Subgroups — C₅×#18764 #11941484

Order 1280.

956 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3482	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1650	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1649	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1627	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1626	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1625	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1552	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#135	—	2	1	no	—
17	C₅×#557	C₂²	1	1	yes	characteristic
18	C₅×#556	—	2	1	no	—
19	C₅×#556	C₂²	1	1	yes	characteristic
20	C₅×#552	—	2	2	no	—
21	C₅×#552	—	2	1	no	—
22	C₅×D₄×Q₈	C₂²	1	1	yes	characteristic
23	C₅×#548	—	2	1	no	—
24	C₅×#547	—	2	1	no	—
25	C₅×#547	C₂²	1	1	yes	characteristic
26	C₅×#533	—	2	1	no	—
27	C₅×#533	C₂²	1	1	yes	characteristic
28	C₅×#532	C₂²	1	1	yes	characteristic
29	C₅×#529	—	2	1	no	—
30	C₅×#529	—	2	2	no	—
31	C₄×C₅×#41	—	2	1	no	—
32	C₄×C₅×#41	C₂²	1	1	yes	characteristic
33	C₄×C₅×#41	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×Q₈	—	2	1	no	—
35	C₅×#500	C₂²	1	1	yes	characteristic
36	C₅×#498	C₂²	1	1	yes	characteristic
37	C₅×#486	C₂²	1	1	yes	characteristic
38	C₅×#486	C₂²	1	1	yes	characteristic
39	C₅×#482	—	2	1	no	—
40	C₅×#482	—	2	2	no	—
41	C₅×#478	—	2	2	no	—
42	C₅×#475	C₂²	1	1	yes	characteristic
43	C₅×#474	C₂²	1	1	yes	characteristic
44	C₅×#464	—	2	1	no	—
45	C₅×#463	—	2	1	no	—
46	C₅×#462	C₂²	1	1	yes	characteristic
47	C₅×#461	C₂²	1	1	yes	characteristic
48	C₅×#460	C₂²	1	1	yes	characteristic
49	C₅×#460	C₂²	1	1	yes	characteristic
50	C₅×#459	C₂²	1	1	yes	characteristic
51	C₅×#459	C₂²	1	1	yes	characteristic
52	C₅×#449	—	2	1	no	—
53	C₅×#442	—	2	1	no	—
54	C₅×#440	—	2	1	no	—
55	C₅×#432	C₂²	1	1	yes	characteristic
56	C₅×#431	C₂²	1	1	yes	characteristic
57	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₅×#416	—	2	1	no	—
59	C₅×#332	C₂²	1	1	yes	characteristic
60	C₅×#332	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×#332	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	#18764	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₅×#41	—	2	1	no	—
73	C₂×C₅×#41	—	2	1	no	—
74	C₂×C₅×#41	D₄	1	1	yes	characteristic
75	C₂×C₅×#41	D₄	1	1	yes	characteristic
76	C₂²×C₅×Q₈	—	2	1	no	—
77	C₅×#135	—	4	4	no	—
78	C₂×C₅×Q₁₆	—	2	1	no	—
79	C₂×C₅×SD₁₆	—	2	1	no	—
80	C₂×C₅×SD₁₆	—	2	1	no	—
81	C₂×C₅×SD₁₆	—	2	1	no	—
82	C₂×C₅×D₈	—	2	1	no	—
83	C₂×C₅×D₈	—	2	1	no	—
84	C₂²×C₅×C₈	—	2	1	no	—
85	C₅×C₄⋊Q₈	—	2	1	no	—
86	C₅×C₄⋊Q₈	—	2	1	no	—
87	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₅×#127	C₂³	1	1	yes	characteristic
91	C₅×#126	—	2	2	no	—
92	C₅×#126	—	2	2	no	—
93	C₅×#126	—	2	2	no	—
94	C₅×#125	—	2	2	no	—
95	C₅×#125	—	2	2	no	—
96	C₅×#125	—	2	1	no	—
97	C₅×#125	D₄	1	2	yes	—
98	C₅×#124	—	2	2	no	—
99	C₅×#124	—	2	1	no	—
100	C₅×#124	—	2	1	no	—
101	C₅×#123	—	2	2	no	—
102	C₅×#123	—	2	2	no	—
103	C₅×#123	—	2	2	no	—
104	C₅×C₂²⋊Q₈	—	2	2	no	—
105	C₅×C₂²⋊Q₈	—	2	1	no	—
106	C₅×C₂²⋊Q₈	—	2	1	no	—
107	C₅×C₂²⋊Q₈	—	2	1	no	—
108	C₅×C₂²⋊Q₈	—	2	1	no	—
109	C₅×C₂²⋊Q₈	—	2	2	no	—
110	C₅×C₂²⋊Q₈	—	2	1	no	—
111	C₅×#121	—	2	2	no	—
112	C₅×#121	—	2	1	no	—
113	C₅×#121	—	2	1	no	—
114	C₅×#121	—	2	1	no	—
115	C₄×C₅×Q₈	—	2	2	no	—
116	C₄×C₅×Q₈	—	2	2	no	—
117	C₄×C₅×Q₈	—	2	1	no	—
118	C₄×C₅×Q₈	—	2	1	no	—
119	C₄×C₅×Q₈	—	2	1	no	—
120	C₄×C₅×Q₈	—	2	1	no	—
121	C₄×C₅×Q₈	—	2	1	no	—
122	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
123	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
124	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
125	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
126	C₄×C₅×D₄	—	2	2	no	—
127	C₄×C₅×D₄	—	2	2	no	—
128	C₄×C₅×D₄	—	2	1	no	—
129	C₄×C₅×D₄	—	2	1	no	—
130	C₄×C₅×D₄	—	2	1	no	—
131	C₄×C₅×D₄	—	2	1	no	—
132	C₄×C₅×D₄	—	2	1	no	—
133	C₄×C₅×D₄	—	2	1	no	—
134	C₄×C₅×D₄	—	2	1	no	—
135	C₄×C₅×D₄	D₄	1	1	yes	characteristic
136	C₄×C₅×D₄	D₄	1	1	yes	characteristic
137	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
138	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
139	C₅×#117	—	2	2	no	—
140	C₅×#117	—	2	2	no	—
141	C₅×#117	—	2	1	no	—
142	C₅×#117	—	2	1	no	—
143	C₅×#117	—	2	1	no	—
144	C₅×#117	D₄	1	1	yes	characteristic
145	C₅×#117	D₄	1	1	yes	characteristic
146	C₅×#117	D₄	1	2	yes	—
147	C₂×C₅×C₄⋊C₄	—	2	2	no	—
148	C₂×C₅×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄²×C₅	—	2	2	no	—
150	C₂×C₄²×C₅	—	2	1	no	—
151	C₂×C₄²×C₅	—	2	1	no	—
152	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
153	C₅×#107	—	2	1	no	—
154	C₅×#107	—	2	2	no	—
155	C₅×#106	—	2	1	no	—
156	C₅×#106	—	2	2	no	—
157	C₅×C₄⋊C₈	—	2	1	no	—
158	C₅×C₄⋊C₈	—	2	1	no	—
159	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₅×#103	—	2	1	no	—
164	C₅×#103	—	2	1	no	—
165	C₅×#103	—	2	1	no	—
166	C₅×#102	—	2	1	no	—
167	C₅×#102	—	2	1	no	—
168	C₅×#102	—	2	1	no	—
169	C₅×#102	—	2	1	no	—
170	C₅×#102	—	2	1	no	—
171	C₅×#102	—	2	2	no	—
172	C₅×#102	—	2	2	no	—
173	C₅×C₂²⋊C₈	—	2	1	no	—
174	C₅×C₂²⋊C₈	—	2	1	no	—
175	C₅×#97	—	2	1	no	—
176	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
177	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
178	#3482	C₂×C₅	1	1	yes	characteristic, complemented
179	#3228	C₂×C₅	1	1	yes	characteristic, complemented
180	#3115	C₂×C₅	1	1	yes	characteristic, complemented
181	#3035	C₂×C₅	1	1	yes	characteristic, complemented
182	#1654	C₂×C₅	1	1	yes	characteristic, complemented
183	#1653	C₂×C₅	1	1	yes	characteristic, complemented
184	#1650	C₂×C₅	1	1	yes	characteristic, complemented
185	#1649	C₂×C₅	1	1	yes	characteristic, complemented
186	#1628	C₂×C₅	1	1	yes	characteristic, complemented
187	#1627	C₂×C₅	1	1	yes	characteristic, complemented
188	#1626	C₂×C₅	1	1	yes	characteristic, complemented
189	#1625	C₂×C₅	1	1	yes	characteristic, complemented
190	#1552	C₂×C₅	1	1	yes	characteristic, complemented
191	#1488	C₂×C₅	1	1	yes	characteristic, complemented
192	#1477	C₂×C₅	1	1	yes	characteristic, complemented
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	2	2	no	—
195	C₅×#41	—	4	1	no	—
196	C₅×#41	—	4	1	no	—
197	C₅×#41	—	4	2	no	—
198	C₅×#41	—	4	2	no	—
199	C₅×#41	—	4	2	no	—
200	C₅×#41	—	4	2	no	—
201	C₂×C₅×Q₈	—	2	2	no	—
202	C₂×C₅×Q₈	—	2	1	no	—
203	C₂×C₅×Q₈	—	2	1	no	—
204	C₂×C₅×Q₈	—	2	1	no	—
205	C₂×C₅×Q₈	—	4	2	no	—
206	C₂×C₅×Q₈	—	4	2	no	—
207	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
208	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
209	C₂×C₅×D₄	—	2	2	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	—	2	1	no	—
213	C₂×C₅×D₄	—	2	1	no	—
214	C₂×C₅×D₄	—	2	1	no	—
215	C₂×C₅×D₄	—	4	1	no	—
216	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
217	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₅	—	2	2	no	—
219	C₂²×C₄×C₅	—	2	2	no	—
220	C₂²×C₄×C₅	—	2	1	no	—
221	C₂²×C₄×C₅	—	2	1	no	—
222	C₂²×C₄×C₅	—	2	1	no	—
223	C₂²×C₄×C₅	—	2	1	no	—
224	C₂²×C₄×C₅	—	2	1	no	—
225	C₂²×C₄×C₅	—	2	1	no	—
226	C₂²×C₄×C₅	—	4	1	no	—
227	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
230	C₅×Q₁₆	—	4	2	no	—
231	C₅×SD₁₆	—	4	2	no	—
232	C₅×SD₁₆	—	4	2	no	—
233	C₅×SD₁₆	—	4	2	no	—
234	C₅×D₈	—	4	2	no	—
235	C₅×D₈	—	4	2	no	—
236	C₂×C₅×C₈	—	2	1	no	—
237	C₂×C₅×C₈	—	2	1	no	—
238	C₂×C₅×C₈	—	2	1	no	—
239	C₂×C₅×C₈	—	2	1	no	—
240	C₂×C₅×C₈	—	2	1	no	—
241	C₂×C₅×C₈	—	2	1	no	—
242	C₂×C₅×C₈	—	4	2	no	—
243	C₅×C₄⋊C₄	—	2	2	no	—
244	C₅×C₄⋊C₄	—	2	2	no	—
245	C₅×C₄⋊C₄	—	2	2	no	—
246	C₅×C₄⋊C₄	—	2	2	no	—
247	C₅×C₄⋊C₄	—	2	2	no	—
248	C₅×C₄⋊C₄	—	2	2	no	—
249	C₅×C₄⋊C₄	—	2	1	no	—
250	C₅×C₄⋊C₄	—	2	2	no	—
251	C₅×C₄⋊C₄	—	2	2	no	—
252	C₅×C₄⋊C₄	—	2	1	no	—
253	C₅×C₄⋊C₄	—	2	1	no	—
254	C₅×C₄⋊C₄	—	2	1	no	—
255	C₅×C₄⋊C₄	—	2	1	no	—
256	C₅×C₄⋊C₄	—	2	1	no	—
257	C₅×C₄⋊C₄	—	2	1	no	—
258	C₅×C₄⋊C₄	—	2	1	no	—
259	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
260	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
261	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
262	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
263	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
264	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
265	C₅×C₂²⋊C₄	—	2	2	no	—
266	C₅×C₂²⋊C₄	—	2	2	no	—
267	C₅×C₂²⋊C₄	—	2	2	no	—
268	C₅×C₂²⋊C₄	—	2	2	no	—
269	C₅×C₂²⋊C₄	—	2	1	no	—
270	C₅×C₂²⋊C₄	—	2	1	no	—
271	C₅×C₂²⋊C₄	—	2	2	no	—
272	C₅×C₂²⋊C₄	—	2	1	no	—
273	C₅×C₂²⋊C₄	—	2	1	no	—
274	C₅×C₂²⋊C₄	—	2	1	no	—
275	C₅×C₂²⋊C₄	—	2	1	no	—
276	C₅×C₂²⋊C₄	—	4	1	no	—
277	C₅×C₂²⋊C₄	—	4	1	no	—
278	C₄²×C₅	—	2	2	no	—
279	C₄²×C₅	—	2	2	no	—
280	C₄²×C₅	—	2	1	no	—
281	C₄²×C₅	—	2	1	no	—
282	C₄²×C₅	—	2	1	no	—
283	C₄²×C₅	—	2	2	no	—
284	C₄²×C₅	—	2	1	no	—
285	C₄²×C₅	—	2	1	no	—
286	C₄²×C₅	—	2	1	no	—
287	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
288	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
289	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
290	C₄²×C₅	C₂⁴	1	1	yes	characteristic
291	C₂×#135	—	2	1	no	—
292	#557	C₂²×C₅	1	1	yes	characteristic
293	#556	—	2	1	no	—
294	#556	C₂²×C₅	1	1	yes	characteristic
295	#552	—	2	2	no	—
296	#552	—	2	1	no	—
297	D₄×Q₈	C₂²×C₅	1	1	yes	characteristic
298	#548	—	2	1	no	—
299	#547	—	2	1	no	—
300	#547	C₂²×C₅	1	1	yes	characteristic
301	#533	—	2	1	no	—
302	#533	C₂²×C₅	1	1	yes	characteristic
303	#532	C₂²×C₅	1	1	yes	characteristic
304	#529	—	2	1	no	—
305	#529	—	2	2	no	—
306	C₄×#41	—	2	1	no	—
307	C₄×#41	C₂²×C₅	1	1	yes	characteristic
308	C₄×#41	C₂²×C₅	1	1	yes	characteristic
309	C₂×C₄×Q₈	—	2	1	no	—
310	#500	C₂²×C₅	1	1	yes	characteristic
311	#498	C₂²×C₅	1	1	yes	characteristic
312	#486	C₂²×C₅	1	1	yes	characteristic
313	#486	C₂²×C₅	1	1	yes	characteristic
314	#482	—	2	1	no	—
315	#482	—	2	2	no	—
316	#478	—	2	2	no	—
317	#475	C₂²×C₅	1	1	yes	characteristic
318	#474	C₂²×C₅	1	1	yes	characteristic
319	#464	—	2	1	no	—
320	#463	—	2	1	no	—
321	#462	C₂²×C₅	1	1	yes	characteristic
322	#461	C₂²×C₅	1	1	yes	characteristic
323	#460	C₂²×C₅	1	1	yes	characteristic
324	#460	C₂²×C₅	1	1	yes	characteristic
325	#459	C₂²×C₅	1	1	yes	characteristic
326	#459	C₂²×C₅	1	1	yes	characteristic
327	#449	—	2	1	no	—
328	#442	—	2	1	no	—
329	#440	—	2	1	no	—
330	#432	C₂²×C₅	1	1	yes	characteristic
331	#431	C₂²×C₅	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
333	#416	—	2	1	no	—
334	#332	C₂²×C₅	1	1	yes	characteristic
335	#332	C₂²×C₅	1	1	yes	characteristic
336	#332	C₂²×C₅	1	1	yes	characteristic
337	#332	C₂²×C₅	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
343	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
345	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
346	C₂³×C₅	—	2	1	no	—
347	C₂³×C₅	—	2	1	no	—
348	C₂³×C₅	—	4	1	no	—
349	C₂³×C₅	#120	1	1	yes	characteristic
350	C₅×Q₈	—	2	1	no	—
351	C₅×Q₈	—	2	1	no	—
352	C₅×Q₈	—	2	1	no	—
353	C₅×Q₈	—	4	4	no	—
354	C₅×Q₈	—	4	1	no	—
355	C₅×Q₈	—	4	1	no	—
356	C₅×Q₈	—	4	1	no	—
357	C₅×Q₈	—	4	1	no	—
358	C₅×Q₈	—	4	1	no	—
359	C₅×Q₈	—	4	1	no	—
360	C₅×Q₈	AGL(1,ℤ/8)	1	2	yes	—
361	C₅×D₄	—	2	1	no	—
362	C₅×D₄	—	2	1	no	—
363	C₅×D₄	—	2	1	no	—
364	C₅×D₄	—	4	2	no	—
365	C₅×D₄	—	4	2	no	—
366	C₅×D₄	—	4	2	no	—
367	C₅×D₄	—	4	4	no	—
368	C₅×D₄	—	4	1	no	—
369	C₅×D₄	—	4	1	no	—
370	C₅×D₄	—	4	1	no	—
371	C₅×D₄	—	4	1	no	—
372	C₅×D₄	—	8	2	no	—
373	C₅×D₄	#135	1	2	yes	complemented
374	C₂×C₄×C₅	—	2	1	no	—
375	C₂×C₄×C₅	—	2	1	no	—
376	C₂×C₄×C₅	—	2	1	no	—
377	C₂×C₄×C₅	—	2	1	no	—
378	C₂×C₄×C₅	—	2	1	no	—
379	C₂×C₄×C₅	—	2	1	no	—
380	C₂×C₄×C₅	—	2	1	no	—
381	C₂×C₄×C₅	—	2	1	no	—
382	C₂×C₄×C₅	—	2	2	no	—
383	C₂×C₄×C₅	—	2	1	no	—
384	C₂×C₄×C₅	—	2	1	no	—
385	C₂×C₄×C₅	—	2	2	no	—
386	C₂×C₄×C₅	—	4	1	no	—
387	C₂×C₄×C₅	—	4	2	no	—
388	C₂×C₄×C₅	—	4	2	no	—
389	C₂×C₄×C₅	—	4	2	no	—
390	C₂×C₄×C₅	—	4	2	no	—
391	C₂×C₄×C₅	—	4	4	no	—
392	C₂×C₄×C₅	—	4	4	no	—
393	C₂×C₄×C₅	—	4	1	no	—
394	C₂×C₄×C₅	—	4	1	no	—
395	C₂×C₄×C₅	—	4	1	no	—
396	C₂×C₄×C₅	—	4	1	no	—
397	C₂×C₄×C₅	—	4	1	no	—
398	C₂×C₄×C₅	—	8	2	no	—
399	C₂×C₄×C₅	#120	1	1	yes	characteristic
400	C₂×C₄×C₅	#120	1	1	yes	characteristic
401	C₂×C₄×C₅	#120	1	1	yes	characteristic
402	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
403	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
404	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
405	C₅×C₈	—	4	1	no	—
406	C₅×C₈	—	4	1	no	—
407	C₅×C₈	—	4	1	no	—
408	C₅×C₈	—	4	1	no	—
409	C₅×C₈	—	4	1	no	—
410	C₅×C₈	—	4	1	no	—
411	C₂×#41	—	2	1	no	—
412	C₂×#41	—	2	1	no	—
413	C₂×#41	C₅×D₄	1	1	yes	characteristic
414	C₂×#41	C₅×D₄	1	1	yes	characteristic
415	C₂²×Q₈	—	2	1	no	—
416	#135	—	4	4	no	—
417	C₂×Q₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×SD₁₆	—	2	1	no	—
421	C₂×D₈	—	2	1	no	—
422	C₂×D₈	—	2	1	no	—
423	C₂²×C₈	—	2	1	no	—
424	C₄⋊Q₈	—	2	1	no	—
425	C₄⋊Q₈	—	2	1	no	—
426	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
427	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
428	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
429	#127	C₂³×C₅	1	1	yes	characteristic
430	#126	—	2	2	no	—
431	#126	—	2	2	no	—
432	#126	—	2	2	no	—
433	#125	—	2	2	no	—
434	#125	—	2	2	no	—
435	#125	—	2	1	no	—
436	#125	C₅×D₄	1	2	yes	—
437	#124	—	2	2	no	—
438	#124	—	2	1	no	—
439	#124	—	2	1	no	—
440	#123	—	2	2	no	—
441	#123	—	2	2	no	—
442	#123	—	2	2	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	1	no	—
445	C₂²⋊Q₈	—	2	1	no	—
446	C₂²⋊Q₈	—	2	1	no	—
447	C₂²⋊Q₈	—	2	1	no	—
448	C₂²⋊Q₈	—	2	2	no	—
449	C₂²⋊Q₈	—	2	1	no	—
450	#121	—	2	2	no	—
451	#121	—	2	1	no	—
452	#121	—	2	1	no	—
453	#121	—	2	1	no	—
454	C₄×Q₈	—	2	2	no	—
455	C₄×Q₈	—	2	2	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×Q₈	—	2	1	no	—
458	C₄×Q₈	—	2	1	no	—
459	C₄×Q₈	—	2	1	no	—
460	C₄×Q₈	—	2	1	no	—
461	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
462	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
463	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
464	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	1	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	—	2	1	no	—
470	C₄×D₄	—	2	1	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	C₅×D₄	1	1	yes	characteristic
475	C₄×D₄	C₅×D₄	1	1	yes	characteristic
476	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
477	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
478	#117	—	2	2	no	—
479	#117	—	2	2	no	—
480	#117	—	2	1	no	—
481	#117	—	2	1	no	—
482	#117	—	2	1	no	—
483	#117	C₅×D₄	1	1	yes	characteristic
484	#117	C₅×D₄	1	1	yes	characteristic
485	#117	C₅×D₄	1	2	yes	—
486	C₂×C₄⋊C₄	—	2	2	no	—
487	C₂×C₄⋊C₄	—	2	1	no	—
488	C₂×C₄²	—	2	2	no	—
489	C₂×C₄²	—	2	1	no	—
490	C₂×C₄²	—	2	1	no	—
491	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
492	#107	—	2	1	no	—
493	#107	—	2	2	no	—
494	#106	—	2	1	no	—
495	#106	—	2	2	no	—
496	C₄⋊C₈	—	2	1	no	—
497	C₄⋊C₈	—	2	1	no	—
498	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
499	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
500	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
501	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
502	#103	—	2	1	no	—
503	#103	—	2	1	no	—
504	#103	—	2	1	no	—
505	#102	—	2	1	no	—
506	#102	—	2	1	no	—
507	#102	—	2	1	no	—
508	#102	—	2	1	no	—
509	#102	—	2	1	no	—
510	#102	—	2	2	no	—
511	#102	—	2	2	no	—
512	C₂²⋊C₈	—	2	1	no	—
513	C₂²⋊C₈	—	2	1	no	—
514	#97	—	2	1	no	—
515	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
516	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
517	C₂²×C₅	—	2	1	no	—
518	C₂²×C₅	—	2	2	no	—
519	C₂²×C₅	—	2	1	no	—
520	C₂²×C₅	—	2	1	no	—
521	C₂²×C₅	—	4	1	no	—
522	C₂²×C₅	—	4	1	no	—
523	C₂²×C₅	—	4	1	no	—
524	C₂²×C₅	—	4	1	no	—
525	C₂²×C₅	—	4	1	no	—
526	C₂²×C₅	—	8	1	no	—
527	C₂²×C₅	—	8	1	no	—
528	C₂²×C₅	—	8	1	no	—
529	C₂²×C₅	C₂×#120	1	1	yes	characteristic
530	C₄×C₅	—	2	1	no	—
531	C₄×C₅	—	2	1	no	—
532	C₄×C₅	—	2	2	no	—
533	C₄×C₅	—	2	1	no	—
534	C₄×C₅	—	2	1	no	—
535	C₄×C₅	—	4	1	no	—
536	C₄×C₅	—	4	1	no	—
537	C₄×C₅	—	4	2	no	—
538	C₄×C₅	—	4	2	no	—
539	C₄×C₅	—	4	1	no	—
540	C₄×C₅	—	4	1	no	—
541	C₄×C₅	—	4	1	no	—
542	C₄×C₅	—	8	1	no	—
543	C₄×C₅	C₂×#135	1	1	yes	characteristic
544	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
545	C₄×C₅	#575	1	1	yes	characteristic
546	C₄×C₅	#575	1	1	yes	characteristic
547	#41	—	2	2	no	—
548	#41	—	2	2	no	—
549	#41	—	4	1	no	—
550	#41	—	4	1	no	—
551	#41	—	4	2	no	—
552	#41	—	4	2	no	—
553	#41	—	4	2	no	—
554	#41	—	4	2	no	—
555	C₂×Q₈	—	2	2	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	4	2	no	—
560	C₂×Q₈	—	4	2	no	—
561	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
562	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
563	C₂×D₄	—	2	2	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	4	1	no	—
570	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
571	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	—	2	2	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	2	1	no	—
579	C₂²×C₄	—	2	1	no	—
580	C₂²×C₄	—	4	1	no	—
581	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
582	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
583	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
584	Q₁₆	—	4	2	no	—
585	SD₁₆	—	4	2	no	—
586	SD₁₆	—	4	2	no	—
587	SD₁₆	—	4	2	no	—
588	D₈	—	4	2	no	—
589	D₈	—	4	2	no	—
590	C₂×C₈	—	2	1	no	—
591	C₂×C₈	—	2	1	no	—
592	C₂×C₈	—	2	1	no	—
593	C₂×C₈	—	2	1	no	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	4	2	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	—	2	2	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
614	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
615	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
616	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
617	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
618	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	—	2	1	no	—
624	C₂²⋊C₄	—	2	1	no	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	1	no	—
627	C₂²⋊C₄	—	2	1	no	—
628	C₂²⋊C₄	—	2	1	no	—
629	C₂²⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	4	1	no	—
631	C₂²⋊C₄	—	4	1	no	—
632	C₄²	—	2	2	no	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	1	no	—
635	C₄²	—	2	1	no	—
636	C₄²	—	2	1	no	—
637	C₄²	—	2	2	no	—
638	C₄²	—	2	1	no	—
639	C₄²	—	2	1	no	—
640	C₄²	—	2	1	no	—
641	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
642	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
643	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
644	C₄²	C₂⁴×C₅	1	1	yes	characteristic
645	C₂×C₅	—	4	1	no	—
646	C₂×C₅	—	4	2	no	—
647	C₂×C₅	—	8	1	no	—
648	C₂×C₅	—	16	1	no	—
649	C₂×C₅	#3002	1	1	yes	characteristic
650	C₂×C₅	#3006	1	1	yes	characteristic
651	C₂×C₅	#3020	1	1	yes	characteristic
652	C₂³	—	2	1	no	—
653	C₂³	—	2	1	no	—
654	C₂³	—	4	1	no	—
655	C₂³	C₅×#120	1	1	yes	characteristic
656	Q₈	—	2	1	no	—
657	Q₈	—	2	1	no	—
658	Q₈	—	2	1	no	—
659	Q₈	—	4	4	no	—
660	Q₈	—	4	1	no	—
661	Q₈	—	4	1	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	1	no	—
664	Q₈	—	4	1	no	—
665	Q₈	—	4	1	no	—
666	Q₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
667	D₄	—	2	1	no	—
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	4	2	no	—
671	D₄	—	4	2	no	—
672	D₄	—	4	2	no	—
673	D₄	—	4	4	no	—
674	D₄	—	4	1	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	4	1	no	—
678	D₄	—	8	2	no	—
679	D₄	C₅×#135	1	2	yes	complemented
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	2	no	—
694	C₂×C₄	—	4	2	no	—
695	C₂×C₄	—	4	4	no	—
696	C₂×C₄	—	4	2	no	—
697	C₂×C₄	—	4	2	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	4	1	no	—
700	C₂×C₄	—	4	1	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	8	2	no	—
705	C₂×C₄	C₅×#120	1	1	yes	characteristic
706	C₂×C₄	C₅×#120	1	1	yes	characteristic
707	C₂×C₄	C₅×#120	1	1	yes	characteristic
708	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
709	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
710	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
711	C₈	—	4	1	no	—
712	C₈	—	4	1	no	—
713	C₈	—	4	1	no	—
714	C₈	—	4	1	no	—
715	C₈	—	4	1	no	—
716	C₈	—	4	1	no	—
717	C₅	#18764	1	1	yes	characteristic, direct product factor, Sylow
718	C₂²	—	2	1	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	2	1	no	—
721	C₂²	—	2	1	no	—
722	C₂²	—	4	1	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	1	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	8	1	no	—
728	C₂²	—	8	1	no	—
729	C₂²	—	8	1	no	—
730	C₂²	C₂×C₅×#120	1	1	yes	characteristic
731	C₄	—	2	1	no	—
732	C₄	—	2	1	no	—
733	C₄	—	2	2	no	—
734	C₄	—	2	1	no	—
735	C₄	—	2	1	no	—
736	C₄	—	4	1	no	—
737	C₄	—	4	1	no	—
738	C₄	—	4	2	no	—
739	C₄	—	4	2	no	—
740	C₄	—	4	1	no	—
741	C₄	—	4	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	8	1	no	—
744	C₄	C₂×C₅×#135	1	1	yes	characteristic
745	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
746	C₄	C₅×#575	1	1	yes	characteristic
747	C₄	C₅×#575	1	1	yes	characteristic
748	C₂	—	4	1	no	—
749	C₂	—	4	2	no	—
750	C₂	—	8	1	no	—
751	C₂	—	16	1	no	—
752	C₂	C₅×#3002	1	1	yes	characteristic
753	C₂	C₅×#3006	1	1	yes	characteristic
754	C₂	C₅×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3482	C₂	semidirect
2. C₅×#3228	C₂	semidirect
3. C₅×#3115	C₂	semidirect
4. C₅×#3035	C₂	semidirect
5. C₅×#1654	C₂	semidirect
6. C₅×#1653	C₂	semidirect
7. C₅×#1650	C₂	semidirect
8. C₅×#1649	C₂	semidirect
9. C₅×#1628	C₂	semidirect
10. C₅×#1627	C₂	semidirect
11. C₅×#1626	C₂	semidirect
12. C₅×#1625	C₂	semidirect
13. C₅×#1552	C₂	semidirect
14. C₅×#1488	C₂	semidirect
15. C₅×#1477	C₂	semidirect
71. #18764	C₅	direct (N × H)
178. #3482	C₂×C₅	semidirect
179. #3228	C₂×C₅	semidirect
180. #3115	C₂×C₅	semidirect
181. #3035	C₂×C₅	semidirect
182. #1654	C₂×C₅	semidirect
183. #1653	C₂×C₅	semidirect
184. #1650	C₂×C₅	semidirect
185. #1649	C₂×C₅	semidirect
186. #1628	C₂×C₅	semidirect
187. #1627	C₂×C₅	semidirect
188. #1626	C₂×C₅	semidirect
189. #1625	C₂×C₅	semidirect
190. #1552	C₂×C₅	semidirect
191. #1488	C₂×C₅	semidirect
192. #1477	C₂×C₅	semidirect
373. C₅×D₄	#135	semidirect
679. D₄	C₅×#135	semidirect
717. C₅	#18764	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×#3482	2
C₅×#3228	2
C₅×#3115	2
C₅×#3035	2
C₅×#1654	2
C₅×#1653	2
C₅×#1650	2
C₅×#1649	2
C₅×#1628	2
C₅×#1627	2
C₅×#1626	2
C₅×#1625	2
C₅×#1552	2
C₅×#1488	2
C₅×#1477	2
#18764	5

Subgroups — C5×#18764 #11941484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#18764 #11941484