Subgroups — C₅×#22222 #11944942

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#488	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
4	C₅×#1991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1958	C₂	1	1	yes	maximal, characteristic
6	C₅×#1928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1927	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1926	C₂	1	1	yes	maximal, characteristic
9	C₅×#1925	C₂	1	1	yes	maximal, characteristic
10	C₅×#1865	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1864	C₂	1	1	yes	maximal, characteristic
12	C₅×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1862	C₂	1	1	yes	maximal, characteristic
14	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#125	C₂²	1	1	yes	characteristic
17	C₂×C₅×#124	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
20	C₅×#488	C₄	1	8	yes	complemented
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#107	C₂²	1	1	yes	characteristic
23	C₂×C₅×#106	C₂²	1	1	yes	characteristic
24	C₂×C₅×#106	C₂²	1	1	yes	characteristic
25	C₂×C₅×#103	C₂²	1	1	yes	characteristic
26	C₂×C₅×#103	C₂²	1	1	yes	characteristic
27	C₂×C₅×#103	C₂²	1	1	yes	characteristic
28	C₂×C₅×#103	C₂²	1	1	yes	characteristic
29	C₂×C₅×#102	C₂²	1	1	yes	characteristic
30	C₂×C₅×#102	C₂²	1	1	yes	characteristic
31	C₂×C₅×#102	C₂²	1	1	yes	characteristic
32	C₂×C₅×#102	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₅×#391	C₂²	1	1	yes	characteristic
36	C₅×#390	C₂²	1	1	yes	characteristic
37	C₅×#389	C₂²	1	1	yes	characteristic
38	C₅×#389	C₂²	1	1	yes	characteristic
39	C₅×#388	—	2	1	no	—
40	C₅×#388	—	2	1	no	—
41	C₅×#387	—	2	1	no	—
42	C₅×#387	—	2	1	no	—
43	C₅×#382	C₂²	1	1	yes	characteristic
44	C₅×#382	C₂²	1	1	yes	characteristic
45	C₄×C₅×C₄⋊C₄	—	2	1	no	—
46	C₄×C₅×C₄⋊C₄	—	2	1	no	—
47	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
50	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
51	C₄³×C₅	C₂²	1	1	yes	characteristic
52	C₅×#340	C₂²	1	1	yes	characteristic
53	C₅×#340	C₂²	1	1	yes	characteristic
54	C₅×#340	C₂²	1	1	yes	characteristic
55	C₅×#340	C₂²	1	1	yes	characteristic
56	C₅×#336	C₂²	1	1	yes	characteristic
57	C₅×#336	C₂²	1	1	yes	characteristic
58	C₅×#336	C₂²	1	1	yes	characteristic
59	C₅×#336	C₂²	1	1	yes	characteristic
60	#22222	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅	—	2	1	no	—
64	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₅×#125	—	2	2	no	—
69	C₅×#125	C₂×C₄	1	4	yes	complemented
70	C₅×#124	—	2	2	no	—
71	C₅×#124	C₂×C₄	1	4	yes	—
72	C₄×C₅×Q₈	—	4	1	no	—
73	C₄×C₅×Q₈	—	4	1	no	—
74	C₄×C₅×D₄	—	4	1	no	—
75	C₄×C₅×D₄	—	4	1	no	—
76	C₂×C₅×C₄⋊C₄	—	2	1	no	—
77	C₂×C₅×C₄⋊C₄	—	2	1	no	—
78	C₂×C₅×C₄⋊C₄	—	2	1	no	—
79	C₂×C₅×C₄⋊C₄	—	2	1	no	—
80	C₂×C₅×C₄⋊C₄	—	2	1	no	—
81	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
88	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
89	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
90	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₄²×C₅	—	2	1	no	—
93	C₂×C₄²×C₅	—	2	1	no	—
94	C₂×C₄²×C₅	—	2	1	no	—
95	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
102	C₅×#107	—	4	1	no	—
103	C₅×#107	—	4	1	no	—
104	C₅×#106	—	4	1	no	—
105	C₅×#106	—	4	1	no	—
106	C₅×#103	—	4	1	no	—
107	C₅×#103	—	4	1	no	—
108	C₅×#103	C₂×C₄	1	4	yes	complemented
109	C₅×#103	C₂×C₄	1	4	yes	complemented
110	C₅×#102	—	4	1	no	—
111	C₅×#102	—	4	1	no	—
112	C₅×#102	C₂×C₄	1	4	yes	—
113	C₅×#102	C₂×C₄	1	4	yes	—
114	C₄×C₅×C₈	—	2	2	no	—
115	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
116	C₅×#95	—	2	1	no	—
117	C₅×#95	—	2	1	no	—
118	C₅×#95	—	2	1	no	—
119	C₅×#95	—	2	1	no	—
120	C₅×#95	—	2	1	no	—
121	C₅×#95	—	2	1	no	—
122	C₅×#95	—	2	1	no	—
123	C₅×#95	—	2	1	no	—
124	C₅×#95	—	2	1	no	—
125	C₅×#95	—	2	1	no	—
126	C₂×#488	C₂×C₅	1	1	yes	characteristic
127	C₄×#125	C₂×C₅	1	1	yes	characteristic, complemented
128	C₄×#124	C₂×C₅	1	1	yes	characteristic
129	#1991	C₂×C₅	1	1	yes	characteristic, complemented
130	#1958	C₂×C₅	1	1	yes	characteristic
131	#1928	C₂×C₅	1	1	yes	characteristic, complemented
132	#1927	C₂×C₅	1	1	yes	characteristic, complemented
133	#1926	C₂×C₅	1	1	yes	characteristic
134	#1925	C₂×C₅	1	1	yes	characteristic
135	#1865	C₂×C₅	1	1	yes	characteristic, complemented
136	#1864	C₂×C₅	1	1	yes	characteristic
137	#1863	C₂×C₅	1	1	yes	characteristic, complemented
138	#1862	C₂×C₅	1	1	yes	characteristic
139	#1846	C₂×C₅	1	1	yes	characteristic, complemented
140	#1744	C₂×C₅	1	1	yes	characteristic, complemented
141	C₂⁴×C₅	—	2	1	no	—
142	C₂×C₅×Q₈	—	2	1	no	—
143	C₂×C₅×Q₈	—	2	1	no	—
144	C₂×C₅×Q₈	—	2	1	no	—
145	C₂×C₅×Q₈	—	2	1	no	—
146	C₂×C₅×Q₈	—	2	1	no	—
147	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
148	C₂×C₅×D₄	—	2	1	no	—
149	C₂×C₅×D₄	—	2	1	no	—
150	C₂×C₅×D₄	—	2	1	no	—
151	C₂×C₅×D₄	—	2	1	no	—
152	C₂×C₅×D₄	—	2	1	no	—
153	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
154	C₂²×C₄×C₅	—	2	1	no	—
155	C₂²×C₄×C₅	—	2	1	no	—
156	C₂²×C₄×C₅	—	2	1	no	—
157	C₂²×C₄×C₅	—	2	1	no	—
158	C₂²×C₄×C₅	—	2	1	no	—
159	C₂²×C₄×C₅	—	2	1	no	—
160	C₂²×C₄×C₅	—	2	1	no	—
161	C₂²×C₄×C₅	—	4	2	no	—
162	C₂²×C₄×C₅	—	4	2	no	—
163	C₂²×C₄×C₅	—	4	2	no	—
164	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
171	C₂×C₅×C₈	—	2	1	no	—
172	C₂×C₅×C₈	—	2	1	no	—
173	C₂×C₅×C₈	—	2	1	no	—
174	C₂×C₅×C₈	—	2	1	no	—
175	C₂×C₅×C₈	—	2	1	no	—
176	C₂×C₅×C₈	—	2	1	no	—
177	C₂×C₅×C₈	—	2	1	no	—
178	C₂×C₅×C₈	—	2	1	no	—
179	C₂×C₅×C₈	—	2	1	no	—
180	C₂×C₅×C₈	—	2	1	no	—
181	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
182	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
183	C₅×C₄⋊C₄	—	2	4	no	—
184	C₅×C₄⋊C₄	—	2	4	no	—
185	C₅×C₄⋊C₄	—	2	1	no	—
186	C₅×C₄⋊C₄	—	2	1	no	—
187	C₅×C₄⋊C₄	—	2	1	no	—
188	C₅×C₄⋊C₄	—	2	1	no	—
189	C₅×C₄⋊C₄	—	2	1	no	—
190	C₅×C₄⋊C₄	—	2	1	no	—
191	C₅×C₄⋊C₄	—	2	1	no	—
192	C₅×C₄⋊C₄	—	2	1	no	—
193	C₅×C₄⋊C₄	—	2	1	no	—
194	C₅×C₄⋊C₄	—	2	1	no	—
195	C₅×C₄⋊C₄	—	4	2	no	—
196	C₅×C₄⋊C₄	—	4	2	no	—
197	C₅×C₄⋊C₄	—	4	2	no	—
198	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
199	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
200	C₅×C₂²⋊C₄	—	2	4	no	—
201	C₅×C₂²⋊C₄	—	2	4	no	—
202	C₅×C₂²⋊C₄	—	4	2	no	—
203	C₅×C₂²⋊C₄	—	4	2	no	—
204	C₅×C₂²⋊C₄	—	4	2	no	—
205	C₄²×C₅	—	2	2	no	—
206	C₄²×C₅	—	2	2	no	—
207	C₄²×C₅	—	2	1	no	—
208	C₄²×C₅	—	2	1	no	—
209	C₄²×C₅	—	2	1	no	—
210	C₄²×C₅	—	2	1	no	—
211	C₄²×C₅	—	4	1	no	—
212	C₄²×C₅	—	4	1	no	—
213	C₄²×C₅	—	4	2	no	—
214	C₄²×C₅	—	4	2	no	—
215	C₄²×C₅	—	4	2	no	—
216	C₄²×C₅	C₂²×C₄	1	2	yes	—
217	C₄²×C₅	#41	1	2	yes	—
218	C₂×#125	C₂²×C₅	1	1	yes	characteristic
219	C₂×#124	C₂²×C₅	1	1	yes	characteristic
220	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
222	#488	C₄×C₅	1	8	yes	complemented
223	C₂×#107	C₂²×C₅	1	1	yes	characteristic
224	C₂×#107	C₂²×C₅	1	1	yes	characteristic
225	C₂×#106	C₂²×C₅	1	1	yes	characteristic
226	C₂×#106	C₂²×C₅	1	1	yes	characteristic
227	C₂×#103	C₂²×C₅	1	1	yes	characteristic
228	C₂×#103	C₂²×C₅	1	1	yes	characteristic
229	C₂×#103	C₂²×C₅	1	1	yes	characteristic
230	C₂×#103	C₂²×C₅	1	1	yes	characteristic
231	C₂×#102	C₂²×C₅	1	1	yes	characteristic
232	C₂×#102	C₂²×C₅	1	1	yes	characteristic
233	C₂×#102	C₂²×C₅	1	1	yes	characteristic
234	C₂×#102	C₂²×C₅	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
237	#391	C₂²×C₅	1	1	yes	characteristic
238	#390	C₂²×C₅	1	1	yes	characteristic
239	#389	C₂²×C₅	1	1	yes	characteristic
240	#389	C₂²×C₅	1	1	yes	characteristic
241	#388	—	2	1	no	—
242	#388	—	2	1	no	—
243	#387	—	2	1	no	—
244	#387	—	2	1	no	—
245	#382	C₂²×C₅	1	1	yes	characteristic
246	#382	C₂²×C₅	1	1	yes	characteristic
247	C₄×C₄⋊C₄	—	2	1	no	—
248	C₄×C₄⋊C₄	—	2	1	no	—
249	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
250	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
251	C₄×C₂²⋊C₄	—	2	1	no	—
252	C₄×C₂²⋊C₄	—	2	1	no	—
253	C₄³	C₂²×C₅	1	1	yes	characteristic
254	#340	C₂²×C₅	1	1	yes	characteristic
255	#340	C₂²×C₅	1	1	yes	characteristic
256	#340	C₂²×C₅	1	1	yes	characteristic
257	#340	C₂²×C₅	1	1	yes	characteristic
258	#336	C₂²×C₅	1	1	yes	characteristic
259	#336	C₂²×C₅	1	1	yes	characteristic
260	#336	C₂²×C₅	1	1	yes	characteristic
261	#336	C₂²×C₅	1	1	yes	characteristic
262	C₂³×C₅	—	2	2	no	—
263	C₂³×C₅	—	4	1	no	—
264	C₂³×C₅	—	4	1	no	—
265	C₂³×C₅	—	4	1	no	—
266	C₂³×C₅	—	4	1	no	—
267	C₂³×C₅	—	4	1	no	—
268	C₂³×C₅	—	4	1	no	—
269	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
270	C₅×Q₈	—	2	2	no	—
271	C₅×Q₈	—	2	2	no	—
272	C₅×Q₈	—	4	1	no	—
273	C₅×Q₈	—	4	1	no	—
274	C₅×D₄	—	2	2	no	—
275	C₅×D₄	—	2	2	no	—
276	C₅×D₄	—	4	1	no	—
277	C₅×D₄	—	4	1	no	—
278	C₂×C₄×C₅	—	2	1	no	—
279	C₂×C₄×C₅	—	2	1	no	—
280	C₂×C₄×C₅	—	2	1	no	—
281	C₂×C₄×C₅	—	2	1	no	—
282	C₂×C₄×C₅	—	2	1	no	—
283	C₂×C₄×C₅	—	2	1	no	—
284	C₂×C₄×C₅	—	2	1	no	—
285	C₂×C₄×C₅	—	2	1	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	2	2	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	2	1	no	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	1	no	—
292	C₂×C₄×C₅	—	2	1	no	—
293	C₂×C₄×C₅	—	2	1	no	—
294	C₂×C₄×C₅	—	2	1	no	—
295	C₂×C₄×C₅	—	4	1	no	—
296	C₂×C₄×C₅	—	4	1	no	—
297	C₂×C₄×C₅	—	4	1	no	—
298	C₂×C₄×C₅	—	4	1	no	—
299	C₂×C₄×C₅	—	4	1	no	—
300	C₂×C₄×C₅	—	4	1	no	—
301	C₂×C₄×C₅	—	4	1	no	—
302	C₂×C₄×C₅	—	4	1	no	—
303	C₂×C₄×C₅	—	4	1	no	—
304	C₂×C₄×C₅	—	4	1	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	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	4	1	no	—
313	C₂×C₄×C₅	—	8	4	no	—
314	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
315	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
316	C₂×C₄×C₅	#135	1	2	yes	—
317	C₂×C₄×C₅	#135	1	2	yes	—
318	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
319	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
320	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
322	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
323	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
324	C₅×C₈	—	2	2	no	—
325	C₅×C₈	—	2	2	no	—
326	C₅×C₈	—	4	1	no	—
327	C₅×C₈	—	4	1	no	—
328	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
329	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
330	C₂³×C₄	—	2	1	no	—
331	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
335	#125	—	2	2	no	—
336	#125	C₂×C₄×C₅	1	4	yes	complemented
337	#124	—	2	2	no	—
338	#124	C₂×C₄×C₅	1	4	yes	—
339	C₄×Q₈	—	4	1	no	—
340	C₄×Q₈	—	4	1	no	—
341	C₄×D₄	—	4	1	no	—
342	C₄×D₄	—	4	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	—	2	1	no	—
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
363	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
364	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
365	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
369	#107	—	4	1	no	—
370	#107	—	4	1	no	—
371	#106	—	4	1	no	—
372	#106	—	4	1	no	—
373	#103	—	4	1	no	—
374	#103	—	4	1	no	—
375	#103	C₂×C₄×C₅	1	4	yes	complemented
376	#103	C₂×C₄×C₅	1	4	yes	complemented
377	#102	—	4	1	no	—
378	#102	—	4	1	no	—
379	#102	C₂×C₄×C₅	1	4	yes	—
380	#102	C₂×C₄×C₅	1	4	yes	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
383	#95	—	2	1	no	—
384	#95	—	2	1	no	—
385	#95	—	2	1	no	—
386	#95	—	2	1	no	—
387	#95	—	2	1	no	—
388	#95	—	2	1	no	—
389	#95	—	2	1	no	—
390	#95	—	2	1	no	—
391	#95	—	2	1	no	—
392	#95	—	2	1	no	—
393	C₂²×C₅	—	4	2	no	—
394	C₂²×C₅	—	4	2	no	—
395	C₂²×C₅	—	4	2	no	—
396	C₂²×C₅	—	8	1	no	—
397	C₂²×C₅	—	8	1	no	—
398	C₂²×C₅	—	8	1	no	—
399	C₂²×C₅	—	8	1	no	—
400	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
401	C₂²×C₅	#535	1	1	yes	characteristic
402	C₂²×C₅	#536	1	1	yes	characteristic
403	C₂²×C₅	C₂×#135	1	1	yes	characteristic
404	C₂²×C₅	C₂×#135	1	1	yes	characteristic
405	C₂²×C₅	#575	1	1	yes	characteristic
406	C₂²×C₅	#575	1	1	yes	characteristic
407	C₄×C₅	—	2	2	no	—
408	C₄×C₅	—	2	2	no	—
409	C₄×C₅	—	2	2	no	—
410	C₄×C₅	—	2	2	no	—
411	C₄×C₅	—	4	2	no	—
412	C₄×C₅	—	4	2	no	—
413	C₄×C₅	—	4	2	no	—
414	C₄×C₅	—	4	1	no	—
415	C₄×C₅	—	4	1	no	—
416	C₄×C₅	—	8	1	no	—
417	C₄×C₅	—	8	1	no	—
418	C₄×C₅	—	8	1	no	—
419	C₄×C₅	—	8	1	no	—
420	C₄×C₅	#520	1	1	yes	characteristic
421	C₄×C₅	#520	1	1	yes	characteristic
422	C₄×C₅	#542	1	1	yes	characteristic
423	C₄×C₅	#542	1	1	yes	characteristic
424	C₂⁴	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
450	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
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	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
465	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	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	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	4	2	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	—	4	2	no	—
481	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
482	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	4	2	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₂²⋊C₄	—	4	2	no	—
488	C₄²	—	2	2	no	—
489	C₄²	—	2	2	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	4	1	no	—
495	C₄²	—	4	1	no	—
496	C₄²	—	4	2	no	—
497	C₄²	—	4	2	no	—
498	C₄²	—	4	2	no	—
499	C₄²	C₂²×C₄×C₅	1	2	yes	—
500	C₄²	C₅×#41	1	2	yes	—
501	C₂×C₅	—	8	2	no	—
502	C₂×C₅	#2326	1	1	yes	characteristic
503	C₂×C₅	C₄×#135	1	1	yes	characteristic
504	C₂×C₅	#2943	1	1	yes	characteristic
505	C₂×C₅	#3211	1	1	yes	characteristic
506	C₂×C₅	#3212	1	1	yes	characteristic
507	C₂×C₅	#3232	1	1	yes	characteristic
508	C₂×C₅	#3233	1	1	yes	characteristic
509	C₂³	—	2	2	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	1	no	—
515	C₂³	—	4	1	no	—
516	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	8	4	no	—
561	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
562	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
563	C₂×C₄	C₅×#135	1	2	yes	—
564	C₂×C₄	C₅×#135	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
566	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
567	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
568	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
569	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
570	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
571	C₈	—	2	2	no	—
572	C₈	—	2	2	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₅	#22222	1	1	yes	characteristic, direct product factor, Sylow
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	8	1	no	—
581	C₂²	—	8	1	no	—
582	C₂²	—	8	1	no	—
583	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
584	C₂²	C₅×#535	1	1	yes	characteristic
585	C₂²	C₅×#536	1	1	yes	characteristic
586	C₂²	C₂×C₅×#135	1	1	yes	characteristic
587	C₂²	C₂×C₅×#135	1	1	yes	characteristic
588	C₂²	C₅×#575	1	1	yes	characteristic
589	C₂²	C₅×#575	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	8	1	no	—
600	C₄	—	8	1	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	C₅×#520	1	1	yes	characteristic
604	C₄	C₅×#520	1	1	yes	characteristic
605	C₄	C₅×#542	1	1	yes	characteristic
606	C₄	C₅×#542	1	1	yes	characteristic
607	C₂	—	8	2	no	—
608	C₂	C₅×#2326	1	1	yes	characteristic
609	C₂	C₄×C₅×#135	1	1	yes	characteristic
610	C₂	C₅×#2943	1	1	yes	characteristic
611	C₂	C₅×#3211	1	1	yes	characteristic
612	C₂	C₅×#3212	1	1	yes	characteristic
613	C₂	C₅×#3232	1	1	yes	characteristic
614	C₂	C₅×#3233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#125	C₂	semidirect
4. C₅×#1991	C₂	semidirect
6. C₅×#1928	C₂	semidirect
7. C₅×#1927	C₂	semidirect
10. C₅×#1865	C₂	semidirect
12. C₅×#1863	C₂	semidirect
14. C₅×#1846	C₂	semidirect
15. C₅×#1744	C₂	semidirect
20. C₅×#488	C₄	semidirect
60. #22222	C₅	direct (N × H)
69. C₅×#125	C₂×C₄	semidirect
108. C₅×#103	C₂×C₄	semidirect
109. C₅×#103	C₂×C₄	semidirect
115. C₄×C₅×C₈	C₂×C₄	semidirect
127. C₄×#125	C₂×C₅	semidirect
129. #1991	C₂×C₅	semidirect
131. #1928	C₂×C₅	semidirect
132. #1927	C₂×C₅	semidirect
135. #1865	C₂×C₅	semidirect
137. #1863	C₂×C₅	semidirect
139. #1846	C₂×C₅	semidirect
140. #1744	C₂×C₅	semidirect
222. #488	C₄×C₅	semidirect
336. #125	C₂×C₄×C₅	semidirect
375. #103	C₂×C₄×C₅	semidirect
376. #103	C₂×C₄×C₅	semidirect
382. C₄×C₈	C₂×C₄×C₅	semidirect
575. C₅	#22222	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#488	2
C₄×C₅×#125	2
C₄×C₅×#124	2
C₅×#1991	2
C₅×#1958	2
C₅×#1928	2
C₅×#1927	2
C₅×#1926	2
C₅×#1925	2
C₅×#1865	2
C₅×#1864	2
C₅×#1863	2
C₅×#1862	2
C₅×#1846	2
C₅×#1744	2
#22222	5

Subgroups — C5×#22222 #11944942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22222 #11944942