Subgroups — C₅×#18520 #11941240

Order 1280.

990 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3480	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1650	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1620	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1540	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1476	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#135	—	2	1	no	—
17	C₅×#559	—	2	1	no	—
18	C₅×#559	C₂²	1	1	yes	characteristic
19	C₅×#557	C₂²	1	1	yes	characteristic
20	C₅×#556	—	2	1	no	—
21	C₅×#556	C₂²	1	1	yes	characteristic
22	C₅×#553	—	2	2	no	—
23	C₅×#553	—	2	1	no	—
24	C₅×#550	C₂²	1	1	yes	characteristic
25	C₅×#548	—	2	1	no	—
26	C₅×#532	—	2	1	no	—
27	C₅×#532	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	—	2	1	no	—
33	C₄×C₅×#41	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
35	C₅×#500	C₂²	1	1	yes	characteristic
36	C₅×#493	C₂²	1	1	yes	characteristic
37	C₅×#486	C₂²	1	1	yes	characteristic
38	C₅×#486	C₂²	1	1	yes	characteristic
39	C₅×#481	—	2	1	no	—
40	C₅×#475	C₂²	1	1	yes	characteristic
41	C₅×#475	C₂²	1	1	yes	characteristic
42	C₅×#466	—	2	2	no	—
43	C₅×#464	—	2	1	no	—
44	C₅×#463	—	2	1	no	—
45	C₅×#462	C₂²	1	1	yes	characteristic
46	C₅×#461	C₂²	1	1	yes	characteristic
47	C₅×#460	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×#448	—	2	1	no	—
52	C₄×C₅×SD₁₆	—	2	1	no	—
53	C₄×C₅×SD₁₆	—	2	1	no	—
54	C₄×C₅×D₈	—	2	2	no	—
55	C₅×#431	C₂²	1	1	yes	characteristic
56	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₅×#416	—	2	1	no	—
58	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
59	C₅×#332	C₂²	1	1	yes	characteristic
60	C₅×#332	C₂²	1	1	yes	characteristic
61	C₅×#331	C₂²	1	1	yes	characteristic
62	C₅×#331	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₅×Q₈⋊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	#18520	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	—	2	1	no	—
75	C₂×C₅×#41	D₄	1	1	yes	characteristic
76	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
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×SD₁₆	—	2	1	no	—
83	C₂×C₅×D₈	—	2	2	no	—
84	C₂×C₅×D₈	—	2	1	no	—
85	C₂²×C₅×C₈	—	2	1	no	—
86	C₂²×C₅×C₈	—	2	1	no	—
87	C₅×C₄⋊Q₈	—	2	1	no	—
88	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₅×#127	—	2	1	no	—
91	C₅×#127	C₂³	1	1	yes	characteristic
92	C₅×#127	C₂³	1	1	yes	characteristic
93	C₅×#126	—	2	2	no	—
94	C₅×#126	—	2	2	no	—
95	C₅×#126	—	2	2	no	—
96	C₅×#125	—	2	2	no	—
97	C₅×#125	—	2	1	no	—
98	C₅×#124	—	2	2	no	—
99	C₅×#124	—	2	1	no	—
100	C₅×#124	—	2	1	no	—
101	C₅×#124	—	2	1	no	—
102	C₅×#124	—	2	1	no	—
103	C₅×#124	—	2	1	no	—
104	C₅×#123	—	2	2	no	—
105	C₅×#123	—	2	2	no	—
106	C₅×#123	—	2	2	no	—
107	C₅×C₂²⋊Q₈	—	2	2	no	—
108	C₅×C₂²⋊Q₈	—	2	1	no	—
109	C₅×#121	—	2	2	no	—
110	C₅×#121	—	2	2	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	C₅×#121	—	2	1	no	—
116	C₅×#121	D₄	1	2	yes	—
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	D₄	1	1	yes	characteristic
125	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
126	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
127	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
128	C₄×C₅×D₄	—	2	2	no	—
129	C₄×C₅×D₄	—	2	2	no	—
130	C₄×C₅×D₄	—	2	2	no	—
131	C₄×C₅×D₄	—	2	2	no	—
132	C₄×C₅×D₄	—	2	2	no	—
133	C₄×C₅×D₄	—	2	1	no	—
134	C₄×C₅×D₄	—	2	1	no	—
135	C₄×C₅×D₄	—	2	1	no	—
136	C₄×C₅×D₄	—	2	1	no	—
137	C₄×C₅×D₄	—	2	1	no	—
138	C₄×C₅×D₄	—	2	1	no	—
139	C₄×C₅×D₄	—	2	1	no	—
140	C₄×C₅×D₄	—	2	1	no	—
141	C₄×C₅×D₄	D₄	1	2	yes	—
142	C₄×C₅×D₄	D₄	1	1	yes	characteristic
143	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
144	C₅×#117	—	2	2	no	—
145	C₅×#117	—	2	2	no	—
146	C₅×#117	—	2	1	no	—
147	C₅×#117	—	2	1	no	—
148	C₅×#117	D₄	1	1	yes	characteristic
149	C₂×C₅×C₄⋊C₄	—	2	1	no	—
150	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
151	C₂×C₄²×C₅	—	2	2	no	—
152	C₂×C₄²×C₅	—	2	1	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×#106	—	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	1	no	—
165	C₅×#103	—	2	1	no	—
166	C₅×#103	—	2	1	no	—
167	C₅×#102	—	2	2	no	—
168	C₅×#102	—	2	1	no	—
169	C₅×#102	—	2	1	no	—
170	C₅×#102	—	2	2	no	—
171	C₅×#102	—	2	1	no	—
172	C₅×#102	—	2	1	no	—
173	C₅×C₂²⋊C₈	—	2	1	no	—
174	C₄×C₅×C₈	—	2	1	no	—
175	C₄×C₅×C₈	—	2	1	no	—
176	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
177	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
178	#3480	C₂×C₅	1	1	yes	characteristic, complemented
179	#3169	C₂×C₅	1	1	yes	characteristic, complemented
180	#3103	C₂×C₅	1	1	yes	characteristic, complemented
181	#3035	C₂×C₅	1	1	yes	characteristic, complemented
182	#1650	C₂×C₅	1	1	yes	characteristic, complemented
183	#1648	C₂×C₅	1	1	yes	characteristic, complemented
184	#1642	C₂×C₅	1	1	yes	characteristic, complemented
185	#1640	C₂×C₅	1	1	yes	characteristic, complemented
186	#1628	C₂×C₅	1	1	yes	characteristic, complemented
187	#1626	C₂×C₅	1	1	yes	characteristic, complemented
188	#1621	C₂×C₅	1	1	yes	characteristic, complemented
189	#1620	C₂×C₅	1	1	yes	characteristic, complemented
190	#1540	C₂×C₅	1	1	yes	characteristic, complemented
191	#1476	C₂×C₅	1	1	yes	characteristic, complemented
192	C₂×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	2	2	no	—
195	C₅×#41	—	4	2	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	C₅×#41	—	4	2	no	—
201	C₂×C₅×Q₈	—	2	1	no	—
202	C₂×C₅×Q₈	—	2	1	no	—
203	C₂×C₅×Q₈	—	2	2	no	—
204	C₂×C₅×Q₈	—	2	1	no	—
205	C₂×C₅×Q₈	—	4	1	no	—
206	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
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	2	no	—
212	C₂×C₅×D₄	—	2	2	no	—
213	C₂×C₅×D₄	—	2	1	no	—
214	C₂×C₅×D₄	—	2	1	no	—
215	C₂×C₅×D₄	—	2	1	no	—
216	C₂×C₅×D₄	—	2	1	no	—
217	C₂×C₅×D₄	—	2	1	no	—
218	C₂×C₅×D₄	—	4	1	no	—
219	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
220	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
221	C₂²×C₄×C₅	—	2	2	no	—
222	C₂²×C₄×C₅	—	2	1	no	—
223	C₂²×C₄×C₅	—	2	2	no	—
224	C₂²×C₄×C₅	—	2	1	no	—
225	C₂²×C₄×C₅	—	2	1	no	—
226	C₂²×C₄×C₅	—	2	2	no	—
227	C₂²×C₄×C₅	—	2	1	no	—
228	C₂²×C₄×C₅	—	2	1	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×Q₁₆	—	4	2	no	—
234	C₅×SD₁₆	—	4	2	no	—
235	C₅×SD₁₆	—	4	2	no	—
236	C₅×SD₁₆	—	4	2	no	—
237	C₅×SD₁₆	—	4	2	no	—
238	C₅×D₈	—	4	4	no	—
239	C₅×D₈	—	4	2	no	—
240	C₂×C₅×C₈	—	2	1	no	—
241	C₂×C₅×C₈	—	2	1	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	—	4	2	no	—
247	C₂×C₅×C₈	—	4	2	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	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	—	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	2	no	—
277	C₅×C₂²⋊C₄	—	2	1	no	—
278	C₅×C₂²⋊C₄	—	4	1	no	—
279	C₅×C₂²⋊C₄	—	4	1	no	—
280	C₄²×C₅	—	2	2	no	—
281	C₄²×C₅	—	2	2	no	—
282	C₄²×C₅	—	2	1	no	—
283	C₄²×C₅	—	2	1	no	—
284	C₄²×C₅	—	2	1	no	—
285	C₄²×C₅	—	2	1	no	—
286	C₄²×C₅	—	2	1	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×#135	—	2	1	no	—
294	#559	—	2	1	no	—
295	#559	C₂²×C₅	1	1	yes	characteristic
296	#557	C₂²×C₅	1	1	yes	characteristic
297	#556	—	2	1	no	—
298	#556	C₂²×C₅	1	1	yes	characteristic
299	#553	—	2	2	no	—
300	#553	—	2	1	no	—
301	#550	C₂²×C₅	1	1	yes	characteristic
302	#548	—	2	1	no	—
303	#532	—	2	1	no	—
304	#532	C₂²×C₅	1	1	yes	characteristic
305	#532	C₂²×C₅	1	1	yes	characteristic
306	#529	—	2	1	no	—
307	#529	—	2	2	no	—
308	C₄×#41	—	2	1	no	—
309	C₄×#41	—	2	1	no	—
310	C₄×#41	C₂²×C₅	1	1	yes	characteristic
311	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
312	#500	C₂²×C₅	1	1	yes	characteristic
313	#493	C₂²×C₅	1	1	yes	characteristic
314	#486	C₂²×C₅	1	1	yes	characteristic
315	#486	C₂²×C₅	1	1	yes	characteristic
316	#481	—	2	1	no	—
317	#475	C₂²×C₅	1	1	yes	characteristic
318	#475	C₂²×C₅	1	1	yes	characteristic
319	#466	—	2	2	no	—
320	#464	—	2	1	no	—
321	#463	—	2	1	no	—
322	#462	C₂²×C₅	1	1	yes	characteristic
323	#461	C₂²×C₅	1	1	yes	characteristic
324	#460	C₂²×C₅	1	1	yes	characteristic
325	#460	C₂²×C₅	1	1	yes	characteristic
326	#460	C₂²×C₅	1	1	yes	characteristic
327	#459	C₂²×C₅	1	1	yes	characteristic
328	#448	—	2	1	no	—
329	C₄×SD₁₆	—	2	1	no	—
330	C₄×SD₁₆	—	2	1	no	—
331	C₄×D₈	—	2	2	no	—
332	#431	C₂²×C₅	1	1	yes	characteristic
333	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
334	#416	—	2	1	no	—
335	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
336	#332	C₂²×C₅	1	1	yes	characteristic
337	#332	C₂²×C₅	1	1	yes	characteristic
338	#331	C₂²×C₅	1	1	yes	characteristic
339	#331	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	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
346	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
347	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
348	C₂³×C₅	—	2	1	no	—
349	C₂³×C₅	—	2	2	no	—
350	C₂³×C₅	—	4	1	no	—
351	C₂³×C₅	#120	1	1	yes	characteristic
352	C₅×Q₈	—	2	1	no	—
353	C₅×Q₈	—	2	1	no	—
354	C₅×Q₈	—	2	1	no	—
355	C₅×Q₈	—	2	1	no	—
356	C₅×Q₈	—	2	1	no	—
357	C₅×Q₈	—	4	2	no	—
358	C₅×Q₈	—	4	1	no	—
359	C₅×Q₈	—	4	1	no	—
360	C₅×Q₈	—	4	1	no	—
361	C₅×Q₈	—	4	1	no	—
362	C₅×Q₈	#135	1	2	yes	complemented
363	C₅×D₄	—	2	1	no	—
364	C₅×D₄	—	2	1	no	—
365	C₅×D₄	—	4	2	no	—
366	C₅×D₄	—	4	4	no	—
367	C₅×D₄	—	4	2	no	—
368	C₅×D₄	—	4	2	no	—
369	C₅×D₄	—	4	4	no	—
370	C₅×D₄	—	4	1	no	—
371	C₅×D₄	—	4	1	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	—	4	1	no	—
377	C₅×D₄	—	4	1	no	—
378	C₅×D₄	—	8	2	no	—
379	C₅×D₄	—	8	2	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	1	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	4	no	—
393	C₂×C₄×C₅	—	4	1	no	—
394	C₂×C₄×C₅	—	4	1	no	—
395	C₂×C₄×C₅	—	4	2	no	—
396	C₂×C₄×C₅	—	4	4	no	—
397	C₂×C₄×C₅	—	4	1	no	—
398	C₂×C₄×C₅	—	8	1	no	—
399	C₂×C₄×C₅	—	8	1	no	—
400	C₂×C₄×C₅	—	8	2	no	—
401	C₂×C₄×C₅	—	8	2	no	—
402	C₂×C₄×C₅	#120	1	1	yes	characteristic
403	C₂×C₄×C₅	#120	1	1	yes	characteristic
404	C₂×C₄×C₅	#120	1	1	yes	characteristic
405	C₂×C₄×C₅	#135	1	2	yes	—
406	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
407	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
410	C₅×C₈	—	2	2	no	—
411	C₅×C₈	—	4	1	no	—
412	C₅×C₈	—	4	1	no	—
413	C₅×C₈	—	4	1	no	—
414	C₅×C₈	—	4	1	no	—
415	C₅×C₈	—	4	1	no	—
416	C₂×#41	—	2	1	no	—
417	C₂×#41	—	2	1	no	—
418	C₂×#41	—	2	1	no	—
419	C₂×#41	C₅×D₄	1	1	yes	characteristic
420	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
421	#135	—	4	4	no	—
422	C₂×Q₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×SD₁₆	—	2	1	no	—
426	C₂×SD₁₆	—	2	1	no	—
427	C₂×D₈	—	2	1	no	—
428	C₂×D₈	—	2	2	no	—
429	C₂²×C₈	—	2	1	no	—
430	C₂²×C₈	—	2	1	no	—
431	C₄⋊Q₈	—	2	1	no	—
432	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
433	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
434	#127	—	2	1	no	—
435	#127	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	#124	—	2	2	no	—
443	#124	—	2	1	no	—
444	#124	—	2	1	no	—
445	#124	—	2	1	no	—
446	#124	—	2	1	no	—
447	#124	—	2	1	no	—
448	#123	—	2	2	no	—
449	#123	—	2	2	no	—
450	#123	—	2	2	no	—
451	C₂²⋊Q₈	—	2	2	no	—
452	C₂²⋊Q₈	—	2	1	no	—
453	#121	—	2	2	no	—
454	#121	—	2	2	no	—
455	#121	—	2	2	no	—
456	#121	—	2	1	no	—
457	#121	—	2	1	no	—
458	#121	—	2	1	no	—
459	#121	—	2	1	no	—
460	#121	C₅×D₄	1	2	yes	—
461	C₄×Q₈	—	2	1	no	—
462	C₄×Q₈	—	2	1	no	—
463	C₄×Q₈	—	2	1	no	—
464	C₄×Q₈	—	2	1	no	—
465	C₄×Q₈	—	2	1	no	—
466	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
467	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
468	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
469	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
470	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
471	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
472	C₄×D₄	—	2	2	no	—
473	C₄×D₄	—	2	2	no	—
474	C₄×D₄	—	2	2	no	—
475	C₄×D₄	—	2	2	no	—
476	C₄×D₄	—	2	2	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	—	2	1	no	—
479	C₄×D₄	—	2	1	no	—
480	C₄×D₄	—	2	1	no	—
481	C₄×D₄	—	2	1	no	—
482	C₄×D₄	—	2	1	no	—
483	C₄×D₄	—	2	1	no	—
484	C₄×D₄	—	2	1	no	—
485	C₄×D₄	C₅×D₄	1	1	yes	characteristic
486	C₄×D₄	C₅×D₄	1	2	yes	—
487	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
488	#117	—	2	2	no	—
489	#117	—	2	2	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	C₅×D₄	1	1	yes	characteristic
495	C₂×C₄²	—	2	2	no	—
496	C₂×C₄²	—	2	1	no	—
497	C₂×C₄²	—	2	1	no	—
498	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
499	#107	—	2	2	no	—
500	#106	—	2	1	no	—
501	#106	—	2	1	no	—
502	C₄⋊C₈	—	2	1	no	—
503	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
504	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
505	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
506	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
507	#103	—	2	1	no	—
508	#103	—	2	1	no	—
509	#103	—	2	1	no	—
510	#103	—	2	1	no	—
511	#102	—	2	2	no	—
512	#102	—	2	1	no	—
513	#102	—	2	1	no	—
514	#102	—	2	2	no	—
515	#102	—	2	1	no	—
516	#102	—	2	1	no	—
517	C₂²⋊C₈	—	2	1	no	—
518	C₄×C₈	—	2	1	no	—
519	C₄×C₈	—	2	1	no	—
520	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
521	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
522	C₂²×C₅	—	2	1	no	—
523	C₂²×C₅	—	2	1	no	—
524	C₂²×C₅	—	4	1	no	—
525	C₂²×C₅	—	4	2	no	—
526	C₂²×C₅	—	4	2	no	—
527	C₂²×C₅	—	4	1	no	—
528	C₂²×C₅	—	4	2	no	—
529	C₂²×C₅	—	4	1	no	—
530	C₂²×C₅	—	8	1	no	—
531	C₂²×C₅	—	8	1	no	—
532	C₂²×C₅	—	8	1	no	—
533	C₂²×C₅	#449	1	2	yes	—
534	C₂²×C₅	C₂×#120	1	1	yes	characteristic
535	C₄×C₅	—	2	1	no	—
536	C₄×C₅	—	2	2	no	—
537	C₄×C₅	—	2	1	no	—
538	C₄×C₅	—	2	1	no	—
539	C₄×C₅	—	4	1	no	—
540	C₄×C₅	—	4	2	no	—
541	C₄×C₅	—	4	1	no	—
542	C₄×C₅	—	4	1	no	—
543	C₄×C₅	—	4	1	no	—
544	C₄×C₅	—	4	1	no	—
545	C₄×C₅	—	4	1	no	—
546	C₄×C₅	—	8	1	no	—
547	C₄×C₅	#449	1	2	yes	—
548	C₄×C₅	C₂×#135	1	1	yes	characteristic
549	C₄×C₅	C₂×#135	1	1	yes	characteristic
550	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
551	C₄×C₅	#575	1	1	yes	characteristic
552	#41	—	2	2	no	—
553	#41	—	2	2	no	—
554	#41	—	4	2	no	—
555	#41	—	4	2	no	—
556	#41	—	4	2	no	—
557	#41	—	4	2	no	—
558	#41	—	4	2	no	—
559	#41	—	4	2	no	—
560	C₂×Q₈	—	2	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	2	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	4	1	no	—
565	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
566	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
567	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
568	C₂×D₄	—	2	2	no	—
569	C₂×D₄	—	2	1	no	—
570	C₂×D₄	—	2	2	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	—	2	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	2	yes	—
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	4	1	no	—
589	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
590	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
591	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
592	Q₁₆	—	4	2	no	—
593	SD₁₆	—	4	2	no	—
594	SD₁₆	—	4	2	no	—
595	SD₁₆	—	4	2	no	—
596	SD₁₆	—	4	2	no	—
597	D₈	—	4	4	no	—
598	D₈	—	4	2	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	2	1	no	—
605	C₂×C₈	—	4	2	no	—
606	C₂×C₈	—	4	2	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
622	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
623	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
624	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
625	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	2	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	1	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	1	no	—
637	C₂²⋊C₄	—	4	1	no	—
638	C₂²⋊C₄	—	4	1	no	—
639	C₄²	—	2	2	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	1	no	—
642	C₄²	—	2	1	no	—
643	C₄²	—	2	1	no	—
644	C₄²	—	2	1	no	—
645	C₄²	—	2	1	no	—
646	C₄²	—	2	1	no	—
647	C₄²	—	2	1	no	—
648	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
649	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
650	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
651	C₄²	C₂⁴×C₅	1	1	yes	characteristic
652	C₂×C₅	—	2	2	no	—
653	C₂×C₅	—	8	1	no	—
654	C₂×C₅	—	8	2	no	—
655	C₂×C₅	—	16	1	no	—
656	C₂×C₅	C₂×#449	1	1	yes	characteristic
657	C₂×C₅	#3002	1	1	yes	characteristic
658	C₂×C₅	#3018	1	1	yes	characteristic
659	C₂³	—	2	1	no	—
660	C₂³	—	2	2	no	—
661	C₂³	—	4	1	no	—
662	C₂³	C₅×#120	1	1	yes	characteristic
663	Q₈	—	2	1	no	—
664	Q₈	—	2	1	no	—
665	Q₈	—	2	1	no	—
666	Q₈	—	2	1	no	—
667	Q₈	—	2	1	no	—
668	Q₈	—	4	2	no	—
669	Q₈	—	4	1	no	—
670	Q₈	—	4	1	no	—
671	Q₈	—	4	1	no	—
672	Q₈	—	4	1	no	—
673	Q₈	C₅×#135	1	2	yes	complemented
674	D₄	—	2	1	no	—
675	D₄	—	2	1	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	4	no	—
678	D₄	—	4	2	no	—
679	D₄	—	4	2	no	—
680	D₄	—	4	4	no	—
681	D₄	—	4	1	no	—
682	D₄	—	4	1	no	—
683	D₄	—	4	1	no	—
684	D₄	—	4	1	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	4	1	no	—
688	D₄	—	4	1	no	—
689	D₄	—	8	2	no	—
690	D₄	—	8	2	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	1	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	2	no	—
703	C₂×C₄	—	4	4	no	—
704	C₂×C₄	—	4	2	no	—
705	C₂×C₄	—	4	4	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	4	1	no	—
709	C₂×C₄	—	8	1	no	—
710	C₂×C₄	—	8	1	no	—
711	C₂×C₄	—	8	2	no	—
712	C₂×C₄	—	8	2	no	—
713	C₂×C₄	C₅×#120	1	1	yes	characteristic
714	C₂×C₄	C₅×#120	1	1	yes	characteristic
715	C₂×C₄	C₅×#120	1	1	yes	characteristic
716	C₂×C₄	C₅×#135	1	2	yes	—
717	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
718	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
719	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
720	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
721	C₈	—	2	2	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₈	—	4	1	no	—
725	C₈	—	4	1	no	—
726	C₈	—	4	1	no	—
727	C₅	#18520	1	1	yes	characteristic, direct product factor, Sylow
728	C₂²	—	2	1	no	—
729	C₂²	—	2	1	no	—
730	C₂²	—	4	1	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	4	2	no	—
733	C₂²	—	4	1	no	—
734	C₂²	—	4	2	no	—
735	C₂²	—	4	1	no	—
736	C₂²	—	8	1	no	—
737	C₂²	—	8	1	no	—
738	C₂²	—	8	1	no	—
739	C₂²	C₅×#449	1	2	yes	—
740	C₂²	C₂×C₅×#120	1	1	yes	characteristic
741	C₄	—	2	1	no	—
742	C₄	—	2	2	no	—
743	C₄	—	2	1	no	—
744	C₄	—	2	1	no	—
745	C₄	—	4	1	no	—
746	C₄	—	4	2	no	—
747	C₄	—	4	1	no	—
748	C₄	—	4	1	no	—
749	C₄	—	4	1	no	—
750	C₄	—	4	1	no	—
751	C₄	—	4	1	no	—
752	C₄	—	8	1	no	—
753	C₄	C₅×#449	1	2	yes	—
754	C₄	C₂×C₅×#135	1	1	yes	characteristic
755	C₄	C₂×C₅×#135	1	1	yes	characteristic
756	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
757	C₄	C₅×#575	1	1	yes	characteristic
758	C₂	—	2	2	no	—
759	C₂	—	8	1	no	—
760	C₂	—	8	2	no	—
761	C₂	—	16	1	no	—
762	C₂	C₂×C₅×#449	1	1	yes	characteristic
763	C₂	C₅×#3002	1	1	yes	characteristic
764	C₂	C₅×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×#120
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×#3480	C₂	semidirect
2. C₅×#3169	C₂	semidirect
3. C₅×#3103	C₂	semidirect
4. C₅×#3035	C₂	semidirect
5. C₅×#1650	C₂	semidirect
6. C₅×#1648	C₂	semidirect
7. C₅×#1642	C₂	semidirect
8. C₅×#1640	C₂	semidirect
9. C₅×#1628	C₂	semidirect
10. C₅×#1626	C₂	semidirect
11. C₅×#1621	C₂	semidirect
12. C₅×#1620	C₂	semidirect
13. C₅×#1540	C₂	semidirect
14. C₅×#1476	C₂	semidirect
15. C₂×C₅×Q₈⋊C₈	C₂	semidirect
71. #18520	C₅	direct (N × H)
178. #3480	C₂×C₅	semidirect
179. #3169	C₂×C₅	semidirect
180. #3103	C₂×C₅	semidirect
181. #3035	C₂×C₅	semidirect
182. #1650	C₂×C₅	semidirect
183. #1648	C₂×C₅	semidirect
184. #1642	C₂×C₅	semidirect
185. #1640	C₂×C₅	semidirect
186. #1628	C₂×C₅	semidirect
187. #1626	C₂×C₅	semidirect
188. #1621	C₂×C₅	semidirect
189. #1620	C₂×C₅	semidirect
190. #1540	C₂×C₅	semidirect
191. #1476	C₂×C₅	semidirect
192. C₂×Q₈⋊C₈	C₂×C₅	semidirect
362. C₅×Q₈	#135	semidirect
673. Q₈	C₅×#135	semidirect
727. C₅	#18520	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×#3480	2
C₅×#3169	2
C₅×#3103	2
C₅×#3035	2
C₅×#1650	2
C₅×#1648	2
C₅×#1642	2
C₅×#1640	2
C₅×#1628	2
C₅×#1626	2
C₅×#1621	2
C₅×#1620	2
C₅×#1540	2
C₅×#1476	2
C₂×C₅×Q₈⋊C₈	2
#18520	5

Subgroups — C5×#18520 #11941240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#18520 #11941240