Subgroups — C₅×#22544 #11945264

Order 1280.

1098 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1983	C₂	1	2	yes	maximal
5	C₅×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1959	C₂	1	2	yes	maximal
7	C₅×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1803	C₂	1	2	yes	maximal, complemented
9	C₅×#1802	C₂	1	1	yes	maximal, characteristic
10	C₅×#1763	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic
12	C₅×#556	—	2	4	no	—
13	C₅×#556	C₄	1	8	yes	complemented
14	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#126	—	2	2	no	—
16	C₂×C₅×#124	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₄×C₅×Q₈	—	2	1	no	—
19	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
20	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₅×#117	—	2	1	no	—
23	C₅×#446	—	2	4	no	—
24	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₅×#103	C₂²	1	2	yes	—
27	C₂×C₅×#103	C₂²	1	2	yes	—
28	C₂×C₅×#103	C₂²	1	2	yes	—
29	C₂×C₅×#103	C₂²	1	2	yes	—
30	C₂×C₅×#102	C₂²	1	1	yes	characteristic
31	C₂×C₅×#102	C₂²	1	1	yes	characteristic
32	C₂×C₅×#102	C₂²	1	2	yes	—
33	C₂×C₅×#97	C₂²	1	2	yes	—
34	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₅×#389	—	2	1	no	—
36	C₅×#389	—	2	2	no	—
37	C₅×#384	C₂²	1	1	yes	characteristic
38	C₅×#384	C₂²	1	2	yes	—
39	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
41	C₅×#340	C₂²	1	1	yes	characteristic
42	C₅×#340	C₂²	1	1	yes	characteristic
43	C₅×#340	C₂²	1	2	yes	—
44	#22544	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
46	C₂²×C₅×Q₈	C₂³	1	2	yes	—
47	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅	—	2	1	no	—
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊Q₈	—	2	2	no	—
53	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
54	C₅×#126	—	2	16	no	—
55	C₅×#124	—	2	4	no	—
56	C₅×#124	C₂×C₄	1	8	yes	—
57	C₅×C₂²⋊Q₈	—	2	8	no	—
58	C₄×C₅×Q₈	—	2	4	no	—
59	C₄×C₅×Q₈	—	4	4	no	—
60	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
61	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
62	C₄×C₅×Q₈	D₄	1	4	yes	—
63	C₄×C₅×D₄	—	2	2	no	—
64	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
65	C₅×#117	—	2	8	no	—
66	C₂×C₅×C₄⋊C₄	—	2	1	no	—
67	C₂×C₅×C₄⋊C₄	—	2	2	no	—
68	C₂×C₅×C₄⋊C₄	—	2	1	no	—
69	C₂×C₅×C₄⋊C₄	—	2	2	no	—
70	C₂×C₅×C₄⋊C₄	—	2	1	no	—
71	C₂×C₅×C₄⋊C₄	—	2	2	no	—
72	C₂×C₅×C₄⋊C₄	—	2	1	no	—
73	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
75	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
77	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
80	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
81	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₄²×C₅	—	2	2	no	—
83	C₂×C₄²×C₅	—	2	2	no	—
84	C₂×C₄²×C₅	—	2	1	no	—
85	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅	C₂³	1	2	yes	—
87	C₅×C₄⋊C₈	—	2	2	no	—
88	C₅×C₄⋊C₈	—	2	4	no	—
89	C₅×#103	—	2	4	no	—
90	C₅×#103	—	2	4	no	—
91	C₅×#103	—	4	2	no	—
92	C₅×#103	—	4	2	no	—
93	C₅×#102	—	2	2	no	—
94	C₅×#102	—	2	2	no	—
95	C₅×#102	—	4	2	no	—
96	C₅×#97	—	2	4	no	—
97	C₄×C₅×C₈	—	2	2	no	—
98	C₅×#95	—	2	2	no	—
99	C₅×#95	—	2	1	no	—
100	C₂×#556	C₂×C₅	1	1	yes	characteristic
101	C₂×#446	C₂×C₅	1	1	yes	characteristic, complemented
102	Q₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
103	#1983	C₂×C₅	1	2	yes	—
104	#1981	C₂×C₅	1	1	yes	characteristic, complemented
105	#1959	C₂×C₅	1	2	yes	—
106	#1956	C₂×C₅	1	1	yes	characteristic, complemented
107	#1803	C₂×C₅	1	2	yes	complemented
108	#1802	C₂×C₅	1	1	yes	characteristic
109	#1763	C₂×C₅	1	2	yes	complemented
110	C₄×#102	C₂×C₅	1	1	yes	characteristic
111	C₂⁴×C₅	—	2	1	no	—
112	C₂×C₅×Q₈	—	2	2	no	—
113	C₂×C₅×Q₈	—	2	2	no	—
114	C₂×C₅×Q₈	—	2	2	no	—
115	C₂×C₅×Q₈	—	2	2	no	—
116	C₂×C₅×Q₈	—	2	2	no	—
117	C₂×C₅×Q₈	—	2	2	no	—
118	C₂×C₅×Q₈	—	2	2	no	—
119	C₂×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
120	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
121	C₂×C₅×D₄	—	2	1	no	—
122	C₂×C₅×D₄	—	2	1	no	—
123	C₂×C₅×D₄	—	2	1	no	—
124	C₂×C₅×D₄	—	2	1	no	—
125	C₂×C₅×D₄	—	2	1	no	—
126	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₅	—	2	1	no	—
128	C₂²×C₄×C₅	—	2	1	no	—
129	C₂²×C₄×C₅	—	2	2	no	—
130	C₂²×C₄×C₅	—	2	4	no	—
131	C₂²×C₄×C₅	—	2	1	no	—
132	C₂²×C₄×C₅	—	2	2	no	—
133	C₂²×C₄×C₅	—	4	2	no	—
134	C₂²×C₄×C₅	—	4	2	no	—
135	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
140	C₂×C₅×C₈	—	2	1	no	—
141	C₂×C₅×C₈	—	2	1	no	—
142	C₂×C₅×C₈	—	2	1	no	—
143	C₂×C₅×C₈	—	2	2	no	—
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	2	2	no	—
146	C₂×C₅×C₈	—	2	2	no	—
147	C₂×C₅×C₈	—	2	1	no	—
148	C₂×C₅×C₈	#41	1	2	yes	—
149	C₅×C₄⋊C₄	—	2	4	no	—
150	C₅×C₄⋊C₄	—	2	8	no	—
151	C₅×C₄⋊C₄	—	2	4	no	—
152	C₅×C₄⋊C₄	—	2	1	no	—
153	C₅×C₄⋊C₄	—	2	1	no	—
154	C₅×C₄⋊C₄	—	2	1	no	—
155	C₅×C₄⋊C₄	—	2	2	no	—
156	C₅×C₄⋊C₄	—	2	2	no	—
157	C₅×C₄⋊C₄	—	2	1	no	—
158	C₅×C₄⋊C₄	—	4	4	no	—
159	C₅×C₄⋊C₄	—	4	2	no	—
160	C₅×C₄⋊C₄	—	8	2	no	—
161	C₅×C₄⋊C₄	—	8	1	no	—
162	C₅×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
163	C₅×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
164	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
165	C₅×C₄⋊C₄	#41	1	2	yes	—
166	C₅×C₂²⋊C₄	—	2	8	no	—
167	C₅×C₂²⋊C₄	—	2	4	no	—
168	C₅×C₂²⋊C₄	—	2	8	no	—
169	C₄²×C₅	—	2	8	no	—
170	C₄²×C₅	—	4	4	no	—
171	C₄²×C₅	—	4	2	no	—
172	C₄²×C₅	C₂²×C₄	1	2	yes	—
173	C₄²×C₅	C₂²×C₄	1	4	yes	—
174	C₄²×C₅	C₂×D₄	1	2	yes	—
175	C₄²×C₅	C₂×D₄	1	4	yes	—
176	#556	—	2	4	no	—
177	#556	C₄×C₅	1	8	yes	complemented
178	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
179	C₂×#126	—	2	2	no	—
180	C₂×#124	C₂²×C₅	1	2	yes	—
181	C₂×C₂²⋊Q₈	—	2	1	no	—
182	C₂×C₄×Q₈	—	2	1	no	—
183	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
184	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
185	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
186	C₂×#117	—	2	1	no	—
187	#446	—	2	4	no	—
188	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
190	C₂×#103	C₂²×C₅	1	2	yes	—
191	C₂×#103	C₂²×C₅	1	2	yes	—
192	C₂×#103	C₂²×C₅	1	2	yes	—
193	C₂×#103	C₂²×C₅	1	2	yes	—
194	C₂×#102	C₂²×C₅	1	1	yes	characteristic
195	C₂×#102	C₂²×C₅	1	1	yes	characteristic
196	C₂×#102	C₂²×C₅	1	2	yes	—
197	C₂×#97	C₂²×C₅	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
199	#389	—	2	1	no	—
200	#389	—	2	2	no	—
201	#384	C₂²×C₅	1	1	yes	characteristic
202	#384	C₂²×C₅	1	2	yes	—
203	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
205	#340	C₂²×C₅	1	2	yes	—
206	#340	C₂²×C₅	1	1	yes	characteristic
207	#340	C₂²×C₅	1	1	yes	characteristic
208	C₂³×C₅	—	2	2	no	—
209	C₂³×C₅	—	4	1	no	—
210	C₂³×C₅	—	4	1	no	—
211	C₂³×C₅	—	4	1	no	—
212	C₂³×C₅	—	4	1	no	—
213	C₂³×C₅	—	4	1	no	—
214	C₂³×C₅	—	4	1	no	—
215	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
216	C₅×Q₈	—	2	8	no	—
217	C₅×Q₈	—	2	4	no	—
218	C₅×Q₈	—	2	4	no	—
219	C₅×Q₈	—	4	2	no	—
220	C₅×Q₈	—	4	2	no	—
221	C₅×D₄	—	2	2	no	—
222	C₅×D₄	—	2	2	no	—
223	C₅×D₄	—	4	1	no	—
224	C₅×D₄	—	4	1	no	—
225	C₂×C₄×C₅	—	2	2	no	—
226	C₂×C₄×C₅	—	2	2	no	—
227	C₂×C₄×C₅	—	2	2	no	—
228	C₂×C₄×C₅	—	2	1	no	—
229	C₂×C₄×C₅	—	2	2	no	—
230	C₂×C₄×C₅	—	2	1	no	—
231	C₂×C₄×C₅	—	2	2	no	—
232	C₂×C₄×C₅	—	2	4	no	—
233	C₂×C₄×C₅	—	2	2	no	—
234	C₂×C₄×C₅	—	4	2	no	—
235	C₂×C₄×C₅	—	4	2	no	—
236	C₂×C₄×C₅	—	4	1	no	—
237	C₂×C₄×C₅	—	4	1	no	—
238	C₂×C₄×C₅	—	4	1	no	—
239	C₂×C₄×C₅	—	4	1	no	—
240	C₂×C₄×C₅	—	4	8	no	—
241	C₂×C₄×C₅	—	4	1	no	—
242	C₂×C₄×C₅	—	4	2	no	—
243	C₂×C₄×C₅	—	4	2	no	—
244	C₂×C₄×C₅	—	4	1	no	—
245	C₂×C₄×C₅	—	4	1	no	—
246	C₂×C₄×C₅	—	4	2	no	—
247	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
248	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
250	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
252	C₂×C₄×C₅	#137	1	2	yes	—
253	C₂×C₄×C₅	C₂³×C₄	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	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
259	C₅×C₈	—	2	2	no	—
260	C₅×C₈	—	4	1	no	—
261	C₅×C₈	—	4	2	no	—
262	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
263	C₂²×Q₈	C₂³×C₅	1	2	yes	—
264	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
265	C₂³×C₄	—	2	1	no	—
266	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₅	1	2	yes	—
268	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
271	#126	—	2	16	no	—
272	#124	—	2	4	no	—
273	#124	C₂×C₄×C₅	1	8	yes	—
274	C₂²⋊Q₈	—	2	8	no	—
275	C₄×Q₈	—	2	4	no	—
276	C₄×Q₈	—	4	4	no	—
277	C₄×Q₈	C₂×C₄×C₅	1	8	yes	—
278	C₄×Q₈	C₂×C₄×C₅	1	4	yes	—
279	C₄×Q₈	C₅×D₄	1	4	yes	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
282	#117	—	2	8	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₅	1	2	yes	—
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	2	4	no	—
306	#103	—	2	4	no	—
307	#103	—	2	4	no	—
308	#103	—	4	2	no	—
309	#103	—	4	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	4	2	no	—
313	#97	—	2	4	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	1	no	—
317	C₂²×C₅	—	4	2	no	—
318	C₂²×C₅	—	4	2	no	—
319	C₂²×C₅	—	4	2	no	—
320	C₂²×C₅	—	8	1	no	—
321	C₂²×C₅	—	8	1	no	—
322	C₂²×C₅	—	8	1	no	—
323	C₂²×C₅	—	8	1	no	—
324	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
325	C₂²×C₅	#548	1	1	yes	characteristic
326	C₂²×C₅	#550	1	1	yes	characteristic
327	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂²×C₅	C₂×#137	1	1	yes	characteristic
329	C₂²×C₅	#577	1	1	yes	characteristic
330	C₂²×C₅	#578	1	1	yes	characteristic
331	C₄×C₅	—	2	4	no	—
332	C₄×C₅	—	2	4	no	—
333	C₄×C₅	—	2	2	no	—
334	C₄×C₅	—	4	4	no	—
335	C₄×C₅	—	4	2	no	—
336	C₄×C₅	—	4	2	no	—
337	C₄×C₅	—	8	1	no	—
338	C₄×C₅	—	8	2	no	—
339	C₄×C₅	#417	1	4	yes	—
340	C₄×C₅	C₄×#41	1	1	yes	characteristic
341	C₄×C₅	#520	1	1	yes	characteristic
342	C₄×C₅	#548	1	1	yes	characteristic
343	C₄×C₅	#550	1	1	yes	characteristic
344	C₂⁴	—	2	1	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
353	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
369	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
370	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
371	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	C₅×#41	1	2	yes	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	8	2	no	—
394	C₄⋊C₄	—	8	1	no	—
395	C₄⋊C₄	C₅×C₂²⋊C₄	1	4	yes	—
396	C₄⋊C₄	C₅×C₂²⋊C₄	1	8	yes	—
397	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
398	C₄⋊C₄	C₅×#41	1	2	yes	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₄²	—	2	8	no	—
403	C₄²	—	4	4	no	—
404	C₄²	—	4	2	no	—
405	C₄²	C₂²×C₄×C₅	1	2	yes	—
406	C₄²	C₂²×C₄×C₅	1	4	yes	—
407	C₄²	C₂×C₅×D₄	1	2	yes	—
408	C₄²	C₂×C₅×D₄	1	4	yes	—
409	C₂×C₅	—	8	2	no	—
410	C₂×C₅	#2341	1	1	yes	characteristic
411	C₂×C₅	C₂×#417	1	1	yes	characteristic
412	C₂×C₅	#2898	1	1	yes	characteristic
413	C₂×C₅	#3367	1	1	yes	characteristic
414	C₂×C₅	#3369	1	1	yes	characteristic
415	C₂×C₅	#3397	1	1	yes	characteristic
416	C₂×C₅	#3398	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	1	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
425	Q₈	—	2	8	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	4	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	4	1	no	—
433	D₄	—	4	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	8	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
457	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
458	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
459	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
460	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
461	C₂×C₄	C₅×#137	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
463	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
464	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
465	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
466	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
467	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	4	1	no	—
470	C₈	—	4	2	no	—
471	C₅	#22544	1	1	yes	characteristic, direct product factor, Sylow
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	8	1	no	—
476	C₂²	—	8	1	no	—
477	C₂²	—	8	1	no	—
478	C₂²	—	8	1	no	—
479	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
480	C₂²	C₅×#548	1	1	yes	characteristic
481	C₂²	C₅×#550	1	1	yes	characteristic
482	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
483	C₂²	C₂×C₅×#137	1	1	yes	characteristic
484	C₂²	C₅×#577	1	1	yes	characteristic
485	C₂²	C₅×#578	1	1	yes	characteristic
486	C₄	—	2	4	no	—
487	C₄	—	2	4	no	—
488	C₄	—	2	2	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	1	no	—
493	C₄	—	8	2	no	—
494	C₄	C₅×#417	1	4	yes	—
495	C₄	C₄×C₅×#41	1	1	yes	characteristic
496	C₄	C₅×#520	1	1	yes	characteristic
497	C₄	C₅×#548	1	1	yes	characteristic
498	C₄	C₅×#550	1	1	yes	characteristic
499	C₂	—	8	2	no	—
500	C₂	C₅×#2341	1	1	yes	characteristic
501	C₂	C₂×C₅×#417	1	1	yes	characteristic
502	C₂	C₅×#2898	1	1	yes	characteristic
503	C₂	C₅×#3367	1	1	yes	characteristic
504	C₂	C₅×#3369	1	1	yes	characteristic
505	C₂	C₅×#3397	1	1	yes	characteristic
506	C₂	C₅×#3398	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 ◁ C₂×C₄×C₅×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#446	C₂	semidirect
3. C₅×Q₈×C₄⋊C₄	C₂	semidirect
5. C₅×#1981	C₂	semidirect
7. C₅×#1956	C₂	semidirect
8. C₅×#1803	C₂	semidirect
10. C₅×#1763	C₂	semidirect
13. C₅×#556	C₄	semidirect
44. #22544	C₅	direct (N × H)
101. C₂×#446	C₂×C₅	semidirect
102. Q₈×C₄⋊C₄	C₂×C₅	semidirect
104. #1981	C₂×C₅	semidirect
106. #1956	C₂×C₅	semidirect
107. #1803	C₂×C₅	semidirect
109. #1763	C₂×C₅	semidirect
177. #556	C₄×C₅	semidirect
471. C₅	#22544	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×#556	2
C₂×C₅×#446	2
C₅×Q₈×C₄⋊C₄	2
C₅×#1983	2
C₅×#1981	2
C₅×#1959	2
C₅×#1956	2
C₅×#1803	2
C₅×#1802	2
C₅×#1763	2
C₄×C₅×#102	2
#22544	5

Subgroups — C5×#22544 #11945264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22544 #11945264