Subgroups — C₅×#80516 #14471582

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5671	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5539	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#463	C₂	1	1	yes	maximal, characteristic
4	C₅×#5162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5153	C₂	1	1	yes	maximal, characteristic
7	C₅×Dic₃×D₈	C₂	1	1	yes	maximal, characteristic
8	C₅×#5056	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#214	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4830	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4829	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#4796	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#4792	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#4768	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#4761	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#3360	—	3	1	no	maximal, Hall
17	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×#124	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	C₅×Q₈×Dic₃	C₂²	1	1	yes	characteristic
22	C₅×#944	C₂²	1	1	yes	characteristic
23	C₅×#936	C₂²	1	1	yes	characteristic
24	C₅×D₄×Dic₃	C₂²	1	1	yes	characteristic
25	C₅×D₄×Dic₃	C₂²	1	1	yes	characteristic
26	C₅×D₄×Dic₃	C₂²	1	1	yes	characteristic
27	C₂×C₅×#214	C₂²	1	1	yes	characteristic
28	C₅×#924	—	2	1	no	—
29	C₂×C₅×Dic₃⋊C₄	—	2	1	no	—
30	C₅×#890	C₂²	1	1	yes	characteristic
31	C₅×#888	C₂²	1	1	yes	characteristic
32	C₅×#879	—	2	1	no	—
33	C₅×#879	—	2	1	no	—
34	C₅×#878	—	2	1	no	—
35	C₅×#878	—	2	1	no	—
36	C₅×#878	—	2	1	no	—
37	C₅×#877	—	2	1	no	—
38	C₅×#870	C₂²	1	1	yes	characteristic
39	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
40	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₅×#103	C₂²	1	1	yes	characteristic
43	C₃×C₅×#102	C₂²	1	1	yes	characteristic
44	C₅×#835	C₂²	1	1	yes	characteristic
45	C₅×#835	C₂²	1	1	yes	characteristic
46	C₅×#832	C₂²	1	1	yes	characteristic
47	C₅×#832	C₂²	1	1	yes	characteristic
48	C₅×#832	C₂²	1	1	yes	characteristic
49	C₅×#818	C₂²	1	1	yes	characteristic
50	C₅×#817	C₂²	1	1	yes	characteristic
51	C₅×#816	C₂²	1	1	yes	characteristic
52	C₅×#816	C₂²	1	1	yes	characteristic
53	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
56	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
57	C₅×#810	C₂²	1	1	yes	characteristic
58	C₅×#808	C₂²	1	1	yes	characteristic
59	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
60	#80516	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
61	C₅×#556	—	3	1	no	—
62	C₅×#554	—	3	1	no	—
63	C₅×#488	—	3	1	no	—
64	C₅×#488	—	3	1	no	—
65	C₅×#487	—	3	1	no	—
66	C₅×#479	—	3	1	no	—
67	C₅×#479	—	3	1	no	—
68	C₅×#476	—	3	1	no	—
69	C₅×#463	—	3	1	no	—
70	C₅×#463	D₃	1	1	yes	characteristic
71	C₅×#461	—	3	1	no	—
72	C₅×C₈×Q₈	—	3	1	no	—
73	C₄×C₅×SD₁₆	—	3	1	no	—
74	C₄×C₅×SD₁₆	—	3	1	no	—
75	C₄×C₅×D₈	—	3	1	no	—
76	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
79	C₂²×C₃×C₄×C₅	—	2	1	no	—
80	C₂²×C₅×Dic₃	—	2	1	no	—
81	C₂²×C₅×Dic₃	—	2	1	no	—
82	C₂²×C₅×Dic₃	—	2	1	no	—
83	C₂×C₅×Dic₆	—	2	1	no	—
84	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
85	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
86	C₃×C₅×SD₁₆	—	2	2	no	—
87	C₃×C₅×D₈	—	2	2	no	—
88	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
93	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
94	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
95	C₅×#217	—	2	1	no	—
96	C₅×#217	—	2	1	no	—
97	C₅×#217	—	2	1	no	—
98	C₅×#217	—	2	1	no	—
99	C₅×#217	—	2	1	no	—
100	C₅×#214	—	2	2	no	—
101	C₅×C₄⋊Dic₃	—	2	1	no	—
102	C₅×C₄⋊Dic₃	—	2	1	no	—
103	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
104	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
106	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	C₅×Dic₃⋊C₄	—	2	1	no	—
112	C₅×Dic₃⋊C₄	—	2	1	no	—
113	C₅×Dic₃⋊C₄	—	2	1	no	—
114	C₅×Dic₃⋊C₄	D₄	1	1	yes	characteristic
115	C₅×Dic₃⋊C₄	D₄	1	1	yes	characteristic
116	C₄×C₅×Dic₃	—	2	1	no	—
117	C₄×C₅×Dic₃	—	2	1	no	—
118	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
119	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
120	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	#5671	C₂×C₅	1	1	yes	characteristic, complemented
123	#5539	C₂×C₅	1	1	yes	characteristic, complemented
124	C₃×#463	C₂×C₅	1	1	yes	characteristic
125	#5162	C₂×C₅	1	1	yes	characteristic, complemented
126	Dic₃×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
127	#5153	C₂×C₅	1	1	yes	characteristic
128	Dic₃×D₈	C₂×C₅	1	1	yes	characteristic
129	#5056	C₂×C₅	1	1	yes	characteristic, complemented
130	C₄×#214	C₂×C₅	1	1	yes	characteristic, complemented
131	#4830	C₂×C₅	1	1	yes	characteristic, complemented
132	#4829	C₂×C₅	1	1	yes	characteristic, complemented
133	#4796	C₂×C₅	1	1	yes	characteristic, complemented
134	#4792	C₂×C₅	1	1	yes	characteristic, complemented
135	#4768	C₂×C₅	1	1	yes	characteristic, complemented
136	#4761	C₂×C₅	1	1	yes	characteristic, complemented
137	C₂×C₅×SD₁₆	—	3	1	no	—
138	C₂×C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic
139	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic
140	C₅×C₄⋊Q₈	—	3	1	no	—
141	C₅×C₄⋊Q₈	—	3	1	no	—
142	C₅×#126	—	6	1	no	—
143	C₅×#126	—	6	1	no	—
144	C₅×#125	—	3	1	no	—
145	C₅×#125	—	3	1	no	—
146	C₅×#124	—	3	1	no	—
147	C₅×#124	C₂×D₃	1	1	yes	characteristic
148	C₅×C₂²⋊Q₈	—	6	1	no	—
149	C₅×C₂²⋊Q₈	—	6	1	no	—
150	C₅×C₂²⋊Q₈	—	6	1	no	—
151	C₅×C₂²⋊Q₈	—	6	1	no	—
152	C₄×C₅×Q₈	—	3	1	no	—
153	C₄×C₅×Q₈	—	3	1	no	—
154	C₄×C₅×Q₈	—	3	1	no	—
155	C₄×C₅×D₄	—	3	1	no	—
156	C₄×C₅×D₄	—	3	1	no	—
157	C₄×C₅×D₄	—	3	1	no	—
158	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
159	C₅×#117	—	6	1	no	—
160	C₂×C₅×C₄⋊C₄	—	6	1	no	—
161	C₅×#107	—	3	1	no	—
162	C₅×#106	—	3	1	no	—
163	C₅×#106	—	3	1	no	—
164	C₅×C₄⋊C₈	—	3	1	no	—
165	C₅×C₄⋊C₈	—	3	1	no	—
166	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
167	C₅×#103	—	3	1	no	—
168	C₅×#103	—	3	1	no	—
169	C₅×#103	—	3	1	no	—
170	C₅×#103	—	3	1	no	—
171	C₅×#103	—	3	1	no	—
172	C₅×#103	C₂×D₃	1	1	yes	characteristic
173	C₅×#102	—	3	1	no	—
174	C₅×#102	—	3	1	no	—
175	C₅×#102	—	3	1	no	—
176	C₅×#102	C₂×D₃	1	1	yes	characteristic
177	C₄×C₅×C₈	—	3	1	no	—
178	C₄×C₅×C₈	—	3	1	no	—
179	C₄×C₅×C₈	—	3	1	no	—
180	#3360	—	3	1	no	Sylow
181	C₂³×C₃×C₅	—	2	1	no	—
182	C₂³×C₃×C₅	—	2	1	no	—
183	C₃×C₅×Q₈	—	2	1	no	—
184	C₃×C₅×Q₈	—	2	1	no	—
185	C₃×C₅×D₄	—	2	1	no	—
186	C₃×C₅×D₄	—	2	1	no	—
187	C₃×C₅×D₄	—	2	1	no	—
188	C₃×C₅×D₄	#41	1	2	yes	—
189	C₂×C₃×C₄×C₅	—	2	1	no	—
190	C₂×C₃×C₄×C₅	—	2	1	no	—
191	C₂×C₃×C₄×C₅	—	8	1	no	—
192	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
193	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
195	C₂×C₅×Dic₃	—	2	1	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	—	4	2	no	—
201	C₂×C₅×Dic₃	—	8	1	no	—
202	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
204	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
205	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
206	C₅×Dic₆	—	2	2	no	—
207	C₅×Dic₆	—	2	1	no	—
208	C₅×Dic₆	—	2	1	no	—
209	C₅×Dic₆	—	8	1	no	—
210	C₃×C₅×C₈	—	2	1	no	—
211	C₃×C₅×C₈	—	2	1	no	—
212	C₅×C₃⋊C₈	—	2	1	no	—
213	C₅×C₃⋊C₈	—	2	1	no	—
214	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic
215	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
216	C₃×#124	C₂²×C₅	1	1	yes	characteristic
217	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
218	Q₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
219	#944	C₂²×C₅	1	1	yes	characteristic
220	#936	C₂²×C₅	1	1	yes	characteristic
221	D₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
222	D₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
223	D₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
224	C₂×#214	C₂²×C₅	1	1	yes	characteristic
225	#924	—	2	1	no	—
226	C₂×Dic₃⋊C₄	—	2	1	no	—
227	#890	C₂²×C₅	1	1	yes	characteristic
228	#888	C₂²×C₅	1	1	yes	characteristic
229	#879	—	2	1	no	—
230	#879	—	2	1	no	—
231	#878	—	2	1	no	—
232	#878	—	2	1	no	—
233	#878	—	2	1	no	—
234	#877	—	2	1	no	—
235	#870	C₂²×C₅	1	1	yes	characteristic
236	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
237	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
238	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
239	C₃×#103	C₂²×C₅	1	1	yes	characteristic
240	C₃×#102	C₂²×C₅	1	1	yes	characteristic
241	#835	C₂²×C₅	1	1	yes	characteristic
242	#835	C₂²×C₅	1	1	yes	characteristic
243	#832	C₂²×C₅	1	1	yes	characteristic
244	#832	C₂²×C₅	1	1	yes	characteristic
245	#832	C₂²×C₅	1	1	yes	characteristic
246	#818	C₂²×C₅	1	1	yes	characteristic
247	#817	C₂²×C₅	1	1	yes	characteristic
248	#816	C₂²×C₅	1	1	yes	characteristic
249	#816	C₂²×C₅	1	1	yes	characteristic
250	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
251	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
252	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
253	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
254	#810	C₂²×C₅	1	1	yes	characteristic
255	#808	C₂²×C₅	1	1	yes	characteristic
256	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
257	C₂×C₅×Q₈	—	3	1	no	—
258	C₂×C₅×Q₈	—	3	1	no	—
259	C₂×C₅×Q₈	—	6	1	no	—
260	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
261	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
262	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
263	C₂²×C₄×C₅	—	2	1	no	—
264	C₂²×C₄×C₅	—	6	1	no	—
265	C₂²×C₄×C₅	—	6	1	no	—
266	C₂²×C₄×C₅	—	6	1	no	—
267	C₅×SD₁₆	—	2	2	no	—
268	C₅×SD₁₆	—	6	2	no	—
269	C₅×D₈	—	2	2	no	—
270	C₂×C₅×C₈	—	3	1	no	—
271	C₂×C₅×C₈	—	3	1	no	—
272	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
273	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
274	C₅×C₄⋊C₄	—	3	1	no	—
275	C₅×C₄⋊C₄	—	3	1	no	—
276	C₅×C₄⋊C₄	—	3	1	no	—
277	C₅×C₄⋊C₄	—	3	1	no	—
278	C₅×C₄⋊C₄	—	3	1	no	—
279	C₅×C₄⋊C₄	—	3	1	no	—
280	C₅×C₄⋊C₄	—	6	1	no	—
281	C₅×C₄⋊C₄	—	6	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	—	6	1	no	—
289	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
290	C₅×C₂²⋊C₄	—	2	1	no	—
291	C₅×C₂²⋊C₄	—	2	1	no	—
292	C₅×C₂²⋊C₄	—	2	1	no	—
293	C₅×C₂²⋊C₄	—	6	1	no	—
294	C₅×C₂²⋊C₄	—	6	1	no	—
295	C₅×C₂²⋊C₄	—	6	1	no	—
296	C₅×C₂²⋊C₄	—	6	1	no	—
297	C₅×C₂²⋊C₄	—	6	1	no	—
298	C₄²×C₅	—	3	1	no	—
299	C₄²×C₅	—	3	1	no	—
300	C₄²×C₅	—	6	1	no	—
301	C₄²×C₅	—	6	1	no	—
302	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
303	#556	—	3	1	no	—
304	#554	—	3	1	no	—
305	#488	—	3	1	no	—
306	#488	—	3	1	no	—
307	#487	—	3	1	no	—
308	#479	—	3	1	no	—
309	#479	—	3	1	no	—
310	#476	—	3	1	no	—
311	#463	—	3	1	no	—
312	#463	C₅×D₃	1	1	yes	characteristic
313	#461	—	3	1	no	—
314	C₈×Q₈	—	3	1	no	—
315	C₄×SD₁₆	—	3	1	no	—
316	C₄×SD₁₆	—	3	1	no	—
317	C₄×D₈	—	3	1	no	—
318	C₂²×C₃×C₅	—	2	2	no	—
319	C₂²×C₃×C₅	—	4	1	no	—
320	C₂²×C₃×C₅	—	4	1	no	—
321	C₂²×C₃×C₅	—	4	1	no	—
322	C₂²×C₃×C₅	—	4	1	no	—
323	C₂²×C₃×C₅	—	4	1	no	—
324	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
325	C₃×C₄×C₅	—	2	1	no	—
326	C₃×C₄×C₅	—	2	1	no	—
327	C₃×C₄×C₅	—	4	1	no	—
328	C₃×C₄×C₅	—	4	1	no	—
329	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
330	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
331	C₅×Dic₃	—	2	1	no	—
332	C₅×Dic₃	—	2	1	no	—
333	C₅×Dic₃	—	2	1	no	—
334	C₅×Dic₃	—	4	1	no	—
335	C₅×Dic₃	—	4	1	no	—
336	C₅×Dic₃	—	4	1	no	—
337	C₅×Dic₃	—	4	1	no	—
338	C₅×Dic₃	#135	1	2	yes	—
339	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
340	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
341	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
342	C₂²×C₃×C₄	—	2	1	no	—
343	C₂²×Dic₃	—	2	1	no	—
344	C₂²×Dic₃	—	2	1	no	—
345	C₂²×Dic₃	—	2	1	no	—
346	C₂×Dic₆	—	2	1	no	—
347	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
348	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
349	C₃×SD₁₆	—	2	2	no	—
350	C₃×D₈	—	2	2	no	—
351	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
352	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
353	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
354	C₃×C₂²⋊C₄	—	2	1	no	—
355	C₃×C₂²⋊C₄	—	2	1	no	—
356	C₃×C₂²⋊C₄	—	2	1	no	—
357	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
358	#217	—	2	1	no	—
359	#217	—	2	1	no	—
360	#217	—	2	1	no	—
361	#217	—	2	1	no	—
362	#217	—	2	1	no	—
363	#214	—	2	2	no	—
364	C₄⋊Dic₃	—	2	1	no	—
365	C₄⋊Dic₃	—	2	1	no	—
366	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
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	Dic₃⋊C₄	—	2	1	no	—
377	Dic₃⋊C₄	C₅×D₄	1	1	yes	characteristic
378	Dic₃⋊C₄	C₅×D₄	1	1	yes	characteristic
379	C₄×Dic₃	—	2	1	no	—
380	C₄×Dic₃	—	2	1	no	—
381	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
382	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
383	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
384	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
385	C₂³×C₅	—	2	1	no	—
386	C₂³×C₅	—	2	1	no	—
387	C₅×Q₈	—	2	1	no	—
388	C₅×Q₈	—	2	1	no	—
389	C₅×Q₈	—	6	2	no	—
390	C₅×Q₈	—	6	1	no	—
391	C₅×Q₈	—	6	1	no	—
392	C₅×Q₈	—	24	1	no	—
393	C₅×D₄	—	2	1	no	—
394	C₅×D₄	—	2	1	no	—
395	C₅×D₄	—	2	1	no	—
396	C₅×D₄	#237	1	2	yes	—
397	C₂×C₄×C₅	—	2	1	no	—
398	C₂×C₄×C₅	—	2	1	no	—
399	C₂×C₄×C₅	—	3	1	no	—
400	C₂×C₄×C₅	—	3	1	no	—
401	C₂×C₄×C₅	—	3	1	no	—
402	C₂×C₄×C₅	—	3	1	no	—
403	C₂×C₄×C₅	—	6	1	no	—
404	C₂×C₄×C₅	—	6	1	no	—
405	C₂×C₄×C₅	—	6	1	no	—
406	C₂×C₄×C₅	—	6	1	no	—
407	C₂×C₄×C₅	—	8	1	no	—
408	C₂×C₄×C₅	—	12	2	no	—
409	C₂×C₄×C₅	—	12	2	no	—
410	C₂×C₄×C₅	—	24	1	no	—
411	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
412	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
413	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
414	C₅×C₈	—	2	1	no	—
415	C₅×C₈	—	2	1	no	—
416	C₅×C₈	—	6	1	no	—
417	C₅×C₈	—	6	1	no	—
418	C₂×SD₁₆	—	3	1	no	—
419	C₂×SD₁₆	C₂×C₅×D₃	1	1	yes	characteristic
420	C₂×D₈	C₂×C₅×D₃	1	1	yes	characteristic
421	C₄⋊Q₈	—	3	1	no	—
422	C₄⋊Q₈	—	3	1	no	—
423	#126	—	6	1	no	—
424	#126	—	6	1	no	—
425	#125	—	3	1	no	—
426	#125	—	3	1	no	—
427	#124	—	3	1	no	—
428	#124	C₂×C₅×D₃	1	1	yes	characteristic
429	C₂²⋊Q₈	—	6	1	no	—
430	C₂²⋊Q₈	—	6	1	no	—
431	C₂²⋊Q₈	—	6	1	no	—
432	C₂²⋊Q₈	—	6	1	no	—
433	C₄×Q₈	—	3	1	no	—
434	C₄×Q₈	—	3	1	no	—
435	C₄×Q₈	—	3	1	no	—
436	C₄×D₄	—	3	1	no	—
437	C₄×D₄	—	3	1	no	—
438	C₄×D₄	—	3	1	no	—
439	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
440	#117	—	6	1	no	—
441	C₂×C₄⋊C₄	—	6	1	no	—
442	#107	—	3	1	no	—
443	#106	—	3	1	no	—
444	#106	—	3	1	no	—
445	C₄⋊C₈	—	3	1	no	—
446	C₄⋊C₈	—	3	1	no	—
447	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
448	#103	—	3	1	no	—
449	#103	—	3	1	no	—
450	#103	—	3	1	no	—
451	#103	—	3	1	no	—
452	#103	—	3	1	no	—
453	#103	C₂×C₅×D₃	1	1	yes	characteristic
454	#102	—	3	1	no	—
455	#102	—	3	1	no	—
456	#102	—	3	1	no	—
457	#102	C₂×C₅×D₃	1	1	yes	characteristic
458	C₄×C₈	—	3	1	no	—
459	C₄×C₈	—	3	1	no	—
460	C₄×C₈	—	3	1	no	—
461	C₂×C₃×C₅	—	4	2	no	—
462	C₂×C₃×C₅	—	8	1	no	—
463	C₂×C₃×C₅	#548	1	1	yes	characteristic
464	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
465	C₂×C₃×C₅	#577	1	1	yes	characteristic
466	C₂³×C₃	—	2	1	no	—
467	C₂³×C₃	—	2	1	no	—
468	C₃×Q₈	—	2	1	no	—
469	C₃×Q₈	—	2	1	no	—
470	C₃×D₄	—	2	1	no	—
471	C₃×D₄	—	2	1	no	—
472	C₃×D₄	—	2	1	no	—
473	C₃×D₄	C₅×#41	1	2	yes	—
474	C₂×C₃×C₄	—	2	1	no	—
475	C₂×C₃×C₄	—	2	1	no	—
476	C₂×C₃×C₄	—	8	1	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	C₂×Dic₃	—	2	1	no	—
481	C₂×Dic₃	—	2	1	no	—
482	C₂×Dic₃	—	2	1	no	—
483	C₂×Dic₃	—	2	1	no	—
484	C₂×Dic₃	—	4	2	no	—
485	C₂×Dic₃	—	4	2	no	—
486	C₂×Dic₃	—	8	1	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	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
491	Dic₆	—	2	2	no	—
492	Dic₆	—	2	1	no	—
493	Dic₆	—	2	1	no	—
494	Dic₆	—	8	1	no	—
495	C₃×C₈	—	2	1	no	—
496	C₃×C₈	—	2	1	no	—
497	C₃⋊C₈	—	2	1	no	—
498	C₃⋊C₈	—	2	1	no	—
499	C₂²×C₅	—	2	2	no	—
500	C₂²×C₅	—	4	1	no	—
501	C₂²×C₅	—	4	1	no	—
502	C₂²×C₅	—	4	1	no	—
503	C₂²×C₅	—	4	1	no	—
504	C₂²×C₅	—	4	1	no	—
505	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
506	C₄×C₅	—	2	1	no	—
507	C₄×C₅	—	2	1	no	—
508	C₄×C₅	—	3	2	no	—
509	C₄×C₅	—	4	1	no	—
510	C₄×C₅	—	4	1	no	—
511	C₄×C₅	—	6	1	no	—
512	C₄×C₅	—	6	1	no	—
513	C₄×C₅	—	6	1	no	—
514	C₄×C₅	—	12	1	no	—
515	C₄×C₅	—	12	1	no	—
516	C₄×C₅	—	12	1	no	—
517	C₄×C₅	—	12	1	no	—
518	C₄×C₅	C₂×#237	1	1	yes	characteristic
519	C₄×C₅	#1010	1	1	yes	characteristic
520	C₂×Q₈	—	3	1	no	—
521	C₂×Q₈	—	3	1	no	—
522	C₂×Q₈	—	6	1	no	—
523	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
524	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
525	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	6	1	no	—
528	C₂²×C₄	—	6	1	no	—
529	C₂²×C₄	—	6	1	no	—
530	SD₁₆	—	2	2	no	—
531	SD₁₆	—	6	2	no	—
532	D₈	—	2	2	no	—
533	C₂×C₈	—	3	1	no	—
534	C₂×C₈	—	3	1	no	—
535	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
536	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
537	C₄⋊C₄	—	3	1	no	—
538	C₄⋊C₄	—	3	1	no	—
539	C₄⋊C₄	—	3	1	no	—
540	C₄⋊C₄	—	3	1	no	—
541	C₄⋊C₄	—	3	1	no	—
542	C₄⋊C₄	—	3	1	no	—
543	C₄⋊C₄	—	6	1	no	—
544	C₄⋊C₄	—	6	1	no	—
545	C₄⋊C₄	—	6	1	no	—
546	C₄⋊C₄	—	6	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	C₂²×C₅×D₃	1	1	yes	characteristic
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	1	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	—	6	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	—	6	1	no	—
564	C₄²	—	6	1	no	—
565	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
566	C₃×C₅	#3360	1	1	yes	characteristic, complemented, Hall
567	C₂²×C₃	—	2	2	no	—
568	C₂²×C₃	—	4	1	no	—
569	C₂²×C₃	—	4	1	no	—
570	C₂²×C₃	—	4	1	no	—
571	C₂²×C₃	—	4	1	no	—
572	C₂²×C₃	—	4	1	no	—
573	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
574	C₃×C₄	—	2	1	no	—
575	C₃×C₄	—	2	1	no	—
576	C₃×C₄	—	4	1	no	—
577	C₃×C₄	—	4	1	no	—
578	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
579	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
580	Dic₃	—	2	1	no	—
581	Dic₃	—	2	1	no	—
582	Dic₃	—	2	1	no	—
583	Dic₃	—	4	1	no	—
584	Dic₃	—	4	1	no	—
585	Dic₃	—	4	1	no	—
586	Dic₃	—	4	1	no	—
587	Dic₃	C₅×#135	1	2	yes	—
588	C₂×C₅	—	4	2	no	—
589	C₂×C₅	—	8	1	no	—
590	C₂×C₅	#5598	1	1	yes	characteristic
591	C₂×C₅	D₃×#135	1	1	yes	characteristic
592	C₂×C₅	#5775	1	1	yes	characteristic
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	Q₈	—	2	1	no	—
596	Q₈	—	2	1	no	—
597	Q₈	—	6	2	no	—
598	Q₈	—	6	1	no	—
599	Q₈	—	6	1	no	—
600	Q₈	—	24	1	no	—
601	D₄	—	2	1	no	—
602	D₄	—	2	1	no	—
603	D₄	—	2	1	no	—
604	D₄	C₅×#237	1	2	yes	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	3	1	no	—
608	C₂×C₄	—	3	1	no	—
609	C₂×C₄	—	3	1	no	—
610	C₂×C₄	—	3	1	no	—
611	C₂×C₄	—	6	1	no	—
612	C₂×C₄	—	6	1	no	—
613	C₂×C₄	—	6	1	no	—
614	C₂×C₄	—	6	1	no	—
615	C₂×C₄	—	8	1	no	—
616	C₂×C₄	—	12	2	no	—
617	C₂×C₄	—	12	2	no	—
618	C₂×C₄	—	24	1	no	—
619	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
620	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
621	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
622	C₈	—	2	1	no	—
623	C₈	—	2	1	no	—
624	C₈	—	6	1	no	—
625	C₈	—	6	1	no	—
626	C₂×C₃	—	4	2	no	—
627	C₂×C₃	—	8	1	no	—
628	C₂×C₃	C₅×#548	1	1	yes	characteristic
629	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
630	C₂×C₃	C₅×#577	1	1	yes	characteristic
631	C₅	#80516	1	1	yes	characteristic, direct product factor, Sylow
632	C₂²	—	2	2	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
639	C₄	—	2	1	no	—
640	C₄	—	2	1	no	—
641	C₄	—	3	2	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	1	no	—
644	C₄	—	6	1	no	—
645	C₄	—	6	1	no	—
646	C₄	—	6	1	no	—
647	C₄	—	12	1	no	—
648	C₄	—	12	1	no	—
649	C₄	—	12	1	no	—
650	C₄	—	12	1	no	—
651	C₄	C₂×C₅×#237	1	1	yes	characteristic
652	C₄	C₅×#1010	1	1	yes	characteristic
653	C₃	C₅×#3360	1	1	yes	characteristic, complemented, Sylow
654	C₂	—	4	2	no	—
655	C₂	—	8	1	no	—
656	C₂	C₅×#5598	1	1	yes	characteristic
657	C₂	C₅×D₃×#135	1	1	yes	characteristic
658	C₂	C₅×#5775	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×#463	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×C₅×D₈	C₂²
hypercenter	C₅×#463	D₃
hypocenter	C₃	C₅×#3360
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#463

Complements

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

N	H ≅ G/N	type
1. C₅×#5671	C₂	semidirect
2. C₅×#5539	C₂	semidirect
4. C₅×#5162	C₂	semidirect
5. C₅×Dic₃×SD₁₆	C₂	semidirect
8. C₅×#5056	C₂	semidirect
9. C₄×C₅×#214	C₂	semidirect
10. C₅×#4830	C₂	semidirect
11. C₅×#4829	C₂	semidirect
12. C₅×#4796	C₂	semidirect
13. C₅×#4792	C₂	semidirect
14. C₅×#4768	C₂	semidirect
15. C₅×#4761	C₂	semidirect
60. #80516	C₅	direct (N × H)
122. #5671	C₂×C₅	semidirect
123. #5539	C₂×C₅	semidirect
125. #5162	C₂×C₅	semidirect
126. Dic₃×SD₁₆	C₂×C₅	semidirect
129. #5056	C₂×C₅	semidirect
130. C₄×#214	C₂×C₅	semidirect
131. #4830	C₂×C₅	semidirect
132. #4829	C₂×C₅	semidirect
133. #4796	C₂×C₅	semidirect
134. #4792	C₂×C₅	semidirect
135. #4768	C₂×C₅	semidirect
136. #4761	C₂×C₅	semidirect
566. C₃×C₅	#3360	semidirect
631. C₅	#80516	direct (N × H)
653. C₃	C₅×#3360	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5671	2
C₅×#5539	2
C₃×C₅×#463	2
C₅×#5162	2
C₅×Dic₃×SD₁₆	2
C₅×#5153	2
C₅×Dic₃×D₈	2
C₅×#5056	2
C₄×C₅×#214	2
C₅×#4830	2
C₅×#4829	2
C₅×#4796	2
C₅×#4792	2
C₅×#4768	2
C₅×#4761	2
C₅×#3360	3
#80516	5

Subgroups — C5×#80516 #14471582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#80516 #14471582