Subgroups — C₅×#10610711 #72223430

Order 3200.

1086 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#66156	C₂	1	1	yes	maximal, characteristic
2	C₅×#66153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#66079	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅²×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#65413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#3937	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#65113	C₂	1	4	yes	maximal, complemented
9	C₅×#65112	C₂	1	4	yes	maximal, complemented
10	C₂×C₅²×#41	C₂²	1	1	yes	characteristic
11	C₂×C₅×#709	C₂²	1	1	yes	characteristic
12	C₂×C₅²×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅²	C₂²	1	1	yes	characteristic
14	C₅×#4097	C₄	1	2	yes	complemented
15	C₅×#4096	C₂²	1	4	yes	—
16	C₅×#4095	C₄	1	2	yes	complemented
17	C₅×#4094	—	2	2	no	—
18	C₅×#4094	C₂²	1	4	yes	complemented
19	C₅×#4088	C₄	1	2	yes	complemented
20	C₅×#4078	C₄	1	2	yes	complemented
21	C₂×C₅×#681	—	2	2	no	—
22	C₂×C₅×#681	C₂²	1	1	yes	characteristic
23	C₂×C₅×#681	C₂²	1	1	yes	characteristic
24	C₂×C₅×#681	C₂²	1	1	yes	characteristic
25	C₅×#4053	C₂²	1	4	yes	complemented
26	C₅×D₅×M₁₆	—	2	2	no	—
27	C₅×#4047	—	2	2	no	—
28	C₅²×#104	C₂²	1	2	yes	complemented
29	C₅²×#104	C₂²	1	2	yes	complemented
30	C₅×#3954	C₂²	1	4	yes	complemented
31	C₅×#3937	C₂²	1	2	yes	complemented
32	C₅×#3937	C₂²	1	2	yes	complemented
33	C₅×#3935	C₂²	1	2	yes	complemented
34	C₄×C₅×#168	C₂²	1	2	yes	complemented
35	C₅×#2956	—	5	1	no	maximal
36	#10610711	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₅²×#41	—	2	1	no	—
38	C₅²×#41	—	2	1	no	—
39	C₅²×#41	C₂×C₄	1	2	yes	—
40	C₅²×#41	C₂³	1	2	yes	—
41	C₂×C₅²×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅²×D₄	—	2	1	no	—
43	C₂×C₅²×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅²	—	2	1	no	—
45	C₂²×C₄×C₅²	—	2	1	no	—
46	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
47	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
48	C₅×#709	—	2	1	no	—
49	C₅×#709	—	2	1	no	—
50	C₅×#709	C₂×C₄	1	2	yes	—
51	C₅×#709	C₂³	1	2	yes	—
52	C₂×C₅×D₂₀	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅×D₅	—	2	1	no	—
54	C₂×C₅×Dic₁₀	C₂×C₄	1	1	yes	characteristic
55	C₅²×M₁₆	—	2	1	no	—
56	C₅²×M₁₆	C₂³	1	2	yes	—
57	C₂×C₅²×C₈	—	2	1	no	—
58	C₄²×C₅²	—	2	1	no	—
59	C₄²×C₅²	C₂³	1	1	yes	characteristic
60	C₄²×C₅²	C₂³	1	1	yes	characteristic
61	C₅×#689	C₂×C₄	1	4	yes	complemented
62	C₅×#688	C₂×C₄	1	4	yes	complemented
63	C₅×#687	C₂×C₄	1	4	yes	complemented
64	C₅×#686	C₂×C₄	1	4	yes	complemented
65	C₅×#681	—	2	1	no	—
66	C₅×#681	—	2	4	no	—
67	C₅×#681	C₂×C₄	1	2	yes	—
68	C₅×#681	C₂×C₄	1	2	yes	—
69	C₅×#681	D₄	1	2	yes	—
70	C₅×#681	D₄	1	2	yes	—
71	C₅×#681	C₂³	1	2	yes	—
72	C₂×C₅×#168	—	2	1	no	—
73	C₂×C₅×#168	—	2	4	no	—
74	C₂×C₅×#168	C₂³	1	2	yes	—
75	C₂×C₅×#168	C₂³	1	2	yes	—
76	C₅×#676	—	2	4	no	—
77	C₅×C₈×D₅	—	2	4	no	—
78	C₅×#575	—	5	1	no	—
79	C₅×#568	—	5	1	no	—
80	C₅×#568	—	5	1	no	—
81	#66156	C₂×C₅	1	1	yes	characteristic
82	#66153	C₂×C₅	1	1	yes	characteristic, complemented
83	#66079	C₂×C₅	1	1	yes	characteristic, complemented
84	C₅×#444	—	5	4	no	—
85	C₅×#443	—	5	4	no	—
86	C₅×#430	—	5	1	no	—
87	C₂×C₅×#104	—	2	2	no	—
88	C₂×C₅×#104	—	5	1	no	—
89	C₂×C₅×#104	D₅	1	1	yes	characteristic, complemented
90	C₂×C₅×#104	C₂×C₅	1	1	yes	characteristic, complemented
91	C₄×C₅×M₁₆	—	5	1	no	—
92	#65413	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂×#3937	C₂×C₅	1	1	yes	characteristic, complemented
94	C₄×#681	C₂×C₅	1	1	yes	characteristic, complemented
95	#65113	C₂×C₅	1	4	yes	complemented
96	#65112	C₂×C₅	1	4	yes	complemented
97	C₂³×C₅²	—	2	1	no	—
98	C₂³×C₅²	#41	1	1	yes	characteristic
99	C₂²×C₅×D₅	—	2	1	no	—
100	C₅²×Q₈	—	2	1	no	—
101	C₅²×Q₈	C₂²×C₄	1	2	yes	—
102	C₅²×D₄	—	2	2	no	—
103	C₅²×D₄	—	2	2	no	—
104	C₅²×D₄	—	2	1	no	—
105	C₅²×D₄	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₅²	—	2	1	no	—
107	C₂×C₄×C₅²	—	2	1	no	—
108	C₂×C₄×C₅²	—	2	2	no	—
109	C₂×C₄×C₅²	—	2	2	no	—
110	C₂×C₄×C₅²	—	2	1	no	—
111	C₂×C₄×C₅²	—	4	2	no	—
112	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅²	#41	1	1	yes	characteristic
117	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
118	C₅×C₂²⋊D₅	—	2	2	no	—
119	C₅×C₂²⋊D₅	—	2	2	no	—
120	C₂×C₅×Dic₅	—	2	1	no	—
121	C₅×D₂₀	—	2	1	no	—
122	C₅×D₂₀	C₂²×C₄	1	2	yes	—
123	C₄×C₅×D₅	—	2	2	no	—
124	C₄×C₅×D₅	—	2	2	no	—
125	C₅×Dic₁₀	—	2	1	no	—
126	C₅×Dic₁₀	C₂²×C₄	1	2	yes	—
127	C₅²×C₈	—	2	2	no	—
128	C₅×#168	—	2	2	no	—
129	C₅×#168	C₂²×C₄	1	4	yes	—
130	C₅×#168	C₂×D₄	1	4	yes	—
131	C₂×C₅×#41	—	2	2	no	—
132	C₂×C₅×#41	—	5	1	no	—
133	C₂×C₅×#41	C₂×D₅	1	1	yes	characteristic
134	C₂×C₅×#41	C₂²×C₅	1	1	yes	characteristic
135	C₂×#709	C₂²×C₅	1	1	yes	characteristic
136	C₅×#137	—	5	2	no	—
137	C₅×#137	—	5	2	no	—
138	C₅×AGL(1,ℤ/8)	—	5	2	no	—
139	C₅×AGL(1,ℤ/8)	—	5	2	no	—
140	C₅×#135	—	5	4	no	—
141	C₅×#131	—	5	4	no	—
142	C₅×#131	—	5	4	no	—
143	C₅×#131	—	10	2	no	—
144	C₅×#131	—	10	2	no	—
145	C₂×C₅×M₁₆	—	2	2	no	—
146	C₂×C₅×M₁₆	—	5	1	no	—
147	C₂×C₅×M₁₆	—	5	1	no	—
148	C₂×C₅×M₁₆	—	5	1	no	—
149	C₂×C₅×M₁₆	—	10	2	no	—
150	C₂×C₅×M₁₆	—	10	2	no	—
151	C₂×C₅×M₁₆	C₂×D₅	1	1	yes	characteristic
152	C₂×C₅×M₁₆	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₄²×C₅	—	2	2	no	—
154	C₂×C₄²×C₅	C₂×D₅	1	1	yes	characteristic
155	C₂×C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
156	#4097	C₄×C₅	1	2	yes	complemented
157	#4096	C₂²×C₅	1	4	yes	—
158	#4095	C₄×C₅	1	2	yes	complemented
159	#4094	—	2	2	no	—
160	#4094	C₂²×C₅	1	4	yes	complemented
161	#4088	C₄×C₅	1	2	yes	complemented
162	#4078	C₄×C₅	1	2	yes	complemented
163	C₂×#681	—	2	2	no	—
164	C₂×#681	C₂²×C₅	1	1	yes	characteristic
165	C₂×#681	C₂²×C₅	1	1	yes	characteristic
166	C₂×#681	C₂²×C₅	1	1	yes	characteristic
167	#4053	C₂²×C₅	1	4	yes	complemented
168	D₅×M₁₆	—	2	2	no	—
169	#4047	—	2	2	no	—
170	C₅×#108	—	5	4	no	—
171	C₅×#104	—	2	4	no	—
172	C₅×#104	—	2	4	no	—
173	C₅×#104	—	5	2	no	—
174	C₅×#104	—	5	2	no	—
175	C₅×#104	C₂×D₅	1	2	yes	complemented
176	C₅×#104	C₂×D₅	1	2	yes	complemented
177	C₅×#104	C₂²×C₅	1	2	yes	complemented
178	C₅×#104	C₂²×C₅	1	2	yes	complemented
179	C₅×#97	—	5	2	no	—
180	C₄×C₅×C₈	—	5	2	no	—
181	#3954	C₂²×C₅	1	4	yes	complemented
182	#3937	C₂²×C₅	1	2	yes	complemented
183	#3937	C₂²×C₅	1	2	yes	complemented
184	#3935	C₂²×C₅	1	2	yes	complemented
185	C₄×#168	C₂²×C₅	1	2	yes	complemented
186	#2956	—	5	1	no	Sylow
187	C₂²×C₅²	—	2	2	no	—
188	C₂²×C₅²	—	4	1	no	—
189	C₂²×C₅²	—	8	1	no	—
190	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
191	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₅²	C₂×#41	1	1	yes	characteristic
193	C₂×C₅×D₅	—	2	2	no	—
194	C₂×C₅×D₅	—	8	1	no	—
195	C₄×C₅²	—	2	2	no	—
196	C₄×C₅²	—	2	2	no	—
197	C₄×C₅²	—	4	1	no	—
198	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
199	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
200	C₄×C₅²	C₂³×C₄	1	1	yes	characteristic
201	C₄×C₅²	C₂²×D₄	1	1	yes	characteristic
202	C₅×Dic₅	—	2	2	no	—
203	C₅×#41	—	2	1	no	—
204	C₅×#41	—	2	1	no	—
205	C₅×#41	—	2	1	no	—
206	C₅×#41	—	2	1	no	—
207	C₅×#41	—	2	4	no	—
208	C₅×#41	—	2	4	no	—
209	C₅×#41	—	4	2	no	—
210	C₅×#41	—	4	2	no	—
211	C₅×#41	—	5	2	no	—
212	C₅×#41	—	5	2	no	—
213	C₅×#41	—	10	1	no	—
214	C₅×#41	—	10	1	no	—
215	C₅×#41	C₄×D₅	1	2	yes	—
216	C₅×#41	C₂×C₄×C₅	1	2	yes	—
217	C₅×#41	C₂²×D₅	1	2	yes	—
218	C₅×#41	C₂³×C₅	1	2	yes	—
219	C₂×C₅×Q₈	—	2	2	no	—
220	C₂×C₅×Q₈	—	5	1	no	—
221	C₂×C₅×Q₈	C₄×D₅	1	1	yes	characteristic
222	C₂×C₅×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
223	C₂×C₅×D₄	—	2	1	no	—
224	C₂×C₅×D₄	—	2	1	no	—
225	C₂×C₅×D₄	—	2	2	no	—
226	C₂×C₅×D₄	—	4	2	no	—
227	C₂×C₅×D₄	—	5	1	no	—
228	C₂×C₅×D₄	—	10	1	no	—
229	C₂×C₅×D₄	C₄×D₅	1	1	yes	characteristic
230	C₂×C₅×D₄	C₂×C₄×C₅	1	1	yes	characteristic
231	C₂²×C₄×C₅	—	2	1	no	—
232	C₂²×C₄×C₅	—	2	1	no	—
233	C₂²×C₄×C₅	—	2	1	no	—
234	C₂²×C₄×C₅	—	2	1	no	—
235	C₂²×C₄×C₅	—	2	2	no	—
236	C₂²×C₄×C₅	—	4	2	no	—
237	C₂²×C₄×C₅	—	4	2	no	—
238	C₂²×C₄×C₅	—	10	1	no	—
239	C₂²×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
240	C₂²×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
241	C₂×C₂²⋊D₅	—	2	1	no	—
242	#709	—	2	1	no	—
243	#709	—	2	1	no	—
244	#709	C₂×C₄×C₅	1	2	yes	—
245	#709	C₂³×C₅	1	2	yes	—
246	C₂×D₂₀	C₂×C₄×C₅	1	1	yes	characteristic
247	C₂×C₄×D₅	—	2	1	no	—
248	C₂×Dic₁₀	C₂×C₄×C₅	1	1	yes	characteristic
249	C₅×Q₁₆	—	5	4	no	—
250	C₅×SD₁₆	—	5	4	no	—
251	C₅×SD₁₆	—	5	4	no	—
252	C₅×D₈	—	5	4	no	—
253	C₅×M₁₆	—	2	1	no	—
254	C₅×M₁₆	—	2	1	no	—
255	C₅×M₁₆	—	2	4	no	—
256	C₅×M₁₆	—	4	2	no	—
257	C₅×M₁₆	—	5	2	no	—
258	C₅×M₁₆	—	5	2	no	—
259	C₅×M₁₆	—	5	2	no	—
260	C₅×M₁₆	—	5	2	no	—
261	C₅×M₁₆	—	5	2	no	—
262	C₅×M₁₆	—	10	4	no	—
263	C₅×M₁₆	—	10	4	no	—
264	C₅×M₁₆	—	10	1	no	—
265	C₅×M₁₆	C₂²×D₅	1	2	yes	—
266	C₅×M₁₆	C₂³×C₅	1	2	yes	—
267	C₂×C₅×C₈	—	2	1	no	—
268	C₂×C₅×C₈	—	2	1	no	—
269	C₂×C₅×C₈	—	4	2	no	—
270	C₂×C₅×C₈	—	5	2	no	—
271	C₂×C₅×C₈	—	5	2	no	—
272	C₂×C₅×C₈	—	10	4	no	—
273	C₂×C₅×C₈	—	10	4	no	—
274	C₂×C₅×C₈	—	10	1	no	—
275	C₄²×C₅	—	2	1	no	—
276	C₄²×C₅	—	2	1	no	—
277	C₄²×C₅	—	2	2	no	—
278	C₄²×C₅	—	2	2	no	—
279	C₄²×C₅	—	4	2	no	—
280	C₄²×C₅	C₂²×D₅	1	1	yes	characteristic
281	C₄²×C₅	C₂²×D₅	1	1	yes	characteristic
282	C₄²×C₅	C₂³×C₅	1	1	yes	characteristic
283	C₄²×C₅	C₂³×C₅	1	1	yes	characteristic
284	#689	C₂×C₄×C₅	1	4	yes	complemented
285	#688	C₂×C₄×C₅	1	4	yes	complemented
286	#687	C₂×C₄×C₅	1	4	yes	complemented
287	#686	C₂×C₄×C₅	1	4	yes	complemented
288	#681	—	2	1	no	—
289	#681	—	2	4	no	—
290	#681	C₂×C₄×C₅	1	2	yes	—
291	#681	C₂×C₄×C₅	1	2	yes	—
292	#681	C₅×D₄	1	2	yes	—
293	#681	C₅×D₄	1	2	yes	—
294	#681	C₂³×C₅	1	2	yes	—
295	C₂×#168	—	2	1	no	—
296	C₂×#168	—	2	4	no	—
297	C₂×#168	C₂³×C₅	1	2	yes	—
298	C₂×#168	C₂³×C₅	1	2	yes	—
299	#676	—	2	4	no	—
300	C₈×D₅	—	2	4	no	—
301	#575	—	5	1	no	—
302	#568	—	5	1	no	—
303	#568	—	5	1	no	—
304	#444	—	5	4	no	—
305	#443	—	5	4	no	—
306	#430	—	5	1	no	—
307	C₂×#104	—	5	1	no	—
308	C₂×#104	C₅×D₅	1	1	yes	characteristic, complemented
309	C₄×M₁₆	—	5	1	no	—
310	C₂×C₅²	—	2	1	no	—
311	C₂×C₅²	—	2	1	no	—
312	C₂×C₅²	—	2	1	no	—
313	C₂×C₅²	—	4	2	no	—
314	C₂×C₅²	C₂×C₄×D₄	1	1	yes	characteristic
315	C₅×D₅	—	4	2	no	—
316	C₂³×C₅	—	2	1	no	—
317	C₂³×C₅	—	2	1	no	—
318	C₂³×C₅	—	2	2	no	—
319	C₂³×C₅	—	4	2	no	—
320	C₂³×C₅	—	10	1	no	—
321	C₂³×C₅	#711	1	1	yes	characteristic
322	C₂³×C₅	C₅×#41	1	1	yes	characteristic
323	C₂²×D₅	—	2	1	no	—
324	C₅×Q₈	—	2	1	no	—
325	C₅×Q₈	—	2	1	no	—
326	C₅×Q₈	—	2	4	no	—
327	C₅×Q₈	—	4	2	no	—
328	C₅×Q₈	—	5	2	no	—
329	C₅×Q₈	—	10	1	no	—
330	C₅×Q₈	C₂×C₄×D₅	1	2	yes	—
331	C₅×Q₈	C₂²×C₄×C₅	1	2	yes	—
332	C₅×D₄	—	2	2	no	—
333	C₅×D₄	—	2	2	no	—
334	C₅×D₄	—	2	2	no	—
335	C₅×D₄	—	2	2	no	—
336	C₅×D₄	—	2	1	no	—
337	C₅×D₄	—	2	1	no	—
338	C₅×D₄	—	2	4	no	—
339	C₅×D₄	—	4	2	no	—
340	C₅×D₄	—	4	4	no	—
341	C₅×D₄	—	4	4	no	—
342	C₅×D₄	—	5	2	no	—
343	C₅×D₄	—	10	2	no	—
344	C₅×D₄	—	10	2	no	—
345	C₅×D₄	—	10	1	no	—
346	C₅×D₄	C₂×C₄×D₅	1	2	yes	—
347	C₅×D₄	C₂²×C₄×C₅	1	2	yes	—
348	C₂×C₄×C₅	—	2	1	no	—
349	C₂×C₄×C₅	—	2	1	no	—
350	C₂×C₄×C₅	—	2	1	no	—
351	C₂×C₄×C₅	—	2	1	no	—
352	C₂×C₄×C₅	—	2	2	no	—
353	C₂×C₄×C₅	—	2	2	no	—
354	C₂×C₄×C₅	—	2	2	no	—
355	C₂×C₄×C₅	—	2	2	no	—
356	C₂×C₄×C₅	—	2	1	no	—
357	C₂×C₄×C₅	—	2	1	no	—
358	C₂×C₄×C₅	—	2	2	no	—
359	C₂×C₄×C₅	—	2	2	no	—
360	C₂×C₄×C₅	—	2	2	no	—
361	C₂×C₄×C₅	—	2	2	no	—
362	C₂×C₄×C₅	—	2	2	no	—
363	C₂×C₄×C₅	—	2	2	no	—
364	C₂×C₄×C₅	—	4	4	no	—
365	C₂×C₄×C₅	—	4	4	no	—
366	C₂×C₄×C₅	—	4	2	no	—
367	C₂×C₄×C₅	—	4	2	no	—
368	C₂×C₄×C₅	—	4	2	no	—
369	C₂×C₄×C₅	—	4	2	no	—
370	C₂×C₄×C₅	—	4	2	no	—
371	C₂×C₄×C₅	—	4	8	no	—
372	C₂×C₄×C₅	—	10	1	no	—
373	C₂×C₄×C₅	—	10	2	no	—
374	C₂×C₄×C₅	—	10	2	no	—
375	C₂×C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
376	C₂×C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
377	C₂×C₄×C₅	D₄×D₅	1	1	yes	characteristic
378	C₂×C₄×C₅	D₄×D₅	1	1	yes	characteristic
379	C₂×C₄×C₅	#711	1	1	yes	characteristic
380	C₂×C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
381	C₂×C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
382	C₂×C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
383	C₂×C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
384	C₂×C₄×C₅	C₅×#41	1	1	yes	characteristic
385	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
386	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
387	C₂²⋊D₅	—	2	2	no	—
388	C₂²⋊D₅	—	2	2	no	—
389	C₂×Dic₅	—	2	1	no	—
390	D₂₀	—	2	1	no	—
391	D₂₀	C₂²×C₄×C₅	1	2	yes	—
392	C₄×D₅	—	2	2	no	—
393	C₄×D₅	—	2	2	no	—
394	Dic₁₀	—	2	1	no	—
395	Dic₁₀	C₂²×C₄×C₅	1	2	yes	—
396	C₅×C₈	—	2	2	no	—
397	C₅×C₈	—	2	2	no	—
398	C₅×C₈	—	4	4	no	—
399	C₅×C₈	—	5	4	no	—
400	C₅×C₈	—	5	4	no	—
401	C₅×C₈	—	10	2	no	—
402	#168	—	2	2	no	—
403	#168	C₂²×C₄×C₅	1	4	yes	—
404	#168	C₂×C₅×D₄	1	4	yes	—
405	C₂×#41	—	5	1	no	—
406	C₂×#41	C₂×C₅×D₅	1	1	yes	characteristic
407	#137	—	5	2	no	—
408	#137	—	5	2	no	—
409	AGL(1,ℤ/8)	—	5	2	no	—
410	AGL(1,ℤ/8)	—	5	2	no	—
411	#135	—	5	4	no	—
412	#131	—	5	4	no	—
413	#131	—	5	4	no	—
414	#131	—	10	2	no	—
415	#131	—	10	2	no	—
416	C₂×M₁₆	—	5	1	no	—
417	C₂×M₁₆	—	5	1	no	—
418	C₂×M₁₆	—	5	1	no	—
419	C₂×M₁₆	—	10	2	no	—
420	C₂×M₁₆	—	10	2	no	—
421	C₂×M₁₆	C₂×C₅×D₅	1	1	yes	characteristic
422	C₂×C₄²	C₂×C₅×D₅	1	1	yes	characteristic
423	#108	—	5	4	no	—
424	#104	—	5	2	no	—
425	#104	—	5	2	no	—
426	#104	C₂×C₅×D₅	1	2	yes	complemented
427	#104	C₂×C₅×D₅	1	2	yes	complemented
428	#97	—	5	2	no	—
429	C₄×C₈	—	5	2	no	—
430	C₅²	#2956	1	1	yes	characteristic, complemented, Sylow
431	C₂²×C₅	—	2	2	no	—
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	2	2	no	—
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	4	4	no	—
437	C₂²×C₅	—	4	1	no	—
438	C₂²×C₅	—	4	1	no	—
439	C₂²×C₅	—	4	4	no	—
440	C₂²×C₅	—	8	1	no	—
441	C₂²×C₅	—	8	1	no	—
442	C₂²×C₅	—	8	4	no	—
443	C₂²×C₅	—	10	2	no	—
444	C₂²×C₅	—	40	1	no	—
445	C₂²×C₅	#4027	1	1	yes	characteristic
446	C₂²×C₅	#4027	1	1	yes	characteristic
447	C₂²×C₅	C₄×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₅	C₄×C₅×D₄	1	1	yes	characteristic
449	C₂²×C₅	C₂×#711	1	1	yes	characteristic
450	C₂²×C₅	C₂×C₅×#41	1	1	yes	characteristic
451	C₂×D₅	—	2	2	no	—
452	C₂×D₅	—	8	1	no	—
453	C₄×C₅	—	2	2	no	—
454	C₄×C₅	—	2	2	no	—
455	C₄×C₅	—	2	2	no	—
456	C₄×C₅	—	2	2	no	—
457	C₄×C₅	—	2	2	no	—
458	C₄×C₅	—	2	2	no	—
459	C₄×C₅	—	2	2	no	—
460	C₄×C₅	—	2	2	no	—
461	C₄×C₅	—	4	4	no	—
462	C₄×C₅	—	4	4	no	—
463	C₄×C₅	—	4	1	no	—
464	C₄×C₅	—	4	1	no	—
465	C₄×C₅	—	4	4	no	—
466	C₄×C₅	—	10	2	no	—
467	C₄×C₅	#4027	1	1	yes	characteristic
468	C₄×C₅	#4027	1	1	yes	characteristic
469	C₄×C₅	C₄×C₅×D₄	1	1	yes	characteristic
470	C₄×C₅	C₄×C₅×D₄	1	1	yes	characteristic
471	C₄×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
472	C₄×C₅	C₂×D₄×D₅	1	1	yes	characteristic
473	C₄×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
474	C₄×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
475	Dic₅	—	2	2	no	—
476	#41	—	2	1	no	—
477	#41	—	2	1	no	—
478	#41	—	5	2	no	—
479	#41	—	5	2	no	—
480	#41	—	10	1	no	—
481	#41	—	10	1	no	—
482	#41	C₄×C₅×D₅	1	2	yes	—
483	#41	C₂²×C₅×D₅	1	2	yes	—
484	C₂×Q₈	—	5	1	no	—
485	C₂×Q₈	C₄×C₅×D₅	1	1	yes	characteristic
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	5	1	no	—
488	C₂×D₄	—	10	1	no	—
489	C₂×D₄	C₄×C₅×D₅	1	1	yes	characteristic
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	10	1	no	—
493	C₂²×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
494	Q₁₆	—	5	4	no	—
495	SD₁₆	—	5	4	no	—
496	SD₁₆	—	5	4	no	—
497	D₈	—	5	4	no	—
498	M₁₆	—	2	1	no	—
499	M₁₆	—	5	2	no	—
500	M₁₆	—	5	2	no	—
501	M₁₆	—	5	2	no	—
502	M₁₆	—	5	2	no	—
503	M₁₆	—	5	2	no	—
504	M₁₆	—	10	4	no	—
505	M₁₆	—	10	4	no	—
506	M₁₆	—	10	1	no	—
507	M₁₆	C₂²×C₅×D₅	1	2	yes	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	5	2	no	—
510	C₂×C₈	—	5	2	no	—
511	C₂×C₈	—	10	4	no	—
512	C₂×C₈	—	10	4	no	—
513	C₂×C₈	—	10	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	C₂²×C₅×D₅	1	1	yes	characteristic
516	C₄²	C₂²×C₅×D₅	1	1	yes	characteristic
517	C₂×C₅	—	2	4	no	—
518	C₂×C₅	—	2	1	no	—
519	C₂×C₅	—	2	1	no	—
520	C₂×C₅	—	2	1	no	—
521	C₂×C₅	—	2	1	no	—
522	C₂×C₅	—	2	1	no	—
523	C₂×C₅	—	2	1	no	—
524	C₂×C₅	—	2	2	no	—
525	C₂×C₅	—	4	2	no	—
526	C₂×C₅	—	4	2	no	—
527	C₂×C₅	—	4	2	no	—
528	C₂×C₅	—	4	2	no	—
529	C₂×C₅	—	8	4	no	—
530	C₂×C₅	—	20	2	no	—
531	C₂×C₅	C₂×#4027	1	1	yes	characteristic
532	C₂×C₅	C₂×C₄×C₅×D₄	1	1	yes	characteristic
533	D₅	—	4	2	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	10	1	no	—
536	C₂³	C₅×#711	1	1	yes	characteristic
537	Q₈	—	2	1	no	—
538	Q₈	—	5	2	no	—
539	Q₈	—	10	1	no	—
540	Q₈	C₂×C₄×C₅×D₅	1	2	yes	—
541	D₄	—	2	2	no	—
542	D₄	—	2	2	no	—
543	D₄	—	2	1	no	—
544	D₄	—	5	2	no	—
545	D₄	—	10	2	no	—
546	D₄	—	10	2	no	—
547	D₄	—	10	1	no	—
548	D₄	C₂×C₄×C₅×D₅	1	2	yes	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	10	1	no	—
556	C₂×C₄	—	10	2	no	—
557	C₂×C₄	—	10	2	no	—
558	C₂×C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
559	C₂×C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
560	C₂×C₄	C₅×D₄×D₅	1	1	yes	characteristic
561	C₂×C₄	C₅×D₄×D₅	1	1	yes	characteristic
562	C₂×C₄	C₅×#711	1	1	yes	characteristic
563	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
564	C₈	—	2	2	no	—
565	C₈	—	5	4	no	—
566	C₈	—	5	4	no	—
567	C₈	—	10	2	no	—
568	C₅	—	2	2	no	—
569	C₅	#10610711	1	1	yes	characteristic, direct product factor
570	C₅	C₅×#2956	1	1	yes	characteristic, complemented
571	C₂²	—	2	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	8	1	no	—
574	C₂²	—	10	2	no	—
575	C₂²	—	40	1	no	—
576	C₂²	C₅×#4027	1	1	yes	characteristic
577	C₂²	C₅×#4027	1	1	yes	characteristic
578	C₂²	C₂×C₅×#711	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	10	2	no	—
583	C₄	C₅×#4027	1	1	yes	characteristic
584	C₄	C₅×#4027	1	1	yes	characteristic
585	C₄	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
586	C₄	C₂×C₅×D₄×D₅	1	1	yes	characteristic
587	C₂	—	2	1	no	—
588	C₂	—	2	1	no	—
589	C₂	—	2	1	no	—
590	C₂	—	4	2	no	—
591	C₂	—	20	2	no	—
592	C₂	C₂×C₅×#4027	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×#104	C₂
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	C₅×#66156	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×#104	D₅
hypocenter	C₅	C₅×#2956
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#104

Complements

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

N	H ≅ G/N	type
2. C₅×#66153	C₂	semidirect
3. C₅×#66079	C₂	semidirect
4. C₂×C₅²×#104	C₂	semidirect
5. C₅×#65413	C₂	semidirect
6. C₂×C₅×#3937	C₂	semidirect
7. C₄×C₅×#681	C₂	semidirect
8. C₅×#65113	C₂	semidirect
9. C₅×#65112	C₂	semidirect
14. C₅×#4097	C₄	semidirect
16. C₅×#4095	C₄	semidirect
18. C₅×#4094	C₂²	semidirect
19. C₅×#4088	C₄	semidirect
20. C₅×#4078	C₄	semidirect
25. C₅×#4053	C₂²	semidirect
28. C₅²×#104	C₂²	semidirect
29. C₅²×#104	C₂²	semidirect
30. C₅×#3954	C₂²	semidirect
31. C₅×#3937	C₂²	semidirect
32. C₅×#3937	C₂²	semidirect
33. C₅×#3935	C₂²	semidirect
34. C₄×C₅×#168	C₂²	semidirect
36. #10610711	C₅	direct (N × H)
61. C₅×#689	C₂×C₄	semidirect
62. C₅×#688	C₂×C₄	semidirect
63. C₅×#687	C₂×C₄	semidirect
64. C₅×#686	C₂×C₄	semidirect
82. #66153	C₂×C₅	semidirect
83. #66079	C₂×C₅	semidirect
89. C₂×C₅×#104	D₅	semidirect
90. C₂×C₅×#104	C₂×C₅	semidirect
92. #65413	C₂×C₅	semidirect
93. C₂×#3937	C₂×C₅	semidirect
94. C₄×#681	C₂×C₅	semidirect
95. #65113	C₂×C₅	semidirect
96. #65112	C₂×C₅	semidirect
156. #4097	C₄×C₅	semidirect
158. #4095	C₄×C₅	semidirect
160. #4094	C₂²×C₅	semidirect
161. #4088	C₄×C₅	semidirect
162. #4078	C₄×C₅	semidirect
167. #4053	C₂²×C₅	semidirect
175. C₅×#104	C₂×D₅	semidirect
176. C₅×#104	C₂×D₅	semidirect
177. C₅×#104	C₂²×C₅	semidirect
178. C₅×#104	C₂²×C₅	semidirect
181. #3954	C₂²×C₅	semidirect
182. #3937	C₂²×C₅	semidirect
183. #3937	C₂²×C₅	semidirect
184. #3935	C₂²×C₅	semidirect
185. C₄×#168	C₂²×C₅	semidirect
284. #689	C₂×C₄×C₅	semidirect
285. #688	C₂×C₄×C₅	semidirect
286. #687	C₂×C₄×C₅	semidirect
287. #686	C₂×C₄×C₅	semidirect
308. C₂×#104	C₅×D₅	semidirect
426. #104	C₂×C₅×D₅	semidirect
427. #104	C₂×C₅×D₅	semidirect
430. C₅²	#2956	semidirect
569. C₅	#10610711	direct (N × H)
570. C₅	C₅×#2956	semidirect

Maximal subgroups

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

subgroup	index
C₅×#66156	2
C₅×#66153	2
C₅×#66079	2
C₂×C₅²×#104	2
C₅×#65413	2
C₂×C₅×#3937	2
C₄×C₅×#681	2
C₅×#65113	2
C₅×#65112	2
C₅×#2956	5
#10610711	5

Subgroups — C5×#10610711 #72223430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10610711 #72223430