C₄	—	2	1	no	—
265	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
266	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
267	C₃×C₂²⋊C₄	—	2	2	no	—
268	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
269	C₃×C₄²	—	2	1	no	—
270	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
271	#217	—	2	2	no	—
272	#217	—	2	2	no	—
273	#216	—	2	2	no	—
274	#214	—	4	2	no	—
275	C₄⋊Dic₃	—	2	1	no	—
276	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
277	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
278	Dic₃⋊C₄	—	2	1	no	—
279	Dic₃⋊C₄	—	2	2	no	—
280	C₄×Dic₃	—	2	1	no	—
281	C₄×Dic₃	—	2	2	no	—
282	C₂×C₃⋊C₈	—	4	2	no	—
283	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
284	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
285	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
286	C₂³×C₅	—	2	2	no	—
287	C₂³×C₅	C₂×#67	1	2	yes	—
288	C₅×Q₈	—	2	1	no	—
289	C₅×Q₈	—	6	1	no	—
290	C₅×Q₈	—	6	1	no	—
291	C₅×Q₈	—	12	2	no	—
292	C₅×Q₈	#237	1	2	yes	—
293	C₅×D₄	—	2	2	no	—
294	C₅×D₄	—	2	2	no	—
295	C₅×D₄	—	2	2	no	—
296	C₅×D₄	—	4	4	no	—
297	C₅×D₄	—	4	2	no	—
298	C₅×D₄	—	8	1	no	—
299	C₅×D₄	—	8	2	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	2	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	4	4	no	—
305	C₂×C₄×C₅	—	6	1	no	—
306	C₂×C₄×C₅	—	6	2	no	—
307	C₂×C₄×C₅	—	6	1	no	—
308	C₂×C₄×C₅	C₂×#67	1	2	yes	—
309	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
312	C₅×C₈	—	6	1	no	—
313	C₅×C₈	—	6	1	no	—
314	C₅×C₈	—	6	2	no	—
315	C₂×#41	C₂×C₅×D₃	1	2	yes	—
316	C₂×Q₁₆	—	3	1	no	—
317	C₂×SD₁₆	—	3	2	no	—
318	C₂²×C₈	—	3	2	no	—
319	C₄⋊Q₈	—	3	1	no	—
320	#127	—	2	1	no	—
321	#127	C₂×C₅×D₃	1	1	yes	characteristic
322	#126	—	6	2	no	—
323	#124	—	6	1	no	—
324	C₂²⋊Q₈	—	3	2	no	—
325	#121	—	2	2	no	—
326	#121	C₂×C₅×D₃	1	2	yes	—
327	C₄×Q₈	—	3	1	no	—
328	C₄×Q₈	—	6	1	no	—
329	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
332	#117	—	3	2	no	—
333	#107	—	3	1	no	—
334	#106	—	3	2	no	—
335	C₄⋊C₈	—	3	1	no	—
336	#103	—	3	1	no	—
337	#103	—	3	2	no	—
338	#103	—	3	1	no	—
339	#102	—	3	2	no	—
340	#102	—	3	2	no	—
341	#102	—	3	2	no	—
342	C₂²⋊C₈	—	3	2	no	—
343	C₄×C₈	—	3	1	no	—
344	C₂×C₃×C₅	—	4	2	no	—
345	C₂×C₃×C₅	—	8	2	no	—
346	C₂×C₃×C₅	#546	1	1	yes	characteristic
347	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
348	C₂×C₃×C₅	#577	1	1	yes	characteristic
349	C₂³×C₃	—	2	2	no	—
350	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
351	C₃×Q₈	—	2	1	no	—
352	C₃×Q₈	C₅×#41	1	2	yes	—
353	C₃×D₄	—	2	2	no	—
354	C₃×D₄	—	2	2	no	—
355	C₃×D₄	—	2	2	no	—
356	C₃×D₄	—	4	4	no	—
357	C₃×D₄	—	4	2	no	—
358	C₃×D₄	—	8	1	no	—
359	C₃×D₄	—	8	2	no	—
360	C₂×C₃×C₄	—	2	1	no	—
361	C₂×C₃×C₄	—	2	2	no	—
362	C₂×C₃×C₄	—	2	2	no	—
363	C₂×C₃×C₄	—	2	1	no	—
364	C₂×C₃×C₄	—	4	4	no	—
365	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
366	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
367	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
368	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
369	C₂×Dic₃	—	2	1	no	—
370	C₂×Dic₃	—	2	1	no	—
371	C₂×Dic₃	—	2	2	no	—
372	Dic₆	—	2	1	no	—
373	Dic₆	—	2	1	no	—
374	Dic₆	—	4	2	no	—
375	C₃⋊C₈	—	2	1	no	—
376	C₃⋊C₈	—	2	1	no	—
377	C₃⋊C₈	—	2	2	no	—
378	C₂²×C₅	—	2	2	no	—
379	C₂²×C₅	—	2	2	no	—
380	C₂²×C₅	—	2	2	no	—
381	C₂²×C₅	—	4	2	no	—
382	C₂²×C₅	—	4	2	no	—
383	C₂²×C₅	—	4	2	no	—
384	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
385	C₄×C₅	—	2	2	no	—
386	C₄×C₅	—	2	2	no	—
387	C₄×C₅	—	2	1	no	—
388	C₄×C₅	—	4	1	no	—
389	C₄×C₅	—	12	1	no	—
390	C₄×C₅	—	12	2	no	—
391	C₄×C₅	—	12	1	no	—
392	C₄×C₅	#950	1	2	yes	—
393	C₄×C₅	C₂×#237	1	1	yes	characteristic
394	C₄×C₅	#1004	1	1	yes	characteristic
395	#41	—	2	4	no	—
396	#41	—	4	2	no	—
397	C₂×Q₈	—	3	1	no	—
398	C₂×Q₈	—	6	1	no	—
399	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	C₅×#67	1	1	yes	characteristic
405	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
408	Q₁₆	—	6	2	no	—
409	SD₁₆	—	12	2	no	—
410	C₂×C₈	—	3	1	no	—
411	C₂×C₈	—	3	1	no	—
412	C₂×C₈	—	3	2	no	—
413	C₂×C₈	—	12	2	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	3	2	no	—
416	C₄⋊C₄	—	3	1	no	—
417	C₄⋊C₄	—	6	1	no	—
418	C₄⋊C₄	—	6	2	no	—
419	C₄⋊C₄	—	6	1	no	—
420	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
421	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
422	C₂²⋊C₄	—	2	2	no	—
423	C₂²⋊C₄	—	6	2	no	—
424	C₂²⋊C₄	—	6	2	no	—
425	C₂²⋊C₄	C₅×#67	1	2	yes	—
426	C₄²	—	2	1	no	—
427	C₄²	—	6	2	no	—
428	C₄²	—	6	1	no	—
429	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
430	C₃×C₅	#3304	1	1	yes	characteristic, complemented, Hall
431	C₂²×C₃	—	2	2	no	—
432	C₂²×C₃	—	2	2	no	—
433	C₂²×C₃	—	2	2	no	—
434	C₂²×C₃	—	4	2	no	—
435	C₂²×C₃	—	4	2	no	—
436	C₂²×C₃	—	4	2	no	—
437	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
438	C₃×C₄	—	2	2	no	—
439	C₃×C₄	—	2	2	no	—
440	C₃×C₄	—	2	1	no	—
441	C₃×C₄	—	4	1	no	—
442	C₃×C₄	C₅×#135	1	2	yes	—
443	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
444	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
445	Dic₃	—	4	1	no	—
446	Dic₃	—	4	2	no	—
447	Dic₃	—	4	1	no	—
448	C₂×C₅	—	4	2	no	—
449	C₂×C₅	—	8	2	no	—
450	C₂×C₅	#5806	1	1	yes	characteristic
451	C₂×C₅	C₂×#950	1	1	yes	characteristic
452	C₂×C₅	#5836	1	1	yes	characteristic
453	C₂³	—	2	2	no	—
454	C₂³	C₂×C₅×#67	1	2	yes	—
455	Q₈	—	2	1	no	—
456	Q₈	—	6	1	no	—
457	Q₈	—	6	1	no	—
458	Q₈	—	12	2	no	—
459	Q₈	C₅×#237	1	2	yes	—
460	D₄	—	2	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	2	no	—
463	D₄	—	4	4	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	1	no	—
466	D₄	—	8	2	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	6	1	no	—
473	C₂×C₄	—	6	2	no	—
474	C₂×C₄	—	6	1	no	—
475	C₂×C₄	C₂×C₅×#67	1	2	yes	—
476	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
477	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
478	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
479	C₈	—	6	1	no	—
480	C₈	—	6	1	no	—
481	C₈	—	6	2	no	—
482	C₂×C₃	—	4	2	no	—
483	C₂×C₃	—	8	2	no	—
484	C₂×C₃	C₅×#546	1	1	yes	characteristic
485	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
486	C₂×C₃	C₅×#577	1	1	yes	characteristic
487	C₅	#83570	1	1	yes	characteristic, direct product factor, Sylow
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
495	C₄	—	2	2	no	—
496	C₄	—	2	2	no	—
497	C₄	—	2	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	12	1	no	—
500	C₄	—	12	2	no	—
501	C₄	—	12	1	no	—
502	C₄	C₅×#950	1	2	yes	—
503	C₄	C₂×C₅×#237	1	1	yes	characteristic
504	C₄	C₅×#1004	1	1	yes	characteristic
505	C₃	C₅×#3304	1	1	yes	characteristic, complemented, Sylow
506	C₂	—	4	2	no	—
507	C₂	—	8	2	no	—
508	C₂	C₅×#5806	1	1	yes	characteristic
509	C₂	C₂×C₅×#950	1	1	yes	characteristic
510	C₂	C₅×#5836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₃×C₅×#550	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#550	D₃
hypocenter	C₃	C₅×#3304
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×#550

Complements

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

N	H ≅ G/N	type
2. C₅×#5542	C₂	semidirect
3. C₅×#5239	C₂	semidirect
4. C₅×#5233	C₂	semidirect
5. C₅×#5075	C₂	semidirect
6. C₅×#5070	C₂	semidirect
7. C₅×#5041	C₂	semidirect
8. C₅×#5035	C₂	semidirect
9. C₄×C₅×#216	C₂	semidirect
10. C₅×#5020	C₂	semidirect
11. C₅×D₄×C₃⋊C₈	C₂	semidirect
42. #83570	C₅	direct (N × H)
90. #5542	C₂×C₅	semidirect
91. #5239	C₂×C₅	semidirect
92. #5233	C₂×C₅	semidirect
93. #5075	C₂×C₅	semidirect
94. #5070	C₂×C₅	semidirect
95. #5041	C₂×C₅	semidirect
96. #5035	C₂×C₅	semidirect
97. C₄×#216	C₂×C₅	semidirect
98. #5020	C₂×C₅	semidirect
99. D₄×C₃⋊C₈	C₂×C₅	semidirect
430. C₃×C₅	#3304	semidirect
487. C₅	#83570	direct (N × H)
505. C₃	C₅×#3304	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#550	2
C₅×#5542	2
C₅×#5239	2
C₅×#5233	2
C₅×#5075	2
C₅×#5070	2
C₅×#5041	2
C₅×#5035	2
C₄×C₅×#216	2
C₅×#5020	2
C₅×D₄×C₃⋊C₈	2
C₅×#3304	3
#83570	5

Subgroups — C5×#83570 #14460267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83570 #14460267