Subgroups — C₅×#19156 #11941876

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3076	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3035	C₂	1	1	yes	maximal, characteristic
4	C₅×#2965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1698	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1697	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1692	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1691	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1686	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1685	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1540	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#135	—	2	1	no	—
17	C₅×#533	C₂²	1	1	yes	characteristic
18	C₅×#532	C₂²	1	1	yes	characteristic
19	C₄×C₅×#41	C₂²	1	1	yes	characteristic
20	C₅×#498	C₂²	1	1	yes	characteristic
21	C₅×#492	C₂²	1	1	yes	characteristic
22	C₅×#487	C₂²	1	1	yes	characteristic
23	C₅×#486	C₂²	1	1	yes	characteristic
24	C₅×#479	—	2	1	no	—
25	C₅×#478	—	2	1	no	—
26	C₅×#477	C₂²	1	1	yes	characteristic
27	C₅×#476	C₂²	1	1	yes	characteristic
28	C₅×#475	C₂²	1	1	yes	characteristic
29	C₅×#474	C₂²	1	1	yes	characteristic
30	C₅×#465	—	2	1	no	—
31	C₅×#465	—	2	1	no	—
32	C₅×#464	—	2	1	no	—
33	C₅×#463	—	2	1	no	—
34	C₅×#462	C₂²	1	1	yes	characteristic
35	C₅×#461	C₂²	1	1	yes	characteristic
36	C₅×#460	C₂²	1	1	yes	characteristic
37	C₅×#459	C₂²	1	1	yes	characteristic
38	C₅×#446	C₂²	1	1	yes	characteristic
39	C₅×#446	C₂²	1	1	yes	characteristic
40	C₄×C₅×SD₁₆	—	2	1	no	—
41	C₄×C₅×SD₁₆	—	2	1	no	—
42	C₅×#436	C₂²	1	1	yes	characteristic
43	C₅×#436	C₂²	1	1	yes	characteristic
44	C₅×#435	—	2	1	no	—
45	C₅×#435	—	2	1	no	—
46	C₅×#431	—	2	1	no	—
47	C₅×#431	C₂²	1	1	yes	characteristic
48	C₅×#427	—	2	1	no	—
49	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₅×#416	—	2	1	no	—
52	C₅×#416	—	2	1	no	—
53	C₄×C₅×M₁₆	—	2	1	no	—
54	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
55	C₅×#335	C₂²	1	1	yes	characteristic
56	C₅×#335	C₂²	1	1	yes	characteristic
57	C₅×#335	C₂²	1	1	yes	characteristic
58	C₅×#335	C₂²	1	1	yes	characteristic
59	C₅×#333	C₂²	1	1	yes	characteristic
60	C₅×#333	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×#332	C₂²	1	1	yes	characteristic
63	C₅×#332	C₂²	1	1	yes	characteristic
64	C₅×#332	C₂²	1	1	yes	characteristic
65	C₅×#331	C₂²	1	1	yes	characteristic
66	C₅×#331	C₂²	1	1	yes	characteristic
67	#19156	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×#41	D₄	1	1	yes	characteristic
70	C₅×#135	—	4	4	no	—
71	C₂×C₅×Q₁₆	—	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×D₈	—	2	1	no	—
77	C₂×C₅×M₁₆	—	2	1	no	—
78	C₂×C₅×M₁₆	—	2	1	no	—
79	C₂²×C₅×C₈	—	2	1	no	—
80	C₂²×C₅×C₈	—	2	1	no	—
81	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
82	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
83	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₅×#127	C₂³	1	1	yes	characteristic
87	C₅×#125	—	2	1	no	—
88	C₅×#124	—	2	1	no	—
89	C₅×C₂²⋊Q₈	—	2	1	no	—
90	C₅×C₂²⋊Q₈	—	2	1	no	—
91	C₅×#121	—	2	1	no	—
92	C₅×#121	—	2	1	no	—
93	C₄×C₅×Q₈	—	2	1	no	—
94	C₄×C₅×Q₈	—	2	1	no	—
95	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
96	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
97	C₄×C₅×D₄	—	2	1	no	—
98	C₄×C₅×D₄	—	2	1	no	—
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	D₄	1	1	yes	characteristic
102	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
103	C₅×#117	—	2	1	no	—
104	C₅×#117	—	2	1	no	—
105	C₅×#117	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₅	—	2	1	no	—
107	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
108	C₅×#107	—	2	1	no	—
109	C₅×#107	—	2	1	no	—
110	C₅×#106	—	2	1	no	—
111	C₅×#106	—	2	1	no	—
112	C₅×#106	—	2	1	no	—
113	C₅×#106	—	2	1	no	—
114	C₅×C₄⋊C₈	—	2	1	no	—
115	C₅×C₄⋊C₈	—	2	1	no	—
116	C₅×C₄⋊C₈	—	2	1	no	—
117	C₅×C₄⋊C₈	—	2	1	no	—
118	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₅×#103	—	2	1	no	—
125	C₅×#103	—	2	1	no	—
126	C₅×#103	—	2	1	no	—
127	C₅×#103	—	2	1	no	—
128	C₅×#103	—	2	1	no	—
129	C₅×#103	—	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×C₂²⋊C₈	—	2	1	no	—
137	C₅×C₂²⋊C₈	—	2	1	no	—
138	C₅×C₂²⋊C₈	—	2	1	no	—
139	C₅×#97	—	2	1	no	—
140	C₅×#97	—	2	1	no	—
141	C₄×C₅×C₈	—	2	1	no	—
142	C₄×C₅×C₈	D₄	1	1	yes	characteristic
143	C₄×C₅×C₈	D₄	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
146	#3167	C₂×C₅	1	1	yes	characteristic, complemented
147	#3076	C₂×C₅	1	1	yes	characteristic, complemented
148	#3035	C₂×C₅	1	1	yes	characteristic
149	#2965	C₂×C₅	1	1	yes	characteristic, complemented
150	#1700	C₂×C₅	1	1	yes	characteristic, complemented
151	#1699	C₂×C₅	1	1	yes	characteristic, complemented
152	#1698	C₂×C₅	1	1	yes	characteristic, complemented
153	#1697	C₂×C₅	1	1	yes	characteristic, complemented
154	#1692	C₂×C₅	1	1	yes	characteristic, complemented
155	#1691	C₂×C₅	1	1	yes	characteristic, complemented
156	#1686	C₂×C₅	1	1	yes	characteristic, complemented
157	#1685	C₂×C₅	1	1	yes	characteristic, complemented
158	C₂×#335	C₂×C₅	1	1	yes	characteristic, complemented
159	#1542	C₂×C₅	1	1	yes	characteristic, complemented
160	#1540	C₂×C₅	1	1	yes	characteristic, complemented
161	C₅×#41	—	2	2	no	—
162	C₅×#41	—	2	2	no	—
163	C₅×#41	—	4	2	no	—
164	C₅×#41	—	4	2	no	—
165	C₂×C₅×Q₈	—	2	1	no	—
166	C₂×C₅×Q₈	—	2	1	no	—
167	C₂×C₅×Q₈	—	2	1	no	—
168	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₅	—	2	1	no	—
176	C₂²×C₄×C₅	—	2	1	no	—
177	C₂²×C₄×C₅	—	2	1	no	—
178	C₂²×C₄×C₅	—	4	1	no	—
179	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅	#41	1	1	yes	characteristic
182	C₅×Q₁₆	—	4	2	no	—
183	C₅×SD₁₆	—	4	2	no	—
184	C₅×SD₁₆	—	4	2	no	—
185	C₅×SD₁₆	—	4	2	no	—
186	C₅×SD₁₆	—	4	2	no	—
187	C₅×D₈	—	4	2	no	—
188	C₅×M₁₆	—	4	2	no	—
189	C₅×M₁₆	—	4	2	no	—
190	C₂×C₅×C₈	—	2	2	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	—	2	1	no	—
197	C₂×C₅×C₈	—	4	1	no	—
198	C₂×C₅×C₈	—	4	2	no	—
199	C₂×C₅×C₈	—	4	2	no	—
200	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
202	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
208	C₅×C₄⋊C₄	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	2	1	no	—
212	C₅×C₄⋊C₄	—	4	1	no	—
213	C₅×C₄⋊C₄	—	4	1	no	—
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	—	2	1	no	—
218	C₅×C₂²⋊C₄	—	2	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	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₅	#41	1	1	yes	characteristic
229	C₄²×C₅	C₂⁴	1	1	yes	characteristic
230	C₂×#135	—	2	1	no	—
231	#533	C₂²×C₅	1	1	yes	characteristic
232	#532	C₂²×C₅	1	1	yes	characteristic
233	C₄×#41	C₂²×C₅	1	1	yes	characteristic
234	#498	C₂²×C₅	1	1	yes	characteristic
235	#492	C₂²×C₅	1	1	yes	characteristic
236	#487	C₂²×C₅	1	1	yes	characteristic
237	#486	C₂²×C₅	1	1	yes	characteristic
238	#479	—	2	1	no	—
239	#478	—	2	1	no	—
240	#477	C₂²×C₅	1	1	yes	characteristic
241	#476	C₂²×C₅	1	1	yes	characteristic
242	#475	C₂²×C₅	1	1	yes	characteristic
243	#474	C₂²×C₅	1	1	yes	characteristic
244	#465	—	2	1	no	—
245	#465	—	2	1	no	—
246	#464	—	2	1	no	—
247	#463	—	2	1	no	—
248	#462	C₂²×C₅	1	1	yes	characteristic
249	#461	C₂²×C₅	1	1	yes	characteristic
250	#460	C₂²×C₅	1	1	yes	characteristic
251	#459	C₂²×C₅	1	1	yes	characteristic
252	#446	C₂²×C₅	1	1	yes	characteristic
253	#446	C₂²×C₅	1	1	yes	characteristic
254	C₄×SD₁₆	—	2	1	no	—
255	C₄×SD₁₆	—	2	1	no	—
256	#436	C₂²×C₅	1	1	yes	characteristic
257	#436	C₂²×C₅	1	1	yes	characteristic
258	#435	—	2	1	no	—
259	#435	—	2	1	no	—
260	#431	—	2	1	no	—
261	#431	C₂²×C₅	1	1	yes	characteristic
262	#427	—	2	1	no	—
263	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
265	#416	—	2	1	no	—
266	#416	—	2	1	no	—
267	C₄×M₁₆	—	2	1	no	—
268	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
269	#335	C₂²×C₅	1	1	yes	characteristic
270	#335	C₂²×C₅	1	1	yes	characteristic
271	#335	C₂²×C₅	1	1	yes	characteristic
272	#335	C₂²×C₅	1	1	yes	characteristic
273	#333	C₂²×C₅	1	1	yes	characteristic
274	#333	C₂²×C₅	1	1	yes	characteristic
275	#332	C₂²×C₅	1	1	yes	characteristic
276	#332	C₂²×C₅	1	1	yes	characteristic
277	#332	C₂²×C₅	1	1	yes	characteristic
278	#332	C₂²×C₅	1	1	yes	characteristic
279	#331	C₂²×C₅	1	1	yes	characteristic
280	#331	C₂²×C₅	1	1	yes	characteristic
281	C₂³×C₅	—	2	1	no	—
282	C₂³×C₅	—	4	1	no	—
283	C₂³×C₅	#121	1	1	yes	characteristic
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	—	4	1	no	—
289	C₅×Q₈	—	4	1	no	—
290	C₅×Q₈	—	8	1	no	—
291	C₅×D₄	—	2	1	no	—
292	C₅×D₄	—	2	1	no	—
293	C₅×D₄	—	4	2	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	—	8	1	no	—
299	C₅×D₄	—	8	2	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	4	1	no	—
306	C₂×C₄×C₅	—	4	2	no	—
307	C₂×C₄×C₅	—	4	1	no	—
308	C₂×C₄×C₅	—	4	1	no	—
309	C₂×C₄×C₅	—	4	2	no	—
310	C₂×C₄×C₅	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	4	1	no	—
313	C₂×C₄×C₅	—	8	2	no	—
314	C₂×C₄×C₅	#121	1	1	yes	characteristic
315	C₂×C₄×C₅	#121	1	1	yes	characteristic
316	C₂×C₄×C₅	#121	1	1	yes	characteristic
317	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
318	C₂×C₄×C₅	#137	1	2	yes	—
319	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
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	C₅×C₈	—	4	1	no	—
331	C₅×C₈	#135	1	2	yes	complemented
332	C₂×#41	—	2	1	no	—
333	C₂×#41	C₅×D₄	1	1	yes	characteristic
334	#135	—	4	4	no	—
335	C₂×Q₁₆	—	2	1	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×SD₁₆	—	2	1	no	—
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	2	1	no	—
340	C₂×D₈	—	2	1	no	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂²×C₈	—	2	1	no	—
344	C₂²×C₈	—	2	1	no	—
345	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
346	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
350	#127	C₂³×C₅	1	1	yes	characteristic
351	#125	—	2	1	no	—
352	#124	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	#121	—	2	1	no	—
356	#121	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
360	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	2	1	no	—
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	C₅×D₄	1	1	yes	characteristic
366	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
367	#117	—	2	1	no	—
368	#117	—	2	1	no	—
369	#117	C₅×D₄	1	1	yes	characteristic
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
372	#107	—	2	1	no	—
373	#107	—	2	1	no	—
374	#106	—	2	1	no	—
375	#106	—	2	1	no	—
376	#106	—	2	1	no	—
377	#106	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	2	1	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
383	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
385	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
386	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
387	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
388	#103	—	2	1	no	—
389	#103	—	2	1	no	—
390	#103	—	2	1	no	—
391	#103	—	2	1	no	—
392	#103	—	2	1	no	—
393	#103	—	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	—	2	1	no	—
400	C₂²⋊C₈	—	2	1	no	—
401	C₂²⋊C₈	—	2	1	no	—
402	C₂²⋊C₈	—	2	1	no	—
403	#97	—	2	1	no	—
404	#97	—	2	1	no	—
405	C₄×C₈	—	2	1	no	—
406	C₄×C₈	C₅×D₄	1	1	yes	characteristic
407	C₄×C₈	C₅×D₄	1	1	yes	characteristic
408	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
409	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
410	C₂²×C₅	—	2	1	no	—
411	C₂²×C₅	—	2	1	no	—
412	C₂²×C₅	—	4	1	no	—
413	C₂²×C₅	—	4	1	no	—
414	C₂²×C₅	—	4	1	no	—
415	C₂²×C₅	—	8	1	no	—
416	C₂²×C₅	—	8	1	no	—
417	C₂²×C₅	—	8	1	no	—
418	C₂²×C₅	#468	1	2	yes	—
419	C₂²×C₅	C₂×#121	1	1	yes	characteristic
420	C₄×C₅	—	2	1	no	—
421	C₄×C₅	—	2	1	no	—
422	C₄×C₅	—	2	1	no	—
423	C₄×C₅	—	4	1	no	—
424	C₄×C₅	—	4	1	no	—
425	C₄×C₅	—	4	1	no	—
426	C₄×C₅	—	8	1	no	—
427	C₄×C₅	—	8	1	no	—
428	C₄×C₅	—	8	1	no	—
429	C₄×C₅	#468	1	2	yes	—
430	C₄×C₅	C₂×#135	1	1	yes	characteristic
431	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
432	C₄×C₅	C₂×#137	1	1	yes	characteristic
433	C₄×C₅	#575	1	1	yes	characteristic
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	—	4	2	no	—
437	#41	—	4	2	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	4	1	no	—
447	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
453	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
454	C₂²×C₄	C₅×#41	1	1	yes	characteristic
455	Q₁₆	—	4	2	no	—
456	SD₁₆	—	4	2	no	—
457	SD₁₆	—	4	2	no	—
458	SD₁₆	—	4	2	no	—
459	SD₁₆	—	4	2	no	—
460	D₈	—	4	2	no	—
461	M₁₆	—	4	2	no	—
462	M₁₆	—	4	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	4	1	no	—
471	C₂×C₈	—	4	2	no	—
472	C₂×C₈	—	4	2	no	—
473	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
474	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
475	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
488	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	4	1	no	—
499	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
500	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
501	C₄²	C₅×#41	1	1	yes	characteristic
502	C₄²	C₂⁴×C₅	1	1	yes	characteristic
503	C₂×C₅	—	2	2	no	—
504	C₂×C₅	—	8	1	no	—
505	C₂×C₅	—	16	1	no	—
506	C₂×C₅	#3035	1	1	yes	characteristic
507	C₂×C₅	C₂×#468	1	1	yes	characteristic
508	C₂×C₅	#3067	1	1	yes	characteristic
509	C₂³	—	2	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	C₅×#121	1	1	yes	characteristic
512	Q₈	—	2	1	no	—
513	Q₈	—	2	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	—	4	1	no	—
518	Q₈	—	8	1	no	—
519	D₄	—	2	1	no	—
520	D₄	—	2	1	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	8	1	no	—
527	D₄	—	8	2	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	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	C₅×#121	1	1	yes	characteristic
543	C₂×C₄	C₅×#121	1	1	yes	characteristic
544	C₂×C₄	C₅×#121	1	1	yes	characteristic
545	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
546	C₂×C₄	C₅×#137	1	2	yes	—
547	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
548	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
549	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	2	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	C₅×#135	1	2	yes	complemented
560	C₅	#19156	1	1	yes	characteristic, direct product factor, Sylow
561	C₂²	—	2	1	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	8	1	no	—
569	C₂²	C₅×#468	1	2	yes	—
570	C₂²	C₂×C₅×#121	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	—	8	1	no	—
580	C₄	C₅×#468	1	2	yes	—
581	C₄	C₂×C₅×#135	1	1	yes	characteristic
582	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
583	C₄	C₂×C₅×#137	1	1	yes	characteristic
584	C₄	C₅×#575	1	1	yes	characteristic
585	C₂	—	2	2	no	—
586	C₂	—	8	1	no	—
587	C₂	—	16	1	no	—
588	C₂	C₅×#3035	1	1	yes	characteristic
589	C₂	C₂×C₅×#468	1	1	yes	characteristic
590	C₂	C₅×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×#121
prime radical	C₅×#3035	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3167	C₂	semidirect
2. C₅×#3076	C₂	semidirect
4. C₅×#2965	C₂	semidirect
5. C₅×#1700	C₂	semidirect
6. C₅×#1699	C₂	semidirect
7. C₅×#1698	C₂	semidirect
8. C₅×#1697	C₂	semidirect
9. C₅×#1692	C₂	semidirect
10. C₅×#1691	C₂	semidirect
11. C₅×#1686	C₂	semidirect
12. C₅×#1685	C₂	semidirect
13. C₂×C₅×#335	C₂	semidirect
14. C₅×#1542	C₂	semidirect
15. C₅×#1540	C₂	semidirect
67. #19156	C₅	direct (N × H)
146. #3167	C₂×C₅	semidirect
147. #3076	C₂×C₅	semidirect
149. #2965	C₂×C₅	semidirect
150. #1700	C₂×C₅	semidirect
151. #1699	C₂×C₅	semidirect
152. #1698	C₂×C₅	semidirect
153. #1697	C₂×C₅	semidirect
154. #1692	C₂×C₅	semidirect
155. #1691	C₂×C₅	semidirect
156. #1686	C₂×C₅	semidirect
157. #1685	C₂×C₅	semidirect
158. C₂×#335	C₂×C₅	semidirect
159. #1542	C₂×C₅	semidirect
160. #1540	C₂×C₅	semidirect
331. C₅×C₈	#135	semidirect
559. C₈	C₅×#135	semidirect
560. C₅	#19156	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×#3167	2
C₅×#3076	2
C₅×#3035	2
C₅×#2965	2
C₅×#1700	2
C₅×#1699	2
C₅×#1698	2
C₅×#1697	2
C₅×#1692	2
C₅×#1691	2
C₅×#1686	2
C₅×#1685	2
C₂×C₅×#335	2
C₅×#1542	2
C₅×#1540	2
#19156	5

Subgroups — C5×#19156 #11941876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19156 #11941876