Subgroups — C₅×#19278 #11941998

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3116	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3115	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1684	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1680	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1679	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1678	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1677	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1554	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1553	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#533	C₂²	1	1	yes	characteristic
17	C₅×#532	C₂²	1	1	yes	characteristic
18	C₄×C₅×#41	C₂²	1	1	yes	characteristic
19	C₅×#488	—	2	1	no	—
20	C₅×#488	—	2	1	no	—
21	C₅×#487	C₂²	1	1	yes	characteristic
22	C₅×#487	C₂²	1	1	yes	characteristic
23	C₅×#486	C₂²	1	1	yes	characteristic
24	C₅×#486	C₂²	1	1	yes	characteristic
25	C₅×#485	—	2	1	no	—
26	C₅×#482	—	2	1	no	—
27	C₅×#477	C₂²	1	1	yes	characteristic
28	C₅×#476	C₂²	1	1	yes	characteristic
29	C₅×#475	C₂²	1	1	yes	characteristic
30	C₅×#474	C₂²	1	1	yes	characteristic
31	C₅×#462	C₂²	1	1	yes	characteristic
32	C₅×#461	C₂²	1	1	yes	characteristic
33	C₅×#460	C₂²	1	1	yes	characteristic
34	C₅×#459	C₂²	1	1	yes	characteristic
35	C₅×#452	—	2	1	no	—
36	C₅×#449	—	2	1	no	—
37	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
38	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
39	C₅×#442	—	2	1	no	—
40	C₅×#441	—	2	1	no	—
41	C₅×#440	—	2	1	no	—
42	C₅×#440	—	2	1	no	—
43	C₅×C₈×D₄	—	2	1	no	—
44	C₅×C₈×D₄	—	2	1	no	—
45	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
46	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
47	C₅×#432	C₂²	1	1	yes	characteristic
48	C₅×#432	C₂²	1	1	yes	characteristic
49	C₅×#431	—	2	1	no	—
50	C₅×#431	C₂²	1	1	yes	characteristic
51	C₅×#416	—	2	1	no	—
52	C₅×#416	—	2	1	no	—
53	C₂×C₄×C₅×C₈	—	2	1	no	—
54	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
55	C₅×#335	C₂²	1	1	yes	characteristic
56	C₅×#335	C₂²	1	1	yes	characteristic
57	C₅×#334	C₂²	1	1	yes	characteristic
58	C₅×#334	C₂²	1	1	yes	characteristic
59	C₅×#333	C₂²	1	1	yes	characteristic
60	C₅×#333	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×#332	C₂²	1	1	yes	characteristic
63	C₅×#332	C₂²	1	1	yes	characteristic
64	C₅×#332	C₂²	1	1	yes	characteristic
65	C₅×#331	C₂²	1	1	yes	characteristic
66	C₅×#331	C₂²	1	1	yes	characteristic
67	#19278	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×#41	D₄	1	1	yes	characteristic
70	C₂×C₅×Q₁₆	—	2	1	no	—
71	C₂×C₅×SD₁₆	—	2	1	no	—
72	C₂×C₅×SD₁₆	—	2	1	no	—
73	C₂×C₅×D₈	—	2	1	no	—
74	C₂²×C₅×C₈	—	2	1	no	—
75	C₂²×C₅×C₈	—	2	1	no	—
76	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
77	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
78	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₅×#127	C₂³	1	1	yes	characteristic
82	C₅×#125	—	2	1	no	—
83	C₅×#124	—	2	1	no	—
84	C₅×C₂²⋊Q₈	—	2	1	no	—
85	C₅×C₂²⋊Q₈	—	2	1	no	—
86	C₅×#121	—	2	1	no	—
87	C₅×#121	—	2	1	no	—
88	C₄×C₅×Q₈	—	2	1	no	—
89	C₄×C₅×Q₈	—	2	1	no	—
90	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
91	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
92	C₄×C₅×D₄	—	2	1	no	—
93	C₄×C₅×D₄	—	2	1	no	—
94	C₄×C₅×D₄	—	2	1	no	—
95	C₄×C₅×D₄	—	2	1	no	—
96	C₄×C₅×D₄	D₄	1	1	yes	characteristic
97	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
98	C₅×#117	—	2	1	no	—
99	C₅×#117	—	2	1	no	—
100	C₅×#117	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₅	—	2	1	no	—
102	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
103	C₅×#107	—	2	1	no	—
104	C₅×#107	—	2	1	no	—
105	C₅×#106	—	2	1	no	—
106	C₅×#106	—	2	1	no	—
107	C₅×C₄⋊C₈	—	2	1	no	—
108	C₅×C₄⋊C₈	—	2	1	no	—
109	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₅×#103	—	2	1	no	—
116	C₅×#103	—	2	1	no	—
117	C₅×#103	—	2	1	no	—
118	C₅×#103	—	2	1	no	—
119	C₅×#103	—	2	1	no	—
120	C₅×#103	—	2	1	no	—
121	C₅×#103	—	2	1	no	—
122	C₅×#103	—	2	1	no	—
123	C₅×#102	—	2	1	no	—
124	C₅×#102	—	2	1	no	—
125	C₅×#102	—	2	1	no	—
126	C₅×#102	—	2	1	no	—
127	C₅×#102	—	2	1	no	—
128	C₅×#102	—	2	1	no	—
129	C₅×#102	—	2	1	no	—
130	C₅×#102	—	2	1	no	—
131	C₅×C₂²⋊C₈	—	2	1	no	—
132	C₅×C₂²⋊C₈	—	2	1	no	—
133	C₅×C₂²⋊C₈	—	2	1	no	—
134	C₅×C₂²⋊C₈	—	2	1	no	—
135	C₅×C₂²⋊C₈	—	2	1	no	—
136	C₅×#97	—	2	1	no	—
137	C₅×#97	—	2	1	no	—
138	C₄×C₅×C₈	—	2	1	no	—
139	C₄×C₅×C₈	—	2	1	no	—
140	C₄×C₅×C₈	—	2	1	no	—
141	C₄×C₅×C₈	D₄	1	1	yes	characteristic
142	C₄×C₅×C₈	D₄	1	1	yes	characteristic
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	#3131	C₂×C₅	1	1	yes	characteristic, complemented
146	#3116	C₂×C₅	1	1	yes	characteristic, complemented
147	#3115	C₂×C₅	1	1	yes	characteristic
148	C₈×#41	C₂×C₅	1	1	yes	characteristic, complemented
149	#1684	C₂×C₅	1	1	yes	characteristic, complemented
150	#1683	C₂×C₅	1	1	yes	characteristic, complemented
151	#1682	C₂×C₅	1	1	yes	characteristic, complemented
152	#1681	C₂×C₅	1	1	yes	characteristic, complemented
153	#1680	C₂×C₅	1	1	yes	characteristic, complemented
154	#1679	C₂×C₅	1	1	yes	characteristic, complemented
155	#1678	C₂×C₅	1	1	yes	characteristic, complemented
156	#1677	C₂×C₅	1	1	yes	characteristic, complemented
157	#1568	C₂×C₅	1	1	yes	characteristic, complemented
158	#1554	C₂×C₅	1	1	yes	characteristic, complemented
159	#1553	C₂×C₅	1	1	yes	characteristic, complemented
160	C₅×#41	—	2	2	no	—
161	C₅×#41	—	2	2	no	—
162	C₅×#41	—	4	2	no	—
163	C₅×#41	—	4	2	no	—
164	C₂×C₅×Q₈	—	2	1	no	—
165	C₂×C₅×Q₈	—	2	1	no	—
166	C₂×C₅×Q₈	—	2	1	no	—
167	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
168	C₂×C₅×D₄	—	2	1	no	—
169	C₂×C₅×D₄	—	2	1	no	—
170	C₂×C₅×D₄	—	2	1	no	—
171	C₂×C₅×D₄	—	2	1	no	—
172	C₂×C₅×D₄	—	4	1	no	—
173	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₅	—	2	1	no	—
175	C₂²×C₄×C₅	—	2	1	no	—
176	C₂²×C₄×C₅	—	2	1	no	—
177	C₂²×C₄×C₅	—	4	1	no	—
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅	#41	1	1	yes	characteristic
181	C₅×Q₁₆	—	4	2	no	—
182	C₅×SD₁₆	—	4	2	no	—
183	C₅×SD₁₆	—	4	2	no	—
184	C₅×D₈	—	4	2	no	—
185	C₂×C₅×C₈	—	2	1	no	—
186	C₂×C₅×C₈	—	2	1	no	—
187	C₂×C₅×C₈	—	2	1	no	—
188	C₂×C₅×C₈	—	2	1	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	4	1	no	—
192	C₂×C₅×C₈	—	4	1	no	—
193	C₂×C₅×C₈	—	4	2	no	—
194	C₂×C₅×C₈	—	4	2	no	—
195	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
196	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
199	C₅×C₄⋊C₄	—	2	1	no	—
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	—	2	1	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	2	1	no	—
207	C₅×C₄⋊C₄	—	4	1	no	—
208	C₅×C₄⋊C₄	—	4	1	no	—
209	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
210	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
211	C₅×C₂²⋊C₄	—	2	1	no	—
212	C₅×C₂²⋊C₄	—	2	1	no	—
213	C₅×C₂²⋊C₄	—	2	1	no	—
214	C₅×C₂²⋊C₄	—	4	1	no	—
215	C₅×C₂²⋊C₄	—	4	1	no	—
216	C₅×C₂²⋊C₄	—	4	1	no	—
217	C₄²×C₅	—	2	1	no	—
218	C₄²×C₅	—	2	1	no	—
219	C₄²×C₅	—	2	1	no	—
220	C₄²×C₅	—	4	1	no	—
221	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₅	#41	1	1	yes	characteristic
224	C₄²×C₅	C₂⁴	1	1	yes	characteristic
225	#533	C₂²×C₅	1	1	yes	characteristic
226	#532	C₂²×C₅	1	1	yes	characteristic
227	C₄×#41	C₂²×C₅	1	1	yes	characteristic
228	#488	—	2	1	no	—
229	#488	—	2	1	no	—
230	#487	C₂²×C₅	1	1	yes	characteristic
231	#487	C₂²×C₅	1	1	yes	characteristic
232	#486	C₂²×C₅	1	1	yes	characteristic
233	#486	C₂²×C₅	1	1	yes	characteristic
234	#485	—	2	1	no	—
235	#482	—	2	1	no	—
236	#477	C₂²×C₅	1	1	yes	characteristic
237	#476	C₂²×C₅	1	1	yes	characteristic
238	#475	C₂²×C₅	1	1	yes	characteristic
239	#474	C₂²×C₅	1	1	yes	characteristic
240	#462	C₂²×C₅	1	1	yes	characteristic
241	#461	C₂²×C₅	1	1	yes	characteristic
242	#460	C₂²×C₅	1	1	yes	characteristic
243	#459	C₂²×C₅	1	1	yes	characteristic
244	#452	—	2	1	no	—
245	#449	—	2	1	no	—
246	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
247	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
248	#442	—	2	1	no	—
249	#441	—	2	1	no	—
250	#440	—	2	1	no	—
251	#440	—	2	1	no	—
252	C₈×D₄	—	2	1	no	—
253	C₈×D₄	—	2	1	no	—
254	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
255	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
256	#432	C₂²×C₅	1	1	yes	characteristic
257	#432	C₂²×C₅	1	1	yes	characteristic
258	#431	—	2	1	no	—
259	#431	C₂²×C₅	1	1	yes	characteristic
260	#416	—	2	1	no	—
261	#416	—	2	1	no	—
262	C₂×C₄×C₈	—	2	1	no	—
263	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
264	#335	C₂²×C₅	1	1	yes	characteristic
265	#335	C₂²×C₅	1	1	yes	characteristic
266	#334	C₂²×C₅	1	1	yes	characteristic
267	#334	C₂²×C₅	1	1	yes	characteristic
268	#333	C₂²×C₅	1	1	yes	characteristic
269	#333	C₂²×C₅	1	1	yes	characteristic
270	#332	C₂²×C₅	1	1	yes	characteristic
271	#332	C₂²×C₅	1	1	yes	characteristic
272	#332	C₂²×C₅	1	1	yes	characteristic
273	#332	C₂²×C₅	1	1	yes	characteristic
274	#331	C₂²×C₅	1	1	yes	characteristic
275	#331	C₂²×C₅	1	1	yes	characteristic
276	C₂³×C₅	—	2	1	no	—
277	C₂³×C₅	—	4	1	no	—
278	C₂³×C₅	#121	1	1	yes	characteristic
279	C₅×Q₈	—	2	1	no	—
280	C₅×Q₈	—	4	1	no	—
281	C₅×Q₈	—	4	1	no	—
282	C₅×Q₈	—	4	1	no	—
283	C₅×Q₈	—	4	1	no	—
284	C₅×Q₈	—	8	1	no	—
285	C₅×Q₈	#135	1	2	yes	—
286	C₅×D₄	—	2	4	no	—
287	C₅×D₄	—	2	1	no	—
288	C₅×D₄	—	4	1	no	—
289	C₅×D₄	—	4	1	no	—
290	C₅×D₄	—	4	1	no	—
291	C₅×D₄	—	4	1	no	—
292	C₅×D₄	—	8	1	no	—
293	C₅×D₄	—	8	2	no	—
294	C₅×D₄	#135	1	2	yes	—
295	C₂×C₄×C₅	—	2	4	no	—
296	C₂×C₄×C₅	—	2	1	no	—
297	C₂×C₄×C₅	—	2	1	no	—
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	2	1	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	4	1	no	—
304	C₂×C₄×C₅	—	4	2	no	—
305	C₂×C₄×C₅	—	4	1	no	—
306	C₂×C₄×C₅	—	4	1	no	—
307	C₂×C₄×C₅	—	4	1	no	—
308	C₂×C₄×C₅	—	4	1	no	—
309	C₂×C₄×C₅	—	4	1	no	—
310	C₂×C₄×C₅	—	8	2	no	—
311	C₂×C₄×C₅	#121	1	1	yes	characteristic
312	C₂×C₄×C₅	#121	1	1	yes	characteristic
313	C₂×C₄×C₅	#121	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
317	C₅×C₈	—	2	1	no	—
318	C₅×C₈	—	2	1	no	—
319	C₅×C₈	—	2	1	no	—
320	C₅×C₈	—	2	1	no	—
321	C₅×C₈	—	4	1	no	—
322	C₅×C₈	—	4	1	no	—
323	C₅×C₈	—	4	1	no	—
324	C₅×C₈	—	4	1	no	—
325	C₅×C₈	—	4	1	no	—
326	C₅×C₈	—	4	1	no	—
327	C₂×#41	—	2	1	no	—
328	C₂×#41	C₅×D₄	1	1	yes	characteristic
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×SD₁₆	—	2	1	no	—
331	C₂×SD₁₆	—	2	1	no	—
332	C₂×D₈	—	2	1	no	—
333	C₂²×C₈	—	2	1	no	—
334	C₂²×C₈	—	2	1	no	—
335	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
336	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
340	#127	C₂³×C₅	1	1	yes	characteristic
341	#125	—	2	1	no	—
342	#124	—	2	1	no	—
343	C₂²⋊Q₈	—	2	1	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	#121	—	2	1	no	—
346	#121	—	2	1	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	—	2	1	no	—
349	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
350	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	C₅×D₄	1	1	yes	characteristic
356	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
357	#117	—	2	1	no	—
358	#117	—	2	1	no	—
359	#117	C₅×D₄	1	1	yes	characteristic
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
362	#107	—	2	1	no	—
363	#107	—	2	1	no	—
364	#106	—	2	1	no	—
365	#106	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
369	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
372	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
373	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
374	#103	—	2	1	no	—
375	#103	—	2	1	no	—
376	#103	—	2	1	no	—
377	#103	—	2	1	no	—
378	#103	—	2	1	no	—
379	#103	—	2	1	no	—
380	#103	—	2	1	no	—
381	#103	—	2	1	no	—
382	#102	—	2	1	no	—
383	#102	—	2	1	no	—
384	#102	—	2	1	no	—
385	#102	—	2	1	no	—
386	#102	—	2	1	no	—
387	#102	—	2	1	no	—
388	#102	—	2	1	no	—
389	#102	—	2	1	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	C₂²⋊C₈	—	2	1	no	—
392	C₂²⋊C₈	—	2	1	no	—
393	C₂²⋊C₈	—	2	1	no	—
394	C₂²⋊C₈	—	2	1	no	—
395	#97	—	2	1	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₄×C₈	—	2	1	no	—
399	C₄×C₈	—	2	1	no	—
400	C₄×C₈	C₅×D₄	1	1	yes	characteristic
401	C₄×C₈	C₅×D₄	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
404	C₂²×C₅	—	2	1	no	—
405	C₂²×C₅	—	2	1	no	—
406	C₂²×C₅	—	2	2	no	—
407	C₂²×C₅	—	2	1	no	—
408	C₂²×C₅	—	4	1	no	—
409	C₂²×C₅	—	4	1	no	—
410	C₂²×C₅	—	8	1	no	—
411	C₂²×C₅	—	8	1	no	—
412	C₂²×C₅	—	8	1	no	—
413	C₂²×C₅	C₂×#121	1	1	yes	characteristic
414	C₄×C₅	—	2	1	no	—
415	C₄×C₅	—	2	1	no	—
416	C₄×C₅	—	2	2	no	—
417	C₄×C₅	—	2	1	no	—
418	C₄×C₅	—	2	1	no	—
419	C₄×C₅	—	4	1	no	—
420	C₄×C₅	—	4	1	no	—
421	C₄×C₅	—	8	1	no	—
422	C₄×C₅	—	8	1	no	—
423	C₄×C₅	—	8	1	no	—
424	C₄×C₅	C₂×#135	1	1	yes	characteristic
425	C₄×C₅	C₂×#135	1	1	yes	characteristic
426	C₄×C₅	#575	1	1	yes	characteristic
427	C₄×C₅	#575	1	1	yes	characteristic
428	#41	—	2	2	no	—
429	#41	—	2	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	2	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₅×#41	1	1	yes	characteristic
449	Q₁₆	—	4	2	no	—
450	SD₁₆	—	4	2	no	—
451	SD₁₆	—	4	2	no	—
452	D₈	—	4	2	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	4	1	no	—
460	C₂×C₈	—	4	1	no	—
461	C₂×C₈	—	4	2	no	—
462	C₂×C₈	—	4	2	no	—
463	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
464	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
465	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
466	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
490	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
491	C₄²	C₅×#41	1	1	yes	characteristic
492	C₄²	C₂⁴×C₅	1	1	yes	characteristic
493	C₂×C₅	—	4	1	no	—
494	C₂×C₅	—	4	2	no	—
495	C₂×C₅	—	16	1	no	—
496	C₂×C₅	#3040	1	1	yes	characteristic
497	C₂×C₅	#3050	1	1	yes	characteristic
498	C₂×C₅	#3067	1	1	yes	characteristic
499	C₂³	—	2	1	no	—
500	C₂³	—	4	1	no	—
501	C₂³	C₅×#121	1	1	yes	characteristic
502	Q₈	—	2	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	8	1	no	—
508	Q₈	C₅×#135	1	2	yes	—
509	D₄	—	2	4	no	—
510	D₄	—	2	1	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	1	no	—
513	D₄	—	4	1	no	—
514	D₄	—	4	1	no	—
515	D₄	—	8	1	no	—
516	D₄	—	8	2	no	—
517	D₄	C₅×#135	1	2	yes	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	8	2	no	—
534	C₂×C₄	C₅×#121	1	1	yes	characteristic
535	C₂×C₄	C₅×#121	1	1	yes	characteristic
536	C₂×C₄	C₅×#121	1	1	yes	characteristic
537	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
538	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
539	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	2	1	no	—
543	C₈	—	2	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₅	#19278	1	1	yes	characteristic, direct product factor, Sylow
551	C₂²	—	2	1	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	C₂×C₅×#121	1	1	yes	characteristic
561	C₄	—	2	1	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	C₂×C₅×#135	1	1	yes	characteristic
572	C₄	C₂×C₅×#135	1	1	yes	characteristic
573	C₄	C₅×#575	1	1	yes	characteristic
574	C₄	C₅×#575	1	1	yes	characteristic
575	C₂	—	4	1	no	—
576	C₂	—	4	2	no	—
577	C₂	—	16	1	no	—
578	C₂	C₅×#3040	1	1	yes	characteristic
579	C₂	C₅×#3050	1	1	yes	characteristic
580	C₂	C₅×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3115	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3131	C₂	semidirect
2. C₅×#3116	C₂	semidirect
4. C₅×C₈×#41	C₂	semidirect
5. C₅×#1684	C₂	semidirect
6. C₅×#1683	C₂	semidirect
7. C₅×#1682	C₂	semidirect
8. C₅×#1681	C₂	semidirect
9. C₅×#1680	C₂	semidirect
10. C₅×#1679	C₂	semidirect
11. C₅×#1678	C₂	semidirect
12. C₅×#1677	C₂	semidirect
13. C₅×#1568	C₂	semidirect
14. C₅×#1554	C₂	semidirect
15. C₅×#1553	C₂	semidirect
67. #19278	C₅	direct (N × H)
145. #3131	C₂×C₅	semidirect
146. #3116	C₂×C₅	semidirect
148. C₈×#41	C₂×C₅	semidirect
149. #1684	C₂×C₅	semidirect
150. #1683	C₂×C₅	semidirect
151. #1682	C₂×C₅	semidirect
152. #1681	C₂×C₅	semidirect
153. #1680	C₂×C₅	semidirect
154. #1679	C₂×C₅	semidirect
155. #1678	C₂×C₅	semidirect
156. #1677	C₂×C₅	semidirect
157. #1568	C₂×C₅	semidirect
158. #1554	C₂×C₅	semidirect
159. #1553	C₂×C₅	semidirect
550. C₅	#19278	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×#3131	2
C₅×#3116	2
C₅×#3115	2
C₅×C₈×#41	2
C₅×#1684	2
C₅×#1683	2
C₅×#1682	2
C₅×#1681	2
C₅×#1680	2
C₅×#1679	2
C₅×#1678	2
C₅×#1677	2
C₅×#1568	2
C₅×#1554	2
C₅×#1553	2
#19278	5

Subgroups — C5×#19278 #11941998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19278 #11941998