Subgroups — C₅×#21555 #11944275

Order 1280.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#472	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#442	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1931	C₂	1	2	yes	maximal, complemented
8	C₅×#1907	C₂	1	2	yes	maximal, complemented
9	C₅×#1815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1778	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×#100	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#131	C₂²	1	1	yes	characteristic
15	C₂×C₅×#131	C₂²	1	1	yes	characteristic
16	C₄×C₅×#41	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
18	C₅×#472	C₄	1	8	yes	complemented
19	C₅×#442	—	2	1	no	—
20	C₅×#442	—	2	1	no	—
21	C₅×#442	—	2	1	no	—
22	C₅×#442	—	2	1	no	—
23	C₅×#442	—	2	2	no	—
24	C₅×#440	—	2	2	no	—
25	C₄×C₅×SD₁₆	—	2	2	no	—
26	C₄×C₅×D₈	—	2	2	no	—
27	C₂×C₅×#108	C₂²	1	1	yes	characteristic
28	C₂×C₅×#108	C₂²	1	1	yes	characteristic
29	C₅×#428	C₂²	1	2	yes	—
30	C₂×C₅×#106	C₂²	1	1	yes	characteristic
31	C₅×#418	C₂²	1	2	yes	—
32	C₅×#417	C₂²	1	2	yes	—
33	C₂×C₅×#102	C₂²	1	2	yes	—
34	C₂×C₅×#100	C₂²	1	2	yes	—
35	C₂×C₅×#100	C₂²	1	2	yes	—
36	C₅×#405	—	2	1	no	—
37	C₅×#405	—	2	1	no	—
38	C₅×#405	C₂²	1	1	yes	characteristic
39	C₅×#405	C₂²	1	1	yes	characteristic
40	C₄×C₅×M₁₆	—	2	1	no	—
41	C₄×C₅×M₁₆	C₂²	1	2	yes	—
42	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
43	C₂×C₅×#97	—	2	1	no	—
44	C₂×C₅×#97	C₂²	1	1	yes	characteristic
45	C₅×#343	C₂²	1	2	yes	—
46	C₅×#343	C₂²	1	2	yes	—
47	C₅×#341	C₂²	1	2	yes	—
48	#21555	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₅×#41	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₄	C₂³	1	2	yes	—
51	C₂³×C₄×C₅	—	2	2	no	—
52	C₅×AGL(1,ℤ/8)	—	2	4	no	—
53	C₅×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
54	C₂×C₅×SD₁₆	—	2	2	no	—
55	C₂×C₅×D₈	—	2	2	no	—
56	C₂×C₅×D₈	—	2	1	no	—
57	C₂×C₅×D₈	—	2	1	no	—
58	C₂×C₅×D₈	—	2	2	no	—
59	C₂×C₅×D₈	C₂×C₄	1	4	yes	complemented
60	C₅×#131	—	2	2	no	—
61	C₅×#131	—	2	2	no	—
62	C₅×#131	—	2	2	no	—
63	C₅×#131	C₂×C₄	1	4	yes	complemented
64	C₂×C₅×M₁₆	—	2	1	no	—
65	C₂×C₅×M₁₆	—	2	1	no	—
66	C₂×C₅×M₁₆	C₂³	1	2	yes	—
67	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₅×M₁₆	C₂³	1	2	yes	—
69	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
70	C₂²×C₅×C₈	—	2	1	no	—
71	C₂²×C₅×C₈	—	2	1	no	—
72	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×Q₈	D₄	1	2	yes	—
75	C₄×C₅×D₄	—	2	2	no	—
76	C₄×C₅×D₄	—	2	2	no	—
77	C₄×C₅×D₄	—	2	2	no	—
78	C₄×C₅×D₄	—	2	2	no	—
79	C₄×C₅×D₄	—	2	2	no	—
80	C₄×C₅×D₄	D₄	1	2	yes	—
81	C₅×#117	—	2	1	no	—
82	C₅×#117	C₂³	1	1	yes	characteristic
83	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
84	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²×C₅	—	2	1	no	—
86	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
87	C₅×#108	—	2	2	no	—
88	C₅×#108	C₂×C₄	1	4	yes	complemented
89	C₅×#107	—	2	2	no	—
90	C₅×#106	—	2	1	no	—
91	C₅×#106	—	2	1	no	—
92	C₅×#106	—	2	2	no	—
93	C₅×#103	—	2	2	no	—
94	C₅×#102	—	2	2	no	—
95	C₅×#102	—	2	2	no	—
96	C₅×#102	—	2	2	no	—
97	C₅×#102	—	2	2	no	—
98	C₅×#102	—	2	2	no	—
99	C₅×#100	—	2	4	no	—
100	C₅×#100	C₂×C₄	1	8	yes	complemented
101	C₅×#97	—	2	1	no	—
102	C₅×#97	—	2	1	no	—
103	C₅×#97	—	2	1	no	—
104	C₅×#97	—	2	2	no	—
105	C₅×#97	D₄	1	1	yes	characteristic
106	C₅×#97	D₄	1	1	yes	characteristic
107	C₅×#97	D₄	1	1	yes	characteristic
108	C₅×#97	D₄	1	1	yes	characteristic
109	C₄×C₅×C₈	—	2	1	no	—
110	C₄×C₅×C₈	—	2	1	no	—
111	C₄×C₅×C₈	—	2	1	no	—
112	C₄×C₅×C₈	—	2	2	no	—
113	C₂×#472	C₂×C₅	1	1	yes	characteristic
114	C₄×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
115	C₂×#442	C₂×C₅	1	1	yes	characteristic, complemented
116	#2875	C₂×C₅	1	1	yes	characteristic, complemented
117	#1978	C₂×C₅	1	1	yes	characteristic, complemented
118	#1951	C₂×C₅	1	1	yes	characteristic, complemented
119	#1931	C₂×C₅	1	2	yes	complemented
120	#1907	C₂×C₅	1	2	yes	complemented
121	#1815	C₂×C₅	1	1	yes	characteristic, complemented
122	#1778	C₂×C₅	1	1	yes	characteristic, complemented
123	C₄×#100	C₂×C₅	1	2	yes	complemented
124	C₂⁴×C₅	—	2	2	no	—
125	C₅×#41	—	2	1	no	—
126	C₅×#41	—	2	1	no	—
127	C₅×#41	C₂²×C₄	1	2	yes	—
128	C₅×#41	#41	1	2	yes	—
129	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅×D₄	—	2	1	no	—
131	C₂×C₅×D₄	—	2	2	no	—
132	C₂×C₅×D₄	—	2	2	no	—
133	C₂×C₅×D₄	—	2	2	no	—
134	C₂×C₅×D₄	—	2	2	no	—
135	C₂×C₅×D₄	—	2	2	no	—
136	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
137	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₅	—	2	1	no	—
139	C₂²×C₄×C₅	—	2	2	no	—
140	C₂²×C₄×C₅	—	2	2	no	—
141	C₂²×C₄×C₅	—	4	2	no	—
142	C₂²×C₄×C₅	—	4	2	no	—
143	C₂²×C₄×C₅	—	8	4	no	—
144	C₂²×C₄×C₅	#41	1	2	yes	—
145	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
146	C₅×SD₁₆	—	2	8	no	—
147	C₅×D₈	—	2	8	no	—
148	C₅×D₈	—	2	4	no	—
149	C₅×D₈	—	2	4	no	—
150	C₅×M₁₆	—	2	4	no	—
151	C₅×M₁₆	—	2	1	no	—
152	C₅×M₁₆	—	2	1	no	—
153	C₅×M₁₆	—	2	2	no	—
154	C₅×M₁₆	—	2	4	no	—
155	C₅×M₁₆	—	2	2	no	—
156	C₅×M₁₆	—	2	2	no	—
157	C₅×M₁₆	C₂²×C₄	1	2	yes	—
158	C₅×M₁₆	C₂²×C₄	1	4	yes	—
159	C₅×M₁₆	#41	1	2	yes	—
160	C₂×C₅×C₈	—	2	4	no	—
161	C₂×C₅×C₈	—	2	1	no	—
162	C₂×C₅×C₈	—	2	1	no	—
163	C₂×C₅×C₈	—	2	2	no	—
164	C₂×C₅×C₈	—	2	1	no	—
165	C₂×C₅×C₈	—	2	1	no	—
166	C₂×C₅×C₈	—	2	4	no	—
167	C₂×C₅×C₈	—	2	2	no	—
168	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
169	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
170	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
171	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₅×C₈	#41	1	2	yes	—
174	C₅×C₄⋊C₄	—	2	2	no	—
175	C₅×C₄⋊C₄	—	2	2	no	—
176	C₅×C₄⋊C₄	—	2	2	no	—
177	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
178	C₅×C₂²⋊C₄	—	2	1	no	—
179	C₅×C₂²⋊C₄	—	2	2	no	—
180	C₅×C₂²⋊C₄	—	4	2	no	—
181	C₅×C₂²⋊C₄	—	4	2	no	—
182	C₄²×C₅	—	2	2	no	—
183	C₄²×C₅	—	2	1	no	—
184	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₅	C₂×D₄	1	2	yes	—
187	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	—
188	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
189	C₂×#131	C₂²×C₅	1	1	yes	characteristic
190	C₂×#131	C₂²×C₅	1	1	yes	characteristic
191	C₄×#41	C₂²×C₅	1	1	yes	characteristic
192	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
193	#472	C₄×C₅	1	8	yes	complemented
194	#442	—	2	2	no	—
195	#442	—	2	1	no	—
196	#442	—	2	1	no	—
197	#442	—	2	1	no	—
198	#442	—	2	1	no	—
199	#440	—	2	2	no	—
200	C₄×SD₁₆	—	2	2	no	—
201	C₄×D₈	—	2	2	no	—
202	C₂×#108	C₂²×C₅	1	1	yes	characteristic
203	C₂×#108	C₂²×C₅	1	1	yes	characteristic
204	#428	C₂²×C₅	1	2	yes	—
205	C₂×#106	C₂²×C₅	1	1	yes	characteristic
206	#418	C₂²×C₅	1	2	yes	—
207	#417	C₂²×C₅	1	2	yes	—
208	C₂×#102	C₂²×C₅	1	2	yes	—
209	C₂×#100	C₂²×C₅	1	2	yes	—
210	C₂×#100	C₂²×C₅	1	2	yes	—
211	#405	—	2	1	no	—
212	#405	—	2	1	no	—
213	#405	C₂²×C₅	1	1	yes	characteristic
214	#405	C₂²×C₅	1	1	yes	characteristic
215	C₄×M₁₆	—	2	1	no	—
216	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
217	C₄×M₁₆	C₂²×C₅	1	2	yes	—
218	C₂×#97	—	2	1	no	—
219	C₂×#97	C₂²×C₅	1	1	yes	characteristic
220	#343	C₂²×C₅	1	2	yes	—
221	#343	C₂²×C₅	1	2	yes	—
222	#341	C₂²×C₅	1	2	yes	—
223	C₂³×C₅	—	2	1	no	—
224	C₂³×C₅	—	2	4	no	—
225	C₂³×C₅	—	4	2	no	—
226	C₂³×C₅	—	4	2	no	—
227	C₂³×C₅	—	8	2	no	—
228	C₂³×C₅	—	8	2	no	—
229	C₂³×C₅	C₂×#41	1	1	yes	characteristic
230	C₅×Q₈	—	2	1	no	—
231	C₅×Q₈	C₄×D₄	1	2	yes	—
232	C₅×D₄	—	2	2	no	—
233	C₅×D₄	—	2	2	no	—
234	C₅×D₄	—	2	4	no	—
235	C₅×D₄	—	2	4	no	—
236	C₅×D₄	—	2	1	no	—
237	C₅×D₄	—	4	2	no	—
238	C₅×D₄	—	4	2	no	—
239	C₅×D₄	C₄×D₄	1	2	yes	—
240	C₂×C₄×C₅	—	2	1	no	—
241	C₂×C₄×C₅	—	2	2	no	—
242	C₂×C₄×C₅	—	2	2	no	—
243	C₂×C₄×C₅	—	2	2	no	—
244	C₂×C₄×C₅	—	4	2	no	—
245	C₂×C₄×C₅	—	4	2	no	—
246	C₂×C₄×C₅	—	8	2	no	—
247	C₂×C₄×C₅	—	8	4	no	—
248	C₂×C₄×C₅	—	8	4	no	—
249	C₂×C₄×C₅	—	8	2	no	—
250	C₂×C₄×C₅	#121	1	2	yes	—
251	C₂×C₄×C₅	#121	1	2	yes	—
252	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
258	C₅×C₈	—	2	2	no	—
259	C₅×C₈	—	2	4	no	—
260	C₅×C₈	—	2	1	no	—
261	C₅×C₈	—	2	2	no	—
262	C₅×C₈	—	2	1	no	—
263	C₅×C₈	—	4	2	no	—
264	C₅×C₈	C₄×D₄	1	2	yes	—
265	C₅×C₈	C₄×D₄	1	2	yes	—
266	C₂×#41	C₂³×C₅	1	1	yes	characteristic
267	C₂²×D₄	C₂³×C₅	1	2	yes	—
268	C₂³×C₄	—	2	2	no	—
269	AGL(1,ℤ/8)	—	2	4	no	—
270	AGL(1,ℤ/8)	C₂×C₄×C₅	1	8	yes	complemented
271	C₂×SD₁₆	—	2	2	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	—	2	1	no	—
274	C₂×D₈	—	2	1	no	—
275	C₂×D₈	—	2	2	no	—
276	C₂×D₈	C₂×C₄×C₅	1	4	yes	complemented
277	#131	—	2	2	no	—
278	#131	—	2	2	no	—
279	#131	—	2	2	no	—
280	#131	C₂×C₄×C₅	1	4	yes	complemented
281	C₂×M₁₆	—	2	1	no	—
282	C₂×M₁₆	—	2	1	no	—
283	C₂×M₁₆	C₂³×C₅	1	2	yes	—
284	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
285	C₂×M₁₆	C₂³×C₅	1	2	yes	—
286	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
287	C₂²×C₈	—	2	1	no	—
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
291	C₄×Q₈	C₅×D₄	1	2	yes	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	C₅×D₄	1	2	yes	—
298	#117	—	2	1	no	—
299	#117	C₂³×C₅	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
304	#108	—	2	2	no	—
305	#108	C₂×C₄×C₅	1	4	yes	complemented
306	#107	—	2	2	no	—
307	#106	—	2	1	no	—
308	#106	—	2	1	no	—
309	#106	—	2	2	no	—
310	#103	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#100	—	2	4	no	—
317	#100	C₂×C₄×C₅	1	8	yes	complemented
318	#97	—	2	1	no	—
319	#97	—	2	1	no	—
320	#97	—	2	1	no	—
321	#97	—	2	2	no	—
322	#97	C₅×D₄	1	1	yes	characteristic
323	#97	C₅×D₄	1	1	yes	characteristic
324	#97	C₅×D₄	1	1	yes	characteristic
325	#97	C₅×D₄	1	1	yes	characteristic
326	C₄×C₈	—	2	1	no	—
327	C₄×C₈	—	2	1	no	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	—	2	2	no	—
330	C₂²×C₅	—	2	2	no	—
331	C₂²×C₅	—	2	2	no	—
332	C₂²×C₅	—	4	2	no	—
333	C₂²×C₅	—	4	4	no	—
334	C₂²×C₅	—	8	4	no	—
335	C₂²×C₅	—	8	4	no	—
336	C₂²×C₅	—	8	2	no	—
337	C₂²×C₅	—	8	2	no	—
338	C₂²×C₅	C₄×#41	1	1	yes	characteristic
339	C₂²×C₅	C₂×#121	1	1	yes	characteristic
340	C₂²×C₅	#529	1	1	yes	characteristic
341	C₄×C₅	—	2	2	no	—
342	C₄×C₅	—	2	2	no	—
343	C₄×C₅	—	2	2	no	—
344	C₄×C₅	—	4	2	no	—
345	C₄×C₅	—	8	2	no	—
346	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
347	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
348	C₄×C₅	#525	1	1	yes	characteristic
349	C₄×C₅	#532	1	1	yes	characteristic
350	C₂⁴	—	2	2	no	—
351	#41	—	2	1	no	—
352	#41	—	2	1	no	—
353	#41	C₂²×C₄×C₅	1	2	yes	—
354	#41	C₅×#41	1	2	yes	—
355	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
363	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	8	4	no	—
370	C₂²×C₄	C₅×#41	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
372	SD₁₆	—	2	8	no	—
373	D₈	—	2	8	no	—
374	D₈	—	2	4	no	—
375	D₈	—	2	4	no	—
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	1	no	—
378	M₁₆	—	2	1	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	4	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	2	2	no	—
383	M₁₆	C₂²×C₄×C₅	1	2	yes	—
384	M₁₆	C₂²×C₄×C₅	1	4	yes	—
385	M₁₆	C₅×#41	1	2	yes	—
386	C₂×C₈	—	2	4	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
395	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
396	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
397	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
398	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
399	C₂×C₈	C₅×#41	1	2	yes	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	4	2	no	—
407	C₂²⋊C₄	—	4	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	1	no	—
410	C₄²	C₂×C₅×D₄	1	2	yes	—
411	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
412	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
413	C₂×C₅	—	2	1	no	—
414	C₂×C₅	—	2	1	no	—
415	C₂×C₅	—	4	2	no	—
416	C₂×C₅	—	8	4	no	—
417	C₂×C₅	C₄×#121	1	1	yes	characteristic
418	C₂×C₅	#3068	1	2	yes	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	8	2	no	—
424	C₂³	—	8	2	no	—
425	C₂³	C₂×C₅×#41	1	1	yes	characteristic
426	Q₈	—	2	1	no	—
427	Q₈	C₄×C₅×D₄	1	2	yes	—
428	D₄	—	2	2	no	—
429	D₄	—	2	2	no	—
430	D₄	—	2	4	no	—
431	D₄	—	2	4	no	—
432	D₄	—	2	1	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	2	no	—
435	D₄	C₄×C₅×D₄	1	2	yes	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	8	2	no	—
443	C₂×C₄	—	8	4	no	—
444	C₂×C₄	—	8	4	no	—
445	C₂×C₄	—	8	2	no	—
446	C₂×C₄	C₅×#121	1	2	yes	—
447	C₂×C₄	C₅×#121	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
449	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
450	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
451	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
452	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
453	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	4	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	2	no	—
458	C₈	—	2	1	no	—
459	C₈	—	4	2	no	—
460	C₈	C₄×C₅×D₄	1	2	yes	—
461	C₈	C₄×C₅×D₄	1	2	yes	—
462	C₅	#21555	1	1	yes	characteristic, direct product factor, Sylow
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	8	4	no	—
468	C₂²	—	8	4	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	8	2	no	—
471	C₂²	C₄×C₅×#41	1	1	yes	characteristic
472	C₂²	C₂×C₅×#121	1	1	yes	characteristic
473	C₂²	C₅×#529	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	8	2	no	—
479	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
480	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
481	C₄	C₅×#525	1	1	yes	characteristic
482	C₄	C₅×#532	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	—	8	4	no	—
487	C₂	C₄×C₅×#121	1	1	yes	characteristic
488	C₂	C₅×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#472	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₅×#442	C₂	semidirect
4. C₅×#2875	C₂	semidirect
5. C₅×#1978	C₂	semidirect
6. C₅×#1951	C₂	semidirect
7. C₅×#1931	C₂	semidirect
8. C₅×#1907	C₂	semidirect
9. C₅×#1815	C₂	semidirect
10. C₅×#1778	C₂	semidirect
11. C₄×C₅×#100	C₂	semidirect
18. C₅×#472	C₄	semidirect
48. #21555	C₅	direct (N × H)
53. C₅×AGL(1,ℤ/8)	C₂×C₄	semidirect
59. C₂×C₅×D₈	C₂×C₄	semidirect
63. C₅×#131	C₂×C₄	semidirect
88. C₅×#108	C₂×C₄	semidirect
100. C₅×#100	C₂×C₄	semidirect
114. C₄×AGL(1,ℤ/8)	C₂×C₅	semidirect
115. C₂×#442	C₂×C₅	semidirect
116. #2875	C₂×C₅	semidirect
117. #1978	C₂×C₅	semidirect
118. #1951	C₂×C₅	semidirect
119. #1931	C₂×C₅	semidirect
120. #1907	C₂×C₅	semidirect
121. #1815	C₂×C₅	semidirect
122. #1778	C₂×C₅	semidirect
123. C₄×#100	C₂×C₅	semidirect
193. #472	C₄×C₅	semidirect
270. AGL(1,ℤ/8)	C₂×C₄×C₅	semidirect
276. C₂×D₈	C₂×C₄×C₅	semidirect
280. #131	C₂×C₄×C₅	semidirect
305. #108	C₂×C₄×C₅	semidirect
317. #100	C₂×C₄×C₅	semidirect
462. C₅	#21555	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×#472	2
C₄×C₅×AGL(1,ℤ/8)	2
C₂×C₅×#442	2
C₅×#2875	2
C₅×#1978	2
C₅×#1951	2
C₅×#1931	2
C₅×#1907	2
C₅×#1815	2
C₅×#1778	2
C₄×C₅×#100	2
#21555	5

Subgroups — C5×#21555 #11944275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21555 #11944275