Subgroups — C₅×#22465 #11945185

Order 1280.

1312 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#578	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2898	C₂	1	2	yes	maximal, complemented
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1971	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1908	C₂	1	2	yes	maximal, complemented
8	C₅×#1882	C₂	1	2	yes	maximal, complemented
9	C₅×#1815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1793	C₂	1	2	yes	maximal, complemented
11	C₅×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₅×#578	—	2	4	no	—
14	C₅×#578	C₄	1	8	yes	complemented
15	C₂×C₅×#137	—	2	2	no	—
16	C₂×C₅×#137	C₂²	1	2	yes	—
17	C₂×C₅×#135	—	2	1	no	—
18	C₂×C₅×#135	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Q₁₆	—	2	1	no	—
20	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₅×#131	C₂²	1	2	yes	—
22	C₂×C₅×#131	C₂²	1	1	yes	characteristic
23	C₅×#520	C₂²	1	2	yes	—
24	C₅×#444	—	2	4	no	—
25	C₅×#442	—	2	1	no	—
26	C₅×#441	—	2	1	no	—
27	C₄×C₅×SD₁₆	—	2	2	no	—
28	C₂×C₅×#108	C₂²	1	1	yes	characteristic
29	C₂×C₅×#106	C₂²	1	1	yes	characteristic
30	C₂×C₅×#104	C₂²	1	1	yes	characteristic
31	C₂×C₅×#104	C₂²	1	2	yes	—
32	C₅×#419	—	2	2	no	—
33	C₅×#419	C₂²	1	2	yes	—
34	C₅×#417	—	2	2	no	—
35	C₅×#417	C₂²	1	2	yes	—
36	C₅×#417	C₂²	1	2	yes	—
37	C₅×#416	—	2	2	no	—
38	C₅×#416	C₂²	1	1	yes	characteristic
39	C₂×C₅×#103	—	2	2	no	—
40	C₂×C₅×#103	C₂²	1	2	yes	—
41	C₂×C₅×#101	C₂²	1	2	yes	—
42	C₅×#405	C₂²	1	1	yes	characteristic
43	C₂×C₅×#97	C₂²	1	1	yes	characteristic
44	C₅×#391	C₂²	1	2	yes	—
45	C₅×#343	C₂²	1	2	yes	—
46	C₅×#341	C₂²	1	1	yes	characteristic
47	C₅×#339	C₂²	1	1	yes	characteristic
48	#22465	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₅×2₋¹⁺⁴	—	2	4	no	—
50	C₅×2₋¹⁺⁴	—	2	4	no	—
51	C₅×2₋¹⁺⁴	—	4	2	no	—
52	C₅×2₋¹⁺⁴	C₂×C₄	1	8	yes	complemented
53	C₂×C₅×#41	—	2	2	no	—
54	C₂×C₅×#41	—	2	2	no	—
55	C₂×C₅×#41	—	4	2	no	—
56	C₂×C₅×#41	C₂³	1	2	yes	—
57	C₂×C₅×#41	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Q₈	—	2	2	no	—
59	C₂²×C₅×Q₈	—	2	2	no	—
60	C₂²×C₅×Q₈	C₂³	1	2	yes	—
61	C₅×#137	—	2	4	no	—
62	C₅×#137	—	2	16	no	—
63	C₅×#137	C₂×C₄	1	8	yes	complemented
64	C₅×#135	—	2	8	no	—
65	C₅×#135	C₂×C₄	1	4	yes	—
66	C₅×#135	D₄	1	4	yes	—
67	C₂×C₅×Q₁₆	—	2	2	no	—
68	C₂×C₅×Q₁₆	—	2	8	no	—
69	C₂×C₅×Q₁₆	—	2	1	no	—
70	C₂×C₅×Q₁₆	—	2	1	no	—
71	C₂×C₅×Q₁₆	—	2	2	no	—
72	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	complemented
73	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
74	C₂×C₅×SD₁₆	—	2	2	no	—
75	C₂×C₅×SD₁₆	D₄	1	2	yes	—
76	C₂×C₅×D₈	D₄	1	1	yes	characteristic
77	C₅×#131	—	2	2	no	—
78	C₅×#131	—	2	4	no	—
79	C₅×#131	—	4	2	no	—
80	C₅×#131	C₂×C₄	1	4	yes	complemented
81	C₂×C₅×M₁₆	—	2	1	no	—
82	C₂×C₅×M₁₆	—	2	2	no	—
83	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₅×M₁₆	C₂³	1	2	yes	—
85	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
86	C₂²×C₅×C₈	—	2	1	no	—
87	C₂²×C₅×C₈	—	2	2	no	—
88	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
89	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₅×Q₈	—	2	2	no	—
91	C₄×C₅×D₄	—	2	2	no	—
92	C₄×C₅×D₄	—	4	2	no	—
93	C₅×#117	—	2	2	no	—
94	C₅×#117	C₂³	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	—	2	2	no	—
96	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
98	C₅×#108	—	2	2	no	—
99	C₅×#106	—	2	1	no	—
100	C₅×#106	—	2	1	no	—
101	C₅×#104	—	2	4	no	—
102	C₅×#104	—	2	2	no	—
103	C₅×#103	—	2	2	no	—
104	C₅×#103	—	2	2	no	—
105	C₅×#103	—	2	2	no	—
106	C₅×#103	—	2	2	no	—
107	C₅×#103	—	2	2	no	—
108	C₅×#103	—	2	2	no	—
109	C₅×#103	D₄	1	2	yes	—
110	C₅×#102	—	2	2	no	—
111	C₅×#102	—	2	2	no	—
112	C₅×#102	D₄	1	2	yes	—
113	C₅×#101	—	4	2	no	—
114	C₅×#97	—	2	2	no	—
115	C₅×#97	—	2	1	no	—
116	C₄×C₅×C₈	—	2	1	no	—
117	C₅×#95	—	4	2	no	—
118	C₂×#578	C₂×C₅	1	1	yes	characteristic
119	C₂×#444	C₂×C₅	1	1	yes	characteristic, complemented
120	#2952	C₂×C₅	1	1	yes	characteristic, complemented
121	#2898	C₂×C₅	1	2	yes	complemented
122	#1979	C₂×C₅	1	1	yes	characteristic, complemented
123	#1971	C₂×C₅	1	1	yes	characteristic, complemented
124	#1908	C₂×C₅	1	2	yes	complemented
125	#1882	C₂×C₅	1	2	yes	complemented
126	#1815	C₂×C₅	1	1	yes	characteristic, complemented
127	#1793	C₂×C₅	1	2	yes	complemented
128	#1765	C₂×C₅	1	1	yes	characteristic, complemented
129	C₅×#41	—	2	1	no	—
130	C₅×#41	—	2	2	no	—
131	C₅×#41	—	2	2	no	—
132	C₅×#41	—	2	1	no	—
133	C₅×#41	—	2	8	no	—
134	C₅×#41	—	2	8	no	—
135	C₅×#41	—	4	8	no	—
136	C₅×#41	—	4	4	no	—
137	C₅×#41	—	4	4	no	—
138	C₅×#41	—	4	8	no	—
139	C₅×#41	C₂²×C₄	1	4	yes	—
140	C₅×#41	C₂²×C₄	1	2	yes	—
141	C₅×#41	C₂×D₄	1	4	yes	—
142	C₅×#41	#41	1	2	yes	—
143	C₂×C₅×Q₈	—	2	2	no	—
144	C₂×C₅×Q₈	—	2	8	no	—
145	C₂×C₅×Q₈	—	2	8	no	—
146	C₂×C₅×Q₈	—	2	2	no	—
147	C₂×C₅×Q₈	—	2	2	no	—
148	C₂×C₅×Q₈	—	2	2	no	—
149	C₂×C₅×Q₈	—	2	2	no	—
150	C₂×C₅×Q₈	—	4	2	no	—
151	C₂×C₅×Q₈	—	4	4	no	—
152	C₂×C₅×Q₈	—	4	4	no	—
153	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
154	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
155	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
156	C₂×C₅×D₄	—	2	2	no	—
157	C₂×C₅×D₄	—	2	1	no	—
158	C₂×C₅×D₄	—	4	2	no	—
159	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
160	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅	—	2	2	no	—
162	C₂²×C₄×C₅	—	2	1	no	—
163	C₂²×C₄×C₅	—	2	2	no	—
164	C₂²×C₄×C₅	—	2	1	no	—
165	C₂²×C₄×C₅	—	2	2	no	—
166	C₂²×C₄×C₅	—	4	2	no	—
167	C₂²×C₄×C₅	—	4	2	no	—
168	C₂²×C₄×C₅	—	4	2	no	—
169	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
170	C₅×Q₁₆	—	2	8	no	—
171	C₅×Q₁₆	—	2	4	no	—
172	C₅×Q₁₆	—	2	4	no	—
173	C₅×Q₁₆	C₂²⋊C₄	1	4	yes	—
174	C₅×SD₁₆	—	2	8	no	—
175	C₅×SD₁₆	C₂²⋊C₄	1	8	yes	—
176	C₅×D₈	C₂²⋊C₄	1	4	yes	—
177	C₅×M₁₆	—	2	1	no	—
178	C₅×M₁₆	—	2	4	no	—
179	C₅×M₁₆	—	2	2	no	—
180	C₅×M₁₆	—	2	1	no	—
181	C₅×M₁₆	—	2	2	no	—
182	C₅×M₁₆	—	2	1	no	—
183	C₅×M₁₆	—	4	4	no	—
184	C₅×M₁₆	C₂²×C₄	1	2	yes	—
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	4	no	—
189	C₂×C₅×C₈	—	2	2	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	4	4	no	—
192	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
193	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
194	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
196	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	2	2	no	—
199	C₅×C₄⋊C₄	—	4	2	no	—
200	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
201	C₅×C₂²⋊C₄	—	2	1	no	—
202	C₅×C₂²⋊C₄	—	4	2	no	—
203	C₄²×C₅	—	2	1	no	—
204	C₄²×C₅	—	2	1	no	—
205	C₄²×C₅	—	2	1	no	—
206	C₄²×C₅	—	4	2	no	—
207	C₂×2₋¹⁺⁴	C₂²×C₅	1	2	yes	—
208	#578	—	2	4	no	—
209	#578	C₄×C₅	1	8	yes	complemented
210	C₂×#137	—	2	2	no	—
211	C₂×#137	C₂²×C₅	1	2	yes	—
212	C₂×#135	—	2	1	no	—
213	C₂×#135	C₂²×C₅	1	1	yes	characteristic
214	C₂²×Q₁₆	—	2	1	no	—
215	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
216	C₂×#131	C₂²×C₅	1	2	yes	—
217	C₂×#131	C₂²×C₅	1	1	yes	characteristic
218	#520	C₂²×C₅	1	2	yes	—
219	#444	—	2	4	no	—
220	#442	—	2	1	no	—
221	#441	—	2	1	no	—
222	C₄×SD₁₆	—	2	2	no	—
223	C₂×#108	C₂²×C₅	1	1	yes	characteristic
224	C₂×#106	C₂²×C₅	1	1	yes	characteristic
225	C₂×#104	C₂²×C₅	1	1	yes	characteristic
226	C₂×#104	C₂²×C₅	1	2	yes	—
227	#419	—	2	2	no	—
228	#419	C₂²×C₅	1	2	yes	—
229	#417	—	2	2	no	—
230	#417	C₂²×C₅	1	2	yes	—
231	#417	C₂²×C₅	1	2	yes	—
232	#416	—	2	2	no	—
233	#416	C₂²×C₅	1	1	yes	characteristic
234	C₂×#103	—	2	2	no	—
235	C₂×#103	C₂²×C₅	1	2	yes	—
236	C₂×#101	C₂²×C₅	1	2	yes	—
237	#405	C₂²×C₅	1	1	yes	characteristic
238	C₂×#97	C₂²×C₅	1	1	yes	characteristic
239	#391	C₂²×C₅	1	2	yes	—
240	#343	C₂²×C₅	1	2	yes	—
241	#341	C₂²×C₅	1	1	yes	characteristic
242	#339	C₂²×C₅	1	1	yes	characteristic
243	C₂³×C₅	—	2	2	no	—
244	C₂³×C₅	—	2	1	no	—
245	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
246	C₅×Q₈	—	2	1	no	—
247	C₅×Q₈	—	2	2	no	—
248	C₅×Q₈	—	2	4	no	—
249	C₅×Q₈	—	2	4	no	—
250	C₅×Q₈	—	4	8	no	—
251	C₅×Q₈	—	4	2	no	—
252	C₅×Q₈	—	4	2	no	—
253	C₅×Q₈	C₂×C₂²⋊C₄	1	4	yes	—
254	C₅×Q₈	C₄×D₄	1	2	yes	—
255	C₅×D₄	—	2	2	no	—
256	C₅×D₄	—	2	4	no	—
257	C₅×D₄	—	2	2	no	—
258	C₅×D₄	—	2	4	no	—
259	C₅×D₄	—	2	1	no	—
260	C₅×D₄	—	2	2	no	—
261	C₅×D₄	—	4	8	no	—
262	C₅×D₄	C₂×C₂²⋊C₄	1	4	yes	—
263	C₅×D₄	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₅	—	2	2	no	—
265	C₂×C₄×C₅	—	2	1	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	4	no	—
268	C₂×C₄×C₅	—	2	4	no	—
269	C₂×C₄×C₅	—	2	2	no	—
270	C₂×C₄×C₅	—	2	4	no	—
271	C₂×C₄×C₅	—	2	2	no	—
272	C₂×C₄×C₅	—	4	8	no	—
273	C₂×C₄×C₅	—	4	8	no	—
274	C₂×C₄×C₅	—	4	2	no	—
275	C₂×C₄×C₅	—	4	2	no	—
276	C₂×C₄×C₅	—	4	2	no	—
277	C₂×C₄×C₅	—	4	2	no	—
278	C₂×C₄×C₅	—	4	1	no	—
279	C₂×C₄×C₅	—	8	2	no	—
280	C₂×C₄×C₅	—	16	2	no	—
281	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
282	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
287	C₅×C₈	—	2	2	no	—
288	C₅×C₈	—	2	2	no	—
289	C₅×C₈	—	4	2	no	—
290	C₅×C₈	—	4	1	no	—
291	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
292	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
293	2₋¹⁺⁴	—	2	4	no	—
294	2₋¹⁺⁴	—	2	4	no	—
295	2₋¹⁺⁴	—	4	2	no	—
296	2₋¹⁺⁴	C₂×C₄×C₅	1	8	yes	complemented
297	C₂×#41	—	2	2	no	—
298	C₂×#41	—	2	2	no	—
299	C₂×#41	—	4	2	no	—
300	C₂×#41	C₂³×C₅	1	2	yes	—
301	C₂×#41	C₂³×C₅	1	1	yes	characteristic
302	C₂²×Q₈	—	2	2	no	—
303	C₂²×Q₈	—	2	2	no	—
304	C₂²×Q₈	C₂³×C₅	1	2	yes	—
305	#137	—	2	4	no	—
306	#137	—	2	16	no	—
307	#137	C₂×C₄×C₅	1	8	yes	complemented
308	#135	—	2	8	no	—
309	#135	C₂×C₄×C₅	1	4	yes	—
310	#135	C₅×D₄	1	4	yes	—
311	C₂×Q₁₆	—	2	2	no	—
312	C₂×Q₁₆	—	2	8	no	—
313	C₂×Q₁₆	—	2	1	no	—
314	C₂×Q₁₆	—	2	1	no	—
315	C₂×Q₁₆	—	2	2	no	—
316	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	complemented
317	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×SD₁₆	C₅×D₄	1	2	yes	—
320	C₂×D₈	C₅×D₄	1	1	yes	characteristic
321	#131	—	2	2	no	—
322	#131	—	2	4	no	—
323	#131	—	4	2	no	—
324	#131	C₂×C₄×C₅	1	4	yes	complemented
325	C₂×M₁₆	—	2	1	no	—
326	C₂×M₁₆	—	2	2	no	—
327	C₂×M₁₆	C₂³×C₅	1	2	yes	—
328	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
329	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
334	C₄×Q₈	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	—	4	2	no	—
337	#117	—	2	2	no	—
338	#117	C₂³×C₅	1	1	yes	characteristic
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
341	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
342	#108	—	2	2	no	—
343	#106	—	2	1	no	—
344	#106	—	2	1	no	—
345	#104	—	2	4	no	—
346	#104	—	2	2	no	—
347	#103	—	2	2	no	—
348	#103	—	2	2	no	—
349	#103	—	2	2	no	—
350	#103	—	2	2	no	—
351	#103	—	2	2	no	—
352	#103	—	2	2	no	—
353	#103	C₅×D₄	1	2	yes	—
354	#102	—	2	2	no	—
355	#102	—	2	2	no	—
356	#102	C₅×D₄	1	2	yes	—
357	#101	—	4	2	no	—
358	#97	—	2	2	no	—
359	#97	—	2	1	no	—
360	C₄×C₈	—	2	1	no	—
361	#95	—	4	2	no	—
362	C₂²×C₅	—	2	2	no	—
363	C₂²×C₅	—	2	2	no	—
364	C₂²×C₅	—	2	4	no	—
365	C₂²×C₅	—	4	2	no	—
366	C₂²×C₅	—	4	2	no	—
367	C₂²×C₅	—	4	2	no	—
368	C₂²×C₅	#518	1	1	yes	characteristic
369	C₂²×C₅	D₄²	1	1	yes	characteristic
370	C₂²×C₅	#546	1	1	yes	characteristic
371	C₄×C₅	—	2	2	no	—
372	C₄×C₅	—	2	4	no	—
373	C₄×C₅	—	4	4	no	—
374	C₄×C₅	—	4	2	no	—
375	C₄×C₅	—	8	1	no	—
376	C₄×C₅	—	8	2	no	—
377	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
378	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
379	C₄×C₅	D₄²	1	1	yes	characteristic
380	C₄×C₅	#546	1	1	yes	characteristic
381	#41	—	2	1	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	1	no	—
385	#41	—	2	8	no	—
386	#41	—	2	8	no	—
387	#41	—	4	8	no	—
388	#41	—	4	4	no	—
389	#41	—	4	4	no	—
390	#41	—	4	8	no	—
391	#41	C₂²×C₄×C₅	1	4	yes	—
392	#41	C₂²×C₄×C₅	1	2	yes	—
393	#41	C₂×C₅×D₄	1	4	yes	—
394	#41	C₅×#41	1	2	yes	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	2	8	no	—
397	C₂×Q₈	—	2	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	4	2	no	—
403	C₂×Q₈	—	4	4	no	—
404	C₂×Q₈	—	4	4	no	—
405	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
406	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
407	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	4	2	no	—
411	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
412	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	2	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
422	Q₁₆	—	2	8	no	—
423	Q₁₆	—	2	4	no	—
424	Q₁₆	—	2	4	no	—
425	Q₁₆	C₅×C₂²⋊C₄	1	4	yes	—
426	SD₁₆	—	2	8	no	—
427	SD₁₆	C₅×C₂²⋊C₄	1	8	yes	—
428	D₈	C₅×C₂²⋊C₄	1	4	yes	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	4	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	1	no	—
433	M₁₆	—	2	1	no	—
434	M₁₆	—	2	2	no	—
435	M₁₆	—	4	4	no	—
436	M₁₆	C₂²×C₄×C₅	1	2	yes	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	4	4	no	—
444	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
445	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
446	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
447	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	2	no	—
452	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
453	C₂²⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	4	2	no	—
455	C₄²	—	2	1	no	—
456	C₄²	—	2	1	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	4	2	no	—
459	C₂×C₅	—	2	1	no	—
460	C₂×C₅	—	2	1	no	—
461	C₂×C₅	—	4	2	no	—
462	C₂×C₅	—	4	4	no	—
463	C₂×C₅	D₄×C₂²⋊C₄	1	1	yes	characteristic
464	C₂×C₅	#3289	1	2	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	1	no	—
467	C₂³	C₅×H(5,𝔽₂)	1	1	yes	characteristic
468	Q₈	—	2	1	no	—
469	Q₈	—	2	2	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	2	4	no	—
472	Q₈	—	4	8	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	C₂×C₅×C₂²⋊C₄	1	4	yes	—
476	Q₈	C₄×C₅×D₄	1	2	yes	—
477	D₄	—	2	2	no	—
478	D₄	—	2	4	no	—
479	D₄	—	2	2	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	1	no	—
482	D₄	—	2	2	no	—
483	D₄	—	4	8	no	—
484	D₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
485	D₄	C₄×C₅×D₄	1	2	yes	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	8	2	no	—
502	C₂×C₄	—	16	2	no	—
503	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
504	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
505	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
506	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
507	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
508	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
509	C₈	—	2	2	no	—
510	C₈	—	2	2	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	1	no	—
513	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
514	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
515	C₅	#22465	1	1	yes	characteristic, direct product factor, Sylow
516	C₂²	—	2	2	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	C₅×#518	1	1	yes	characteristic
523	C₂²	C₅×D₄²	1	1	yes	characteristic
524	C₂²	C₅×#546	1	1	yes	characteristic
525	C₄	—	2	2	no	—
526	C₄	—	2	4	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	4	no	—
529	C₄	—	8	1	no	—
530	C₄	—	8	2	no	—
531	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
532	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
533	C₄	C₅×D₄²	1	1	yes	characteristic
534	C₄	C₅×#546	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	1	no	—
537	C₂	—	4	2	no	—
538	C₂	—	4	4	no	—
539	C₂	C₅×D₄×C₂²⋊C₄	1	1	yes	characteristic
540	C₂	C₅×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#444	C₂	semidirect
3. C₅×#2952	C₂	semidirect
4. C₅×#2898	C₂	semidirect
5. C₅×#1979	C₂	semidirect
6. C₅×#1971	C₂	semidirect
7. C₅×#1908	C₂	semidirect
8. C₅×#1882	C₂	semidirect
9. C₅×#1815	C₂	semidirect
10. C₅×#1793	C₂	semidirect
11. C₅×#1765	C₂	semidirect
14. C₅×#578	C₄	semidirect
48. #22465	C₅	direct (N × H)
52. C₅×2₋¹⁺⁴	C₂×C₄	semidirect
63. C₅×#137	C₂×C₄	semidirect
72. C₂×C₅×Q₁₆	C₂×C₄	semidirect
80. C₅×#131	C₂×C₄	semidirect
119. C₂×#444	C₂×C₅	semidirect
120. #2952	C₂×C₅	semidirect
121. #2898	C₂×C₅	semidirect
122. #1979	C₂×C₅	semidirect
123. #1971	C₂×C₅	semidirect
124. #1908	C₂×C₅	semidirect
125. #1882	C₂×C₅	semidirect
126. #1815	C₂×C₅	semidirect
127. #1793	C₂×C₅	semidirect
128. #1765	C₂×C₅	semidirect
209. #578	C₄×C₅	semidirect
296. 2₋¹⁺⁴	C₂×C₄×C₅	semidirect
307. #137	C₂×C₄×C₅	semidirect
316. C₂×Q₁₆	C₂×C₄×C₅	semidirect
324. #131	C₂×C₄×C₅	semidirect
515. C₅	#22465	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×#578	2
C₂×C₅×#444	2
C₅×#2952	2
C₅×#2898	2
C₅×#1979	2
C₅×#1971	2
C₅×#1908	2
C₅×#1882	2
C₅×#1815	2
C₅×#1793	2
C₅×#1765	2
#22465	5

Subgroups — C5×#22465 #11945185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22465 #11945185