Subgroups — C₅×#81578 #14460264

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	C₅×#5732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5718	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5094	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5075	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4784	C₂	1	2	yes	maximal, complemented
7	C₅×#4783	C₂	1	2	yes	maximal, complemented
8	C₅×#4778	C₂	1	2	yes	maximal, complemented
9	C₅×#4772	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3304	—	3	1	no	maximal, Hall
13	C₂×C₅×#237	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#127	C₂²	1	1	yes	characteristic
16	C₅×#947	—	2	1	no	—
17	C₅×Q₈×Dic₃	—	2	1	no	—
18	C₅×#940	—	2	1	no	—
19	C₅×#939	—	2	2	no	—
20	C₅×#938	C₂²	1	2	yes	—
21	C₅×D₄×Dic₃	—	2	1	no	—
22	C₂×C₅×#214	C₂²	1	2	yes	—
23	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
24	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
25	C₅×#898	—	2	2	no	—
26	C₅×#897	C₂²	1	2	yes	—
27	C₅×#892	C₂²	1	2	yes	—
28	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
31	C₃×C₅×#102	C₂²	1	2	yes	—
32	C₃×C₅×#102	C₂²	1	2	yes	—
33	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₅×#832	C₂²	1	2	yes	—
35	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
36	C₅×#818	C₂²	1	1	yes	characteristic
37	C₅×#816	C₂²	1	1	yes	characteristic
38	C₅×#816	C₂²	1	1	yes	characteristic
39	C₅×#810	C₂²	1	2	yes	—
40	C₅×#808	C₂²	1	2	yes	—
41	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#81578	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#556	—	3	1	no	—
44	C₅×#550	—	3	1	no	—
45	C₅×#486	D₃	1	1	yes	characteristic, complemented
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	C₂×C₃×C₅×Q₈	—	2	1	no	—
55	C₂×C₃×C₅×D₄	—	2	1	no	—
56	C₂×C₃×C₅×D₄	—	2	1	no	—
57	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
58	C₂×C₅×#67	—	2	2	no	—
59	C₂×C₅×#67	—	2	2	no	—
60	C₂²×C₅×Dic₃	—	2	2	no	—
61	C₅×#237	—	2	4	no	—
62	C₅×#237	—	4	2	no	—
63	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
64	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
65	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
66	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
72	C₅×#217	—	2	2	no	—
73	C₅×#214	—	4	2	no	—
74	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
75	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
76	C₅×(C₂×D₃)⋊C₄	D₄	1	2	yes	—
77	C₅×C₄⋊Dic₃	—	2	2	no	—
78	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
79	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
81	C₅×Dic₃⋊C₄	—	2	1	no	—
82	C₅×Dic₃⋊C₄	—	2	1	no	—
83	C₄×C₅×Dic₃	—	2	1	no	—
84	C₄×C₅×Dic₃	—	2	1	no	—
85	C₄×C₅×Dic₃	—	2	2	no	—
86	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
87	C₅×Dic₁₂	—	2	2	no	—
88	C₅×D₃×C₈	—	4	2	no	—
89	#5732	C₂×C₅	1	1	yes	characteristic, complemented
90	#5718	C₂×C₅	1	1	yes	characteristic
91	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
92	#5094	C₂×C₅	1	1	yes	characteristic, complemented
93	#5075	C₂×C₅	1	1	yes	characteristic, complemented
94	#4784	C₂×C₅	1	2	yes	complemented
95	#4783	C₂×C₅	1	2	yes	complemented
96	#4778	C₂×C₅	1	2	yes	complemented
97	#4772	C₂×C₅	1	2	yes	complemented
98	C₄×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
99	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×C₅×#41	—	3	2	no	—
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	C₂×D₃	1	1	yes	characteristic
105	C₅×#127	—	6	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	—	3	2	no	—
111	C₅×#121	—	6	2	no	—
112	C₄×C₅×Q₈	—	3	1	no	—
113	C₄×C₅×Q₈	—	3	1	no	—
114	C₄×C₅×Q₈	—	6	1	no	—
115	C₄×C₅×D₄	—	3	1	no	—
116	C₄×C₅×D₄	—	6	1	no	—
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	1	no	—
123	C₅×#103	—	3	2	no	—
124	C₅×#102	—	3	2	no	—
125	C₅×#102	C₂×D₃	1	2	yes	—
126	C₅×#102	C₂×D₃	1	2	yes	—
127	C₅×C₂²⋊C₈	—	3	2	no	—
128	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
129	#3304	—	3	1	no	Sylow
130	C₂³×C₃×C₅	—	2	2	no	—
131	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
132	C₃×C₅×Q₈	—	4	2	no	—
133	C₃×C₅×D₄	—	2	2	no	—
134	C₃×C₅×D₄	—	2	2	no	—
135	C₃×C₅×D₄	—	4	2	no	—
136	C₃×C₅×D₄	—	8	1	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
141	C₅×#67	—	4	4	no	—
142	C₅×#67	—	8	2	no	—
143	C₂×C₅×Dic₃	—	2	1	no	—
144	C₂×C₅×Dic₃	—	2	1	no	—
145	C₂×C₅×Dic₃	—	2	1	no	—
146	C₂×C₅×Dic₃	—	2	1	no	—
147	C₂×C₅×Dic₃	—	4	4	no	—
148	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
149	C₅×D₁₂	—	2	2	no	—
150	C₄×C₅×D₃	—	2	2	no	—
151	C₄×C₅×D₃	—	2	2	no	—
152	C₅×Dic₆	—	2	1	no	—
153	C₅×Dic₆	—	2	1	no	—
154	C₅×Dic₆	—	2	1	no	—
155	C₅×Dic₆	#41	1	2	yes	—
156	C₃×C₅×C₈	—	2	1	no	—
157	C₃×C₅×C₈	—	2	1	no	—
158	C₅×C₃⋊C₈	—	2	2	no	—
159	C₂×#237	C₂²×C₅	1	2	yes	—
160	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₃×#127	C₂²×C₅	1	1	yes	characteristic
162	#947	—	2	1	no	—
163	Q₈×Dic₃	—	2	1	no	—
164	#940	—	2	1	no	—
165	#939	—	2	2	no	—
166	#938	C₂²×C₅	1	2	yes	—
167	D₄×Dic₃	—	2	1	no	—
168	C₂×#214	C₂²×C₅	1	2	yes	—
169	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
170	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
171	#898	—	2	2	no	—
172	#897	C₂²×C₅	1	2	yes	—
173	#892	C₂²×C₅	1	2	yes	—
174	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
175	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
176	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
177	C₃×#102	C₂²×C₅	1	2	yes	—
178	C₃×#102	C₂²×C₅	1	2	yes	—
179	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
180	#832	C₂²×C₅	1	2	yes	—
181	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	—
182	#818	C₂²×C₅	1	1	yes	characteristic
183	#816	C₂²×C₅	1	1	yes	characteristic
184	#816	C₂²×C₅	1	1	yes	characteristic
185	#810	C₂²×C₅	1	2	yes	—
186	#808	C₂²×C₅	1	2	yes	—
187	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
188	C₅×#41	—	6	4	no	—
189	C₅×#41	—	12	2	no	—
190	C₂×C₅×Q₈	—	2	1	no	—
191	C₂×C₅×Q₈	—	3	1	no	—
192	C₂×C₅×Q₈	—	3	1	no	—
193	C₂×C₅×D₄	—	2	1	no	—
194	C₂×C₅×D₄	—	2	1	no	—
195	C₂×C₅×D₄	—	3	1	no	—
196	C₂×C₅×D₄	—	6	2	no	—
197	C₂×C₅×D₄	—	6	2	no	—
198	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
199	C₂²×C₄×C₅	—	3	2	no	—
200	C₂²×C₄×C₅	—	6	2	no	—
201	C₅×Q₁₆	—	6	2	no	—
202	C₅×SD₁₆	—	12	2	no	—
203	C₂×C₅×C₈	—	3	2	no	—
204	C₂×C₅×C₈	—	12	2	no	—
205	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
206	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	3	1	no	—
209	C₅×C₄⋊C₄	—	3	1	no	—
210	C₅×C₄⋊C₄	—	3	1	no	—
211	C₅×C₄⋊C₄	—	6	2	no	—
212	C₅×C₄⋊C₄	—	6	1	no	—
213	C₅×C₄⋊C₄	—	6	1	no	—
214	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
215	C₅×C₂²⋊C₄	—	3	2	no	—
216	C₅×C₂²⋊C₄	—	6	2	no	—
217	C₅×C₂²⋊C₄	—	6	2	no	—
218	C₅×C₂²⋊C₄	—	6	2	no	—
219	C₄²×C₅	—	6	2	no	—
220	C₄²×C₅	—	6	1	no	—
221	C₄²×C₅	—	6	1	no	—
222	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
223	#556	—	3	1	no	—
224	#550	—	3	1	no	—
225	#486	C₅×D₃	1	1	yes	characteristic, complemented
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	—	4	2	no	—
235	C₂²×C₃×C₅	—	4	2	no	—
236	C₂²×C₃×C₅	—	4	2	no	—
237	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₅×D₃	—	2	2	no	—
239	C₂×C₅×D₃	—	2	2	no	—
240	C₂×C₅×D₃	—	2	2	no	—
241	C₃×C₄×C₅	—	2	1	no	—
242	C₃×C₄×C₅	—	4	2	no	—
243	C₃×C₄×C₅	#135	1	2	yes	—
244	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
245	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
246	C₅×Dic₃	—	2	2	no	—
247	C₅×Dic₃	—	2	2	no	—
248	C₅×Dic₃	—	4	1	no	—
249	C₅×Dic₃	—	4	1	no	—
250	C₅×Dic₃	—	4	1	no	—
251	C₂×C₃×Q₈	—	2	1	no	—
252	C₂×C₃×D₄	—	2	1	no	—
253	C₂×C₃×D₄	—	2	1	no	—
254	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
255	C₂×#67	—	2	2	no	—
256	C₂×#67	—	2	2	no	—
257	C₂²×Dic₃	—	2	2	no	—
258	#237	—	2	4	no	—
259	#237	—	4	2	no	—
260	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
261	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
262	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
263	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
264	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
265	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
266	C₃×C₄⋊C₄	—	2	1	no	—
267	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
268	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
269	#217	—	2	2	no	—
270	#214	—	4	2	no	—
271	(C₂×D₃)⋊C₄	—	2	2	no	—
272	(C₂×D₃)⋊C₄	—	2	2	no	—
273	(C₂×D₃)⋊C₄	C₅×D₄	1	2	yes	—
274	C₄⋊Dic₃	—	2	2	no	—
275	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
276	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
277	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
278	Dic₃⋊C₄	—	2	1	no	—
279	Dic₃⋊C₄	—	2	1	no	—
280	C₄×Dic₃	—	2	1	no	—
281	C₄×Dic₃	—	2	1	no	—
282	C₄×Dic₃	—	2	2	no	—
283	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
284	Dic₁₂	—	2	2	no	—
285	D₃×C₈	—	4	2	no	—
286	C₂³×C₅	—	2	2	no	—
287	C₂³×C₅	—	3	2	no	—
288	C₅×Q₈	—	3	2	no	—
289	C₅×Q₈	—	4	2	no	—
290	C₅×Q₈	—	6	1	no	—
291	C₅×Q₈	—	6	1	no	—
292	C₅×Q₈	—	6	1	no	—
293	C₅×D₄	—	2	2	no	—
294	C₅×D₄	—	2	2	no	—
295	C₅×D₄	—	4	2	no	—
296	C₅×D₄	—	6	2	no	—
297	C₅×D₄	—	8	1	no	—
298	C₅×D₄	—	12	4	no	—
299	C₅×D₄	—	24	2	no	—
300	C₂×C₄×C₅	—	2	2	no	—
301	C₂×C₄×C₅	—	3	2	no	—
302	C₂×C₄×C₅	—	6	1	no	—
303	C₂×C₄×C₅	—	6	2	no	—
304	C₂×C₄×C₅	—	6	2	no	—
305	C₂×C₄×C₅	—	6	1	no	—
306	C₂×C₄×C₅	—	6	1	no	—
307	C₂×C₄×C₅	—	6	1	no	—
308	C₂×C₄×C₅	—	12	4	no	—
309	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
310	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
312	C₅×C₈	—	2	1	no	—
313	C₅×C₈	—	2	1	no	—
314	C₅×C₈	—	6	2	no	—
315	C₂×#41	—	3	2	no	—
316	C₂×Q₁₆	—	3	1	no	—
317	C₂×SD₁₆	—	3	2	no	—
318	C₂²×C₈	—	3	2	no	—
319	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
320	#127	—	6	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	—	3	2	no	—
326	#121	—	6	2	no	—
327	C₄×Q₈	—	3	1	no	—
328	C₄×Q₈	—	3	1	no	—
329	C₄×Q₈	—	6	1	no	—
330	C₄×D₄	—	3	1	no	—
331	C₄×D₄	—	6	1	no	—
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	1	no	—
338	#103	—	3	2	no	—
339	#102	—	3	2	no	—
340	#102	C₂×C₅×D₃	1	2	yes	—
341	#102	C₂×C₅×D₃	1	2	yes	—
342	C₂²⋊C₈	—	3	2	no	—
343	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
344	C₂×C₃×C₅	—	8	2	no	—
345	C₂×C₃×C₅	#546	1	1	yes	characteristic
346	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
347	C₂×C₃×C₅	#577	1	1	yes	characteristic
348	C₅×D₃	—	4	2	no	—
349	C₂³×C₃	—	2	2	no	—
350	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
351	C₃×Q₈	—	4	2	no	—
352	C₃×D₄	—	2	2	no	—
353	C₃×D₄	—	2	2	no	—
354	C₃×D₄	—	4	2	no	—
355	C₃×D₄	—	8	1	no	—
356	C₂×C₃×C₄	—	2	2	no	—
357	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
358	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
359	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
360	#67	—	4	4	no	—
361	#67	—	8	2	no	—
362	C₂×Dic₃	—	2	1	no	—
363	C₂×Dic₃	—	2	1	no	—
364	C₂×Dic₃	—	2	1	no	—
365	C₂×Dic₃	—	2	1	no	—
366	C₂×Dic₃	—	4	4	no	—
367	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
368	D₁₂	—	2	2	no	—
369	C₄×D₃	—	2	2	no	—
370	C₄×D₃	—	2	2	no	—
371	Dic₆	—	2	1	no	—
372	Dic₆	—	2	1	no	—
373	Dic₆	—	2	1	no	—
374	Dic₆	C₅×#41	1	2	yes	—
375	C₃×C₈	—	2	1	no	—
376	C₃×C₈	—	2	1	no	—
377	C₃⋊C₈	—	2	2	no	—
378	C₂²×C₅	—	4	2	no	—
379	C₂²×C₅	—	4	2	no	—
380	C₂²×C₅	—	4	2	no	—
381	C₂²×C₅	—	6	2	no	—
382	C₂²×C₅	—	6	2	no	—
383	C₂²×C₅	—	6	2	no	—
384	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
385	C₄×C₅	—	2	1	no	—
386	C₄×C₅	—	4	2	no	—
387	C₄×C₅	—	6	2	no	—
388	C₄×C₅	—	6	2	no	—
389	C₄×C₅	—	12	1	no	—
390	C₄×C₅	—	12	1	no	—
391	C₄×C₅	—	12	1	no	—
392	C₄×C₅	#912	1	2	yes	—
393	C₄×C₅	D₃×#41	1	1	yes	characteristic
394	C₄×C₅	#1009	1	1	yes	characteristic
395	#41	—	6	4	no	—
396	#41	—	12	2	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	3	1	no	—
399	C₂×Q₈	—	3	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	3	1	no	—
403	C₂×D₄	—	6	2	no	—
404	C₂×D₄	—	6	2	no	—
405	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
406	C₂²×C₄	—	3	2	no	—
407	C₂²×C₄	—	6	2	no	—
408	Q₁₆	—	6	2	no	—
409	SD₁₆	—	12	2	no	—
410	C₂×C₈	—	3	2	no	—
411	C₂×C₈	—	12	2	no	—
412	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
413	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	3	1	no	—
416	C₄⋊C₄	—	3	1	no	—
417	C₄⋊C₄	—	3	1	no	—
418	C₄⋊C₄	—	6	2	no	—
419	C₄⋊C₄	—	6	1	no	—
420	C₄⋊C₄	—	6	1	no	—
421	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
422	C₂²⋊C₄	—	3	2	no	—
423	C₂²⋊C₄	—	6	2	no	—
424	C₂²⋊C₄	—	6	2	no	—
425	C₂²⋊C₄	—	6	2	no	—
426	C₄²	—	6	2	no	—
427	C₄²	—	6	1	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	—	4	2	no	—
432	C₂²×C₃	—	4	2	no	—
433	C₂²×C₃	—	4	2	no	—
434	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
435	C₂×D₃	—	2	2	no	—
436	C₂×D₃	—	2	2	no	—
437	C₂×D₃	—	2	2	no	—
438	C₃×C₄	—	2	1	no	—
439	C₃×C₄	—	4	2	no	—
440	C₃×C₄	C₅×#135	1	2	yes	—
441	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
442	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
443	Dic₃	—	2	2	no	—
444	Dic₃	—	2	2	no	—
445	Dic₃	—	4	1	no	—
446	Dic₃	—	4	1	no	—
447	Dic₃	—	4	1	no	—
448	C₂×C₅	—	8	2	no	—
449	C₂×C₅	—	12	2	no	—
450	C₂×C₅	#5676	1	1	yes	characteristic
451	C₂×C₅	C₂×#912	1	1	yes	characteristic
452	C₂×C₅	#5762	1	1	yes	characteristic
453	C₂³	—	2	2	no	—
454	C₂³	—	3	2	no	—
455	Q₈	—	3	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	6	1	no	—
458	Q₈	—	6	1	no	—
459	Q₈	—	6	1	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	6	2	no	—
464	D₄	—	8	1	no	—
465	D₄	—	12	4	no	—
466	D₄	—	24	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	3	2	no	—
469	C₂×C₄	—	6	1	no	—
470	C₂×C₄	—	6	2	no	—
471	C₂×C₄	—	6	2	no	—
472	C₂×C₄	—	6	1	no	—
473	C₂×C₄	—	6	1	no	—
474	C₂×C₄	—	6	1	no	—
475	C₂×C₄	—	12	4	no	—
476	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
477	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
478	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
479	C₈	—	2	1	no	—
480	C₈	—	2	1	no	—
481	C₈	—	6	2	no	—
482	C₂×C₃	—	8	2	no	—
483	C₂×C₃	C₅×#546	1	1	yes	characteristic
484	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
485	C₂×C₃	C₅×#577	1	1	yes	characteristic
486	D₃	—	4	2	no	—
487	C₅	#81578	1	1	yes	characteristic, direct product factor, Sylow
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	6	2	no	—
492	C₂²	—	6	2	no	—
493	C₂²	—	6	2	no	—
494	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	4	2	no	—
497	C₄	—	6	2	no	—
498	C₄	—	6	2	no	—
499	C₄	—	12	1	no	—
500	C₄	—	12	1	no	—
501	C₄	—	12	1	no	—
502	C₄	C₅×#912	1	2	yes	—
503	C₄	C₅×D₃×#41	1	1	yes	characteristic
504	C₄	C₅×#1009	1	1	yes	characteristic
505	C₃	C₅×#3304	1	1	yes	characteristic, complemented, Sylow
506	C₂	—	8	2	no	—
507	C₂	—	12	2	no	—
508	C₂	C₅×#5676	1	1	yes	characteristic
509	C₂	C₂×C₅×#912	1	1	yes	characteristic
510	C₂	C₅×#5762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#486	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5718	C₂
hypercenter	C₅×#486	D₃
hypocenter	C₃	C₅×#3304
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#486

Complements

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

N	H ≅ G/N	type
1. C₅×#5732	C₂	semidirect
3. C₃×C₅×#486	C₂	semidirect
4. C₅×#5094	C₂	semidirect
5. C₅×#5075	C₂	semidirect
6. C₅×#4784	C₂	semidirect
7. C₅×#4783	C₂	semidirect
8. C₅×#4778	C₂	semidirect
9. C₅×#4772	C₂	semidirect
10. C₄×C₅×Dic₁₂	C₂	semidirect
11. C₅×C₈×D₁₂	C₂	semidirect
42. #81578	C₅	direct (N × H)
45. C₅×#486	D₃	semidirect
89. #5732	C₂×C₅	semidirect
91. C₃×#486	C₂×C₅	semidirect
92. #5094	C₂×C₅	semidirect
93. #5075	C₂×C₅	semidirect
94. #4784	C₂×C₅	semidirect
95. #4783	C₂×C₅	semidirect
96. #4778	C₂×C₅	semidirect
97. #4772	C₂×C₅	semidirect
98. C₄×Dic₁₂	C₂×C₅	semidirect
99. C₈×D₁₂	C₂×C₅	semidirect
225. #486	C₅×D₃	semidirect
430. C₃×C₅	#3304	semidirect
487. C₅	#81578	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
C₅×#5732	2
C₅×#5718	2
C₃×C₅×#486	2
C₅×#5094	2
C₅×#5075	2
C₅×#4784	2
C₅×#4783	2
C₅×#4778	2
C₅×#4772	2
C₄×C₅×Dic₁₂	2
C₅×C₈×D₁₂	2
C₅×#3304	3
#81578	5

Subgroups — C5×#81578 #14460264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81578 #14460264