Subgroups — C₅×#10612597 #72242547

Order 3200.

1098 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#4058	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#4053	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#65942	C₂	1	2	yes	maximal, complemented
4	C₅²×#469	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#65448	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#65426	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#65330	C₂	1	2	yes	maximal, complemented
8	C₅×#65157	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#65067	C₂	1	2	yes	maximal, complemented
11	C₅×#65060	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#711	C₂²	1	2	yes	—
13	C₂×C₅×#709	C₂²	1	1	yes	characteristic
14	C₂×C₅²×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₅²×M₁₆	C₂²	1	1	yes	characteristic
16	C₅²×#121	C₂²	1	2	yes	—
17	C₅×#4085	—	2	2	no	—
18	C₅×#4084	—	2	2	no	—
19	C₅×#4084	C₂²	1	2	yes	—
20	C₂×C₅×#687	C₂²	1	2	yes	—
21	C₅×#4070	C₂²	1	2	yes	—
22	C₂²×C₅×#168	C₂²	1	1	yes	characteristic
23	C₅×#4058	—	2	4	no	—
24	C₅×#4053	—	2	4	no	—
25	C₂×C₅×Dic₂₀	C₂²	1	1	yes	characteristic
26	C₂×C₅×#676	C₂²	1	1	yes	characteristic
27	C₂×C₅×#676	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₈×D₅	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₈×D₅	C₂²	1	1	yes	characteristic
30	C₅×#4043	C₂²	1	2	yes	—
31	C₅×D₅×C₄⋊C₄	C₂²	1	2	yes	—
32	C₅×#4030	—	2	2	no	—
33	C₅×#4024	—	2	2	no	—
34	C₅²×#106	C₂²	1	1	yes	characteristic
35	C₅²×#102	C₂²	1	2	yes	—
36	C₅×#3964	C₂²	1	2	yes	—
37	C₅×#3949	C₂²	1	1	yes	characteristic
38	C₅×#3948	C₂²	1	2	yes	—
39	C₅×#3942	C₂²	1	2	yes	—
40	C₅×#3940	C₂²	1	2	yes	—
41	C₅×#3060	—	5	1	no	maximal
42	#10612597	C₅	1	1	yes	maximal, characteristic, direct product factor
43	C₂×C₅²×D₄	—	2	2	no	—
44	C₂×C₅²×D₄	C₂³	1	2	yes	—
45	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₂²⋊D₅	—	2	2	no	—
47	C₂×C₅×C₂²⋊D₅	—	2	2	no	—
48	C₂×C₅×C₂²⋊D₅	D₄	1	1	yes	characteristic
49	C₂×C₅×C₂²⋊D₅	D₄	1	1	yes	characteristic
50	C₂²×C₅×Dic₅	—	2	2	no	—
51	C₅×#711	—	2	4	no	—
52	C₅×#711	—	2	4	no	—
53	C₅×#709	—	2	2	no	—
54	C₅×#709	—	2	2	no	—
55	C₂×C₅×D₂₀	D₄	1	1	yes	characteristic
56	C₂×C₄×C₅×D₅	—	2	2	no	—
57	C₂×C₄×C₅×D₅	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅×D₅	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₁₀	D₄	1	1	yes	characteristic
60	C₂×C₅×Dic₁₀	C₂³	1	2	yes	—
61	C₅²×D₈	—	2	2	no	—
62	C₅²×M₁₆	—	2	2	no	—
63	C₂×C₅²×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₅²×C₈	C₂³	1	1	yes	characteristic
65	C₅²×C₄⋊C₄	C₂³	1	2	yes	—
66	C₅²×C₂²⋊C₄	—	2	2	no	—
67	C₅×C₂²⋊Dic₅	—	2	2	no	—
68	C₅×C₂²⋊Dic₅	—	2	2	no	—
69	C₅×#687	—	2	4	no	—
70	C₅×#685	—	2	2	no	—
71	C₅×C₄⋊Dic₅	—	2	2	no	—
72	C₅×C₄⋊Dic₅	C₂³	1	2	yes	—
73	C₅×#683	—	2	2	no	—
74	C₅×#683	—	2	2	no	—
75	C₂×C₅×#168	—	2	2	no	—
76	C₂×C₅×#168	C₂³	1	1	yes	characteristic
77	C₂×C₅×#168	C₂³	1	1	yes	characteristic
78	C₅×Dic₂₀	—	2	2	no	—
79	C₅×#676	—	2	2	no	—
80	C₅×#676	—	2	2	no	—
81	C₅×C₈×D₅	—	2	2	no	—
82	C₅×C₈×D₅	D₄	1	4	yes	complemented
83	C₂×C₅×#135	—	5	1	no	—
84	C₂×C₅×#131	—	5	1	no	—
85	C₅×#537	—	5	2	no	—
86	C₂×#4058	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂×#4053	C₂×C₅	1	1	yes	characteristic, complemented
88	#65942	C₂×C₅	1	2	yes	complemented
89	C₅×#470	—	5	1	no	—
90	C₅×#470	—	5	1	no	—
91	C₅×#469	—	2	2	no	—
92	C₅×#469	—	5	1	no	—
93	C₅×#469	D₅	1	1	yes	characteristic, complemented
94	C₅×#469	C₂×C₅	1	1	yes	characteristic, complemented
95	C₅×#465	—	5	2	no	—
96	C₅×#465	—	5	2	no	—
97	C₂×C₅×#106	—	5	1	no	—
98	C₅×#417	—	5	2	no	—
99	#65448	C₂×C₅	1	1	yes	characteristic, complemented
100	#65426	C₂×C₅	1	1	yes	characteristic, complemented
101	#65330	C₂×C₅	1	2	yes	complemented
102	#65157	C₂×C₅	1	1	yes	characteristic, complemented
103	D₅×#106	C₂×C₅	1	1	yes	characteristic, complemented
104	#65067	C₂×C₅	1	2	yes	complemented
105	#65060	C₂×C₅	1	2	yes	complemented
106	C₂³×C₅²	—	2	2	no	—
107	C₂³×C₅²	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₅×D₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₅×D₅	C₂×D₄	1	1	yes	characteristic
110	C₅²×D₄	—	2	2	no	—
111	C₅²×D₄	—	2	2	no	—
112	C₅²×D₄	—	8	2	no	—
113	C₂×C₄×C₅²	—	2	1	no	—
114	C₂×C₄×C₅²	—	2	1	no	—
115	C₂×C₄×C₅²	—	2	2	no	—
116	C₂×C₄×C₅²	C₂×D₄	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	C₅×C₂²⋊D₅	—	4	4	no	—
121	C₅×C₂²⋊D₅	—	8	2	no	—
122	C₂×C₅×Dic₅	—	2	2	no	—
123	C₂×C₅×Dic₅	—	2	2	no	—
124	C₂×C₅×Dic₅	—	4	4	no	—
125	C₂×C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
126	C₂×C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
127	C₅×D₂₀	—	2	2	no	—
128	C₄×C₅×D₅	—	2	1	no	—
129	C₄×C₅×D₅	—	2	1	no	—
130	C₄×C₅×D₅	—	8	2	no	—
131	C₄×C₅×D₅	C₂×D₄	1	2	yes	—
132	C₄×C₅×D₅	#41	1	2	yes	—
133	C₅×Dic₁₀	—	2	2	no	—
134	C₅×Dic₁₀	—	2	2	no	—
135	C₅×Dic₁₀	—	2	2	no	—
136	C₅²×C₈	—	2	1	no	—
137	C₅²×C₈	C₂×D₄	1	2	yes	—
138	C₅×#168	—	2	1	no	—
139	C₅×#168	C₂×D₄	1	2	yes	—
140	C₂×C₅×#41	—	5	1	no	—
141	C₂×C₅×#41	—	5	2	no	—
142	C₂×#711	C₂²×C₅	1	2	yes	—
143	C₂×#709	C₂²×C₅	1	1	yes	characteristic
144	C₅×#135	—	10	4	no	—
145	C₂×C₅×Q₁₆	—	5	1	no	—
146	C₂×C₅×SD₁₆	—	5	2	no	—
147	C₂×C₅×D₈	—	2	2	no	—
148	C₂×C₅×D₈	C₂×D₅	1	1	yes	characteristic
149	C₂×C₅×D₈	C₂²×C₅	1	1	yes	characteristic
150	C₅×#131	—	10	4	no	—
151	C₂×C₅×M₁₆	—	2	2	no	—
152	C₂×C₅×M₁₆	—	5	1	no	—
153	C₂×C₅×M₁₆	—	5	1	no	—
154	C₂×C₅×M₁₆	C₂×D₅	1	1	yes	characteristic
155	C₂×C₅×M₁₆	C₂²×C₅	1	1	yes	characteristic
156	C₂²×C₅×C₈	—	5	1	no	—
157	C₂²×C₅×C₈	—	5	1	no	—
158	C₂²×C₅×C₈	—	5	1	no	—
159	C₅×C₂²⋊Q₈	—	5	2	no	—
160	C₅×C₂²⋊Q₈	—	5	2	no	—
161	C₅×C₂²⋊Q₈	—	10	2	no	—
162	C₅×#121	—	2	4	no	—
163	C₅×#121	—	5	2	no	—
164	C₅×#121	—	10	2	no	—
165	C₅×#121	—	10	2	no	—
166	C₅×#121	—	10	2	no	—
167	C₅×#121	C₂×D₅	1	2	yes	—
168	C₅×#121	C₂²×C₅	1	2	yes	—
169	C₂×C₅×C₄⋊C₄	—	5	2	no	—
170	#4085	—	2	2	no	—
171	#4084	—	2	2	no	—
172	#4084	C₂²×C₅	1	2	yes	—
173	C₂×#687	C₂²×C₅	1	2	yes	—
174	#4070	C₂²×C₅	1	2	yes	—
175	C₂²×#168	C₂²×C₅	1	1	yes	characteristic
176	#4058	—	2	4	no	—
177	#4053	—	2	4	no	—
178	C₂×Dic₂₀	C₂²×C₅	1	1	yes	characteristic
179	C₂×#676	C₂²×C₅	1	1	yes	characteristic
180	C₂×#676	C₂²×C₅	1	1	yes	characteristic
181	C₂×C₈×D₅	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₈×D₅	C₂²×C₅	1	1	yes	characteristic
183	#4043	C₂²×C₅	1	2	yes	—
184	D₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
185	#4030	—	2	2	no	—
186	#4024	—	2	2	no	—
187	C₅×#106	—	2	2	no	—
188	C₅×#106	—	5	1	no	—
189	C₅×#106	—	5	2	no	—
190	C₅×#106	C₂×D₅	1	1	yes	characteristic
191	C₅×#106	C₂²×C₅	1	1	yes	characteristic
192	C₅×#103	—	5	2	no	—
193	C₅×#103	—	5	2	no	—
194	C₅×#102	—	2	4	no	—
195	C₅×#102	—	5	2	no	—
196	C₅×#102	C₂×D₅	1	2	yes	—
197	C₅×#102	C₂²×C₅	1	2	yes	—
198	#3964	C₂²×C₅	1	2	yes	—
199	#3949	C₂²×C₅	1	1	yes	characteristic
200	#3948	C₂²×C₅	1	2	yes	—
201	#3942	C₂²×C₅	1	2	yes	—
202	#3940	C₂²×C₅	1	2	yes	—
203	#3060	—	5	1	no	Sylow
204	C₂²×C₅²	—	2	1	no	—
205	C₂²×C₅²	—	2	2	no	—
206	C₂²×C₅²	—	4	2	no	—
207	C₂²×C₅²	—	4	2	no	—
208	C₂²×C₅²	—	4	2	no	—
209	C₂²×C₅²	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₅×D₅	—	2	1	no	—
211	C₂×C₅×D₅	—	2	2	no	—
212	C₂×C₅×D₅	—	2	2	no	—
213	C₂×C₅×D₅	#121	1	2	yes	—
214	C₄×C₅²	—	2	1	no	—
215	C₄×C₅²	—	4	2	no	—
216	C₄×C₅²	C₂²×D₄	1	1	yes	characteristic
217	C₄×C₅²	C₂×#41	1	1	yes	characteristic
218	C₅×Dic₅	—	2	1	no	—
219	C₅×Dic₅	—	4	2	no	—
220	C₅×Dic₅	—	4	2	no	—
221	C₅×Dic₅	#121	1	2	yes	—
222	C₅×#41	—	10	2	no	—
223	C₅×#41	—	10	2	no	—
224	C₅×#41	—	10	4	no	—
225	C₅×#41	—	10	4	no	—
226	C₂×C₅×Q₈	—	5	1	no	—
227	C₂×C₅×Q₈	—	5	2	no	—
228	C₂×C₅×D₄	—	2	2	no	—
229	C₂×C₅×D₄	—	2	2	no	—
230	C₂×C₅×D₄	—	2	4	no	—
231	C₂×C₅×D₄	—	4	4	no	—
232	C₂×C₅×D₄	—	5	1	no	—
233	C₂×C₅×D₄	—	5	1	no	—
234	C₂×C₅×D₄	—	5	1	no	—
235	C₂×C₅×D₄	—	10	2	no	—
236	C₂×C₅×D₄	—	10	2	no	—
237	C₂×C₅×D₄	C₂²×D₅	1	2	yes	—
238	C₂×C₅×D₄	C₂³×C₅	1	2	yes	—
239	C₂²×C₄×C₅	—	2	2	no	—
240	C₂²×C₄×C₅	—	5	1	no	—
241	C₂²×C₄×C₅	—	5	1	no	—
242	C₂²×C₄×C₅	—	10	2	no	—
243	C₂²×C₄×C₅	—	10	2	no	—
244	C₂²×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
245	C₂²×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
246	C₂×C₂²⋊D₅	—	2	2	no	—
247	C₂×C₂²⋊D₅	—	2	2	no	—
248	C₂×C₂²⋊D₅	C₅×D₄	1	1	yes	characteristic
249	C₂×C₂²⋊D₅	C₅×D₄	1	1	yes	characteristic
250	C₂²×Dic₅	—	2	2	no	—
251	#711	—	2	4	no	—
252	#711	—	2	4	no	—
253	#709	—	2	2	no	—
254	#709	—	2	2	no	—
255	C₂×D₂₀	C₅×D₄	1	1	yes	characteristic
256	C₂×C₄×D₅	—	2	2	no	—
257	C₂×C₄×D₅	C₂³×C₅	1	1	yes	characteristic
258	C₂×C₄×D₅	C₂³×C₅	1	1	yes	characteristic
259	C₂×Dic₁₀	C₅×D₄	1	1	yes	characteristic
260	C₂×Dic₁₀	C₂³×C₅	1	2	yes	—
261	C₅×Q₁₆	—	10	2	no	—
262	C₅×SD₁₆	—	10	4	no	—
263	C₅×D₈	—	2	2	no	—
264	C₅×D₈	—	2	2	no	—
265	C₅×D₈	—	4	4	no	—
266	C₅×M₁₆	—	2	2	no	—
267	C₅×M₁₆	—	2	2	no	—
268	C₅×M₁₆	—	4	4	no	—
269	C₅×M₁₆	—	10	2	no	—
270	C₅×M₁₆	—	10	2	no	—
271	C₂×C₅×C₈	—	2	2	no	—
272	C₂×C₅×C₈	—	2	2	no	—
273	C₂×C₅×C₈	—	5	1	no	—
274	C₂×C₅×C₈	—	5	4	no	—
275	C₂×C₅×C₈	—	5	1	no	—
276	C₂×C₅×C₈	—	10	2	no	—
277	C₂×C₅×C₈	—	10	2	no	—
278	C₂×C₅×C₈	C₂²×D₅	1	1	yes	characteristic
279	C₂×C₅×C₈	C₂²×D₅	1	1	yes	characteristic
280	C₂×C₅×C₈	C₂³×C₅	1	1	yes	characteristic
281	C₂×C₅×C₈	C₂³×C₅	1	1	yes	characteristic
282	C₅×C₄⋊C₄	—	2	4	no	—
283	C₅×C₄⋊C₄	—	5	2	no	—
284	C₅×C₄⋊C₄	—	10	2	no	—
285	C₅×C₄⋊C₄	—	10	2	no	—
286	C₅×C₄⋊C₄	—	10	2	no	—
287	C₅×C₄⋊C₄	C₂²×D₅	1	2	yes	—
288	C₅×C₄⋊C₄	C₂³×C₅	1	2	yes	—
289	C₅×C₂²⋊C₄	—	2	2	no	—
290	C₅×C₂²⋊C₄	—	2	2	no	—
291	C₅×C₂²⋊C₄	—	4	4	no	—
292	C₅×C₂²⋊C₄	—	10	2	no	—
293	C₅×C₂²⋊C₄	—	10	2	no	—
294	C₅×C₂²⋊C₄	—	10	2	no	—
295	C₂²⋊Dic₅	—	2	2	no	—
296	C₂²⋊Dic₅	—	2	2	no	—
297	#687	—	2	4	no	—
298	#685	—	2	2	no	—
299	C₄⋊Dic₅	—	2	2	no	—
300	C₄⋊Dic₅	C₂³×C₅	1	2	yes	—
301	#683	—	2	2	no	—
302	#683	—	2	2	no	—
303	C₂×#168	—	2	2	no	—
304	C₂×#168	C₂³×C₅	1	1	yes	characteristic
305	C₂×#168	C₂³×C₅	1	1	yes	characteristic
306	Dic₂₀	—	2	2	no	—
307	#676	—	2	2	no	—
308	#676	—	2	2	no	—
309	C₈×D₅	—	2	2	no	—
310	C₈×D₅	C₅×D₄	1	4	yes	complemented
311	C₂×#135	—	5	1	no	—
312	C₂×#131	—	5	1	no	—
313	#537	—	5	2	no	—
314	#470	—	5	1	no	—
315	#470	—	5	1	no	—
316	#469	—	5	1	no	—
317	#469	C₅×D₅	1	1	yes	characteristic, complemented
318	#465	—	5	2	no	—
319	#465	—	5	2	no	—
320	C₂×#106	—	5	1	no	—
321	#417	—	5	2	no	—
322	C₂×C₅²	—	4	1	no	—
323	C₂×C₅²	—	8	2	no	—
324	C₂×C₅²	C₂×#121	1	1	yes	characteristic
325	C₂×C₅²	#577	1	2	yes	—
326	C₅×D₅	—	2	2	no	—
327	C₅×D₅	—	4	1	no	—
328	C₂³×C₅	—	2	2	no	—
329	C₂³×C₅	—	2	2	no	—
330	C₂³×C₅	—	2	2	no	—
331	C₂³×C₅	—	4	4	no	—
332	C₂³×C₅	—	5	1	no	—
333	C₂³×C₅	—	5	1	no	—
334	C₂³×C₅	D₄×D₅	1	1	yes	characteristic
335	C₂³×C₅	C₂×C₅×D₄	1	1	yes	characteristic
336	C₂²×D₅	C₂×C₅×D₄	1	1	yes	characteristic
337	C₂²×D₅	C₂×C₅×D₄	1	1	yes	characteristic
338	C₅×Q₈	—	10	2	no	—
339	C₅×Q₈	—	10	2	no	—
340	C₅×Q₈	—	10	2	no	—
341	C₅×D₄	—	2	2	no	—
342	C₅×D₄	—	2	2	no	—
343	C₅×D₄	—	2	2	no	—
344	C₅×D₄	—	2	2	no	—
345	C₅×D₄	—	4	4	no	—
346	C₅×D₄	—	4	4	no	—
347	C₅×D₄	—	8	2	no	—
348	C₅×D₄	—	8	2	no	—
349	C₅×D₄	—	8	4	no	—
350	C₅×D₄	—	8	4	no	—
351	C₅×D₄	—	10	2	no	—
352	C₅×D₄	—	10	2	no	—
353	C₅×D₄	—	10	2	no	—
354	C₅×D₄	—	20	4	no	—
355	C₅×D₄	—	40	2	no	—
356	C₂×C₄×C₅	—	2	1	no	—
357	C₂×C₄×C₅	—	2	1	no	—
358	C₂×C₄×C₅	—	2	1	no	—
359	C₂×C₄×C₅	—	2	1	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	2	no	—
365	C₂×C₄×C₅	—	4	2	no	—
366	C₂×C₄×C₅	—	4	4	no	—
367	C₂×C₄×C₅	—	5	1	no	—
368	C₂×C₄×C₅	—	5	2	no	—
369	C₂×C₄×C₅	—	5	2	no	—
370	C₂×C₄×C₅	—	5	1	no	—
371	C₂×C₄×C₅	—	10	1	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	—	20	4	no	—
376	C₂×C₄×C₅	—	40	2	no	—
377	C₂×C₄×C₅	D₄×D₅	1	1	yes	characteristic
378	C₂×C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
379	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
380	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
381	C₂²⋊D₅	—	2	2	no	—
382	C₂²⋊D₅	—	2	2	no	—
383	C₂²⋊D₅	—	4	4	no	—
384	C₂²⋊D₅	—	8	2	no	—
385	C₂×Dic₅	—	2	2	no	—
386	C₂×Dic₅	—	2	2	no	—
387	C₂×Dic₅	—	4	4	no	—
388	C₂×Dic₅	C₂×C₅×D₄	1	1	yes	characteristic
389	C₂×Dic₅	C₂×C₅×D₄	1	1	yes	characteristic
390	D₂₀	—	2	2	no	—
391	C₄×D₅	—	2	1	no	—
392	C₄×D₅	—	2	1	no	—
393	C₄×D₅	—	8	2	no	—
394	C₄×D₅	C₂×C₅×D₄	1	2	yes	—
395	C₄×D₅	C₅×#41	1	2	yes	—
396	Dic₁₀	—	2	2	no	—
397	Dic₁₀	—	2	2	no	—
398	Dic₁₀	—	2	2	no	—
399	C₅×C₈	—	2	1	no	—
400	C₅×C₈	—	2	1	no	—
401	C₅×C₈	—	2	4	no	—
402	C₅×C₈	—	4	2	no	—
403	C₅×C₈	—	5	2	no	—
404	C₅×C₈	—	10	1	no	—
405	C₅×C₈	D₄×D₅	1	2	yes	—
406	C₅×C₈	C₂×C₅×D₄	1	2	yes	—
407	#168	—	2	1	no	—
408	#168	C₂×C₅×D₄	1	2	yes	—
409	C₂×#41	—	5	1	no	—
410	C₂×#41	—	5	2	no	—
411	#135	—	10	4	no	—
412	C₂×Q₁₆	—	5	1	no	—
413	C₂×SD₁₆	—	5	2	no	—
414	C₂×D₈	C₂×C₅×D₅	1	1	yes	characteristic
415	#131	—	10	4	no	—
416	C₂×M₁₆	—	5	1	no	—
417	C₂×M₁₆	—	5	1	no	—
418	C₂×M₁₆	C₂×C₅×D₅	1	1	yes	characteristic
419	C₂²×C₈	—	5	1	no	—
420	C₂²×C₈	—	5	1	no	—
421	C₂²×C₈	—	5	1	no	—
422	C₂²⋊Q₈	—	5	2	no	—
423	C₂²⋊Q₈	—	5	2	no	—
424	C₂²⋊Q₈	—	10	2	no	—
425	#121	—	5	2	no	—
426	#121	—	10	2	no	—
427	#121	—	10	2	no	—
428	#121	—	10	2	no	—
429	#121	C₂×C₅×D₅	1	2	yes	—
430	C₂×C₄⋊C₄	—	5	2	no	—
431	#106	—	5	1	no	—
432	#106	—	5	2	no	—
433	#106	C₂×C₅×D₅	1	1	yes	characteristic
434	#103	—	5	2	no	—
435	#103	—	5	2	no	—
436	#102	—	5	2	no	—
437	#102	C₂×C₅×D₅	1	2	yes	—
438	C₅²	#3060	1	1	yes	characteristic, complemented, Sylow
439	C₂²×C₅	—	2	2	no	—
440	C₂²×C₅	—	2	1	no	—
441	C₂²×C₅	—	2	1	no	—
442	C₂²×C₅	—	2	2	no	—
443	C₂²×C₅	—	2	2	no	—
444	C₂²×C₅	—	4	2	no	—
445	C₂²×C₅	—	4	4	no	—
446	C₂²×C₅	—	4	2	no	—
447	C₂²×C₅	—	4	2	no	—
448	C₂²×C₅	—	4	2	no	—
449	C₂²×C₅	—	4	2	no	—
450	C₂²×C₅	—	4	2	no	—
451	C₂²×C₅	—	4	2	no	—
452	C₂²×C₅	—	5	2	no	—
453	C₂²×C₅	—	8	4	no	—
454	C₂²×C₅	—	8	4	no	—
455	C₂²×C₅	—	8	4	no	—
456	C₂²×C₅	—	10	2	no	—
457	C₂²×C₅	—	10	2	no	—
458	C₂²×C₅	—	10	1	no	—
459	C₂²×C₅	C₂×D₄×D₅	1	1	yes	characteristic
460	C₂²×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
461	C₂×D₅	—	2	2	no	—
462	C₂×D₅	—	2	1	no	—
463	C₂×D₅	—	2	2	no	—
464	C₂×D₅	C₅×#121	1	2	yes	—
465	C₄×C₅	—	2	1	no	—
466	C₄×C₅	—	2	1	no	—
467	C₄×C₅	—	2	2	no	—
468	C₄×C₅	—	2	2	no	—
469	C₄×C₅	—	4	2	no	—
470	C₄×C₅	—	4	2	no	—
471	C₄×C₅	—	4	2	no	—
472	C₄×C₅	—	5	2	no	—
473	C₄×C₅	—	8	4	no	—
474	C₄×C₅	—	10	1	no	—
475	C₄×C₅	—	20	2	no	—
476	C₄×C₅	—	20	2	no	—
477	C₄×C₅	C₂×D₄×D₅	1	1	yes	characteristic
478	C₄×C₅	D₅×#41	1	1	yes	characteristic
479	C₄×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
480	C₄×C₅	C₂×C₅×#41	1	1	yes	characteristic
481	Dic₅	—	2	1	no	—
482	Dic₅	—	4	2	no	—
483	Dic₅	—	4	2	no	—
484	Dic₅	C₅×#121	1	2	yes	—
485	#41	—	10	2	no	—
486	#41	—	10	2	no	—
487	#41	—	10	4	no	—
488	#41	—	10	4	no	—
489	C₂×Q₈	—	5	1	no	—
490	C₂×Q₈	—	5	2	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	5	1	no	—
493	C₂×D₄	—	5	1	no	—
494	C₂×D₄	—	5	1	no	—
495	C₂×D₄	—	10	2	no	—
496	C₂×D₄	—	10	2	no	—
497	C₂×D₄	C₂²×C₅×D₅	1	2	yes	—
498	C₂²×C₄	—	5	1	no	—
499	C₂²×C₄	—	5	1	no	—
500	C₂²×C₄	—	10	2	no	—
501	C₂²×C₄	—	10	2	no	—
502	C₂²×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
503	Q₁₆	—	10	2	no	—
504	SD₁₆	—	10	4	no	—
505	D₈	—	2	2	no	—
506	M₁₆	—	2	2	no	—
507	M₁₆	—	10	2	no	—
508	M₁₆	—	10	2	no	—
509	C₂×C₈	—	5	1	no	—
510	C₂×C₈	—	5	4	no	—
511	C₂×C₈	—	5	1	no	—
512	C₂×C₈	—	10	2	no	—
513	C₂×C₈	—	10	2	no	—
514	C₂×C₈	C₂²×C₅×D₅	1	1	yes	characteristic
515	C₂×C₈	C₂²×C₅×D₅	1	1	yes	characteristic
516	C₄⋊C₄	—	5	2	no	—
517	C₄⋊C₄	—	10	2	no	—
518	C₄⋊C₄	—	10	2	no	—
519	C₄⋊C₄	—	10	2	no	—
520	C₄⋊C₄	C₂²×C₅×D₅	1	2	yes	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	10	2	no	—
523	C₂²⋊C₄	—	10	2	no	—
524	C₂²⋊C₄	—	10	2	no	—
525	C₂×C₅	—	2	2	no	—
526	C₂×C₅	—	2	4	no	—
527	C₂×C₅	—	4	1	no	—
528	C₂×C₅	—	4	1	no	—
529	C₂×C₅	—	8	2	no	—
530	C₂×C₅	—	8	2	no	—
531	C₂×C₅	—	8	2	no	—
532	C₂×C₅	—	10	2	no	—
533	C₂×C₅	—	16	4	no	—
534	C₂×C₅	—	20	1	no	—
535	C₂×C₅	D₅×#121	1	1	yes	characteristic
536	C₂×C₅	#66109	1	2	yes	—
537	C₂×C₅	C₂×C₅×#121	1	1	yes	characteristic
538	C₂×C₅	C₅×#577	1	2	yes	—
539	D₅	—	2	2	no	—
540	D₅	—	4	1	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	5	1	no	—
543	C₂³	—	5	1	no	—
544	C₂³	C₅×D₄×D₅	1	1	yes	characteristic
545	Q₈	—	10	2	no	—
546	Q₈	—	10	2	no	—
547	Q₈	—	10	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	2	2	no	—
550	D₄	—	8	2	no	—
551	D₄	—	10	2	no	—
552	D₄	—	10	2	no	—
553	D₄	—	10	2	no	—
554	D₄	—	20	4	no	—
555	D₄	—	40	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	5	1	no	—
560	C₂×C₄	—	5	2	no	—
561	C₂×C₄	—	5	2	no	—
562	C₂×C₄	—	5	1	no	—
563	C₂×C₄	—	10	1	no	—
564	C₂×C₄	—	10	1	no	—
565	C₂×C₄	—	10	2	no	—
566	C₂×C₄	—	10	2	no	—
567	C₂×C₄	—	20	4	no	—
568	C₂×C₄	—	40	2	no	—
569	C₂×C₄	C₅×D₄×D₅	1	1	yes	characteristic
570	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
571	C₈	—	2	1	no	—
572	C₈	—	5	2	no	—
573	C₈	—	10	1	no	—
574	C₈	C₅×D₄×D₅	1	2	yes	—
575	C₅	—	2	2	no	—
576	C₅	#10612597	1	1	yes	characteristic, direct product factor
577	C₅	C₅×#3060	1	1	yes	characteristic, complemented
578	C₂²	—	2	1	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	5	2	no	—
584	C₂²	—	10	2	no	—
585	C₂²	—	10	1	no	—
586	C₂²	—	10	2	no	—
587	C₂²	C₂×C₅×D₄×D₅	1	1	yes	characteristic
588	C₄	—	2	1	no	—
589	C₄	—	4	2	no	—
590	C₄	—	5	2	no	—
591	C₄	—	10	1	no	—
592	C₄	—	20	2	no	—
593	C₄	—	20	2	no	—
594	C₄	C₂×C₅×D₄×D₅	1	1	yes	characteristic
595	C₄	C₅×D₅×#41	1	1	yes	characteristic
596	C₂	—	4	1	no	—
597	C₂	—	8	2	no	—
598	C₂	—	10	2	no	—
599	C₂	—	20	1	no	—
600	C₂	C₅×D₅×#121	1	1	yes	characteristic
601	C₂	C₅×#66109	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₅
Fitting	C₅²×#469	C₂
socle	C₂²×C₅²	C₂²×D₄
hypercenter	C₅×#469	D₅
hypocenter	C₅	C₅×#3060
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#469

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#4058	C₂	semidirect
2. C₂×C₅×#4053	C₂	semidirect
3. C₅×#65942	C₂	semidirect
4. C₅²×#469	C₂	semidirect
5. C₅×#65448	C₂	semidirect
6. C₅×#65426	C₂	semidirect
7. C₅×#65330	C₂	semidirect
8. C₅×#65157	C₂	semidirect
9. C₅×D₅×#106	C₂	semidirect
10. C₅×#65067	C₂	semidirect
11. C₅×#65060	C₂	semidirect
42. #10612597	C₅	direct (N × H)
82. C₅×C₈×D₅	D₄	semidirect
86. C₂×#4058	C₂×C₅	semidirect
87. C₂×#4053	C₂×C₅	semidirect
88. #65942	C₂×C₅	semidirect
93. C₅×#469	D₅	semidirect
94. C₅×#469	C₂×C₅	semidirect
99. #65448	C₂×C₅	semidirect
100. #65426	C₂×C₅	semidirect
101. #65330	C₂×C₅	semidirect
102. #65157	C₂×C₅	semidirect
103. D₅×#106	C₂×C₅	semidirect
104. #65067	C₂×C₅	semidirect
105. #65060	C₂×C₅	semidirect
310. C₈×D₅	C₅×D₄	semidirect
317. #469	C₅×D₅	semidirect
438. C₅²	#3060	semidirect
576. C₅	#10612597	direct (N × H)
577. C₅	C₅×#3060	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#4058	2
C₂×C₅×#4053	2
C₅×#65942	2
C₅²×#469	2
C₅×#65448	2
C₅×#65426	2
C₅×#65330	2
C₅×#65157	2
C₅×D₅×#106	2
C₅×#65067	2
C₅×#65060	2
C₅×#3060	5
#10612597	5

Subgroups — C5×#10612597 #72242547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10612597 #72242547