Subgroups — C₅×#19833 #11942553

Order 1280.

608 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3392	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1717	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1701	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1678	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1636	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1621	C₂	1	1	yes	maximal, characteristic
13	C₅×#1620	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#559	C₂²	1	1	yes	characteristic
17	C₅×#557	C₂²	1	1	yes	characteristic
18	C₅×#554	C₂²	1	1	yes	characteristic
19	C₅×#550	C₂²	1	1	yes	characteristic
20	C₅×#493	C₂²	1	1	yes	characteristic
21	C₅×#492	—	2	1	no	—
22	C₅×#492	C₂²	1	1	yes	characteristic
23	C₅×#488	—	2	1	no	—
24	C₅×#488	—	2	1	no	—
25	C₅×#487	C₂²	1	1	yes	characteristic
26	C₅×#486	C₂²	1	1	yes	characteristic
27	C₅×#486	C₂²	1	1	yes	characteristic
28	C₅×#486	C₂²	1	1	yes	characteristic
29	C₅×#480	—	2	1	no	—
30	C₅×#478	—	2	1	no	—
31	C₅×#478	—	2	1	no	—
32	C₅×#477	C₂²	1	1	yes	characteristic
33	C₅×#477	C₂²	1	1	yes	characteristic
34	C₅×#475	C₂²	1	1	yes	characteristic
35	C₅×#468	—	2	1	no	—
36	C₅×#464	—	2	1	no	—
37	C₅×#463	—	2	1	no	—
38	C₅×#461	—	2	1	no	—
39	C₅×#461	C₂²	1	1	yes	characteristic
40	C₅×#460	—	2	1	no	—
41	C₅×#460	C₂²	1	1	yes	characteristic
42	C₅×#460	C₂²	1	1	yes	characteristic
43	C₅×#460	C₂²	1	1	yes	characteristic
44	C₅×#459	C₂²	1	1	yes	characteristic
45	C₅×#452	—	2	1	no	—
46	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
47	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
50	C₄×C₅×SD₁₆	—	2	1	no	—
51	C₄×C₅×SD₁₆	—	2	1	no	—
52	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
53	C₄×C₅×D₈	—	2	1	no	—
54	C₅×#436	C₂²	1	1	yes	characteristic
55	C₅×#417	—	2	1	no	—
56	C₅×#334	C₂²	1	1	yes	characteristic
57	C₅×#334	C₂²	1	1	yes	characteristic
58	C₅×#333	C₂²	1	1	yes	characteristic
59	C₅×#332	C₂²	1	1	yes	characteristic
60	C₅×#331	C₂²	1	1	yes	characteristic
61	C₅×#331	C₂²	1	1	yes	characteristic
62	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×#322	C₂²	1	1	yes	characteristic
67	#19833	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
70	C₂×C₅×SD₁₆	—	2	1	no	—
71	C₂×C₅×SD₁₆	—	2	1	no	—
72	C₂×C₅×SD₁₆	—	2	1	no	—
73	C₂×C₅×SD₁₆	—	2	1	no	—
74	C₂×C₅×SD₁₆	—	2	1	no	—
75	C₂×C₅×SD₁₆	—	2	1	no	—
76	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
77	C₂×C₅×D₈	—	2	1	no	—
78	C₂×C₅×D₈	—	4	1	no	—
79	C₂×C₅×M₁₆	—	2	1	no	—
80	C₅×C₄⋊Q₈	—	2	1	no	—
81	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₅×#127	—	2	1	no	—
84	C₅×#127	C₂³	1	1	yes	characteristic
85	C₅×#127	C₂³	1	1	yes	characteristic
86	C₅×#125	—	2	1	no	—
87	C₅×#125	—	2	1	no	—
88	C₅×#125	—	4	1	no	—
89	C₅×#124	—	2	1	no	—
90	C₅×#124	—	2	1	no	—
91	C₅×#123	—	4	1	no	—
92	C₅×C₂²⋊Q₈	—	2	1	no	—
93	C₅×C₂²⋊Q₈	—	4	1	no	—
94	C₅×#121	—	2	1	no	—
95	C₅×#121	—	4	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	—	2	1	no	—
98	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
99	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
100	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	—	2	1	no	—
105	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
106	C₂×C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×#107	—	2	1	no	—
108	C₅×#107	—	2	1	no	—
109	C₅×#107	D₄	1	1	yes	characteristic
110	C₅×#106	—	2	1	no	—
111	C₅×#106	D₄	1	1	yes	characteristic
112	C₅×C₄⋊C₈	—	2	1	no	—
113	C₅×C₄⋊C₈	—	2	1	no	—
114	C₅×C₄⋊C₈	—	2	1	no	—
115	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₅×#103	—	2	1	no	—
120	C₅×#103	—	2	1	no	—
121	C₅×#103	—	2	1	no	—
122	C₅×#103	—	2	1	no	—
123	C₅×#103	—	4	1	no	—
124	C₅×#103	D₄	1	1	yes	characteristic
125	C₅×#103	D₄	1	1	yes	characteristic
126	C₅×#103	D₄	1	1	yes	characteristic
127	C₅×#102	—	2	1	no	—
128	C₅×#102	—	2	1	no	—
129	C₅×#102	—	2	1	no	—
130	C₅×#102	—	2	1	no	—
131	C₅×#102	—	2	1	no	—
132	C₅×#102	—	2	1	no	—
133	C₅×#102	—	2	1	no	—
134	C₅×#102	—	2	1	no	—
135	C₅×#102	—	2	1	no	—
136	C₅×#102	—	2	1	no	—
137	C₅×#102	D₄	1	1	yes	characteristic
138	C₅×C₂²⋊C₈	—	2	1	no	—
139	C₄×C₅×C₈	—	2	1	no	—
140	C₄×C₅×C₈	—	2	1	no	—
141	C₄×C₅×C₈	—	2	1	no	—
142	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	#3392	C₂×C₅	1	1	yes	characteristic, complemented
146	#3365	C₂×C₅	1	1	yes	characteristic, complemented
147	#3361	C₂×C₅	1	1	yes	characteristic, complemented
148	#3333	C₂×C₅	1	1	yes	characteristic, complemented
149	#1717	C₂×C₅	1	1	yes	characteristic, complemented
150	#1701	C₂×C₅	1	1	yes	characteristic, complemented
151	#1678	C₂×C₅	1	1	yes	characteristic, complemented
152	#1678	C₂×C₅	1	1	yes	characteristic, complemented
153	#1666	C₂×C₅	1	1	yes	characteristic, complemented
154	#1643	C₂×C₅	1	1	yes	characteristic, complemented
155	#1636	C₂×C₅	1	1	yes	characteristic, complemented
156	#1621	C₂×C₅	1	1	yes	characteristic
157	#1620	C₂×C₅	1	1	yes	characteristic, complemented
158	#1583	C₂×C₅	1	1	yes	characteristic, complemented
159	#1579	C₂×C₅	1	1	yes	characteristic, complemented
160	C₅×#41	—	4	2	no	—
161	C₅×#41	—	8	1	no	—
162	C₂×C₅×Q₈	—	2	1	no	—
163	C₂×C₅×Q₈	—	2	1	no	—
164	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
165	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
166	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
167	C₂×C₅×D₄	—	2	1	no	—
168	C₂×C₅×D₄	—	2	1	no	—
169	C₂×C₅×D₄	—	2	1	no	—
170	C₂×C₅×D₄	—	2	1	no	—
171	C₂×C₅×D₄	—	2	1	no	—
172	C₂×C₅×D₄	—	2	1	no	—
173	C₂×C₅×D₄	—	4	1	no	—
174	C₂×C₅×D₄	—	4	1	no	—
175	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅	—	2	1	no	—
177	C₂²×C₄×C₅	—	4	1	no	—
178	C₂²×C₄×C₅	—	4	1	no	—
179	C₂²×C₄×C₅	—	4	1	no	—
180	C₅×Q₁₆	—	2	2	no	—
181	C₅×SD₁₆	—	4	1	no	—
182	C₅×SD₁₆	—	4	2	no	—
183	C₅×SD₁₆	—	4	2	no	—
184	C₅×SD₁₆	—	8	1	no	—
185	C₅×SD₁₆	—	8	1	no	—
186	C₅×SD₁₆	—	8	1	no	—
187	C₅×SD₁₆	—	8	1	no	—
188	C₅×D₈	—	4	2	no	—
189	C₅×D₈	—	8	2	no	—
190	C₅×M₁₆	—	8	1	no	—
191	C₂×C₅×C₈	—	2	1	no	—
192	C₂×C₅×C₈	—	2	1	no	—
193	C₂×C₅×C₈	—	2	1	no	—
194	C₂×C₅×C₈	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	—	2	1	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	2	1	no	—
207	C₅×C₄⋊C₄	—	4	1	no	—
208	C₅×C₄⋊C₄	—	4	1	no	—
209	C₅×C₄⋊C₄	—	4	1	no	—
210	C₅×C₄⋊C₄	—	4	1	no	—
211	C₅×C₄⋊C₄	—	4	1	no	—
212	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
214	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
215	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
216	C₅×C₂²⋊C₄	—	2	1	no	—
217	C₅×C₂²⋊C₄	—	4	1	no	—
218	C₅×C₂²⋊C₄	—	4	1	no	—
219	C₅×C₂²⋊C₄	—	4	1	no	—
220	C₅×C₂²⋊C₄	—	4	1	no	—
221	C₅×C₂²⋊C₄	—	4	1	no	—
222	C₄²×C₅	—	2	1	no	—
223	C₄²×C₅	—	2	1	no	—
224	C₄²×C₅	—	2	1	no	—
225	C₄²×C₅	—	4	1	no	—
226	C₄²×C₅	C₂⁴	1	1	yes	characteristic
227	#559	C₂²×C₅	1	1	yes	characteristic
228	#557	C₂²×C₅	1	1	yes	characteristic
229	#554	C₂²×C₅	1	1	yes	characteristic
230	#550	C₂²×C₅	1	1	yes	characteristic
231	#493	C₂²×C₅	1	1	yes	characteristic
232	#492	—	2	1	no	—
233	#492	C₂²×C₅	1	1	yes	characteristic
234	#488	—	2	1	no	—
235	#488	—	2	1	no	—
236	#487	C₂²×C₅	1	1	yes	characteristic
237	#486	C₂²×C₅	1	1	yes	characteristic
238	#486	C₂²×C₅	1	1	yes	characteristic
239	#486	C₂²×C₅	1	1	yes	characteristic
240	#480	—	2	1	no	—
241	#478	—	2	1	no	—
242	#478	—	2	1	no	—
243	#477	C₂²×C₅	1	1	yes	characteristic
244	#477	C₂²×C₅	1	1	yes	characteristic
245	#475	C₂²×C₅	1	1	yes	characteristic
246	#468	—	2	1	no	—
247	#464	—	2	1	no	—
248	#463	—	2	1	no	—
249	#461	—	2	1	no	—
250	#461	C₂²×C₅	1	1	yes	characteristic
251	#460	—	2	1	no	—
252	#460	C₂²×C₅	1	1	yes	characteristic
253	#460	C₂²×C₅	1	1	yes	characteristic
254	#460	C₂²×C₅	1	1	yes	characteristic
255	#459	C₂²×C₅	1	1	yes	characteristic
256	#452	—	2	1	no	—
257	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
258	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
259	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
260	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
261	C₄×SD₁₆	—	2	1	no	—
262	C₄×SD₁₆	—	2	1	no	—
263	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
264	C₄×D₈	—	2	1	no	—
265	#436	C₂²×C₅	1	1	yes	characteristic
266	#417	—	2	1	no	—
267	#334	C₂²×C₅	1	1	yes	characteristic
268	#334	C₂²×C₅	1	1	yes	characteristic
269	#333	C₂²×C₅	1	1	yes	characteristic
270	#332	C₂²×C₅	1	1	yes	characteristic
271	#331	C₂²×C₅	1	1	yes	characteristic
272	#331	C₂²×C₅	1	1	yes	characteristic
273	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
274	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
275	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
276	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
277	#322	C₂²×C₅	1	1	yes	characteristic
278	C₂³×C₅	—	2	1	no	—
279	C₂³×C₅	—	4	1	no	—
280	C₂³×C₅	—	4	1	no	—
281	C₅×Q₈	—	2	1	no	—
282	C₅×Q₈	—	2	1	no	—
283	C₅×Q₈	—	2	1	no	—
284	C₅×Q₈	—	2	1	no	—
285	C₅×Q₈	—	2	1	no	—
286	C₅×Q₈	—	4	1	no	—
287	C₅×Q₈	—	4	1	no	—
288	C₅×Q₈	—	8	1	no	—
289	C₅×Q₈	#135	1	2	yes	—
290	C₅×D₄	—	2	1	no	—
291	C₅×D₄	—	2	1	no	—
292	C₅×D₄	—	4	1	no	—
293	C₅×D₄	—	4	1	no	—
294	C₅×D₄	—	4	1	no	—
295	C₅×D₄	—	4	1	no	—
296	C₅×D₄	—	4	1	no	—
297	C₅×D₄	—	4	1	no	—
298	C₅×D₄	—	4	1	no	—
299	C₅×D₄	—	4	1	no	—
300	C₅×D₄	—	8	2	no	—
301	C₅×D₄	—	8	1	no	—
302	C₅×D₄	—	8	1	no	—
303	C₅×D₄	—	16	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	2	1	no	—
306	C₂×C₄×C₅	—	2	1	no	—
307	C₂×C₄×C₅	—	2	1	no	—
308	C₂×C₄×C₅	—	2	1	no	—
309	C₂×C₄×C₅	—	2	1	no	—
310	C₂×C₄×C₅	—	2	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	4	1	no	—
313	C₂×C₄×C₅	—	4	1	no	—
314	C₂×C₄×C₅	—	4	1	no	—
315	C₂×C₄×C₅	—	8	2	no	—
316	C₂×C₄×C₅	—	16	1	no	—
317	C₂×C₄×C₅	—	16	1	no	—
318	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
321	C₅×C₈	—	2	1	no	—
322	C₅×C₈	—	2	1	no	—
323	C₅×C₈	—	2	1	no	—
324	C₅×C₈	—	2	1	no	—
325	C₅×C₈	—	4	1	no	—
326	C₅×C₈	—	4	1	no	—
327	C₅×C₈	—	4	1	no	—
328	C₅×C₈	—	4	1	no	—
329	C₅×C₈	—	4	1	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
332	C₂×SD₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	1	no	—
334	C₂×SD₁₆	—	2	1	no	—
335	C₂×SD₁₆	—	2	1	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×SD₁₆	—	2	1	no	—
338	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
339	C₂×D₈	—	2	1	no	—
340	C₂×D₈	—	4	1	no	—
341	C₂×M₁₆	—	2	1	no	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
344	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
345	#127	—	2	1	no	—
346	#127	C₂³×C₅	1	1	yes	characteristic
347	#127	C₂³×C₅	1	1	yes	characteristic
348	#125	—	2	1	no	—
349	#125	—	2	1	no	—
350	#125	—	4	1	no	—
351	#124	—	2	1	no	—
352	#124	—	2	1	no	—
353	#123	—	4	1	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	—	4	1	no	—
356	#121	—	2	1	no	—
357	#121	—	4	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
361	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
362	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₄⋊C₄	—	2	1	no	—
369	#107	—	2	1	no	—
370	#107	—	2	1	no	—
371	#107	C₅×D₄	1	1	yes	characteristic
372	#106	—	2	1	no	—
373	#106	C₅×D₄	1	1	yes	characteristic
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
381	#103	—	2	1	no	—
382	#103	—	2	1	no	—
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	—	4	1	no	—
386	#103	C₅×D₄	1	1	yes	characteristic
387	#103	C₅×D₄	1	1	yes	characteristic
388	#103	C₅×D₄	1	1	yes	characteristic
389	#102	—	2	1	no	—
390	#102	—	2	1	no	—
391	#102	—	2	1	no	—
392	#102	—	2	1	no	—
393	#102	—	2	1	no	—
394	#102	—	2	1	no	—
395	#102	—	2	1	no	—
396	#102	—	2	1	no	—
397	#102	—	2	1	no	—
398	#102	—	2	1	no	—
399	#102	C₅×D₄	1	1	yes	characteristic
400	C₂²⋊C₈	—	2	1	no	—
401	C₄×C₈	—	2	1	no	—
402	C₄×C₈	—	2	1	no	—
403	C₄×C₈	—	2	1	no	—
404	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
405	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
407	C₂²×C₅	—	4	1	no	—
408	C₂²×C₅	—	4	1	no	—
409	C₂²×C₅	—	4	1	no	—
410	C₂²×C₅	—	8	1	no	—
411	C₂²×C₅	—	8	1	no	—
412	C₂²×C₅	—	8	1	no	—
413	C₂²×C₅	—	8	1	no	—
414	C₂²×C₅	—	8	1	no	—
415	C₂²×C₅	—	8	1	no	—
416	C₂²×C₅	D₄²	1	1	yes	characteristic
417	C₄×C₅	—	2	2	no	—
418	C₄×C₅	—	2	1	no	—
419	C₄×C₅	—	4	1	no	—
420	C₄×C₅	—	4	1	no	—
421	C₄×C₅	—	4	1	no	—
422	C₄×C₅	—	4	1	no	—
423	C₄×C₅	—	4	1	no	—
424	C₄×C₅	—	4	1	no	—
425	C₄×C₅	—	8	1	no	—
426	C₄×C₅	—	8	1	no	—
427	C₄×C₅	C₂×#135	1	1	yes	characteristic
428	C₄×C₅	#575	1	1	yes	characteristic
429	C₄×C₅	#577	1	1	yes	characteristic
430	C₄×C₅	#577	1	1	yes	characteristic
431	#41	—	4	2	no	—
432	#41	—	8	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
436	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
437	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	Q₁₆	—	2	2	no	—
452	SD₁₆	—	4	1	no	—
453	SD₁₆	—	4	2	no	—
454	SD₁₆	—	4	2	no	—
455	SD₁₆	—	8	1	no	—
456	SD₁₆	—	8	1	no	—
457	SD₁₆	—	8	1	no	—
458	SD₁₆	—	8	1	no	—
459	D₈	—	4	2	no	—
460	D₈	—	8	2	no	—
461	M₁₆	—	8	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
468	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
469	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
470	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
484	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
485	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
486	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	C₂⁴×C₅	1	1	yes	characteristic
498	C₂×C₅	—	8	1	no	—
499	C₂×C₅	—	16	1	no	—
500	C₂×C₅	—	16	1	no	—
501	C₂×C₅	#3271	1	1	yes	characteristic
502	C₂×C₅	#3282	1	1	yes	characteristic
503	C₂×C₅	#3288	1	1	yes	characteristic
504	C₂³	—	2	1	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	1	no	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	8	1	no	—
515	Q₈	C₅×#135	1	2	yes	—
516	D₄	—	2	1	no	—
517	D₄	—	2	1	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	8	2	no	—
527	D₄	—	8	1	no	—
528	D₄	—	8	1	no	—
529	D₄	—	16	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	8	2	no	—
542	C₂×C₄	—	16	1	no	—
543	C₂×C₄	—	16	1	no	—
544	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
545	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
546	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	2	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₅	#19833	1	1	yes	characteristic, direct product factor, Sylow
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	C₅×D₄²	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	2	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	C₂×C₅×#135	1	1	yes	characteristic
578	C₄	C₅×#575	1	1	yes	characteristic
579	C₄	C₅×#577	1	1	yes	characteristic
580	C₄	C₅×#577	1	1	yes	characteristic
581	C₂	—	8	1	no	—
582	C₂	—	16	1	no	—
583	C₂	—	16	1	no	—
584	C₂	C₅×#3271	1	1	yes	characteristic
585	C₂	C₅×#3282	1	1	yes	characteristic
586	C₂	C₅×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#1621	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3392	C₂	semidirect
2. C₅×#3365	C₂	semidirect
3. C₅×#3361	C₂	semidirect
4. C₅×#3333	C₂	semidirect
5. C₅×#1717	C₂	semidirect
6. C₅×#1701	C₂	semidirect
7. C₅×#1678	C₂	semidirect
8. C₅×#1678	C₂	semidirect
9. C₅×#1666	C₂	semidirect
10. C₅×#1643	C₂	semidirect
11. C₅×#1636	C₂	semidirect
13. C₅×#1620	C₂	semidirect
14. C₅×#1583	C₂	semidirect
15. C₅×#1579	C₂	semidirect
67. #19833	C₅	direct (N × H)
145. #3392	C₂×C₅	semidirect
146. #3365	C₂×C₅	semidirect
147. #3361	C₂×C₅	semidirect
148. #3333	C₂×C₅	semidirect
149. #1717	C₂×C₅	semidirect
150. #1701	C₂×C₅	semidirect
151. #1678	C₂×C₅	semidirect
152. #1678	C₂×C₅	semidirect
153. #1666	C₂×C₅	semidirect
154. #1643	C₂×C₅	semidirect
155. #1636	C₂×C₅	semidirect
157. #1620	C₂×C₅	semidirect
158. #1583	C₂×C₅	semidirect
159. #1579	C₂×C₅	semidirect
556. C₅	#19833	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3392	2
C₅×#3365	2
C₅×#3361	2
C₅×#3333	2
C₅×#1717	2
C₅×#1701	2
C₅×#1678	2
C₅×#1666	2
C₅×#1643	2
C₅×#1636	2
C₅×#1621	2
C₅×#1620	2
C₅×#1583	2
C₅×#1579	2
#19833	5

Subgroups — C5×#19833 #11942553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19833 #11942553