Subgroups — C₅×#19801 #11942521

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3392	C₂	1	1	yes	maximal, characteristic
2	C₅×#3374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3304	C₂	1	1	yes	maximal, characteristic
5	C₅×#1719	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1702	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1679	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1650	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1641	C₂	1	1	yes	maximal, characteristic
13	C₅×#1626	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#557	C₂²	1	1	yes	characteristic
17	C₅×#557	C₂²	1	1	yes	characteristic
18	C₅×#556	C₂²	1	1	yes	characteristic
19	C₅×#550	C₂²	1	1	yes	characteristic
20	C₅×#500	C₂²	1	1	yes	characteristic
21	C₅×#492	—	2	1	no	—
22	C₅×#492	C₂²	1	1	yes	characteristic
23	C₅×#491	—	2	1	no	—
24	C₅×#488	—	2	1	no	—
25	C₅×#487	C₂²	1	1	yes	characteristic
26	C₅×#487	C₂²	1	1	yes	characteristic
27	C₅×#486	C₂²	1	1	yes	characteristic
28	C₅×#486	C₂²	1	1	yes	characteristic
29	C₅×#482	—	2	1	no	—
30	C₅×#479	—	2	1	no	—
31	C₅×#479	—	2	1	no	—
32	C₅×#475	C₂²	1	1	yes	characteristic
33	C₅×#475	C₂²	1	1	yes	characteristic
34	C₅×#474	C₂²	1	1	yes	characteristic
35	C₅×#465	—	2	1	no	—
36	C₅×#464	—	2	1	no	—
37	C₅×#462	—	2	1	no	—
38	C₅×#462	C₂²	1	1	yes	characteristic
39	C₅×#462	C₂²	1	1	yes	characteristic
40	C₅×#461	—	2	1	no	—
41	C₅×#461	C₂²	1	1	yes	characteristic
42	C₅×#460	—	2	1	no	—
43	C₅×#460	C₂²	1	1	yes	characteristic
44	C₅×#460	C₂²	1	1	yes	characteristic
45	C₅×#452	—	2	1	no	—
46	C₅×#446	C₂²	1	1	yes	characteristic
47	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₅×#441	—	2	1	no	—
50	C₄×C₅×Q₁₆	—	2	1	no	—
51	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₅×SD₁₆	—	2	1	no	—
53	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
54	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
55	C₅×#416	—	2	1	no	—
56	C₅×#334	C₂²	1	1	yes	characteristic
57	C₅×#334	C₂²	1	1	yes	characteristic
58	C₅×#332	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×Q₈⋊C₈	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×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×#322	C₂²	1	1	yes	characteristic
67	#19801	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×Q₁₆	—	2	1	no	—
70	C₂×C₅×Q₁₆	—	2	1	no	—
71	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
72	C₂×C₅×SD₁₆	—	2	1	no	—
73	C₂×C₅×SD₁₆	—	2	1	no	—
74	C₂×C₅×SD₁₆	—	2	1	no	—
75	C₂×C₅×SD₁₆	—	2	1	no	—
76	C₂×C₅×SD₁₆	—	2	1	no	—
77	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
78	C₂²×C₅×C₈	—	2	1	no	—
79	C₅×C₄⋊Q₈	—	2	1	no	—
80	C₅×C₄⋊Q₈	—	2	1	no	—
81	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₅×#127	—	2	1	no	—
85	C₅×#127	C₂³	1	1	yes	characteristic
86	C₅×#126	—	4	1	no	—
87	C₅×#125	—	2	1	no	—
88	C₅×#125	—	2	1	no	—
89	C₅×#125	—	4	1	no	—
90	C₅×#125	—	4	1	no	—
91	C₅×#124	—	2	1	no	—
92	C₅×C₂²⋊Q₈	—	2	1	no	—
93	C₅×#121	—	2	1	no	—
94	C₅×#121	—	4	1	no	—
95	C₄×C₅×Q₈	—	2	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	—	2	1	no	—
98	C₄×C₅×Q₈	—	2	1	no	—
99	C₄×C₅×Q₈	—	2	1	no	—
100	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
101	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
102	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
105	C₅×#117	—	2	1	no	—
106	C₅×#107	—	2	1	no	—
107	C₅×#107	—	4	1	no	—
108	C₅×#107	D₄	1	1	yes	characteristic
109	C₅×#106	—	2	1	no	—
110	C₅×#106	—	2	1	no	—
111	C₅×#106	D₄	1	1	yes	characteristic
112	C₅×C₄⋊C₈	—	2	1	no	—
113	C₅×C₄⋊C₈	—	2	1	no	—
114	C₅×C₄⋊C₈	—	2	1	no	—
115	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
124	C₅×#103	—	2	1	no	—
125	C₅×#103	—	2	1	no	—
126	C₅×#103	—	2	1	no	—
127	C₅×#103	—	2	1	no	—
128	C₅×#103	D₄	1	1	yes	characteristic
129	C₅×#103	D₄	1	1	yes	characteristic
130	C₅×#103	D₄	1	1	yes	characteristic
131	C₅×#102	—	2	1	no	—
132	C₅×#102	—	2	1	no	—
133	C₅×#102	—	2	1	no	—
134	C₅×#102	—	2	1	no	—
135	C₅×#102	—	2	1	no	—
136	C₅×#102	—	4	1	no	—
137	C₅×#102	D₄	1	1	yes	characteristic
138	C₅×C₂²⋊C₈	—	2	1	no	—
139	C₅×#97	—	2	1	no	—
140	C₄×C₅×C₈	—	2	1	no	—
141	C₄×C₅×C₈	—	2	1	no	—
142	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	#3392	C₂×C₅	1	1	yes	characteristic
146	#3374	C₂×C₅	1	1	yes	characteristic, complemented
147	#3364	C₂×C₅	1	1	yes	characteristic, complemented
148	#3304	C₂×C₅	1	1	yes	characteristic
149	#1719	C₂×C₅	1	1	yes	characteristic, complemented
150	#1702	C₂×C₅	1	1	yes	characteristic, complemented
151	#1680	C₂×C₅	1	1	yes	characteristic, complemented
152	#1679	C₂×C₅	1	1	yes	characteristic, complemented
153	#1666	C₂×C₅	1	1	yes	characteristic, complemented
154	#1650	C₂×C₅	1	1	yes	characteristic, complemented
155	#1648	C₂×C₅	1	1	yes	characteristic, complemented
156	#1641	C₂×C₅	1	1	yes	characteristic
157	#1626	C₂×C₅	1	1	yes	characteristic, complemented
158	#1582	C₂×C₅	1	1	yes	characteristic, complemented
159	#1579	C₂×C₅	1	1	yes	characteristic, complemented
160	C₅×#41	—	4	2	no	—
161	C₅×#41	—	8	1	no	—
162	C₂×C₅×Q₈	—	2	1	no	—
163	C₂×C₅×Q₈	—	2	1	no	—
164	C₂×C₅×Q₈	—	2	1	no	—
165	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
166	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
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	—	4	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	—	4	1	no	—
176	C₅×Q₁₆	—	2	2	no	—
177	C₅×Q₁₆	—	4	2	no	—
178	C₅×Q₁₆	—	4	2	no	—
179	C₅×SD₁₆	—	4	1	no	—
180	C₅×SD₁₆	—	4	2	no	—
181	C₅×SD₁₆	—	8	1	no	—
182	C₅×SD₁₆	—	8	1	no	—
183	C₅×SD₁₆	—	8	1	no	—
184	C₅×SD₁₆	—	8	1	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	—	8	1	no	—
191	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
192	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
193	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
195	C₅×C₄⋊C₄	—	2	1	no	—
196	C₅×C₄⋊C₄	—	2	1	no	—
197	C₅×C₄⋊C₄	—	2	1	no	—
198	C₅×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
205	C₅×C₄⋊C₄	—	4	1	no	—
206	C₅×C₄⋊C₄	—	4	1	no	—
207	C₅×C₄⋊C₄	—	4	1	no	—
208	C₅×C₄⋊C₄	—	4	1	no	—
209	C₅×C₄⋊C₄	—	4	1	no	—
210	C₅×C₄⋊C₄	—	4	1	no	—
211	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
214	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
215	C₅×C₂²⋊C₄	—	2	1	no	—
216	C₅×C₂²⋊C₄	—	4	1	no	—
217	C₅×C₂²⋊C₄	—	4	1	no	—
218	C₅×C₂²⋊C₄	—	4	1	no	—
219	C₄²×C₅	—	2	1	no	—
220	C₄²×C₅	—	2	1	no	—
221	C₄²×C₅	—	2	1	no	—
222	C₄²×C₅	—	4	1	no	—
223	C₄²×C₅	—	4	1	no	—
224	C₄²×C₅	—	4	1	no	—
225	C₄²×C₅	C₂⁴	1	1	yes	characteristic
226	#557	C₂²×C₅	1	1	yes	characteristic
227	#557	C₂²×C₅	1	1	yes	characteristic
228	#556	C₂²×C₅	1	1	yes	characteristic
229	#550	C₂²×C₅	1	1	yes	characteristic
230	#500	C₂²×C₅	1	1	yes	characteristic
231	#492	—	2	1	no	—
232	#492	C₂²×C₅	1	1	yes	characteristic
233	#491	—	2	1	no	—
234	#488	—	2	1	no	—
235	#487	C₂²×C₅	1	1	yes	characteristic
236	#487	C₂²×C₅	1	1	yes	characteristic
237	#486	C₂²×C₅	1	1	yes	characteristic
238	#486	C₂²×C₅	1	1	yes	characteristic
239	#482	—	2	1	no	—
240	#479	—	2	1	no	—
241	#479	—	2	1	no	—
242	#475	C₂²×C₅	1	1	yes	characteristic
243	#475	C₂²×C₅	1	1	yes	characteristic
244	#474	C₂²×C₅	1	1	yes	characteristic
245	#465	—	2	1	no	—
246	#464	—	2	1	no	—
247	#462	—	2	1	no	—
248	#462	C₂²×C₅	1	1	yes	characteristic
249	#462	C₂²×C₅	1	1	yes	characteristic
250	#461	—	2	1	no	—
251	#461	C₂²×C₅	1	1	yes	characteristic
252	#460	—	2	1	no	—
253	#460	C₂²×C₅	1	1	yes	characteristic
254	#460	C₂²×C₅	1	1	yes	characteristic
255	#452	—	2	1	no	—
256	#446	C₂²×C₅	1	1	yes	characteristic
257	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
258	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
259	#441	—	2	1	no	—
260	C₄×Q₁₆	—	2	1	no	—
261	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
262	C₄×SD₁₆	—	2	1	no	—
263	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
264	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
265	#416	—	2	1	no	—
266	#334	C₂²×C₅	1	1	yes	characteristic
267	#334	C₂²×C₅	1	1	yes	characteristic
268	#332	C₂²×C₅	1	1	yes	characteristic
269	#332	C₂²×C₅	1	1	yes	characteristic
270	#332	C₂²×C₅	1	1	yes	characteristic
271	#331	C₂²×C₅	1	1	yes	characteristic
272	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
273	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
274	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
275	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
276	#322	C₂²×C₅	1	1	yes	characteristic
277	C₂³×C₅	—	2	1	no	—
278	C₂³×C₅	—	4	1	no	—
279	C₅×Q₈	—	2	1	no	—
280	C₅×Q₈	—	2	1	no	—
281	C₅×Q₈	—	2	1	no	—
282	C₅×Q₈	—	2	1	no	—
283	C₅×Q₈	—	2	1	no	—
284	C₅×Q₈	—	4	1	no	—
285	C₅×Q₈	—	4	1	no	—
286	C₅×Q₈	—	4	1	no	—
287	C₅×Q₈	—	4	1	no	—
288	C₅×Q₈	—	4	1	no	—
289	C₅×Q₈	—	4	1	no	—
290	C₅×Q₈	#135	1	2	yes	—
291	C₅×D₄	—	2	1	no	—
292	C₅×D₄	—	2	1	no	—
293	C₅×D₄	—	4	1	no	—
294	C₅×D₄	—	4	1	no	—
295	C₅×D₄	—	4	1	no	—
296	C₅×D₄	—	4	1	no	—
297	C₅×D₄	—	8	2	no	—
298	C₅×D₄	—	8	1	no	—
299	C₅×D₄	—	16	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	—	2	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	2	1	no	—
306	C₂×C₄×C₅	—	2	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	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	8	2	no	—
313	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
316	C₅×C₈	—	2	1	no	—
317	C₅×C₈	—	2	1	no	—
318	C₅×C₈	—	2	1	no	—
319	C₅×C₈	—	2	1	no	—
320	C₅×C₈	—	4	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	C₂×#41	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
329	C₂×SD₁₆	—	2	1	no	—
330	C₂×SD₁₆	—	2	1	no	—
331	C₂×SD₁₆	—	2	1	no	—
332	C₂×SD₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	1	no	—
334	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
341	#127	—	2	1	no	—
342	#127	C₂³×C₅	1	1	yes	characteristic
343	#126	—	4	1	no	—
344	#125	—	2	1	no	—
345	#125	—	2	1	no	—
346	#125	—	4	1	no	—
347	#125	—	4	1	no	—
348	#124	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	#121	—	2	1	no	—
351	#121	—	4	1	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
362	#117	—	2	1	no	—
363	#107	—	2	1	no	—
364	#107	—	4	1	no	—
365	#107	C₅×D₄	1	1	yes	characteristic
366	#106	—	2	1	no	—
367	#106	—	2	1	no	—
368	#106	C₅×D₄	1	1	yes	characteristic
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
373	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
374	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
375	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
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	#103	—	2	1	no	—
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	C₅×D₄	1	1	yes	characteristic
386	#103	C₅×D₄	1	1	yes	characteristic
387	#103	C₅×D₄	1	1	yes	characteristic
388	#102	—	2	1	no	—
389	#102	—	2	1	no	—
390	#102	—	2	1	no	—
391	#102	—	2	1	no	—
392	#102	—	2	1	no	—
393	#102	—	4	1	no	—
394	#102	C₅×D₄	1	1	yes	characteristic
395	C₂²⋊C₈	—	2	1	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₄×C₈	—	2	1	no	—
399	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
402	C₂²×C₅	—	4	1	no	—
403	C₂²×C₅	—	4	1	no	—
404	C₂²×C₅	—	4	1	no	—
405	C₂²×C₅	—	8	1	no	—
406	C₂²×C₅	—	8	1	no	—
407	C₂²×C₅	—	8	1	no	—
408	C₂²×C₅	D₄²	1	1	yes	characteristic
409	C₄×C₅	—	2	2	no	—
410	C₄×C₅	—	2	1	no	—
411	C₄×C₅	—	4	1	no	—
412	C₄×C₅	—	4	1	no	—
413	C₄×C₅	—	4	1	no	—
414	C₄×C₅	—	4	1	no	—
415	C₄×C₅	—	4	1	no	—
416	C₄×C₅	—	4	1	no	—
417	C₄×C₅	—	8	1	no	—
418	C₄×C₅	—	8	1	no	—
419	C₄×C₅	—	8	1	no	—
420	C₄×C₅	C₂×#135	1	1	yes	characteristic
421	C₄×C₅	#575	1	1	yes	characteristic
422	C₄×C₅	#577	1	1	yes	characteristic
423	C₄×C₅	#577	1	1	yes	characteristic
424	#41	—	4	2	no	—
425	#41	—	8	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
430	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
431	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	—	4	1	no	—
437	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	Q₁₆	—	2	2	no	—
441	Q₁₆	—	4	2	no	—
442	Q₁₆	—	4	2	no	—
443	SD₁₆	—	4	1	no	—
444	SD₁₆	—	4	2	no	—
445	SD₁₆	—	8	1	no	—
446	SD₁₆	—	8	1	no	—
447	SD₁₆	—	8	1	no	—
448	SD₁₆	—	8	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	8	1	no	—
455	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
456	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
457	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
458	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
476	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
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	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	4	1	no	—
487	C₄²	—	4	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	C₂⁴×C₅	1	1	yes	characteristic
490	C₂×C₅	—	8	1	no	—
491	C₂×C₅	—	16	1	no	—
492	C₂×C₅	#3271	1	1	yes	characteristic
493	C₂×C₅	#3282	1	1	yes	characteristic
494	C₂×C₅	#3289	1	1	yes	characteristic
495	C₂³	—	2	1	no	—
496	C₂³	—	4	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	2	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	C₅×#135	1	2	yes	—
509	D₄	—	2	1	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	2	no	—
516	D₄	—	8	1	no	—
517	D₄	—	16	1	no	—
518	C₂×C₄	—	2	1	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	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	8	2	no	—
531	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
532	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
533	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	2	1	no	—
537	C₈	—	2	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	1	no	—
541	C₈	—	4	1	no	—
542	C₈	—	4	1	no	—
543	C₅	#19801	1	1	yes	characteristic, direct product factor, Sylow
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	—	8	1	no	—
549	C₂²	—	8	1	no	—
550	C₂²	C₅×D₄²	1	1	yes	characteristic
551	C₄	—	2	2	no	—
552	C₄	—	2	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	8	1	no	—
560	C₄	—	8	1	no	—
561	C₄	—	8	1	no	—
562	C₄	C₂×C₅×#135	1	1	yes	characteristic
563	C₄	C₅×#575	1	1	yes	characteristic
564	C₄	C₅×#577	1	1	yes	characteristic
565	C₄	C₅×#577	1	1	yes	characteristic
566	C₂	—	8	1	no	—
567	C₂	—	16	1	no	—
568	C₂	C₅×#3271	1	1	yes	characteristic
569	C₂	C₅×#3282	1	1	yes	characteristic
570	C₂	C₅×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3374	C₂	semidirect
3. C₅×#3364	C₂	semidirect
5. C₅×#1719	C₂	semidirect
6. C₅×#1702	C₂	semidirect
7. C₅×#1680	C₂	semidirect
8. C₅×#1679	C₂	semidirect
9. C₅×#1666	C₂	semidirect
10. C₅×#1650	C₂	semidirect
11. C₅×#1648	C₂	semidirect
13. C₅×#1626	C₂	semidirect
14. C₅×#1582	C₂	semidirect
15. C₅×#1579	C₂	semidirect
67. #19801	C₅	direct (N × H)
146. #3374	C₂×C₅	semidirect
147. #3364	C₂×C₅	semidirect
149. #1719	C₂×C₅	semidirect
150. #1702	C₂×C₅	semidirect
151. #1680	C₂×C₅	semidirect
152. #1679	C₂×C₅	semidirect
153. #1666	C₂×C₅	semidirect
154. #1650	C₂×C₅	semidirect
155. #1648	C₂×C₅	semidirect
157. #1626	C₂×C₅	semidirect
158. #1582	C₂×C₅	semidirect
159. #1579	C₂×C₅	semidirect
543. C₅	#19801	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×#3392	2
C₅×#3374	2
C₅×#3364	2
C₅×#3304	2
C₅×#1719	2
C₅×#1702	2
C₅×#1680	2
C₅×#1679	2
C₅×#1666	2
C₅×#1650	2
C₅×#1648	2
C₅×#1641	2
C₅×#1626	2
C₅×#1582	2
C₅×#1579	2
#19801	5

Subgroups — C5×#19801 #11942521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19801 #11942521