Subgroups — C₅×#17547 #11940267

Order 1280.

1166 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3519	C₂	1	1	yes	maximal, characteristic
2	C₅×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1507	C₂	1	2	yes	maximal, complemented
6	C₅×#1505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1504	C₂	1	2	yes	maximal, complemented
8	C₅×#1502	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1501	C₂	1	2	yes	maximal, complemented
10	C₅×#1500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#327	C₂	1	2	yes	maximal, complemented
12	C₅×#555	C₄	1	1	yes	characteristic, complemented
13	C₅×#554	C₄	1	1	yes	characteristic, complemented
14	C₅×#553	—	2	1	no	—
15	C₅×#553	C₄	1	2	yes	complemented
16	C₅×#550	C₄	1	1	yes	characteristic, complemented
17	C₅×#546	—	2	2	no	—
18	C₅×#546	C₄	1	2	yes	complemented
19	C₅×D₄²	—	2	1	no	—
20	C₅×D₄²	C₄	1	1	yes	characteristic, complemented
21	C₅×#540	—	2	1	no	—
22	C₅×#538	—	2	2	no	—
23	C₅×#532	C₂²	1	1	yes	characteristic
24	C₅×#529	C₂²	1	1	yes	characteristic
25	C₅×#525	C₂²	1	2	yes	—
26	C₂×C₅×#121	C₂²	1	2	yes	—
27	C₅×#518	—	2	1	no	—
28	C₂×C₄×C₅×D₄	—	2	1	no	—
29	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
30	C₅×#436	—	2	1	no	—
31	C₅×#435	—	2	2	no	—
32	C₅×C₈×D₄	—	2	1	no	—
33	C₅×#432	C₂²	1	1	yes	characteristic, complemented
34	C₅×#431	C₂²	1	1	yes	characteristic, complemented
35	C₅×#407	C₂²	1	2	yes	complemented
36	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
37	C₄×C₅×C₄⋊C₄	—	2	1	no	—
38	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
41	C₅×#376	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₅×#95	C₂²	1	2	yes	complemented
43	C₅×#328	C₂²	1	1	yes	characteristic, complemented
44	C₅×#328	C₂²	1	1	yes	characteristic, complemented
45	C₅×#328	C₂²	1	2	yes	complemented
46	C₅×#327	C₂²	1	1	yes	characteristic
47	C₅×#327	C₂²	1	1	yes	characteristic
48	C₅×#327	C₂²	1	2	yes	—
49	C₅×#327	C₂²	1	2	yes	complemented
50	C₅×#327	C₂²	1	2	yes	complemented
51	C₅×#327	C₂²	1	2	yes	complemented
52	C₅×#327	C₂²	1	2	yes	complemented
53	#17547	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₅×#41	—	2	1	no	—
55	C₂×C₅×#41	C₂×C₄	1	2	yes	complemented
56	C₂²×C₅×D₄	—	2	2	no	—
57	C₂²×C₅×D₄	—	2	1	no	—
58	C₂²×C₅×D₄	C₂×C₄	1	2	yes	complemented
59	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₅	—	2	2	no	—
61	C₂³×C₄×C₅	C₂³	1	2	yes	—
62	C₂×C₅×M₁₆	—	2	2	no	—
63	C₂²×C₅×C₈	—	2	2	no	—
64	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
65	C₅×#127	—	2	1	no	—
66	C₅×#127	C₂×C₄	1	1	yes	characteristic, complemented
67	C₅×#126	—	2	2	no	—
68	C₅×#126	—	2	1	no	—
69	C₅×#125	—	2	1	no	—
70	C₅×#125	C₂×C₄	1	1	yes	characteristic, complemented
71	C₅×#124	—	2	2	no	—
72	C₅×#124	—	2	1	no	—
73	C₅×#124	C₂×C₄	1	1	yes	characteristic, complemented
74	C₅×#123	—	2	2	no	—
75	C₅×#123	—	2	1	no	—
76	C₅×#123	—	2	2	no	—
77	C₅×#123	—	2	2	no	—
78	C₅×C₂²⋊Q₈	—	2	2	no	—
79	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
80	C₅×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
81	C₅×#121	—	2	2	no	—
82	C₅×#121	—	2	1	no	—
83	C₅×#121	—	2	2	no	—
84	C₅×#121	—	2	2	no	—
85	C₅×#121	—	2	1	no	—
86	C₅×#121	—	2	1	no	—
87	C₅×#121	—	2	2	no	—
88	C₅×#121	—	2	2	no	—
89	C₅×#121	C₂×C₄	1	2	yes	complemented
90	C₅×#121	C₂×C₄	1	2	yes	complemented
91	C₅×#121	C₂×C₄	1	2	yes	complemented
92	C₅×#121	C₂³	1	1	yes	characteristic
93	C₅×#121	C₂³	1	2	yes	—
94	C₅×#120	—	2	2	no	—
95	C₅×#120	—	2	2	no	—
96	C₅×#120	—	2	2	no	—
97	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
98	C₄×C₅×D₄	—	2	8	no	—
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	—	2	2	no	—
101	C₄×C₅×D₄	—	2	2	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	—	2	2	no	—
105	C₄×C₅×D₄	—	2	1	no	—
106	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
107	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
108	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
109	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic, complemented
110	C₄×C₅×D₄	C₂×C₄	1	2	yes	complemented
111	C₄×C₅×D₄	D₄	1	1	yes	characteristic
112	C₄×C₅×D₄	D₄	1	2	yes	—
113	C₄×C₅×D₄	D₄	1	1	yes	characteristic
114	C₅×#117	—	2	1	no	—
115	C₅×#117	—	2	1	no	—
116	C₅×#117	C₂×C₄	1	2	yes	complemented
117	C₂×C₅×C₄⋊C₄	—	2	1	no	—
118	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
119	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
120	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
121	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
122	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
123	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
124	C₂×C₄²×C₅	—	2	1	no	—
125	C₂×C₄²×C₅	—	2	1	no	—
126	C₂×C₄²×C₅	—	2	2	no	—
127	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
128	C₅×C₄⋊C₈	—	2	1	no	—
129	C₅×C₄⋊C₈	—	2	1	no	—
130	C₅×C₂²⋊C₈	—	2	2	no	—
131	C₅×C₂²⋊C₈	—	2	2	no	—
132	C₅×C₂²⋊C₈	C₂³	1	1	yes	characteristic
133	C₅×C₂²⋊C₈	C₂³	1	2	yes	—
134	C₅×C₂²⋊C₈	C₂³	1	1	yes	characteristic
135	C₅×#97	—	2	1	no	—
136	C₄×C₅×C₈	—	2	1	no	—
137	C₅×#95	—	2	2	no	—
138	C₅×#95	—	2	1	no	—
139	C₅×#95	C₂³	1	1	yes	characteristic
140	C₅×#95	C₂³	1	2	yes	—
141	C₅×#95	C₂³	1	1	yes	characteristic
142	#3519	C₂×C₅	1	1	yes	characteristic
143	#2974	C₂×C₅	1	1	yes	characteristic, complemented
144	#2275	C₂×C₅	1	1	yes	characteristic, complemented
145	#1509	C₂×C₅	1	1	yes	characteristic, complemented
146	#1507	C₂×C₅	1	2	yes	complemented
147	#1505	C₂×C₅	1	1	yes	characteristic, complemented
148	#1504	C₂×C₅	1	2	yes	complemented
149	#1502	C₂×C₅	1	1	yes	characteristic, complemented
150	#1501	C₂×C₅	1	2	yes	complemented
151	#1500	C₂×C₅	1	1	yes	characteristic, complemented
152	C₂×#327	C₂×C₅	1	2	yes	complemented
153	C₂⁴×C₅	—	2	2	no	—
154	C₂⁴×C₅	C₂×D₄	1	2	yes	—
155	C₅×#41	—	8	1	no	—
156	C₅×#41	—	8	1	no	—
157	C₅×#41	—	8	2	no	—
158	C₂×C₅×Q₈	—	2	1	no	—
159	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₅×D₄	—	2	4	no	—
161	C₂×C₅×D₄	—	2	2	no	—
162	C₂×C₅×D₄	—	2	2	no	—
163	C₂×C₅×D₄	—	2	1	no	—
164	C₂×C₅×D₄	—	2	2	no	—
165	C₂×C₅×D₄	—	2	2	no	—
166	C₂×C₅×D₄	—	2	2	no	—
167	C₂×C₅×D₄	—	2	1	no	—
168	C₂×C₅×D₄	—	2	1	no	—
169	C₂×C₅×D₄	—	2	2	no	—
170	C₂×C₅×D₄	—	4	4	no	—
171	C₂×C₅×D₄	—	8	2	no	—
172	C₂×C₅×D₄	—	8	2	no	—
173	C₂×C₅×D₄	—	8	2	no	—
174	C₂×C₅×D₄	C₂²⋊C₄	1	2	yes	—
175	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
179	C₂²×C₄×C₅	—	2	1	no	—
180	C₂²×C₄×C₅	—	2	2	no	—
181	C₂²×C₄×C₅	—	2	1	no	—
182	C₂²×C₄×C₅	—	2	8	no	—
183	C₂²×C₄×C₅	—	2	2	no	—
184	C₂²×C₄×C₅	—	2	1	no	—
185	C₂²×C₄×C₅	—	2	2	no	—
186	C₂²×C₄×C₅	—	2	2	no	—
187	C₂²×C₄×C₅	—	8	2	no	—
188	C₂²×C₄×C₅	—	8	2	no	—
189	C₂²×C₄×C₅	—	8	2	no	—
190	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
191	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
192	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
198	C₅×M₁₆	—	8	2	no	—
199	C₂×C₅×C₈	—	2	1	no	—
200	C₂×C₅×C₈	—	2	2	no	—
201	C₂×C₅×C₈	—	2	1	no	—
202	C₂×C₅×C₈	—	8	2	no	—
203	C₅×C₄⋊C₄	—	2	4	no	—
204	C₅×C₄⋊C₄	—	2	2	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	2	1	no	—
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	2	2	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
212	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
213	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
214	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
215	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
216	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
217	C₅×C₂²⋊C₄	—	2	8	no	—
218	C₅×C₂²⋊C₄	—	2	2	no	—
219	C₅×C₂²⋊C₄	—	2	2	no	—
220	C₅×C₂²⋊C₄	—	2	1	no	—
221	C₅×C₂²⋊C₄	—	2	1	no	—
222	C₅×C₂²⋊C₄	—	2	2	no	—
223	C₅×C₂²⋊C₄	—	2	2	no	—
224	C₅×C₂²⋊C₄	—	2	1	no	—
225	C₅×C₂²⋊C₄	—	2	2	no	—
226	C₅×C₂²⋊C₄	—	2	1	no	—
227	C₅×C₂²⋊C₄	—	2	2	no	—
228	C₅×C₂²⋊C₄	—	2	1	no	—
229	C₅×C₂²⋊C₄	—	2	2	no	—
230	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
231	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
232	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
233	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
234	C₄²×C₅	—	2	4	no	—
235	C₄²×C₅	—	2	2	no	—
236	C₄²×C₅	—	2	1	no	—
237	C₄²×C₅	—	8	1	no	—
238	C₄²×C₅	—	8	2	no	—
239	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
240	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
241	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
242	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
243	#555	C₄×C₅	1	1	yes	characteristic, complemented
244	#554	C₄×C₅	1	1	yes	characteristic, complemented
245	#553	—	2	1	no	—
246	#553	C₄×C₅	1	2	yes	complemented
247	#550	C₄×C₅	1	1	yes	characteristic, complemented
248	#546	—	2	2	no	—
249	#546	C₄×C₅	1	2	yes	complemented
250	D₄²	—	2	1	no	—
251	D₄²	C₄×C₅	1	1	yes	characteristic, complemented
252	#540	—	2	1	no	—
253	#538	—	2	2	no	—
254	#532	C₂²×C₅	1	1	yes	characteristic
255	#529	C₂²×C₅	1	1	yes	characteristic
256	#525	C₂²×C₅	1	2	yes	—
257	C₂×#121	C₂²×C₅	1	2	yes	—
258	#518	—	2	1	no	—
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
261	#436	—	2	1	no	—
262	#435	—	2	2	no	—
263	C₈×D₄	—	2	1	no	—
264	#432	C₂²×C₅	1	1	yes	characteristic, complemented
265	#431	C₂²×C₅	1	1	yes	characteristic, complemented
266	#407	C₂²×C₅	1	2	yes	complemented
267	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
268	C₄×C₄⋊C₄	—	2	1	no	—
269	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
270	C₄×C₂²⋊C₄	—	2	2	no	—
271	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
272	#376	C₂²×C₅	1	1	yes	characteristic, complemented
273	C₂×#95	C₂²×C₅	1	2	yes	complemented
274	#328	C₂²×C₅	1	1	yes	characteristic, complemented
275	#328	C₂²×C₅	1	1	yes	characteristic, complemented
276	#328	C₂²×C₅	1	2	yes	complemented
277	#327	C₂²×C₅	1	1	yes	characteristic
278	#327	C₂²×C₅	1	2	yes	—
279	#327	C₂²×C₅	1	1	yes	characteristic
280	#327	C₂²×C₅	1	2	yes	complemented
281	#327	C₂²×C₅	1	2	yes	complemented
282	#327	C₂²×C₅	1	2	yes	complemented
283	#327	C₂²×C₅	1	2	yes	complemented
284	C₂³×C₅	—	2	2	no	—
285	C₂³×C₅	—	2	2	no	—
286	C₂³×C₅	—	2	4	no	—
287	C₂³×C₅	—	2	1	no	—
288	C₂³×C₅	—	2	2	no	—
289	C₂³×C₅	—	4	2	no	—
290	C₂³×C₅	—	4	4	no	—
291	C₂³×C₅	—	8	2	no	—
292	C₂³×C₅	—	8	2	no	—
293	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
294	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
295	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
296	C₅×Q₈	—	4	1	no	—
297	C₅×Q₈	—	8	1	no	—
298	C₅×D₄	—	2	4	no	—
299	C₅×D₄	—	2	2	no	—
300	C₅×D₄	—	4	1	no	—
301	C₅×D₄	—	4	2	no	—
302	C₅×D₄	—	4	1	no	—
303	C₅×D₄	—	4	2	no	—
304	C₅×D₄	—	4	1	no	—
305	C₅×D₄	—	4	2	no	—
306	C₅×D₄	—	4	4	no	—
307	C₅×D₄	—	4	2	no	—
308	C₅×D₄	—	8	2	no	—
309	C₅×D₄	—	8	1	no	—
310	C₅×D₄	—	8	2	no	—
311	C₅×D₄	—	8	2	no	—
312	C₅×D₄	—	8	2	no	—
313	C₅×D₄	#104	1	4	yes	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	2	1	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	2	no	—
318	C₂×C₄×C₅	—	2	2	no	—
319	C₂×C₄×C₅	—	2	4	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	2	1	no	—
324	C₂×C₄×C₅	—	4	2	no	—
325	C₂×C₄×C₅	—	4	2	no	—
326	C₂×C₄×C₅	—	4	2	no	—
327	C₂×C₄×C₅	—	4	1	no	—
328	C₂×C₄×C₅	—	4	2	no	—
329	C₂×C₄×C₅	—	4	2	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	2	no	—
332	C₂×C₄×C₅	—	4	1	no	—
333	C₂×C₄×C₅	—	4	2	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	4	2	no	—
337	C₂×C₄×C₅	—	8	2	no	—
338	C₂×C₄×C₅	—	8	2	no	—
339	C₂×C₄×C₅	—	8	2	no	—
340	C₂×C₄×C₅	—	8	8	no	—
341	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
342	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
343	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
344	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
345	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
346	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
347	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
348	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
349	C₅×C₈	—	4	1	no	—
350	C₅×C₈	—	4	2	no	—
351	C₅×C₈	—	4	1	no	—
352	C₂×#41	—	2	1	no	—
353	C₂×#41	C₂×C₄×C₅	1	2	yes	complemented
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	C₂×C₄×C₅	1	2	yes	complemented
357	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
358	C₂³×C₄	—	2	2	no	—
359	C₂³×C₄	C₂³×C₅	1	2	yes	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂²×C₈	—	2	2	no	—
362	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
363	#127	—	2	1	no	—
364	#127	C₂×C₄×C₅	1	1	yes	characteristic, complemented
365	#126	—	2	2	no	—
366	#126	—	2	1	no	—
367	#125	—	2	1	no	—
368	#125	C₂×C₄×C₅	1	1	yes	characteristic, complemented
369	#124	—	2	2	no	—
370	#124	—	2	1	no	—
371	#124	C₂×C₄×C₅	1	1	yes	characteristic, complemented
372	#123	—	2	2	no	—
373	#123	—	2	1	no	—
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	complemented
378	C₂²⋊Q₈	C₂³×C₅	1	1	yes	characteristic
379	#121	—	2	2	no	—
380	#121	—	2	1	no	—
381	#121	—	2	2	no	—
382	#121	—	2	2	no	—
383	#121	—	2	1	no	—
384	#121	—	2	1	no	—
385	#121	—	2	2	no	—
386	#121	—	2	2	no	—
387	#121	C₂×C₄×C₅	1	2	yes	complemented
388	#121	C₂×C₄×C₅	1	2	yes	complemented
389	#121	C₂×C₄×C₅	1	2	yes	complemented
390	#121	C₂³×C₅	1	1	yes	characteristic
391	#121	C₂³×C₅	1	2	yes	—
392	#120	—	2	2	no	—
393	#120	—	2	2	no	—
394	#120	—	2	2	no	—
395	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
405	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
406	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
407	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic, complemented
408	C₄×D₄	C₂×C₄×C₅	1	2	yes	complemented
409	C₄×D₄	C₅×D₄	1	1	yes	characteristic
410	C₄×D₄	C₅×D₄	1	2	yes	—
411	C₄×D₄	C₅×D₄	1	1	yes	characteristic
412	#117	—	2	1	no	—
413	#117	—	2	1	no	—
414	#117	C₂×C₄×C₅	1	2	yes	complemented
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	complemented
417	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
418	C₂×C₂²⋊C₄	—	2	2	no	—
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
422	C₂×C₄²	—	2	1	no	—
423	C₂×C₄²	—	2	1	no	—
424	C₂×C₄²	—	2	2	no	—
425	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
426	C₄⋊C₈	—	2	1	no	—
427	C₄⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	2	no	—
429	C₂²⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	C₂³×C₅	1	1	yes	characteristic
431	C₂²⋊C₈	C₂³×C₅	1	2	yes	—
432	C₂²⋊C₈	C₂³×C₅	1	1	yes	characteristic
433	#97	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	#95	—	2	2	no	—
436	#95	—	2	1	no	—
437	#95	C₂³×C₅	1	1	yes	characteristic
438	#95	C₂³×C₅	1	2	yes	—
439	#95	C₂³×C₅	1	1	yes	characteristic
440	C₂²×C₅	—	2	1	no	—
441	C₂²×C₅	—	2	2	no	—
442	C₂²×C₅	—	2	2	no	—
443	C₂²×C₅	—	2	1	no	—
444	C₂²×C₅	—	2	2	no	—
445	C₂²×C₅	—	2	2	no	—
446	C₂²×C₅	—	2	2	no	—
447	C₂²×C₅	—	4	4	no	—
448	C₂²×C₅	—	4	4	no	—
449	C₂²×C₅	—	4	2	no	—
450	C₂²×C₅	—	4	1	no	—
451	C₂²×C₅	—	4	2	no	—
452	C₂²×C₅	—	4	1	no	—
453	C₂²×C₅	—	4	1	no	—
454	C₂²×C₅	—	4	2	no	—
455	C₂²×C₅	—	16	4	no	—
456	C₂²×C₅	C₂×#104	1	4	yes	—
457	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
458	C₂²×C₅	#576	1	1	yes	characteristic
459	C₂²×C₅	#577	1	1	yes	characteristic
460	C₄×C₅	—	2	1	no	—
461	C₄×C₅	—	2	1	no	—
462	C₄×C₅	—	2	2	no	—
463	C₄×C₅	—	2	2	no	—
464	C₄×C₅	—	2	1	no	—
465	C₄×C₅	—	2	1	no	—
466	C₄×C₅	—	2	1	no	—
467	C₄×C₅	—	4	2	no	—
468	C₄×C₅	—	4	4	no	—
469	C₄×C₅	—	4	2	no	—
470	C₄×C₅	—	4	1	no	—
471	C₄×C₅	—	4	1	no	—
472	C₄×C₅	—	4	2	no	—
473	C₄×C₅	—	4	1	no	—
474	C₄×C₅	C₂×#104	1	2	yes	—
475	C₂⁴	—	2	2	no	—
476	C₂⁴	C₂×C₅×D₄	1	2	yes	—
477	#41	—	8	1	no	—
478	#41	—	8	1	no	—
479	#41	—	8	2	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	2	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	2	no	—
487	C₂×D₄	—	2	2	no	—
488	C₂×D₄	—	2	2	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	4	4	no	—
493	C₂×D₄	—	8	2	no	—
494	C₂×D₄	—	8	2	no	—
495	C₂×D₄	—	8	2	no	—
496	C₂×D₄	C₅×C₂²⋊C₄	1	2	yes	—
497	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
498	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
499	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
500	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	8	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	2	no	—
509	C₂²×C₄	—	8	2	no	—
510	C₂²×C₄	—	8	2	no	—
511	C₂²×C₄	—	8	2	no	—
512	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
513	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
514	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
515	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
516	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
517	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
518	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
520	M₁₆	—	8	2	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	2	no	—
524	C₂×C₈	—	8	2	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
534	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
535	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
536	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
537	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
538	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	1	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	1	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
553	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
554	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
555	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	8	1	no	—
560	C₄²	—	8	2	no	—
561	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
562	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
563	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
564	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
565	C₂×C₅	—	2	1	no	—
566	C₂×C₅	—	2	1	no	—
567	C₂×C₅	—	2	4	no	—
568	C₂×C₅	—	4	2	no	—
569	C₂×C₅	—	8	2	no	—
570	C₂×C₅	—	8	1	no	—
571	C₂×C₅	#2883	1	1	yes	characteristic
572	C₂×C₅	#2897	1	1	yes	characteristic
573	C₂×C₅	C₂²×#104	1	1	yes	characteristic
574	C₂³	—	2	4	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	1	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	4	2	no	—
580	C₂³	—	4	4	no	—
581	C₂³	—	8	2	no	—
582	C₂³	—	8	2	no	—
583	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
584	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
585	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
586	Q₈	—	4	1	no	—
587	Q₈	—	8	1	no	—
588	D₄	—	2	4	no	—
589	D₄	—	2	2	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	2	no	—
592	D₄	—	4	4	no	—
593	D₄	—	4	1	no	—
594	D₄	—	4	2	no	—
595	D₄	—	4	1	no	—
596	D₄	—	4	2	no	—
597	D₄	—	4	2	no	—
598	D₄	—	8	2	no	—
599	D₄	—	8	1	no	—
600	D₄	—	8	2	no	—
601	D₄	—	8	2	no	—
602	D₄	—	8	2	no	—
603	D₄	C₅×#104	1	4	yes	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	8	2	no	—
628	C₂×C₄	—	8	2	no	—
629	C₂×C₄	—	8	2	no	—
630	C₂×C₄	—	8	8	no	—
631	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
632	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
633	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
634	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
635	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
636	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
637	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
638	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
639	C₈	—	4	1	no	—
640	C₈	—	4	2	no	—
641	C₈	—	4	1	no	—
642	C₅	#17547	1	1	yes	characteristic, direct product factor, Sylow
643	C₂²	—	2	1	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	1	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	2	no	—
650	C₂²	—	4	4	no	—
651	C₂²	—	4	4	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	1	no	—
656	C₂²	—	4	1	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	16	4	no	—
659	C₂²	C₂×C₅×#104	1	4	yes	—
660	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
661	C₂²	C₅×#576	1	1	yes	characteristic
662	C₂²	C₅×#577	1	1	yes	characteristic
663	C₄	—	2	1	no	—
664	C₄	—	2	1	no	—
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	1	no	—
668	C₄	—	2	1	no	—
669	C₄	—	2	1	no	—
670	C₄	—	4	2	no	—
671	C₄	—	4	4	no	—
672	C₄	—	4	2	no	—
673	C₄	—	4	1	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	1	no	—
677	C₄	C₂×C₅×#104	1	2	yes	—
678	C₂	—	2	1	no	—
679	C₂	—	2	1	no	—
680	C₂	—	2	4	no	—
681	C₂	—	4	2	no	—
682	C₂	—	8	1	no	—
683	C₂	—	8	2	no	—
684	C₂	C₅×#2883	1	1	yes	characteristic
685	C₂	C₅×#2897	1	1	yes	characteristic
686	C₂	C₂²×C₅×#104	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#3519	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2974	C₂	semidirect
3. C₅×#2275	C₂	semidirect
4. C₅×#1509	C₂	semidirect
5. C₅×#1507	C₂	semidirect
6. C₅×#1505	C₂	semidirect
7. C₅×#1504	C₂	semidirect
8. C₅×#1502	C₂	semidirect
9. C₅×#1501	C₂	semidirect
10. C₅×#1500	C₂	semidirect
11. C₂×C₅×#327	C₂	semidirect
12. C₅×#555	C₄	semidirect
13. C₅×#554	C₄	semidirect
15. C₅×#553	C₄	semidirect
16. C₅×#550	C₄	semidirect
18. C₅×#546	C₄	semidirect
20. C₅×D₄²	C₄	semidirect
33. C₅×#432	C₂²	semidirect
34. C₅×#431	C₂²	semidirect
35. C₅×#407	C₂²	semidirect
36. C₂×C₅×C₂²⋊C₈	C₂²	semidirect
38. C₄×C₅×C₄⋊C₄	C₂²	semidirect
40. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
41. C₅×#376	C₂²	semidirect
42. C₂×C₅×#95	C₂²	semidirect
43. C₅×#328	C₂²	semidirect
44. C₅×#328	C₂²	semidirect
45. C₅×#328	C₂²	semidirect
49. C₅×#327	C₂²	semidirect
50. C₅×#327	C₂²	semidirect
51. C₅×#327	C₂²	semidirect
52. C₅×#327	C₂²	semidirect
53. #17547	C₅	direct (N × H)
55. C₂×C₅×#41	C₂×C₄	semidirect
58. C₂²×C₅×D₄	C₂×C₄	semidirect
64. C₅×C₄⋊Q₈	C₂×C₄	semidirect
66. C₅×#127	C₂×C₄	semidirect
70. C₅×#125	C₂×C₄	semidirect
73. C₅×#124	C₂×C₄	semidirect
79. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
89. C₅×#121	C₂×C₄	semidirect
90. C₅×#121	C₂×C₄	semidirect
91. C₅×#121	C₂×C₄	semidirect
97. C₄×C₅×Q₈	C₂×C₄	semidirect
109. C₄×C₅×D₄	C₂×C₄	semidirect
110. C₄×C₅×D₄	C₂×C₄	semidirect
116. C₅×#117	C₂×C₄	semidirect
118. C₂×C₅×C₄⋊C₄	C₂×C₄	semidirect
143. #2974	C₂×C₅	semidirect
144. #2275	C₂×C₅	semidirect
145. #1509	C₂×C₅	semidirect
146. #1507	C₂×C₅	semidirect
147. #1505	C₂×C₅	semidirect
148. #1504	C₂×C₅	semidirect
149. #1502	C₂×C₅	semidirect
150. #1501	C₂×C₅	semidirect
151. #1500	C₂×C₅	semidirect
152. C₂×#327	C₂×C₅	semidirect
243. #555	C₄×C₅	semidirect
244. #554	C₄×C₅	semidirect
246. #553	C₄×C₅	semidirect
247. #550	C₄×C₅	semidirect
249. #546	C₄×C₅	semidirect
251. D₄²	C₄×C₅	semidirect
264. #432	C₂²×C₅	semidirect
265. #431	C₂²×C₅	semidirect
266. #407	C₂²×C₅	semidirect
267. C₂×C₂²⋊C₈	C₂²×C₅	semidirect
269. C₄×C₄⋊C₄	C₂²×C₅	semidirect
271. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
272. #376	C₂²×C₅	semidirect
273. C₂×#95	C₂²×C₅	semidirect
274. #328	C₂²×C₅	semidirect
275. #328	C₂²×C₅	semidirect
276. #328	C₂²×C₅	semidirect
280. #327	C₂²×C₅	semidirect
281. #327	C₂²×C₅	semidirect
282. #327	C₂²×C₅	semidirect
283. #327	C₂²×C₅	semidirect
353. C₂×#41	C₂×C₄×C₅	semidirect
356. C₂²×D₄	C₂×C₄×C₅	semidirect
362. C₄⋊Q₈	C₂×C₄×C₅	semidirect
364. #127	C₂×C₄×C₅	semidirect
368. #125	C₂×C₄×C₅	semidirect
371. #124	C₂×C₄×C₅	semidirect
377. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
387. #121	C₂×C₄×C₅	semidirect
388. #121	C₂×C₄×C₅	semidirect
389. #121	C₂×C₄×C₅	semidirect
395. C₄×Q₈	C₂×C₄×C₅	semidirect
407. C₄×D₄	C₂×C₄×C₅	semidirect
408. C₄×D₄	C₂×C₄×C₅	semidirect
414. #117	C₂×C₄×C₅	semidirect
416. C₂×C₄⋊C₄	C₂×C₄×C₅	semidirect
642. C₅	#17547	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×#3519	2
C₅×#2974	2
C₅×#2275	2
C₅×#1509	2
C₅×#1507	2
C₅×#1505	2
C₅×#1504	2
C₅×#1502	2
C₅×#1501	2
C₅×#1500	2
C₂×C₅×#327	2
#17547	5

Subgroups — C5×#17547 #11940267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17547 #11940267