Subgroups — C₅×#81165 #14430216

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5688	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#237	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#478	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5063	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5009	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4880	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4875	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4860	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4854	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4829	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#4772	C₂	1	1	yes	maximal, characteristic
13	C₅×#4772	C₂	1	1	yes	maximal, characteristic
14	C₅×#4761	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#4759	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#3076	—	3	1	no	maximal, Hall
17	C₂×C₅×#237	C₂²	1	1	yes	characteristic
18	C₃×C₅×#125	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
20	C₅×D₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₄×C₅×#67	—	2	1	no	—
22	C₂×C₄×C₅×Dic₃	—	2	1	no	—
23	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
25	C₅×#896	—	2	1	no	—
26	C₅×#896	—	2	1	no	—
27	C₅×#892	C₂²	1	1	yes	characteristic
28	C₅×#892	C₂²	1	1	yes	characteristic
29	C₅×#892	C₂²	1	1	yes	characteristic
30	C₅×#888	C₂²	1	1	yes	characteristic
31	C₅×#888	C₂²	1	1	yes	characteristic
32	C₅×#887	—	2	1	no	—
33	C₅×#887	—	2	1	no	—
34	C₅×#887	C₂²	1	1	yes	characteristic
35	C₅×#67⋊C₄	—	2	1	no	—
36	C₅×#877	—	2	1	no	—
37	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
38	C₅×#870	C₂²	1	1	yes	characteristic
39	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
40	C₃×C₅×#107	C₂²	1	1	yes	characteristic
41	C₃×C₅×#106	C₂²	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₅×#102	C₂²	1	1	yes	characteristic
44	C₃×C₅×#102	C₂²	1	1	yes	characteristic
45	C₅×#832	C₂²	1	1	yes	characteristic
46	C₅×#832	C₂²	1	1	yes	characteristic
47	C₅×#818	C₂²	1	1	yes	characteristic
48	C₅×#817	C₂²	1	1	yes	characteristic
49	C₅×#816	C₂²	1	1	yes	characteristic
50	C₅×#816	C₂²	1	1	yes	characteristic
51	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	C₅×#810	C₂²	1	1	yes	characteristic
54	C₅×#810	C₂²	1	1	yes	characteristic
55	C₅×#808	C₂²	1	1	yes	characteristic
56	C₅×#808	C₂²	1	1	yes	characteristic
57	C₅×#807	C₂²	1	1	yes	characteristic
58	C₅×#807	C₂²	1	1	yes	characteristic
59	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
60	#81165	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
61	C₅×#533	—	3	1	no	—
62	C₄×C₅×#41	—	3	1	no	—
63	C₅×#479	—	3	1	no	—
64	C₅×#479	—	3	1	no	—
65	C₅×#478	—	3	1	no	—
66	C₅×#478	D₃	1	1	yes	characteristic, complemented
67	C₅×#477	—	3	1	no	—
68	C₅×#476	—	3	1	no	—
69	C₅×#475	—	3	1	no	—
70	C₅×#474	—	3	1	no	—
71	C₅×#427	—	3	1	no	—
72	C₅×#427	—	3	1	no	—
73	C₂×C₅×C₄⋊C₈	—	3	1	no	—
74	C₅×#416	—	3	1	no	—
75	C₅×#416	—	3	1	no	—
76	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₅×#67	—	2	1	no	—
79	C₂²×C₅×Dic₃	—	2	1	no	—
80	C₂²×C₅×Dic₃	—	2	1	no	—
81	C₅×#237	—	2	2	no	—
82	C₅×#237	Q₈	1	4	yes	complemented
83	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
84	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
85	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
86	C₂×C₅×Dic₆	—	2	1	no	—
87	C₂×C₅×Dic₆	—	2	1	no	—
88	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
91	C₃×C₅×C₄⋊C₄	—	2	1	no	—
92	C₃×C₅×C₄⋊C₄	—	2	1	no	—
93	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
96	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
97	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
98	C₅×#217	—	2	1	no	—
99	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
100	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
101	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
102	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
103	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
104	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₅×Dic₃⋊C₄	—	2	1	no	—
106	C₅×Dic₃⋊C₄	—	2	1	no	—
107	C₅×Dic₃⋊C₄	—	2	1	no	—
108	C₅×Dic₃⋊C₄	—	2	1	no	—
109	C₅×Dic₃⋊C₄	—	2	1	no	—
110	C₅×Dic₃⋊C₄	—	2	1	no	—
111	C₄×C₅×Dic₃	—	2	1	no	—
112	C₄×C₅×Dic₃	—	2	1	no	—
113	C₄×C₅×Dic₃	—	2	1	no	—
114	C₄×C₅×Dic₃	—	2	1	no	—
115	C₄×C₅×Dic₃	—	2	1	no	—
116	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
117	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
118	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
119	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₅×D₃×C₈	—	2	2	no	—
122	C₅×D₃×C₈	—	2	2	no	—
123	#5688	C₂×C₅	1	1	yes	characteristic, complemented
124	C₄×#237	C₂×C₅	1	1	yes	characteristic
125	C₃×#478	C₂×C₅	1	1	yes	characteristic, complemented
126	#5063	C₂×C₅	1	1	yes	characteristic, complemented
127	#5009	C₂×C₅	1	1	yes	characteristic, complemented
128	#4880	C₂×C₅	1	1	yes	characteristic, complemented
129	#4875	C₂×C₅	1	1	yes	characteristic, complemented
130	#4860	C₂×C₅	1	1	yes	characteristic, complemented
131	#4854	C₂×C₅	1	1	yes	characteristic, complemented
132	D₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
133	#4829	C₂×C₅	1	1	yes	characteristic, complemented
134	#4772	C₂×C₅	1	1	yes	characteristic
135	#4772	C₂×C₅	1	1	yes	characteristic
136	#4761	C₂×C₅	1	1	yes	characteristic, complemented
137	#4759	C₂×C₅	1	1	yes	characteristic, complemented
138	C₂×C₅×#41	—	3	1	no	—
139	C₂²×C₅×C₈	—	3	1	no	—
140	C₂²×C₅×C₈	—	3	1	no	—
141	C₅×C₄⋊Q₈	—	3	1	no	—
142	C₅×C₄⋊Q₈	—	3	1	no	—
143	C₅×#125	—	3	1	no	—
144	C₅×#125	C₂×D₃	1	1	yes	characteristic
145	C₅×C₂²⋊Q₈	—	6	1	no	—
146	C₅×C₂²⋊Q₈	—	6	1	no	—
147	C₄×C₅×Q₈	—	3	1	no	—
148	C₄×C₅×Q₈	—	3	1	no	—
149	C₄×C₅×Q₈	—	6	1	no	—
150	C₄×C₅×Q₈	—	6	1	no	—
151	C₄×C₅×D₄	—	3	1	no	—
152	C₄×C₅×D₄	—	6	1	no	—
153	C₄×C₅×D₄	—	6	1	no	—
154	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
155	C₅×#117	—	3	1	no	—
156	C₅×#117	—	3	1	no	—
157	C₅×#117	—	3	1	no	—
158	C₅×#117	—	6	1	no	—
159	C₂×C₄²×C₅	—	3	1	no	—
160	C₂×C₄²×C₅	—	6	1	no	—
161	C₅×#107	—	3	1	no	—
162	C₅×#107	—	3	1	no	—
163	C₅×#107	—	3	1	no	—
164	C₅×#107	C₂×D₃	1	1	yes	characteristic
165	C₅×#106	—	3	1	no	—
166	C₅×#106	—	3	1	no	—
167	C₅×#106	—	3	1	no	—
168	C₅×#106	C₂×D₃	1	1	yes	characteristic
169	C₅×C₄⋊C₈	—	3	1	no	—
170	C₅×C₄⋊C₈	—	3	1	no	—
171	C₅×C₄⋊C₈	—	3	1	no	—
172	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
173	C₅×#103	—	3	1	no	—
174	C₅×#103	—	3	1	no	—
175	C₅×#103	—	3	1	no	—
176	C₅×#103	—	3	1	no	—
177	C₅×#102	—	3	1	no	—
178	C₅×#102	—	3	1	no	—
179	C₅×#102	C₂×D₃	1	1	yes	characteristic
180	C₅×#102	C₂×D₃	1	1	yes	characteristic
181	#3076	—	3	1	no	Sylow
182	C₂³×C₃×C₅	—	2	1	no	—
183	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
184	C₃×C₅×D₄	—	2	1	no	—
185	C₃×C₅×D₄	C₂×Q₈	1	2	yes	—
186	C₂×C₃×C₄×C₅	—	2	1	no	—
187	C₂×C₃×C₄×C₅	—	2	1	no	—
188	C₂×C₃×C₄×C₅	—	2	1	no	—
189	C₂×C₃×C₄×C₅	—	4	2	no	—
190	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
193	C₅×#67	—	2	4	no	—
194	C₂×C₅×Dic₃	—	2	1	no	—
195	C₂×C₅×Dic₃	—	2	4	no	—
196	C₂×C₅×Dic₃	—	2	1	no	—
197	C₂×C₅×Dic₃	—	2	1	no	—
198	C₂×C₅×Dic₃	—	2	1	no	—
199	C₂×C₅×Dic₃	—	4	2	no	—
200	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
202	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
203	C₄×C₅×D₃	—	2	2	no	—
204	C₄×C₅×D₃	—	4	1	no	—
205	C₄×C₅×D₃	C₂×Q₈	1	2	yes	—
206	C₄×C₅×D₃	#41	1	2	yes	—
207	C₅×Dic₆	—	2	1	no	—
208	C₅×Dic₆	—	4	2	no	—
209	C₅×Dic₆	—	4	2	no	—
210	C₅×Dic₆	C₂×Q₈	1	2	yes	—
211	C₃×C₅×C₈	—	2	1	no	—
212	C₃×C₅×C₈	—	2	1	no	—
213	C₅×C₃⋊C₈	—	2	1	no	—
214	C₅×C₃⋊C₈	—	2	1	no	—
215	C₂×#237	C₂²×C₅	1	1	yes	characteristic
216	C₃×#125	C₂²×C₅	1	1	yes	characteristic
217	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
218	D₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
219	C₄×#67	—	2	1	no	—
220	C₂×C₄×Dic₃	—	2	1	no	—
221	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
222	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
223	#896	—	2	1	no	—
224	#896	—	2	1	no	—
225	#892	C₂²×C₅	1	1	yes	characteristic
226	#892	C₂²×C₅	1	1	yes	characteristic
227	#892	C₂²×C₅	1	1	yes	characteristic
228	#888	C₂²×C₅	1	1	yes	characteristic
229	#888	C₂²×C₅	1	1	yes	characteristic
230	#887	—	2	1	no	—
231	#887	—	2	1	no	—
232	#887	C₂²×C₅	1	1	yes	characteristic
233	#67⋊C₄	—	2	1	no	—
234	#877	—	2	1	no	—
235	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
236	#870	C₂²×C₅	1	1	yes	characteristic
237	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
238	C₃×#107	C₂²×C₅	1	1	yes	characteristic
239	C₃×#106	C₂²×C₅	1	1	yes	characteristic
240	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
241	C₃×#102	C₂²×C₅	1	1	yes	characteristic
242	C₃×#102	C₂²×C₅	1	1	yes	characteristic
243	#832	C₂²×C₅	1	1	yes	characteristic
244	#832	C₂²×C₅	1	1	yes	characteristic
245	#818	C₂²×C₅	1	1	yes	characteristic
246	#817	C₂²×C₅	1	1	yes	characteristic
247	#816	C₂²×C₅	1	1	yes	characteristic
248	#816	C₂²×C₅	1	1	yes	characteristic
249	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
250	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
251	#810	C₂²×C₅	1	1	yes	characteristic
252	#810	C₂²×C₅	1	1	yes	characteristic
253	#808	C₂²×C₅	1	1	yes	characteristic
254	#808	C₂²×C₅	1	1	yes	characteristic
255	#807	C₂²×C₅	1	1	yes	characteristic
256	#807	C₂²×C₅	1	1	yes	characteristic
257	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
258	C₅×#41	—	3	4	no	—
259	C₅×#41	—	6	2	no	—
260	C₂×C₅×Q₈	—	3	1	no	—
261	C₂×C₅×Q₈	—	6	1	no	—
262	C₂×C₅×Q₈	—	6	1	no	—
263	C₂×C₅×D₄	—	6	1	no	—
264	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
265	C₂²×C₄×C₅	—	2	1	no	—
266	C₂²×C₄×C₅	—	3	1	no	—
267	C₂²×C₄×C₅	—	3	1	no	—
268	C₂²×C₄×C₅	—	3	1	no	—
269	C₂²×C₄×C₅	—	6	1	no	—
270	C₂²×C₄×C₅	—	6	1	no	—
271	C₂×C₅×C₈	—	3	1	no	—
272	C₂×C₅×C₈	—	3	1	no	—
273	C₂×C₅×C₈	—	6	2	no	—
274	C₂×C₅×C₈	—	6	2	no	—
275	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
276	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
277	C₅×C₄⋊C₄	—	2	1	no	—
278	C₅×C₄⋊C₄	—	2	1	no	—
279	C₅×C₄⋊C₄	—	3	1	no	—
280	C₅×C₄⋊C₄	—	3	1	no	—
281	C₅×C₄⋊C₄	—	3	1	no	—
282	C₅×C₄⋊C₄	—	6	1	no	—
283	C₅×C₄⋊C₄	—	6	1	no	—
284	C₅×C₄⋊C₄	—	6	1	no	—
285	C₅×C₄⋊C₄	—	6	1	no	—
286	C₅×C₄⋊C₄	—	6	1	no	—
287	C₅×C₄⋊C₄	—	6	1	no	—
288	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
289	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
290	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
291	C₅×C₂²⋊C₄	—	2	1	no	—
292	C₅×C₂²⋊C₄	—	6	1	no	—
293	C₅×C₂²⋊C₄	—	6	1	no	—
294	C₅×C₂²⋊C₄	—	6	1	no	—
295	C₅×C₂²⋊C₄	—	6	1	no	—
296	C₄²×C₅	—	3	1	no	—
297	C₄²×C₅	—	3	1	no	—
298	C₄²×C₅	—	3	1	no	—
299	C₄²×C₅	—	6	1	no	—
300	C₄²×C₅	—	6	1	no	—
301	C₄²×C₅	—	6	1	no	—
302	C₄²×C₅	—	6	1	no	—
303	C₄²×C₅	—	6	1	no	—
304	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
305	#533	—	3	1	no	—
306	C₄×#41	—	3	1	no	—
307	#479	—	3	1	no	—
308	#479	—	3	1	no	—
309	#478	—	3	1	no	—
310	#478	C₅×D₃	1	1	yes	characteristic, complemented
311	#477	—	3	1	no	—
312	#476	—	3	1	no	—
313	#475	—	3	1	no	—
314	#474	—	3	1	no	—
315	#427	—	3	1	no	—
316	#427	—	3	1	no	—
317	C₂×C₄⋊C₈	—	3	1	no	—
318	#416	—	3	1	no	—
319	#416	—	3	1	no	—
320	C₂²×C₃×C₅	—	2	2	no	—
321	C₂²×C₃×C₅	—	4	1	no	—
322	C₂²×C₃×C₅	—	4	1	no	—
323	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₅×D₃	—	2	1	no	—
325	C₂×C₅×D₃	—	2	1	no	—
326	C₂×C₅×D₃	C₂²⋊Q₈	1	2	yes	—
327	C₃×C₄×C₅	—	2	1	no	—
328	C₃×C₄×C₅	—	2	1	no	—
329	C₃×C₄×C₅	—	4	1	no	—
330	C₃×C₄×C₅	—	4	1	no	—
331	C₃×C₄×C₅	—	4	1	no	—
332	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
333	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
334	C₅×Dic₃	—	2	2	no	—
335	C₅×Dic₃	—	2	1	no	—
336	C₅×Dic₃	—	4	1	no	—
337	C₅×Dic₃	—	4	1	no	—
338	C₅×Dic₃	—	4	1	no	—
339	C₅×Dic₃	C₂²⋊Q₈	1	2	yes	—
340	C₅×Dic₃	#135	1	2	yes	—
341	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
342	C₂²×C₃×C₄	—	2	1	no	—
343	C₂×#67	—	2	1	no	—
344	C₂²×Dic₃	—	2	1	no	—
345	C₂²×Dic₃	—	2	1	no	—
346	#237	—	2	2	no	—
347	#237	C₅×Q₈	1	4	yes	complemented
348	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
349	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
350	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
351	C₂×Dic₆	—	2	1	no	—
352	C₂×Dic₆	—	2	1	no	—
353	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
354	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
355	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
356	C₃×C₄⋊C₄	—	2	1	no	—
357	C₃×C₄⋊C₄	—	2	1	no	—
358	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
359	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
360	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
361	C₃×C₂²⋊C₄	—	2	1	no	—
362	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
363	#217	—	2	1	no	—
364	(C₂×D₃)⋊C₄	—	2	1	no	—
365	(C₂×D₃)⋊C₄	—	2	1	no	—
366	(C₂×D₃)⋊C₄	—	2	1	no	—
367	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
368	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
369	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
370	Dic₃⋊C₄	—	2	1	no	—
371	Dic₃⋊C₄	—	2	1	no	—
372	Dic₃⋊C₄	—	2	1	no	—
373	Dic₃⋊C₄	—	2	1	no	—
374	Dic₃⋊C₄	—	2	1	no	—
375	Dic₃⋊C₄	—	2	1	no	—
376	C₄×Dic₃	—	2	1	no	—
377	C₄×Dic₃	—	2	1	no	—
378	C₄×Dic₃	—	2	1	no	—
379	C₄×Dic₃	—	2	1	no	—
380	C₄×Dic₃	—	2	1	no	—
381	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
382	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
383	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
384	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
385	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
386	D₃×C₈	—	2	2	no	—
387	D₃×C₈	—	2	2	no	—
388	C₂³×C₅	—	2	1	no	—
389	C₂³×C₅	—	3	1	no	—
390	C₅×Q₈	—	3	2	no	—
391	C₅×Q₈	—	6	1	no	—
392	C₅×Q₈	—	12	2	no	—
393	C₅×Q₈	—	12	2	no	—
394	C₅×D₄	—	2	1	no	—
395	C₅×D₄	—	6	4	no	—
396	C₅×D₄	D₃×Q₈	1	2	yes	—
397	C₂×C₄×C₅	—	2	1	no	—
398	C₂×C₄×C₅	—	2	1	no	—
399	C₂×C₄×C₅	—	2	1	no	—
400	C₂×C₄×C₅	—	3	1	no	—
401	C₂×C₄×C₅	—	3	1	no	—
402	C₂×C₄×C₅	—	3	2	no	—
403	C₂×C₄×C₅	—	3	2	no	—
404	C₂×C₄×C₅	—	3	1	no	—
405	C₂×C₄×C₅	—	4	2	no	—
406	C₂×C₄×C₅	—	6	2	no	—
407	C₂×C₄×C₅	—	6	4	no	—
408	C₂×C₄×C₅	—	6	1	no	—
409	C₂×C₄×C₅	—	6	1	no	—
410	C₂×C₄×C₅	—	6	1	no	—
411	C₂×C₄×C₅	—	6	1	no	—
412	C₂×C₄×C₅	—	12	1	no	—
413	C₂×C₄×C₅	—	12	2	no	—
414	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
415	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
416	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
417	C₅×C₈	—	2	1	no	—
418	C₅×C₈	—	2	1	no	—
419	C₅×C₈	—	6	1	no	—
420	C₅×C₈	—	6	1	no	—
421	C₂×#41	—	3	1	no	—
422	C₂²×C₈	—	3	1	no	—
423	C₂²×C₈	—	3	1	no	—
424	C₄⋊Q₈	—	3	1	no	—
425	C₄⋊Q₈	—	3	1	no	—
426	#125	—	3	1	no	—
427	#125	C₂×C₅×D₃	1	1	yes	characteristic
428	C₂²⋊Q₈	—	6	1	no	—
429	C₂²⋊Q₈	—	6	1	no	—
430	C₄×Q₈	—	3	1	no	—
431	C₄×Q₈	—	3	1	no	—
432	C₄×Q₈	—	6	1	no	—
433	C₄×Q₈	—	6	1	no	—
434	C₄×D₄	—	3	1	no	—
435	C₄×D₄	—	6	1	no	—
436	C₄×D₄	—	6	1	no	—
437	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
438	#117	—	3	1	no	—
439	#117	—	3	1	no	—
440	#117	—	3	1	no	—
441	#117	—	6	1	no	—
442	C₂×C₄²	—	3	1	no	—
443	C₂×C₄²	—	6	1	no	—
444	#107	—	3	1	no	—
445	#107	—	3	1	no	—
446	#107	—	3	1	no	—
447	#107	C₂×C₅×D₃	1	1	yes	characteristic
448	#106	—	3	1	no	—
449	#106	—	3	1	no	—
450	#106	—	3	1	no	—
451	#106	C₂×C₅×D₃	1	1	yes	characteristic
452	C₄⋊C₈	—	3	1	no	—
453	C₄⋊C₈	—	3	1	no	—
454	C₄⋊C₈	—	3	1	no	—
455	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
456	#103	—	3	1	no	—
457	#103	—	3	1	no	—
458	#103	—	3	1	no	—
459	#103	—	3	1	no	—
460	#102	—	3	1	no	—
461	#102	—	3	1	no	—
462	#102	C₂×C₅×D₃	1	1	yes	characteristic
463	#102	C₂×C₅×D₃	1	1	yes	characteristic
464	C₂×C₃×C₅	—	4	2	no	—
465	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
466	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
467	C₂×C₃×C₅	#575	1	1	yes	characteristic
468	C₅×D₃	—	2	2	no	—
469	C₂³×C₃	—	2	1	no	—
470	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
471	C₃×D₄	—	2	1	no	—
472	C₃×D₄	C₂×C₅×Q₈	1	2	yes	—
473	C₂×C₃×C₄	—	2	1	no	—
474	C₂×C₃×C₄	—	2	1	no	—
475	C₂×C₃×C₄	—	2	1	no	—
476	C₂×C₃×C₄	—	4	2	no	—
477	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
479	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
480	#67	—	2	4	no	—
481	C₂×Dic₃	—	2	1	no	—
482	C₂×Dic₃	—	2	4	no	—
483	C₂×Dic₃	—	2	1	no	—
484	C₂×Dic₃	—	2	1	no	—
485	C₂×Dic₃	—	2	1	no	—
486	C₂×Dic₃	—	4	2	no	—
487	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
488	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
489	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
490	C₄×D₃	—	2	2	no	—
491	C₄×D₃	—	4	1	no	—
492	C₄×D₃	C₂×C₅×Q₈	1	2	yes	—
493	C₄×D₃	C₅×#41	1	2	yes	—
494	Dic₆	—	2	1	no	—
495	Dic₆	—	4	2	no	—
496	Dic₆	—	4	2	no	—
497	Dic₆	C₂×C₅×Q₈	1	2	yes	—
498	C₃×C₈	—	2	1	no	—
499	C₃×C₈	—	2	1	no	—
500	C₃⋊C₈	—	2	1	no	—
501	C₃⋊C₈	—	2	1	no	—
502	C₂²×C₅	—	2	2	no	—
503	C₂²×C₅	—	3	2	no	—
504	C₂²×C₅	—	4	1	no	—
505	C₂²×C₅	—	4	1	no	—
506	C₂²×C₅	—	6	1	no	—
507	C₂²×C₅	—	6	1	no	—
508	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
509	C₄×C₅	—	2	1	no	—
510	C₄×C₅	—	2	1	no	—
511	C₄×C₅	—	3	2	no	—
512	C₄×C₅	—	3	2	no	—
513	C₄×C₅	—	4	1	no	—
514	C₄×C₅	—	4	1	no	—
515	C₄×C₅	—	4	1	no	—
516	C₄×C₅	—	6	1	no	—
517	C₄×C₅	—	6	2	no	—
518	C₄×C₅	—	12	1	no	—
519	C₄×C₅	—	12	1	no	—
520	C₄×C₅	—	12	1	no	—
521	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
522	C₄×C₅	D₃×#41	1	1	yes	characteristic
523	#41	—	3	4	no	—
524	#41	—	6	2	no	—
525	C₂×Q₈	—	3	1	no	—
526	C₂×Q₈	—	6	1	no	—
527	C₂×Q₈	—	6	1	no	—
528	C₂×D₄	—	6	1	no	—
529	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	3	1	no	—
532	C₂²×C₄	—	3	1	no	—
533	C₂²×C₄	—	3	1	no	—
534	C₂²×C₄	—	6	1	no	—
535	C₂²×C₄	—	6	1	no	—
536	C₂×C₈	—	3	1	no	—
537	C₂×C₈	—	3	1	no	—
538	C₂×C₈	—	6	2	no	—
539	C₂×C₈	—	6	2	no	—
540	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
541	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	3	1	no	—
545	C₄⋊C₄	—	3	1	no	—
546	C₄⋊C₄	—	3	1	no	—
547	C₄⋊C₄	—	6	1	no	—
548	C₄⋊C₄	—	6	1	no	—
549	C₄⋊C₄	—	6	1	no	—
550	C₄⋊C₄	—	6	1	no	—
551	C₄⋊C₄	—	6	1	no	—
552	C₄⋊C₄	—	6	1	no	—
553	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
554	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
555	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
556	C₂²⋊C₄	—	2	1	no	—
557	C₂²⋊C₄	—	6	1	no	—
558	C₂²⋊C₄	—	6	1	no	—
559	C₂²⋊C₄	—	6	1	no	—
560	C₂²⋊C₄	—	6	1	no	—
561	C₄²	—	3	1	no	—
562	C₄²	—	3	1	no	—
563	C₄²	—	3	1	no	—
564	C₄²	—	6	1	no	—
565	C₄²	—	6	1	no	—
566	C₄²	—	6	1	no	—
567	C₄²	—	6	1	no	—
568	C₄²	—	6	1	no	—
569	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
570	C₃×C₅	#3076	1	1	yes	characteristic, complemented, Hall
571	C₂²×C₃	—	2	2	no	—
572	C₂²×C₃	—	4	1	no	—
573	C₂²×C₃	—	4	1	no	—
574	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
575	C₂×D₃	—	2	1	no	—
576	C₂×D₃	—	2	1	no	—
577	C₂×D₃	C₅×C₂²⋊Q₈	1	2	yes	—
578	C₃×C₄	—	2	1	no	—
579	C₃×C₄	—	2	1	no	—
580	C₃×C₄	—	4	1	no	—
581	C₃×C₄	—	4	1	no	—
582	C₃×C₄	—	4	1	no	—
583	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
584	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
585	Dic₃	—	2	2	no	—
586	Dic₃	—	2	1	no	—
587	Dic₃	—	4	1	no	—
588	Dic₃	—	4	1	no	—
589	Dic₃	—	4	1	no	—
590	Dic₃	C₅×C₂²⋊Q₈	1	2	yes	—
591	Dic₃	C₅×#135	1	2	yes	—
592	C₂×C₅	—	4	2	no	—
593	C₂×C₅	—	6	2	no	—
594	C₂×C₅	D₃×C₂²⋊Q₈	1	1	yes	characteristic
595	C₂×C₅	D₃×#135	1	1	yes	characteristic
596	C₂×C₅	#5773	1	1	yes	characteristic
597	C₂³	—	2	1	no	—
598	C₂³	—	3	1	no	—
599	Q₈	—	3	2	no	—
600	Q₈	—	6	1	no	—
601	Q₈	—	12	2	no	—
602	Q₈	—	12	2	no	—
603	D₄	—	2	1	no	—
604	D₄	—	6	4	no	—
605	D₄	C₅×D₃×Q₈	1	2	yes	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	3	1	no	—
610	C₂×C₄	—	3	1	no	—
611	C₂×C₄	—	3	2	no	—
612	C₂×C₄	—	3	2	no	—
613	C₂×C₄	—	3	1	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	6	2	no	—
616	C₂×C₄	—	6	4	no	—
617	C₂×C₄	—	6	1	no	—
618	C₂×C₄	—	6	1	no	—
619	C₂×C₄	—	6	1	no	—
620	C₂×C₄	—	6	1	no	—
621	C₂×C₄	—	12	1	no	—
622	C₂×C₄	—	12	2	no	—
623	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
624	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
625	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
626	C₈	—	2	1	no	—
627	C₈	—	2	1	no	—
628	C₈	—	6	1	no	—
629	C₈	—	6	1	no	—
630	C₂×C₃	—	4	2	no	—
631	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
632	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
633	C₂×C₃	C₅×#575	1	1	yes	characteristic
634	D₃	—	2	2	no	—
635	C₅	#81165	1	1	yes	characteristic, direct product factor, Sylow
636	C₂²	—	2	2	no	—
637	C₂²	—	3	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	6	1	no	—
641	C₂²	—	6	1	no	—
642	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
643	C₄	—	2	1	no	—
644	C₄	—	2	1	no	—
645	C₄	—	3	2	no	—
646	C₄	—	3	2	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	1	no	—
649	C₄	—	4	1	no	—
650	C₄	—	6	1	no	—
651	C₄	—	6	2	no	—
652	C₄	—	12	1	no	—
653	C₄	—	12	1	no	—
654	C₄	—	12	1	no	—
655	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
656	C₄	C₅×D₃×#41	1	1	yes	characteristic
657	C₃	C₅×#3076	1	1	yes	characteristic, complemented, Sylow
658	C₂	—	4	2	no	—
659	C₂	—	6	2	no	—
660	C₂	C₅×D₃×C₂²⋊Q₈	1	1	yes	characteristic
661	C₂	C₅×D₃×#135	1	1	yes	characteristic
662	C₂	C₅×#5773	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#478	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#237	C₂²
hypercenter	C₅×#478	D₃
hypocenter	C₃	C₅×#3076
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#478

Complements

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

N	H ≅ G/N	type
1. C₅×#5688	C₂	semidirect
3. C₃×C₅×#478	C₂	semidirect
4. C₅×#5063	C₂	semidirect
5. C₅×#5009	C₂	semidirect
6. C₅×#4880	C₂	semidirect
7. C₅×#4875	C₂	semidirect
8. C₅×#4860	C₂	semidirect
9. C₅×#4854	C₂	semidirect
10. C₅×D₃×C₄⋊C₈	C₂	semidirect
11. C₅×#4829	C₂	semidirect
14. C₅×#4761	C₂	semidirect
15. C₅×#4759	C₂	semidirect
60. #81165	C₅	direct (N × H)
66. C₅×#478	D₃	semidirect
82. C₅×#237	Q₈	semidirect
123. #5688	C₂×C₅	semidirect
125. C₃×#478	C₂×C₅	semidirect
126. #5063	C₂×C₅	semidirect
127. #5009	C₂×C₅	semidirect
128. #4880	C₂×C₅	semidirect
129. #4875	C₂×C₅	semidirect
130. #4860	C₂×C₅	semidirect
131. #4854	C₂×C₅	semidirect
132. D₃×C₄⋊C₈	C₂×C₅	semidirect
133. #4829	C₂×C₅	semidirect
136. #4761	C₂×C₅	semidirect
137. #4759	C₂×C₅	semidirect
310. #478	C₅×D₃	semidirect
347. #237	C₅×Q₈	semidirect
570. C₃×C₅	#3076	semidirect
635. C₅	#81165	direct (N × H)
657. C₃	C₅×#3076	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#5688	2
C₄×C₅×#237	2
C₃×C₅×#478	2
C₅×#5063	2
C₅×#5009	2
C₅×#4880	2
C₅×#4875	2
C₅×#4860	2
C₅×#4854	2
C₅×D₃×C₄⋊C₈	2
C₅×#4829	2
C₅×#4772	2
C₅×#4761	2
C₅×#4759	2
C₅×#3076	3
#81165	5

Subgroups — C5×#81165 #14430216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81165 #14430216