Subgroups — C₅×#18521 #11941241

Order 1280.

936 conjugacy classes of subgroups.

The subgroup lattice has 734 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
2	C₅×#3167	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1652	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1641	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1638	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1636	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1631	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1630	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	C₅×#559	—	2	1	no	—
17	C₅×#559	C₂²	1	1	yes	characteristic
18	C₅×#557	C₂²	1	1	yes	characteristic
19	C₅×#556	—	2	1	no	—
20	C₅×#556	C₂²	1	1	yes	characteristic
21	C₅×#553	—	2	2	no	—
22	C₅×#553	—	2	1	no	—
23	C₅×#550	C₂²	1	1	yes	characteristic
24	C₅×#548	—	2	1	no	—
25	C₅×#533	C₂²	1	1	yes	characteristic
26	C₅×#532	—	2	1	no	—
27	C₅×#532	C₂²	1	1	yes	characteristic
28	C₅×#529	—	2	1	no	—
29	C₅×#529	—	2	2	no	—
30	C₄×C₅×#41	—	2	1	no	—
31	C₄×C₅×#41	—	2	1	no	—
32	C₄×C₅×#41	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
34	C₅×#498	C₂²	1	1	yes	characteristic
35	C₅×#492	C₂²	1	1	yes	characteristic
36	C₅×#487	C₂²	1	1	yes	characteristic
37	C₅×#486	C₂²	1	1	yes	characteristic
38	C₅×#484	—	2	1	no	—
39	C₅×#479	—	2	1	no	—
40	C₅×#478	—	2	1	no	—
41	C₅×#477	C₂²	1	1	yes	characteristic
42	C₅×#477	C₂²	1	1	yes	characteristic
43	C₅×#477	C₂²	1	1	yes	characteristic
44	C₅×#476	C₂²	1	1	yes	characteristic
45	C₅×#475	C₂²	1	1	yes	characteristic
46	C₅×#474	C₂²	1	1	yes	characteristic
47	C₅×#465	—	2	2	no	—
48	C₅×#462	C₂²	1	1	yes	characteristic
49	C₅×#462	C₂²	1	1	yes	characteristic
50	C₅×#451	—	2	1	no	—
51	C₄×C₅×Q₁₆	—	2	1	no	—
52	C₄×C₅×Q₁₆	—	2	1	no	—
53	C₄×C₅×SD₁₆	—	2	2	no	—
54	C₅×#431	C₂²	1	1	yes	characteristic
55	C₅×#427	—	2	1	no	—
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	#18521	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×Q₈	D₄	1	1	yes	characteristic
76	C₂×C₅×Q₁₆	—	2	1	no	—
77	C₂×C₅×Q₁₆	—	2	1	no	—
78	C₂×C₅×SD₁₆	—	2	2	no	—
79	C₂²×C₅×C₈	—	2	1	no	—
80	C₂²×C₅×C₈	—	2	1	no	—
81	C₅×C₄⋊Q₈	—	2	1	no	—
82	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₅×#127	—	2	1	no	—
86	C₅×#127	C₂³	1	1	yes	characteristic
87	C₅×#126	—	2	2	no	—
88	C₅×#126	—	2	2	no	—
89	C₅×#126	—	2	2	no	—
90	C₅×#125	—	2	2	no	—
91	C₅×#125	—	2	1	no	—
92	C₅×#125	—	2	1	no	—
93	C₅×#124	—	2	2	no	—
94	C₅×#124	—	2	1	no	—
95	C₅×#124	—	2	1	no	—
96	C₅×#124	—	2	1	no	—
97	C₅×#124	—	2	1	no	—
98	C₅×#123	—	2	2	no	—
99	C₅×#123	—	2	2	no	—
100	C₅×#123	—	2	2	no	—
101	C₅×C₂²⋊Q₈	—	2	2	no	—
102	C₅×C₂²⋊Q₈	—	2	1	no	—
103	C₅×C₂²⋊Q₈	—	2	1	no	—
104	C₅×C₂²⋊Q₈	—	2	1	no	—
105	C₅×#121	—	2	2	no	—
106	C₅×#121	—	2	2	no	—
107	C₅×#121	—	2	2	no	—
108	C₅×#121	—	2	1	no	—
109	C₅×#121	—	2	1	no	—
110	C₅×#121	D₄	1	2	yes	—
111	C₄×C₅×Q₈	—	2	1	no	—
112	C₄×C₅×Q₈	—	2	1	no	—
113	C₄×C₅×Q₈	—	2	1	no	—
114	C₄×C₅×Q₈	—	2	1	no	—
115	C₄×C₅×Q₈	—	2	1	no	—
116	C₄×C₅×Q₈	—	2	1	no	—
117	C₄×C₅×Q₈	—	2	1	no	—
118	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
119	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
120	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
121	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
122	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
123	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
124	C₄×C₅×D₄	—	2	2	no	—
125	C₄×C₅×D₄	—	2	2	no	—
126	C₄×C₅×D₄	—	2	2	no	—
127	C₄×C₅×D₄	—	2	2	no	—
128	C₄×C₅×D₄	—	2	2	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	2	yes	—
136	C₄×C₅×D₄	D₄	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	1	no	—
141	C₅×#117	—	2	1	no	—
142	C₅×#117	—	2	1	no	—
143	C₅×#117	D₄	1	1	yes	characteristic
144	C₂×C₅×C₄⋊C₄	—	2	1	no	—
145	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
146	C₂×C₄²×C₅	—	2	2	no	—
147	C₂×C₄²×C₅	—	2	1	no	—
148	C₂×C₄²×C₅	—	2	1	no	—
149	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
150	C₅×#107	—	2	1	no	—
151	C₅×#107	—	2	1	no	—
152	C₅×#107	—	2	1	no	—
153	C₅×#107	—	2	1	no	—
154	C₅×#106	—	2	1	no	—
155	C₅×#106	—	2	1	no	—
156	C₅×#106	—	2	2	no	—
157	C₅×C₄⋊C₈	—	2	1	no	—
158	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
163	C₅×#103	—	2	1	no	—
164	C₅×#103	—	2	1	no	—
165	C₅×#103	—	2	1	no	—
166	C₅×#103	—	2	2	no	—
167	C₅×#103	—	2	1	no	—
168	C₅×#103	—	2	1	no	—
169	C₅×#102	—	2	1	no	—
170	C₅×#102	—	2	1	no	—
171	C₅×#102	—	2	2	no	—
172	C₅×C₂²⋊C₈	—	2	1	no	—
173	C₄×C₅×C₈	—	2	1	no	—
174	C₄×C₅×C₈	—	2	1	no	—
175	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
176	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
177	#3480	C₂×C₅	1	1	yes	characteristic
178	#3167	C₂×C₅	1	1	yes	characteristic, complemented
179	#3104	C₂×C₅	1	1	yes	characteristic, complemented
180	#3076	C₂×C₅	1	1	yes	characteristic, complemented
181	#1653	C₂×C₅	1	1	yes	characteristic, complemented
182	#1652	C₂×C₅	1	1	yes	characteristic, complemented
183	#1641	C₂×C₅	1	1	yes	characteristic, complemented
184	#1640	C₂×C₅	1	1	yes	characteristic, complemented
185	#1638	C₂×C₅	1	1	yes	characteristic, complemented
186	#1636	C₂×C₅	1	1	yes	characteristic, complemented
187	#1631	C₂×C₅	1	1	yes	characteristic, complemented
188	#1630	C₂×C₅	1	1	yes	characteristic, complemented
189	#1540	C₂×C₅	1	1	yes	characteristic, complemented
190	#1476	C₂×C₅	1	1	yes	characteristic, complemented
191	C₂×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
192	C₅×#41	—	2	2	no	—
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	4	2	no	—
195	C₅×#41	—	4	2	no	—
196	C₅×#41	—	4	2	no	—
197	C₅×#41	—	4	2	no	—
198	C₂×C₅×Q₈	—	2	1	no	—
199	C₂×C₅×Q₈	—	2	1	no	—
200	C₂×C₅×Q₈	—	2	2	no	—
201	C₂×C₅×Q₈	—	2	1	no	—
202	C₂×C₅×Q₈	—	2	1	no	—
203	C₂×C₅×Q₈	—	4	1	no	—
204	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
207	C₂×C₅×D₄	—	2	2	no	—
208	C₂×C₅×D₄	—	2	1	no	—
209	C₂×C₅×D₄	—	2	2	no	—
210	C₂×C₅×D₄	—	2	2	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	—	2	1	no	—
213	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₅	—	2	2	no	—
216	C₂²×C₄×C₅	—	2	1	no	—
217	C₂²×C₄×C₅	—	2	2	no	—
218	C₂²×C₄×C₅	—	2	1	no	—
219	C₂²×C₄×C₅	—	2	1	no	—
220	C₂²×C₄×C₅	—	2	2	no	—
221	C₂²×C₄×C₅	—	2	1	no	—
222	C₂²×C₄×C₅	—	2	1	no	—
223	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
226	C₅×Q₁₆	—	4	2	no	—
227	C₅×Q₁₆	—	4	2	no	—
228	C₅×SD₁₆	—	4	4	no	—
229	C₂×C₅×C₈	—	2	1	no	—
230	C₂×C₅×C₈	—	2	1	no	—
231	C₂×C₅×C₈	—	2	1	no	—
232	C₂×C₅×C₈	—	2	1	no	—
233	C₂×C₅×C₈	—	2	1	no	—
234	C₂×C₅×C₈	—	2	1	no	—
235	C₂×C₅×C₈	—	4	2	no	—
236	C₂×C₅×C₈	—	4	2	no	—
237	C₅×C₄⋊C₄	—	2	2	no	—
238	C₅×C₄⋊C₄	—	2	2	no	—
239	C₅×C₄⋊C₄	—	2	2	no	—
240	C₅×C₄⋊C₄	—	2	2	no	—
241	C₅×C₄⋊C₄	—	2	1	no	—
242	C₅×C₄⋊C₄	—	2	1	no	—
243	C₅×C₄⋊C₄	—	2	1	no	—
244	C₅×C₄⋊C₄	—	2	1	no	—
245	C₅×C₄⋊C₄	—	2	1	no	—
246	C₅×C₄⋊C₄	—	2	1	no	—
247	C₅×C₄⋊C₄	—	2	1	no	—
248	C₅×C₄⋊C₄	—	2	1	no	—
249	C₅×C₄⋊C₄	—	2	1	no	—
250	C₅×C₄⋊C₄	—	2	1	no	—
251	C₅×C₄⋊C₄	—	2	1	no	—
252	C₅×C₄⋊C₄	—	2	1	no	—
253	C₅×C₄⋊C₄	—	4	1	no	—
254	C₅×C₄⋊C₄	—	4	1	no	—
255	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
258	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
259	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
260	C₅×C₂²⋊C₄	—	2	2	no	—
261	C₅×C₂²⋊C₄	—	2	2	no	—
262	C₅×C₂²⋊C₄	—	2	2	no	—
263	C₅×C₂²⋊C₄	—	2	2	no	—
264	C₅×C₂²⋊C₄	—	2	2	no	—
265	C₅×C₂²⋊C₄	—	2	2	no	—
266	C₅×C₂²⋊C₄	—	2	1	no	—
267	C₅×C₂²⋊C₄	—	2	2	no	—
268	C₅×C₂²⋊C₄	—	2	1	no	—
269	C₅×C₂²⋊C₄	—	2	2	no	—
270	C₅×C₂²⋊C₄	—	2	1	no	—
271	C₅×C₂²⋊C₄	—	4	1	no	—
272	C₄²×C₅	—	2	2	no	—
273	C₄²×C₅	—	2	2	no	—
274	C₄²×C₅	—	2	1	no	—
275	C₄²×C₅	—	2	1	no	—
276	C₄²×C₅	—	2	1	no	—
277	C₄²×C₅	—	2	1	no	—
278	C₄²×C₅	—	2	1	no	—
279	C₄²×C₅	—	2	1	no	—
280	C₄²×C₅	—	2	1	no	—
281	C₄²×C₅	—	4	1	no	—
282	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
285	C₄²×C₅	C₂⁴	1	1	yes	characteristic
286	#559	—	2	1	no	—
287	#559	C₂²×C₅	1	1	yes	characteristic
288	#557	C₂²×C₅	1	1	yes	characteristic
289	#556	—	2	1	no	—
290	#556	C₂²×C₅	1	1	yes	characteristic
291	#553	—	2	2	no	—
292	#553	—	2	1	no	—
293	#550	C₂²×C₅	1	1	yes	characteristic
294	#548	—	2	1	no	—
295	#533	C₂²×C₅	1	1	yes	characteristic
296	#532	—	2	1	no	—
297	#532	C₂²×C₅	1	1	yes	characteristic
298	#529	—	2	1	no	—
299	#529	—	2	2	no	—
300	C₄×#41	—	2	1	no	—
301	C₄×#41	—	2	1	no	—
302	C₄×#41	C₂²×C₅	1	1	yes	characteristic
303	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
304	#498	C₂²×C₅	1	1	yes	characteristic
305	#492	C₂²×C₅	1	1	yes	characteristic
306	#487	C₂²×C₅	1	1	yes	characteristic
307	#486	C₂²×C₅	1	1	yes	characteristic
308	#484	—	2	1	no	—
309	#479	—	2	1	no	—
310	#478	—	2	1	no	—
311	#477	C₂²×C₅	1	1	yes	characteristic
312	#477	C₂²×C₅	1	1	yes	characteristic
313	#477	C₂²×C₅	1	1	yes	characteristic
314	#476	C₂²×C₅	1	1	yes	characteristic
315	#475	C₂²×C₅	1	1	yes	characteristic
316	#474	C₂²×C₅	1	1	yes	characteristic
317	#465	—	2	2	no	—
318	#462	C₂²×C₅	1	1	yes	characteristic
319	#462	C₂²×C₅	1	1	yes	characteristic
320	#451	—	2	1	no	—
321	C₄×Q₁₆	—	2	1	no	—
322	C₄×Q₁₆	—	2	1	no	—
323	C₄×SD₁₆	—	2	2	no	—
324	#431	C₂²×C₅	1	1	yes	characteristic
325	#427	—	2	1	no	—
326	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
327	#416	—	2	1	no	—
328	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
329	#332	C₂²×C₅	1	1	yes	characteristic
330	#332	C₂²×C₅	1	1	yes	characteristic
331	#331	C₂²×C₅	1	1	yes	characteristic
332	#331	C₂²×C₅	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
336	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
340	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
341	C₂³×C₅	—	2	1	no	—
342	C₂³×C₅	—	2	2	no	—
343	C₂³×C₅	#120	1	1	yes	characteristic
344	C₅×Q₈	—	2	1	no	—
345	C₅×Q₈	—	2	1	no	—
346	C₅×Q₈	—	2	1	no	—
347	C₅×Q₈	—	2	1	no	—
348	C₅×Q₈	—	2	1	no	—
349	C₅×Q₈	—	4	2	no	—
350	C₅×Q₈	—	4	2	no	—
351	C₅×Q₈	—	4	1	no	—
352	C₅×Q₈	—	4	1	no	—
353	C₅×Q₈	—	4	1	no	—
354	C₅×Q₈	—	4	1	no	—
355	C₅×Q₈	#135	1	2	yes	—
356	C₅×D₄	—	2	1	no	—
357	C₅×D₄	—	2	1	no	—
358	C₅×D₄	—	4	2	no	—
359	C₅×D₄	—	4	4	no	—
360	C₅×D₄	—	4	2	no	—
361	C₅×D₄	—	4	4	no	—
362	C₅×D₄	—	4	1	no	—
363	C₅×D₄	—	4	1	no	—
364	C₅×D₄	—	4	1	no	—
365	C₅×D₄	—	4	1	no	—
366	C₅×D₄	—	8	2	no	—
367	C₂×C₄×C₅	—	2	1	no	—
368	C₂×C₄×C₅	—	2	1	no	—
369	C₂×C₄×C₅	—	2	1	no	—
370	C₂×C₄×C₅	—	2	1	no	—
371	C₂×C₄×C₅	—	2	1	no	—
372	C₂×C₄×C₅	—	2	1	no	—
373	C₂×C₄×C₅	—	2	1	no	—
374	C₂×C₄×C₅	—	2	1	no	—
375	C₂×C₄×C₅	—	2	2	no	—
376	C₂×C₄×C₅	—	2	1	no	—
377	C₂×C₄×C₅	—	4	1	no	—
378	C₂×C₄×C₅	—	4	2	no	—
379	C₂×C₄×C₅	—	4	4	no	—
380	C₂×C₄×C₅	—	4	1	no	—
381	C₂×C₄×C₅	—	4	1	no	—
382	C₂×C₄×C₅	—	4	2	no	—
383	C₂×C₄×C₅	—	4	4	no	—
384	C₂×C₄×C₅	—	4	1	no	—
385	C₂×C₄×C₅	—	4	1	no	—
386	C₂×C₄×C₅	—	8	1	no	—
387	C₂×C₄×C₅	—	8	1	no	—
388	C₂×C₄×C₅	—	8	2	no	—
389	C₂×C₄×C₅	#120	1	1	yes	characteristic
390	C₂×C₄×C₅	#120	1	1	yes	characteristic
391	C₂×C₄×C₅	#120	1	1	yes	characteristic
392	C₂×C₄×C₅	#135	1	2	yes	—
393	C₂×C₄×C₅	#137	1	2	yes	—
394	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
395	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
396	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
397	C₅×C₈	—	2	2	no	—
398	C₅×C₈	—	4	1	no	—
399	C₅×C₈	—	4	1	no	—
400	C₅×C₈	—	4	1	no	—
401	C₅×C₈	—	4	1	no	—
402	C₅×C₈	—	4	1	no	—
403	C₂×#41	—	2	1	no	—
404	C₂×#41	—	2	1	no	—
405	C₂×#41	C₅×D₄	1	1	yes	characteristic
406	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
407	C₂×Q₁₆	—	2	1	no	—
408	C₂×Q₁₆	—	2	1	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂²×C₈	—	2	1	no	—
411	C₂²×C₈	—	2	1	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
414	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
415	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
416	#127	—	2	1	no	—
417	#127	C₂³×C₅	1	1	yes	characteristic
418	#126	—	2	2	no	—
419	#126	—	2	2	no	—
420	#126	—	2	2	no	—
421	#125	—	2	2	no	—
422	#125	—	2	1	no	—
423	#125	—	2	1	no	—
424	#124	—	2	2	no	—
425	#124	—	2	1	no	—
426	#124	—	2	1	no	—
427	#124	—	2	1	no	—
428	#124	—	2	1	no	—
429	#123	—	2	2	no	—
430	#123	—	2	2	no	—
431	#123	—	2	2	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	1	no	—
434	C₂²⋊Q₈	—	2	1	no	—
435	C₂²⋊Q₈	—	2	1	no	—
436	#121	—	2	2	no	—
437	#121	—	2	2	no	—
438	#121	—	2	2	no	—
439	#121	—	2	1	no	—
440	#121	—	2	1	no	—
441	#121	C₅×D₄	1	2	yes	—
442	C₄×Q₈	—	2	1	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
450	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
451	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
452	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
453	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
454	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	2	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	—	2	1	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	—	2	1	no	—
465	C₄×D₄	—	2	1	no	—
466	C₄×D₄	C₅×D₄	1	1	yes	characteristic
467	C₄×D₄	C₅×D₄	1	2	yes	—
468	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
469	#117	—	2	2	no	—
470	#117	—	2	2	no	—
471	#117	—	2	1	no	—
472	#117	—	2	1	no	—
473	#117	—	2	1	no	—
474	#117	C₅×D₄	1	1	yes	characteristic
475	C₂×C₄⋊C₄	—	2	1	no	—
476	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
477	C₂×C₄²	—	2	2	no	—
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	—	2	1	no	—
480	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
481	#107	—	2	1	no	—
482	#107	—	2	1	no	—
483	#107	—	2	1	no	—
484	#107	—	2	1	no	—
485	#106	—	2	1	no	—
486	#106	—	2	1	no	—
487	#106	—	2	2	no	—
488	C₄⋊C₈	—	2	1	no	—
489	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
490	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
491	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
492	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
493	#103	—	2	1	no	—
494	#103	—	2	1	no	—
495	#103	—	2	1	no	—
496	#103	—	2	1	no	—
497	#103	—	2	2	no	—
498	#103	—	2	1	no	—
499	#103	—	2	1	no	—
500	#102	—	2	1	no	—
501	#102	—	2	1	no	—
502	#102	—	2	2	no	—
503	C₂²⋊C₈	—	2	1	no	—
504	C₄×C₈	—	2	1	no	—
505	C₄×C₈	—	2	1	no	—
506	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
507	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
508	C₂²×C₅	—	2	1	no	—
509	C₂²×C₅	—	2	1	no	—
510	C₂²×C₅	—	4	1	no	—
511	C₂²×C₅	—	4	2	no	—
512	C₂²×C₅	—	4	2	no	—
513	C₂²×C₅	—	4	1	no	—
514	C₂²×C₅	—	4	2	no	—
515	C₂²×C₅	—	4	1	no	—
516	C₂²×C₅	#452	1	2	yes	—
517	C₂²×C₅	C₂×#120	1	1	yes	characteristic
518	C₄×C₅	—	2	1	no	—
519	C₄×C₅	—	2	2	no	—
520	C₄×C₅	—	2	1	no	—
521	C₄×C₅	—	2	1	no	—
522	C₄×C₅	—	4	1	no	—
523	C₄×C₅	—	4	2	no	—
524	C₄×C₅	—	4	1	no	—
525	C₄×C₅	—	4	1	no	—
526	C₄×C₅	—	4	1	no	—
527	C₄×C₅	—	4	1	no	—
528	C₄×C₅	—	4	1	no	—
529	C₄×C₅	—	8	1	no	—
530	C₄×C₅	—	8	1	no	—
531	C₄×C₅	#452	1	2	yes	—
532	C₄×C₅	C₂×#135	1	1	yes	characteristic
533	C₄×C₅	C₂×#135	1	1	yes	characteristic
534	C₄×C₅	C₂×#137	1	1	yes	characteristic
535	C₄×C₅	#575	1	1	yes	characteristic
536	#41	—	2	2	no	—
537	#41	—	2	2	no	—
538	#41	—	4	2	no	—
539	#41	—	4	2	no	—
540	#41	—	4	2	no	—
541	#41	—	4	2	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	2	1	no	—
544	C₂×Q₈	—	2	2	no	—
545	C₂×Q₈	—	2	1	no	—
546	C₂×Q₈	—	2	1	no	—
547	C₂×Q₈	—	4	1	no	—
548	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
549	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
550	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	2	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
558	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	2	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	2	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
568	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
569	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
570	Q₁₆	—	4	2	no	—
571	Q₁₆	—	4	2	no	—
572	SD₁₆	—	4	4	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	4	2	no	—
580	C₂×C₈	—	4	2	no	—
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	1	no	—
597	C₄⋊C₄	—	4	1	no	—
598	C₄⋊C₄	—	4	1	no	—
599	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
600	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
601	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
602	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
603	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	2	2	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	1	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	2	2	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	4	1	no	—
616	C₄²	—	2	2	no	—
617	C₄²	—	2	2	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	2	1	no	—
621	C₄²	—	2	1	no	—
622	C₄²	—	2	1	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	4	1	no	—
626	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
627	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
628	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
629	C₄²	C₂⁴×C₅	1	1	yes	characteristic
630	C₂×C₅	—	2	2	no	—
631	C₂×C₅	—	8	1	no	—
632	C₂×C₅	—	8	2	no	—
633	C₂×C₅	C₂×#452	1	1	yes	characteristic
634	C₂×C₅	#3002	1	1	yes	characteristic
635	C₂×C₅	#3018	1	1	yes	characteristic
636	C₂³	—	2	1	no	—
637	C₂³	—	2	2	no	—
638	C₂³	C₅×#120	1	1	yes	characteristic
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	2	1	no	—
642	Q₈	—	2	1	no	—
643	Q₈	—	2	1	no	—
644	Q₈	—	4	2	no	—
645	Q₈	—	4	2	no	—
646	Q₈	—	4	1	no	—
647	Q₈	—	4	1	no	—
648	Q₈	—	4	1	no	—
649	Q₈	—	4	1	no	—
650	Q₈	C₅×#135	1	2	yes	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	4	2	no	—
654	D₄	—	4	4	no	—
655	D₄	—	4	2	no	—
656	D₄	—	4	4	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	4	1	no	—
660	D₄	—	4	1	no	—
661	D₄	—	8	2	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	2	no	—
674	C₂×C₄	—	4	4	no	—
675	C₂×C₄	—	4	2	no	—
676	C₂×C₄	—	4	4	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	8	1	no	—
682	C₂×C₄	—	8	1	no	—
683	C₂×C₄	—	8	2	no	—
684	C₂×C₄	C₅×#120	1	1	yes	characteristic
685	C₂×C₄	C₅×#120	1	1	yes	characteristic
686	C₂×C₄	C₅×#120	1	1	yes	characteristic
687	C₂×C₄	C₅×#135	1	2	yes	—
688	C₂×C₄	C₅×#137	1	2	yes	—
689	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
690	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
691	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
692	C₈	—	2	2	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₈	—	4	1	no	—
697	C₈	—	4	1	no	—
698	C₅	#18521	1	1	yes	characteristic, direct product factor, Sylow
699	C₂²	—	2	1	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	—	4	1	no	—
705	C₂²	—	4	2	no	—
706	C₂²	—	4	1	no	—
707	C₂²	C₅×#452	1	2	yes	—
708	C₂²	C₂×C₅×#120	1	1	yes	characteristic
709	C₄	—	2	1	no	—
710	C₄	—	2	2	no	—
711	C₄	—	2	1	no	—
712	C₄	—	2	1	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	1	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	8	1	no	—
721	C₄	—	8	1	no	—
722	C₄	C₅×#452	1	2	yes	—
723	C₄	C₂×C₅×#135	1	1	yes	characteristic
724	C₄	C₂×C₅×#135	1	1	yes	characteristic
725	C₄	C₂×C₅×#137	1	1	yes	characteristic
726	C₄	C₅×#575	1	1	yes	characteristic
727	C₂	—	2	2	no	—
728	C₂	—	8	1	no	—
729	C₂	—	8	2	no	—
730	C₂	C₂×C₅×#452	1	1	yes	characteristic
731	C₂	C₅×#3002	1	1	yes	characteristic
732	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
prime radical	C₅×#3480	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
2. C₅×#3167	C₂	semidirect
3. C₅×#3104	C₂	semidirect
4. C₅×#3076	C₂	semidirect
5. C₅×#1653	C₂	semidirect
6. C₅×#1652	C₂	semidirect
7. C₅×#1641	C₂	semidirect
8. C₅×#1640	C₂	semidirect
9. C₅×#1638	C₂	semidirect
10. C₅×#1636	C₂	semidirect
11. C₅×#1631	C₂	semidirect
12. C₅×#1630	C₂	semidirect
13. C₅×#1540	C₂	semidirect
14. C₅×#1476	C₂	semidirect
15. C₂×C₅×Q₈⋊C₈	C₂	semidirect
71. #18521	C₅	direct (N × H)
178. #3167	C₂×C₅	semidirect
179. #3104	C₂×C₅	semidirect
180. #3076	C₂×C₅	semidirect
181. #1653	C₂×C₅	semidirect
182. #1652	C₂×C₅	semidirect
183. #1641	C₂×C₅	semidirect
184. #1640	C₂×C₅	semidirect
185. #1638	C₂×C₅	semidirect
186. #1636	C₂×C₅	semidirect
187. #1631	C₂×C₅	semidirect
188. #1630	C₂×C₅	semidirect
189. #1540	C₂×C₅	semidirect
190. #1476	C₂×C₅	semidirect
191. C₂×Q₈⋊C₈	C₂×C₅	semidirect
698. C₅	#18521	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×#3167	2
C₅×#3104	2
C₅×#3076	2
C₅×#1653	2
C₅×#1652	2
C₅×#1641	2
C₅×#1640	2
C₅×#1638	2
C₅×#1636	2
C₅×#1631	2
C₅×#1630	2
C₅×#1540	2
C₅×#1476	2
C₂×C₅×Q₈⋊C₈	2
#18521	5

Subgroups — C5×#18521 #11941241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#18521 #11941241