Subgroups — C₅×#18560 #11941280

Order 1280.

986 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3182	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1650	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1646	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1625	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1624	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1559	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1490	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×AGL(1,ℤ/8)	—	2	1	no	—
17	C₅×#556	C₂²	1	1	yes	characteristic
18	C₅×#555	C₂²	1	1	yes	characteristic
19	C₅×#554	C₂²	1	1	yes	characteristic
20	C₅×#553	—	2	1	no	—
21	C₅×#552	—	2	1	no	—
22	C₅×#552	—	2	1	no	—
23	C₅×#551	—	2	2	no	—
24	C₅×#547	C₂²	1	1	yes	characteristic
25	C₅×#546	—	2	1	no	—
26	C₅×#533	C₂²	1	1	yes	characteristic
27	C₅×#532	C₂²	1	1	yes	characteristic
28	C₅×#529	—	2	2	no	—
29	C₅×#529	—	2	1	no	—
30	C₅×#525	—	2	1	no	—
31	C₄×C₅×#41	—	2	1	no	—
32	C₄×C₅×#41	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₅×#117	—	2	1	no	—
35	C₅×#500	C₂²	1	1	yes	characteristic, complemented
36	C₅×#492	C₂²	1	1	yes	characteristic, complemented
37	C₅×#487	C₂²	1	1	yes	characteristic, complemented
38	C₅×#486	C₂²	1	1	yes	characteristic, complemented
39	C₅×#482	—	2	1	no	—
40	C₅×#475	C₂²	1	1	yes	characteristic, complemented
41	C₅×#474	C₂²	1	1	yes	characteristic, complemented
42	C₅×#468	—	2	2	no	—
43	C₅×#463	—	2	1	no	—
44	C₅×#463	—	2	1	no	—
45	C₅×#461	C₂²	1	1	yes	characteristic
46	C₅×#461	C₂²	1	1	yes	characteristic
47	C₅×#460	C₂²	1	1	yes	characteristic, complemented
48	C₅×#459	C₂²	1	1	yes	characteristic
49	C₅×#459	C₂²	1	1	yes	characteristic
50	C₅×#459	C₂²	1	1	yes	characteristic, complemented
51	C₅×#449	—	2	1	no	—
52	C₅×#442	—	2	1	no	—
53	C₅×#440	—	2	1	no	—
54	C₅×#440	—	2	2	no	—
55	C₅×#432	C₂²	1	1	yes	characteristic
56	C₅×#423	C₂²	1	1	yes	characteristic
57	C₅×#417	—	2	1	no	—
58	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
59	C₅×#333	C₂²	1	1	yes	characteristic, complemented
60	C₅×#333	C₂²	1	1	yes	characteristic, complemented
61	C₅×#332	C₂²	1	1	yes	characteristic, complemented
62	C₅×#332	C₂²	1	1	yes	characteristic, complemented
63	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×D₄⋊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, complemented
70	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
71	#18560	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	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₅	—	2	1	no	—
77	C₅×AGL(1,ℤ/8)	—	4	4	no	—
78	C₂×C₅×SD₁₆	—	2	1	no	—
79	C₂×C₅×SD₁₆	—	2	2	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	C₂×C₅×D₈	—	2	1	no	—
85	C₂×C₅×M₁₆	—	2	1	no	—
86	C₂×C₅×M₁₆	—	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	1	no	—
96	C₅×#125	—	2	1	no	—
97	C₅×#124	—	2	2	no	—
98	C₅×#124	—	2	1	no	—
99	C₅×#124	—	2	1	no	—
100	C₅×#123	—	2	2	no	—
101	C₅×#123	—	2	2	no	—
102	C₅×#123	—	2	2	no	—
103	C₅×#123	—	2	2	no	—
104	C₅×C₂²⋊Q₈	—	2	1	no	—
105	C₅×C₂²⋊Q₈	—	2	1	no	—
106	C₅×C₂²⋊Q₈	—	2	2	no	—
107	C₅×C₂²⋊Q₈	—	2	1	no	—
108	C₅×C₂²⋊Q₈	—	2	1	no	—
109	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
110	C₅×#121	—	2	2	no	—
111	C₅×#121	—	2	1	no	—
112	C₅×#121	—	2	1	no	—
113	C₅×#121	—	2	1	no	—
114	C₅×#121	—	2	1	no	—
115	C₅×#121	—	2	1	no	—
116	C₅×#120	—	2	2	no	—
117	C₄×C₅×Q₈	—	2	1	no	—
118	C₄×C₅×Q₈	—	2	1	no	—
119	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
120	C₄×C₅×Q₈	D₄	1	2	yes	—
121	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
122	C₄×C₅×D₄	—	2	2	no	—
123	C₄×C₅×D₄	—	2	2	no	—
124	C₄×C₅×D₄	—	2	2	no	—
125	C₄×C₅×D₄	—	2	1	no	—
126	C₄×C₅×D₄	—	2	1	no	—
127	C₄×C₅×D₄	—	2	1	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	D₄	1	1	yes	characteristic
133	C₄×C₅×D₄	D₄	1	1	yes	characteristic
134	C₄×C₅×D₄	D₄	1	1	yes	characteristic
135	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
136	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
137	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
138	C₅×#117	—	2	2	no	—
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	—	2	1	no	—
145	C₅×#117	—	2	1	no	—
146	C₅×#117	D₄	1	1	yes	characteristic
147	C₂×C₅×C₄⋊C₄	—	2	1	no	—
148	C₂×C₅×C₄⋊C₄	—	2	1	no	—
149	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
150	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
151	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
152	C₂×C₄²×C₅	—	2	2	no	—
153	C₂×C₄²×C₅	—	2	1	no	—
154	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
155	C₅×#107	—	2	2	no	—
156	C₅×#107	—	2	1	no	—
157	C₅×#106	—	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	2	no	—
165	C₅×#103	—	2	1	no	—
166	C₅×#103	—	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	1	no	—
173	C₅×C₂²⋊C₈	—	2	1	no	—
174	C₅×#97	—	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	#3475	C₂×C₅	1	1	yes	characteristic, complemented
179	#3182	C₂×C₅	1	1	yes	characteristic, complemented
180	#3115	C₂×C₅	1	1	yes	characteristic, complemented
181	#3036	C₂×C₅	1	1	yes	characteristic, complemented
182	#1650	C₂×C₅	1	1	yes	characteristic, complemented
183	#1647	C₂×C₅	1	1	yes	characteristic, complemented
184	#1646	C₂×C₅	1	1	yes	characteristic, complemented
185	#1643	C₂×C₅	1	1	yes	characteristic, complemented
186	#1628	C₂×C₅	1	1	yes	characteristic, complemented
187	#1625	C₂×C₅	1	1	yes	characteristic, complemented
188	#1624	C₂×C₅	1	1	yes	characteristic, complemented
189	#1622	C₂×C₅	1	1	yes	characteristic, complemented
190	#1559	C₂×C₅	1	1	yes	characteristic, complemented
191	#1490	C₂×C₅	1	1	yes	characteristic, complemented
192	#1489	C₂×C₅	1	1	yes	characteristic, complemented
193	C₂⁴×C₅	—	2	1	no	—
194	C₅×#41	—	4	1	no	—
195	C₅×#41	—	4	1	no	—
196	C₅×#41	—	4	2	no	—
197	C₅×#41	—	4	2	no	—
198	C₅×#41	—	4	2	no	—
199	C₅×#41	—	4	2	no	—
200	C₂×C₅×Q₈	—	2	1	no	—
201	C₂×C₅×Q₈	—	2	1	no	—
202	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
204	C₂×C₅×D₄	—	2	2	no	—
205	C₂×C₅×D₄	—	2	2	no	—
206	C₂×C₅×D₄	—	2	1	no	—
207	C₂×C₅×D₄	—	2	2	no	—
208	C₂×C₅×D₄	—	2	1	no	—
209	C₂×C₅×D₄	—	2	1	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	—	4	1	no	—
213	C₂×C₅×D₄	—	4	1	no	—
214	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
215	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
216	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₅	—	2	2	no	—
218	C₂²×C₄×C₅	—	2	1	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	—	2	1	no	—
227	C₂²×C₄×C₅	—	4	2	no	—
228	C₂²×C₄×C₅	—	4	2	no	—
229	C₂²×C₄×C₅	—	4	1	no	—
230	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
231	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
232	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
233	C₅×SD₁₆	—	4	4	no	—
234	C₅×SD₁₆	—	4	2	no	—
235	C₅×SD₁₆	—	4	2	no	—
236	C₅×SD₁₆	—	4	2	no	—
237	C₅×D₈	—	4	2	no	—
238	C₅×D₈	—	4	2	no	—
239	C₅×D₈	—	4	2	no	—
240	C₅×M₁₆	—	4	2	no	—
241	C₅×M₁₆	—	4	2	no	—
242	C₂×C₅×C₈	—	2	1	no	—
243	C₂×C₅×C₈	—	2	1	no	—
244	C₂×C₅×C₈	—	2	1	no	—
245	C₂×C₅×C₈	—	2	1	no	—
246	C₂×C₅×C₈	—	2	1	no	—
247	C₂×C₅×C₈	—	2	1	no	—
248	C₅×C₄⋊C₄	—	2	2	no	—
249	C₅×C₄⋊C₄	—	2	2	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	2	no	—
254	C₅×C₄⋊C₄	—	2	1	no	—
255	C₅×C₄⋊C₄	—	2	2	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	—	2	1	no	—
260	C₅×C₄⋊C₄	—	2	1	no	—
261	C₅×C₄⋊C₄	—	2	1	no	—
262	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
263	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
264	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
265	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
266	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
267	C₅×C₂²⋊C₄	—	2	2	no	—
268	C₅×C₂²⋊C₄	—	2	2	no	—
269	C₅×C₂²⋊C₄	—	2	2	no	—
270	C₅×C₂²⋊C₄	—	2	2	no	—
271	C₅×C₂²⋊C₄	—	2	2	no	—
272	C₅×C₂²⋊C₄	—	2	2	no	—
273	C₅×C₂²⋊C₄	—	2	1	no	—
274	C₅×C₂²⋊C₄	—	2	2	no	—
275	C₅×C₂²⋊C₄	—	2	1	no	—
276	C₅×C₂²⋊C₄	—	2	1	no	—
277	C₅×C₂²⋊C₄	—	2	1	no	—
278	C₅×C₂²⋊C₄	—	2	1	no	—
279	C₅×C₂²⋊C₄	—	2	1	no	—
280	C₅×C₂²⋊C₄	—	2	1	no	—
281	C₅×C₂²⋊C₄	—	4	1	no	—
282	C₅×C₂²⋊C₄	—	4	1	no	—
283	C₄²×C₅	—	2	2	no	—
284	C₄²×C₅	—	2	2	no	—
285	C₄²×C₅	—	2	1	no	—
286	C₄²×C₅	—	2	2	no	—
287	C₄²×C₅	—	2	1	no	—
288	C₄²×C₅	—	2	1	no	—
289	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
290	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
291	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
292	C₄²×C₅	C₂⁴	1	1	yes	characteristic
293	C₂×AGL(1,ℤ/8)	—	2	1	no	—
294	#556	C₂²×C₅	1	1	yes	characteristic
295	#555	C₂²×C₅	1	1	yes	characteristic
296	#554	C₂²×C₅	1	1	yes	characteristic
297	#553	—	2	1	no	—
298	#552	—	2	1	no	—
299	#552	—	2	1	no	—
300	#551	—	2	2	no	—
301	#547	C₂²×C₅	1	1	yes	characteristic
302	#546	—	2	1	no	—
303	#533	C₂²×C₅	1	1	yes	characteristic
304	#532	C₂²×C₅	1	1	yes	characteristic
305	#529	—	2	2	no	—
306	#529	—	2	1	no	—
307	#525	—	2	1	no	—
308	C₄×#41	—	2	1	no	—
309	C₄×#41	C₂²×C₅	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
311	C₂×#117	—	2	1	no	—
312	#500	C₂²×C₅	1	1	yes	characteristic, complemented
313	#492	C₂²×C₅	1	1	yes	characteristic, complemented
314	#487	C₂²×C₅	1	1	yes	characteristic, complemented
315	#486	C₂²×C₅	1	1	yes	characteristic, complemented
316	#482	—	2	1	no	—
317	#475	C₂²×C₅	1	1	yes	characteristic, complemented
318	#474	C₂²×C₅	1	1	yes	characteristic, complemented
319	#468	—	2	2	no	—
320	#463	—	2	1	no	—
321	#463	—	2	1	no	—
322	#461	C₂²×C₅	1	1	yes	characteristic
323	#461	C₂²×C₅	1	1	yes	characteristic
324	#460	C₂²×C₅	1	1	yes	characteristic, complemented
325	#459	C₂²×C₅	1	1	yes	characteristic
326	#459	C₂²×C₅	1	1	yes	characteristic
327	#459	C₂²×C₅	1	1	yes	characteristic, complemented
328	#449	—	2	1	no	—
329	#442	—	2	1	no	—
330	#440	—	2	1	no	—
331	#440	—	2	2	no	—
332	#432	C₂²×C₅	1	1	yes	characteristic
333	#423	C₂²×C₅	1	1	yes	characteristic
334	#417	—	2	1	no	—
335	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
336	#333	C₂²×C₅	1	1	yes	characteristic, complemented
337	#333	C₂²×C₅	1	1	yes	characteristic, complemented
338	#332	C₂²×C₅	1	1	yes	characteristic, complemented
339	#332	C₂²×C₅	1	1	yes	characteristic, complemented
340	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
341	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
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	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
347	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
348	C₂³×C₅	—	2	1	no	—
349	C₂³×C₅	—	2	1	no	—
350	C₂³×C₅	—	2	1	no	—
351	C₂³×C₅	—	4	2	no	—
352	C₂³×C₅	—	4	2	no	—
353	C₂³×C₅	—	4	1	no	—
354	C₂³×C₅	—	8	1	no	—
355	C₂³×C₅	#120	1	1	yes	characteristic
356	C₅×Q₈	—	2	1	no	—
357	C₅×Q₈	—	2	1	no	—
358	C₅×Q₈	—	4	2	no	—
359	C₅×Q₈	—	4	1	no	—
360	C₅×Q₈	—	4	1	no	—
361	C₅×Q₈	—	4	1	no	—
362	C₅×Q₈	—	4	1	no	—
363	C₅×D₄	—	2	1	no	—
364	C₅×D₄	—	2	1	no	—
365	C₅×D₄	—	2	1	no	—
366	C₅×D₄	—	2	1	no	—
367	C₅×D₄	—	2	1	no	—
368	C₅×D₄	—	4	2	no	—
369	C₅×D₄	—	4	2	no	—
370	C₅×D₄	—	4	2	no	—
371	C₅×D₄	—	4	4	no	—
372	C₅×D₄	—	4	1	no	—
373	C₅×D₄	—	4	1	no	—
374	C₅×D₄	—	4	1	no	—
375	C₅×D₄	—	4	1	no	—
376	C₅×D₄	—	8	2	no	—
377	C₅×D₄	—	8	2	no	—
378	C₅×D₄	AGL(1,ℤ/8)	1	2	yes	—
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	1	no	—
383	C₂×C₄×C₅	—	2	1	no	—
384	C₂×C₄×C₅	—	2	1	no	—
385	C₂×C₄×C₅	—	2	1	no	—
386	C₂×C₄×C₅	—	2	1	no	—
387	C₂×C₄×C₅	—	2	2	no	—
388	C₂×C₄×C₅	—	2	2	no	—
389	C₂×C₄×C₅	—	2	1	no	—
390	C₂×C₄×C₅	—	4	1	no	—
391	C₂×C₄×C₅	—	4	2	no	—
392	C₂×C₄×C₅	—	4	2	no	—
393	C₂×C₄×C₅	—	4	2	no	—
394	C₂×C₄×C₅	—	4	2	no	—
395	C₂×C₄×C₅	—	4	4	no	—
396	C₂×C₄×C₅	—	4	1	no	—
397	C₂×C₄×C₅	—	4	1	no	—
398	C₂×C₄×C₅	—	4	1	no	—
399	C₂×C₄×C₅	—	4	1	no	—
400	C₂×C₄×C₅	—	4	1	no	—
401	C₂×C₄×C₅	—	4	1	no	—
402	C₂×C₄×C₅	—	4	1	no	—
403	C₂×C₄×C₅	—	8	1	no	—
404	C₂×C₄×C₅	—	8	2	no	—
405	C₂×C₄×C₅	—	8	2	no	—
406	C₂×C₄×C₅	#120	1	1	yes	characteristic
407	C₂×C₄×C₅	#120	1	1	yes	characteristic
408	C₂×C₄×C₅	#120	1	1	yes	characteristic
409	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
412	C₅×C₈	—	2	2	no	—
413	C₅×C₈	—	4	1	no	—
414	C₅×C₈	—	4	1	no	—
415	C₅×C₈	—	4	1	no	—
416	C₅×C₈	—	4	1	no	—
417	C₅×C₈	—	4	1	no	—
418	C₂×#41	—	2	1	no	—
419	C₂×#41	—	2	1	no	—
420	C₂×#41	C₅×D₄	1	1	yes	characteristic
421	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
422	C₂³×C₄	—	2	1	no	—
423	AGL(1,ℤ/8)	—	4	4	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×SD₁₆	—	2	2	no	—
426	C₂×SD₁₆	—	2	1	no	—
427	C₂×SD₁₆	—	2	1	no	—
428	C₂×D₈	—	2	1	no	—
429	C₂×D₈	—	2	1	no	—
430	C₂×D₈	—	2	1	no	—
431	C₂×M₁₆	—	2	1	no	—
432	C₂×M₁₆	—	2	1	no	—
433	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
435	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
436	#127	C₂³×C₅	1	1	yes	characteristic
437	#126	—	2	2	no	—
438	#126	—	2	2	no	—
439	#126	—	2	2	no	—
440	#125	—	2	2	no	—
441	#125	—	2	1	no	—
442	#125	—	2	1	no	—
443	#124	—	2	2	no	—
444	#124	—	2	1	no	—
445	#124	—	2	1	no	—
446	#123	—	2	2	no	—
447	#123	—	2	2	no	—
448	#123	—	2	2	no	—
449	#123	—	2	2	no	—
450	C₂²⋊Q₈	—	2	1	no	—
451	C₂²⋊Q₈	—	2	1	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	1	no	—
454	C₂²⋊Q₈	—	2	1	no	—
455	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
456	#121	—	2	2	no	—
457	#121	—	2	1	no	—
458	#121	—	2	1	no	—
459	#121	—	2	1	no	—
460	#121	—	2	1	no	—
461	#121	—	2	1	no	—
462	#120	—	2	2	no	—
463	C₄×Q₈	—	2	1	no	—
464	C₄×Q₈	—	2	1	no	—
465	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
466	C₄×Q₈	C₅×D₄	1	2	yes	—
467	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
468	C₄×D₄	—	2	2	no	—
469	C₄×D₄	—	2	2	no	—
470	C₄×D₄	—	2	2	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	—	2	1	no	—
475	C₄×D₄	—	2	1	no	—
476	C₄×D₄	—	2	1	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	C₅×D₄	1	1	yes	characteristic
479	C₄×D₄	C₅×D₄	1	1	yes	characteristic
480	C₄×D₄	C₅×D₄	1	1	yes	characteristic
481	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
482	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
483	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
484	#117	—	2	2	no	—
485	#117	—	2	2	no	—
486	#117	—	2	2	no	—
487	#117	—	2	1	no	—
488	#117	—	2	1	no	—
489	#117	—	2	1	no	—
490	#117	—	2	1	no	—
491	#117	—	2	1	no	—
492	#117	C₅×D₄	1	1	yes	characteristic
493	C₂×C₄⋊C₄	—	2	1	no	—
494	C₂×C₄⋊C₄	—	2	1	no	—
495	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
496	C₂×C₂²⋊C₄	—	2	2	no	—
497	C₂×C₂²⋊C₄	—	2	1	no	—
498	C₂×C₄²	—	2	2	no	—
499	C₂×C₄²	—	2	1	no	—
500	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
501	#107	—	2	2	no	—
502	#107	—	2	1	no	—
503	#106	—	2	1	no	—
504	C₄⋊C₈	—	2	1	no	—
505	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
506	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
507	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
508	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
509	#103	—	2	1	no	—
510	#103	—	2	2	no	—
511	#103	—	2	1	no	—
512	#103	—	2	1	no	—
513	#102	—	2	1	no	—
514	#102	—	2	1	no	—
515	#102	—	2	1	no	—
516	#102	—	2	1	no	—
517	#102	—	2	2	no	—
518	#102	—	2	1	no	—
519	C₂²⋊C₈	—	2	1	no	—
520	#97	—	2	1	no	—
521	#97	—	2	1	no	—
522	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
523	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
524	C₂²×C₅	—	2	1	no	—
525	C₂²×C₅	—	2	1	no	—
526	C₂²×C₅	—	2	2	no	—
527	C₂²×C₅	—	2	1	no	—
528	C₂²×C₅	—	4	1	no	—
529	C₂²×C₅	—	4	1	no	—
530	C₂²×C₅	—	4	1	no	—
531	C₂²×C₅	—	4	1	no	—
532	C₂²×C₅	—	4	1	no	—
533	C₂²×C₅	—	4	1	no	—
534	C₂²×C₅	—	4	1	no	—
535	C₂²×C₅	—	4	1	no	—
536	C₂²×C₅	—	8	4	no	—
537	C₂²×C₅	—	8	1	no	—
538	C₂²×C₅	—	8	1	no	—
539	C₂²×C₅	—	8	1	no	—
540	C₂²×C₅	C₂×#120	1	1	yes	characteristic
541	C₄×C₅	—	2	1	no	—
542	C₄×C₅	—	2	1	no	—
543	C₄×C₅	—	2	1	no	—
544	C₄×C₅	—	2	1	no	—
545	C₄×C₅	—	4	1	no	—
546	C₄×C₅	—	4	1	no	—
547	C₄×C₅	—	4	1	no	—
548	C₄×C₅	—	4	1	no	—
549	C₄×C₅	—	4	2	no	—
550	C₄×C₅	—	4	1	no	—
551	C₄×C₅	—	4	2	no	—
552	C₄×C₅	—	8	1	no	—
553	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
554	C₄×C₅	#575	1	1	yes	characteristic
555	C₄×C₅	#575	1	1	yes	characteristic
556	C₄×C₅	#575	1	1	yes	characteristic
557	C₂⁴	—	2	1	no	—
558	#41	—	4	1	no	—
559	#41	—	4	1	no	—
560	#41	—	4	2	no	—
561	#41	—	4	2	no	—
562	#41	—	4	2	no	—
563	#41	—	4	2	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
567	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
568	C₂×D₄	—	2	2	no	—
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	—	2	1	no	—
571	C₂×D₄	—	2	2	no	—
572	C₂×D₄	—	2	1	no	—
573	C₂×D₄	—	2	1	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	4	1	no	—
577	C₂×D₄	—	4	1	no	—
578	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
579	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
580	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
581	C₂²×C₄	—	2	2	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	2	1	no	—
590	C₂²×C₄	—	2	1	no	—
591	C₂²×C₄	—	4	2	no	—
592	C₂²×C₄	—	4	2	no	—
593	C₂²×C₄	—	4	1	no	—
594	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
595	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
596	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
597	SD₁₆	—	4	4	no	—
598	SD₁₆	—	4	2	no	—
599	SD₁₆	—	4	2	no	—
600	SD₁₆	—	4	2	no	—
601	D₈	—	4	2	no	—
602	D₈	—	4	2	no	—
603	D₈	—	4	2	no	—
604	M₁₆	—	4	2	no	—
605	M₁₆	—	4	2	no	—
606	C₂×C₈	—	2	1	no	—
607	C₂×C₈	—	2	1	no	—
608	C₂×C₈	—	2	1	no	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	2	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
627	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
628	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
629	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
630	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	1	no	—
638	C₂²⋊C₄	—	2	2	no	—
639	C₂²⋊C₄	—	2	1	no	—
640	C₂²⋊C₄	—	2	1	no	—
641	C₂²⋊C₄	—	2	1	no	—
642	C₂²⋊C₄	—	2	1	no	—
643	C₂²⋊C₄	—	2	1	no	—
644	C₂²⋊C₄	—	2	1	no	—
645	C₂²⋊C₄	—	4	1	no	—
646	C₂²⋊C₄	—	4	1	no	—
647	C₄²	—	2	2	no	—
648	C₄²	—	2	2	no	—
649	C₄²	—	2	1	no	—
650	C₄²	—	2	2	no	—
651	C₄²	—	2	1	no	—
652	C₄²	—	2	1	no	—
653	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
654	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
655	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
656	C₄²	C₂⁴×C₅	1	1	yes	characteristic
657	C₂×C₅	—	4	1	no	—
658	C₂×C₅	—	4	2	no	—
659	C₂×C₅	—	8	1	no	—
660	C₂×C₅	—	8	1	no	—
661	C₂×C₅	—	16	1	no	—
662	C₂×C₅	#3005	1	1	yes	characteristic
663	C₂×C₅	#3006	1	1	yes	characteristic
664	C₂×C₅	#3018	1	1	yes	characteristic
665	C₂³	—	2	1	no	—
666	C₂³	—	2	1	no	—
667	C₂³	—	2	1	no	—
668	C₂³	—	4	2	no	—
669	C₂³	—	4	2	no	—
670	C₂³	—	4	1	no	—
671	C₂³	—	8	1	no	—
672	C₂³	C₅×#120	1	1	yes	characteristic
673	Q₈	—	2	1	no	—
674	Q₈	—	2	1	no	—
675	Q₈	—	4	2	no	—
676	Q₈	—	4	1	no	—
677	Q₈	—	4	1	no	—
678	Q₈	—	4	1	no	—
679	Q₈	—	4	1	no	—
680	D₄	—	2	1	no	—
681	D₄	—	2	1	no	—
682	D₄	—	2	1	no	—
683	D₄	—	2	1	no	—
684	D₄	—	2	1	no	—
685	D₄	—	4	2	no	—
686	D₄	—	4	2	no	—
687	D₄	—	4	2	no	—
688	D₄	—	4	4	no	—
689	D₄	—	4	1	no	—
690	D₄	—	4	1	no	—
691	D₄	—	4	1	no	—
692	D₄	—	4	1	no	—
693	D₄	—	8	2	no	—
694	D₄	—	8	2	no	—
695	D₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	1	no	—
701	C₂×C₄	—	2	1	no	—
702	C₂×C₄	—	2	1	no	—
703	C₂×C₄	—	2	1	no	—
704	C₂×C₄	—	2	2	no	—
705	C₂×C₄	—	2	2	no	—
706	C₂×C₄	—	2	1	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	4	2	no	—
709	C₂×C₄	—	4	2	no	—
710	C₂×C₄	—	4	2	no	—
711	C₂×C₄	—	4	2	no	—
712	C₂×C₄	—	4	4	no	—
713	C₂×C₄	—	4	1	no	—
714	C₂×C₄	—	4	1	no	—
715	C₂×C₄	—	4	1	no	—
716	C₂×C₄	—	4	1	no	—
717	C₂×C₄	—	4	1	no	—
718	C₂×C₄	—	4	1	no	—
719	C₂×C₄	—	4	1	no	—
720	C₂×C₄	—	8	1	no	—
721	C₂×C₄	—	8	2	no	—
722	C₂×C₄	—	8	2	no	—
723	C₂×C₄	C₅×#120	1	1	yes	characteristic
724	C₂×C₄	C₅×#120	1	1	yes	characteristic
725	C₂×C₄	C₅×#120	1	1	yes	characteristic
726	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
727	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
728	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
729	C₈	—	2	2	no	—
730	C₈	—	4	1	no	—
731	C₈	—	4	1	no	—
732	C₈	—	4	1	no	—
733	C₈	—	4	1	no	—
734	C₈	—	4	1	no	—
735	C₅	#18560	1	1	yes	characteristic, direct product factor, Sylow
736	C₂²	—	2	1	no	—
737	C₂²	—	2	1	no	—
738	C₂²	—	2	2	no	—
739	C₂²	—	2	1	no	—
740	C₂²	—	4	1	no	—
741	C₂²	—	4	1	no	—
742	C₂²	—	4	1	no	—
743	C₂²	—	4	1	no	—
744	C₂²	—	4	1	no	—
745	C₂²	—	4	1	no	—
746	C₂²	—	4	1	no	—
747	C₂²	—	4	1	no	—
748	C₂²	—	8	4	no	—
749	C₂²	—	8	1	no	—
750	C₂²	—	8	1	no	—
751	C₂²	—	8	1	no	—
752	C₂²	C₂×C₅×#120	1	1	yes	characteristic
753	C₄	—	2	1	no	—
754	C₄	—	2	1	no	—
755	C₄	—	2	1	no	—
756	C₄	—	2	1	no	—
757	C₄	—	4	1	no	—
758	C₄	—	4	1	no	—
759	C₄	—	4	1	no	—
760	C₄	—	4	1	no	—
761	C₄	—	4	2	no	—
762	C₄	—	4	1	no	—
763	C₄	—	4	2	no	—
764	C₄	—	8	1	no	—
765	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
766	C₄	C₅×#575	1	1	yes	characteristic
767	C₄	C₅×#575	1	1	yes	characteristic
768	C₄	C₅×#575	1	1	yes	characteristic
769	C₂	—	4	1	no	—
770	C₂	—	4	2	no	—
771	C₂	—	8	1	no	—
772	C₂	—	8	1	no	—
773	C₂	—	16	1	no	—
774	C₂	C₅×#3005	1	1	yes	characteristic
775	C₂	C₅×#3006	1	1	yes	characteristic
776	C₂	C₅×#3018	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×#3475	C₂	semidirect
2. C₅×#3182	C₂	semidirect
3. C₅×#3115	C₂	semidirect
4. C₅×#3036	C₂	semidirect
5. C₅×#1650	C₂	semidirect
6. C₅×#1647	C₂	semidirect
7. C₅×#1646	C₂	semidirect
8. C₅×#1643	C₂	semidirect
9. C₅×#1628	C₂	semidirect
10. C₅×#1625	C₂	semidirect
11. C₅×#1624	C₂	semidirect
12. C₅×#1622	C₂	semidirect
13. C₅×#1559	C₂	semidirect
14. C₅×#1490	C₂	semidirect
15. C₅×#1489	C₂	semidirect
35. C₅×#500	C₂²	semidirect
36. C₅×#492	C₂²	semidirect
37. C₅×#487	C₂²	semidirect
38. C₅×#486	C₂²	semidirect
40. C₅×#475	C₂²	semidirect
41. C₅×#474	C₂²	semidirect
47. C₅×#460	C₂²	semidirect
50. C₅×#459	C₂²	semidirect
59. C₅×#333	C₂²	semidirect
60. C₅×#333	C₂²	semidirect
61. C₅×#332	C₂²	semidirect
62. C₅×#332	C₂²	semidirect
63. C₅×Q₈⋊C₈	C₂²	semidirect
64. C₅×Q₈⋊C₈	C₂²	semidirect
69. C₅×D₄⋊C₈	C₂²	semidirect
70. C₅×D₄⋊C₈	C₂²	semidirect
71. #18560	C₅	direct (N × H)
178. #3475	C₂×C₅	semidirect
179. #3182	C₂×C₅	semidirect
180. #3115	C₂×C₅	semidirect
181. #3036	C₂×C₅	semidirect
182. #1650	C₂×C₅	semidirect
183. #1647	C₂×C₅	semidirect
184. #1646	C₂×C₅	semidirect
185. #1643	C₂×C₅	semidirect
186. #1628	C₂×C₅	semidirect
187. #1625	C₂×C₅	semidirect
188. #1624	C₂×C₅	semidirect
189. #1622	C₂×C₅	semidirect
190. #1559	C₂×C₅	semidirect
191. #1490	C₂×C₅	semidirect
192. #1489	C₂×C₅	semidirect
312. #500	C₂²×C₅	semidirect
313. #492	C₂²×C₅	semidirect
314. #487	C₂²×C₅	semidirect
315. #486	C₂²×C₅	semidirect
317. #475	C₂²×C₅	semidirect
318. #474	C₂²×C₅	semidirect
324. #460	C₂²×C₅	semidirect
327. #459	C₂²×C₅	semidirect
336. #333	C₂²×C₅	semidirect
337. #333	C₂²×C₅	semidirect
338. #332	C₂²×C₅	semidirect
339. #332	C₂²×C₅	semidirect
340. Q₈⋊C₈	C₂²×C₅	semidirect
341. Q₈⋊C₈	C₂²×C₅	semidirect
346. D₄⋊C₈	C₂²×C₅	semidirect
347. D₄⋊C₈	C₂²×C₅	semidirect
735. C₅	#18560	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×#3475	2
C₅×#3182	2
C₅×#3115	2
C₅×#3036	2
C₅×#1650	2
C₅×#1647	2
C₅×#1646	2
C₅×#1643	2
C₅×#1628	2
C₅×#1625	2
C₅×#1624	2
C₅×#1622	2
C₅×#1559	2
C₅×#1490	2
C₅×#1489	2
#18560	5

Subgroups — C5×#18560 #11941280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#18560 #11941280