Subgroups — C₅×#23025 #11945745

Order 1280.

1150 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2361	C₂	1	2	yes	maximal, complemented
5	C₅×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1925	C₂	1	2	yes	maximal, complemented
8	C₅×#1874	C₂	1	2	yes	maximal, complemented
9	C₅×#1862	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#121	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₄	—	2	2	no	—
16	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
17	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₅×#466	C₄	1	8	yes	complemented
19	C₄×C₅×D₈	—	2	4	no	—
20	C₂×C₅×#107	—	2	4	no	—
21	C₂×C₅×#107	C₂²	1	2	yes	—
22	C₂×C₅×#107	C₂²	1	1	yes	characteristic
23	C₂×C₅×#107	C₂²	1	1	yes	characteristic
24	C₂×C₅×#107	C₂²	1	1	yes	characteristic
25	C₂×C₅×#102	C₂²	1	2	yes	—
26	C₂×C₅×#102	C₂²	1	2	yes	—
27	C₂×C₅×#102	C₂²	1	2	yes	—
28	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
31	C₅×#390	C₂²	1	2	yes	—
32	C₅×#389	C₂²	1	2	yes	—
33	C₅×#388	—	2	2	no	—
34	C₅×#386	—	2	2	no	—
35	C₅×#385	—	2	2	no	—
36	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
37	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₅×#340	C₂²	1	2	yes	—
39	C₅×#340	C₂²	1	2	yes	—
40	C₅×#336	C₂²	1	1	yes	characteristic
41	C₅×#336	C₂²	1	1	yes	characteristic
42	#23025	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	C₂³	1	2	yes	—
45	C₂³×C₄×C₅	—	2	2	no	—
46	C₂³×C₄×C₅	—	2	2	no	—
47	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×D₈	—	2	2	no	—
49	C₂×C₅×D₈	—	2	2	no	—
50	C₂×C₅×D₈	C₂×C₄	1	4	yes	complemented
51	C₂²×C₅×C₈	—	2	2	no	—
52	C₂²×C₅×C₈	—	2	2	no	—
53	C₂²×C₅×C₈	C₂×C₄	1	4	yes	complemented
54	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₅×#121	—	2	4	no	—
59	C₅×#121	C₂×C₄	1	8	yes	complemented
60	C₄×C₅×D₄	—	2	4	no	—
61	C₄×C₅×D₄	—	4	2	no	—
62	C₄×C₅×D₄	—	4	8	no	—
63	C₂×C₅×C₄⋊C₄	—	2	4	no	—
64	C₂×C₅×C₄⋊C₄	—	2	2	no	—
65	C₂×C₅×C₄⋊C₄	—	2	2	no	—
66	C₂×C₅×C₄⋊C₄	—	4	2	no	—
67	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
68	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
73	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
74	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
75	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₅	—	2	2	no	—
77	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₅×#107	—	2	2	no	—
80	C₅×#107	—	2	4	no	—
81	C₅×#107	C₂×C₄	1	4	yes	complemented
82	C₅×#107	D₄	1	4	yes	complemented
83	C₅×#102	—	2	4	no	—
84	C₅×#102	—	4	2	no	—
85	C₅×#102	C₂×C₄	1	8	yes	complemented
86	C₅×C₂²⋊C₈	—	2	2	no	—
87	C₄×C₅×C₈	—	2	2	no	—
88	C₄×C₅×C₈	—	2	2	no	—
89	C₅×#95	—	2	2	no	—
90	C₅×#95	—	2	2	no	—
91	C₅×#95	D₄	1	1	yes	characteristic
92	C₅×#95	D₄	1	1	yes	characteristic
93	C₂×#466	C₂×C₅	1	1	yes	characteristic
94	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂²×#107	C₂×C₅	1	1	yes	characteristic, complemented
96	#2361	C₂×C₅	1	2	yes	complemented
97	#1970	C₂×C₅	1	1	yes	characteristic, complemented
98	#1942	C₂×C₅	1	1	yes	characteristic, complemented
99	#1925	C₂×C₅	1	2	yes	complemented
100	#1874	C₂×C₅	1	2	yes	complemented
101	#1862	C₂×C₅	1	2	yes	complemented
102	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
103	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
104	C₂⁴×C₅	—	2	2	no	—
105	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×D₄	—	2	8	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	2	2	no	—
111	C₂×C₅×D₄	—	2	2	no	—
112	C₂×C₅×D₄	—	4	4	no	—
113	C₂×C₅×D₄	—	4	4	no	—
114	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
115	C₂²×C₄×C₅	—	2	1	no	—
116	C₂²×C₄×C₅	—	2	1	no	—
117	C₂²×C₄×C₅	—	2	1	no	—
118	C₂²×C₄×C₅	—	2	1	no	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	—	2	2	no	—
121	C₂²×C₄×C₅	—	2	2	no	—
122	C₂²×C₄×C₅	—	4	4	no	—
123	C₂²×C₄×C₅	—	4	4	no	—
124	C₂²×C₄×C₅	—	4	4	no	—
125	C₂²×C₄×C₅	—	4	4	no	—
126	C₂²×C₄×C₅	—	8	2	no	—
127	C₂²×C₄×C₅	—	8	2	no	—
128	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅	#41	1	2	yes	—
135	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
136	C₅×D₈	—	2	8	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	2	8	no	—
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	1	no	—
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	2	1	no	—
146	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
147	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
148	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
149	C₂×C₅×C₈	#41	1	2	yes	—
150	C₅×C₄⋊C₄	—	2	2	no	—
151	C₅×C₄⋊C₄	—	2	2	no	—
152	C₅×C₄⋊C₄	—	2	2	no	—
153	C₅×C₄⋊C₄	—	2	2	no	—
154	C₅×C₄⋊C₄	—	4	4	no	—
155	C₅×C₄⋊C₄	—	4	4	no	—
156	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
157	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
158	C₅×C₂²⋊C₄	—	2	2	no	—
159	C₅×C₂²⋊C₄	—	2	8	no	—
160	C₅×C₂²⋊C₄	—	4	4	no	—
161	C₅×C₂²⋊C₄	—	4	4	no	—
162	C₅×C₂²⋊C₄	—	8	2	no	—
163	C₅×C₂²⋊C₄	D₈	1	4	yes	complemented
164	C₄²×C₅	—	2	1	no	—
165	C₄²×C₅	—	2	1	no	—
166	C₄²×C₅	—	2	1	no	—
167	C₄²×C₅	—	2	1	no	—
168	C₄²×C₅	—	4	4	no	—
169	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
170	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
171	C₂×#121	C₂²×C₅	1	2	yes	—
172	C₂×C₄×D₄	—	2	2	no	—
173	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
175	#466	C₄×C₅	1	8	yes	complemented
176	C₄×D₈	—	2	4	no	—
177	C₂×#107	—	2	4	no	—
178	C₂×#107	C₂²×C₅	1	2	yes	—
179	C₂×#107	C₂²×C₅	1	1	yes	characteristic
180	C₂×#107	C₂²×C₅	1	1	yes	characteristic
181	C₂×#107	C₂²×C₅	1	1	yes	characteristic
182	C₂×#102	C₂²×C₅	1	2	yes	—
183	C₂×#102	C₂²×C₅	1	2	yes	—
184	C₂×#102	C₂²×C₅	1	2	yes	—
185	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
188	#390	C₂²×C₅	1	2	yes	—
189	#389	C₂²×C₅	1	2	yes	—
190	#388	—	2	2	no	—
191	#386	—	2	2	no	—
192	#385	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
194	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
195	#340	C₂²×C₅	1	2	yes	—
196	#340	C₂²×C₅	1	2	yes	—
197	#336	C₂²×C₅	1	1	yes	characteristic
198	#336	C₂²×C₅	1	1	yes	characteristic
199	C₂³×C₅	—	2	1	no	—
200	C₂³×C₅	—	2	1	no	—
201	C₂³×C₅	—	2	1	no	—
202	C₂³×C₅	—	2	1	no	—
203	C₂³×C₅	—	2	4	no	—
204	C₂³×C₅	—	4	2	no	—
205	C₂³×C₅	—	4	2	no	—
206	C₂³×C₅	—	4	2	no	—
207	C₂³×C₅	—	4	2	no	—
208	C₂³×C₅	—	4	2	no	—
209	C₂³×C₅	—	4	2	no	—
210	C₂³×C₅	C₄×D₄	1	2	yes	—
211	C₂³×C₅	C₂×D₈	1	2	yes	—
212	C₂³×C₅	#135	1	2	yes	—
213	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
214	C₅×D₄	—	2	4	no	—
215	C₅×D₄	—	2	4	no	—
216	C₅×D₄	—	4	16	no	—
217	C₅×D₄	—	4	2	no	—
218	C₅×D₄	—	4	2	no	—
219	C₂×C₄×C₅	—	2	1	no	—
220	C₂×C₄×C₅	—	2	1	no	—
221	C₂×C₄×C₅	—	2	1	no	—
222	C₂×C₄×C₅	—	2	1	no	—
223	C₂×C₄×C₅	—	2	1	no	—
224	C₂×C₄×C₅	—	2	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	2	no	—
229	C₂×C₄×C₅	—	2	4	no	—
230	C₂×C₄×C₅	—	2	1	no	—
231	C₂×C₄×C₅	—	2	4	no	—
232	C₂×C₄×C₅	—	2	1	no	—
233	C₂×C₄×C₅	—	2	1	no	—
234	C₂×C₄×C₅	—	2	1	no	—
235	C₂×C₄×C₅	—	2	1	no	—
236	C₂×C₄×C₅	—	2	1	no	—
237	C₂×C₄×C₅	—	4	2	no	—
238	C₂×C₄×C₅	—	4	2	no	—
239	C₂×C₄×C₅	—	4	2	no	—
240	C₂×C₄×C₅	—	4	2	no	—
241	C₂×C₄×C₅	—	4	2	no	—
242	C₂×C₄×C₅	—	4	2	no	—
243	C₂×C₄×C₅	—	4	2	no	—
244	C₂×C₄×C₅	—	8	8	no	—
245	C₂×C₄×C₅	—	8	8	no	—
246	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
247	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
248	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
249	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
255	C₅×C₈	—	2	2	no	—
256	C₅×C₈	—	4	1	no	—
257	C₅×C₈	—	4	1	no	—
258	C₅×C₈	C₄×D₄	1	4	yes	complemented
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	C₂³×C₅	1	2	yes	—
261	C₂³×C₄	—	2	2	no	—
262	C₂³×C₄	—	2	2	no	—
263	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂×C₄×C₅	1	4	yes	complemented
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	C₂×C₄×C₅	1	4	yes	complemented
270	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
274	#121	—	2	4	no	—
275	#121	C₂×C₄×C₅	1	8	yes	complemented
276	C₄×D₄	—	2	4	no	—
277	C₄×D₄	—	4	2	no	—
278	C₄×D₄	—	4	8	no	—
279	C₂×C₄⋊C₄	—	2	4	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	4	2	no	—
283	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
295	#107	—	2	2	no	—
296	#107	—	2	4	no	—
297	#107	C₂×C₄×C₅	1	4	yes	complemented
298	#107	C₅×D₄	1	4	yes	complemented
299	#102	—	2	4	no	—
300	#102	—	4	2	no	—
301	#102	C₂×C₄×C₅	1	8	yes	complemented
302	C₂²⋊C₈	—	2	2	no	—
303	C₄×C₈	—	2	2	no	—
304	C₄×C₈	—	2	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	C₅×D₄	1	1	yes	characteristic
308	#95	C₅×D₄	1	1	yes	characteristic
309	C₂²×C₅	—	2	2	no	—
310	C₂²×C₅	—	2	2	no	—
311	C₂²×C₅	—	2	2	no	—
312	C₂²×C₅	—	2	2	no	—
313	C₂²×C₅	—	2	2	no	—
314	C₂²×C₅	—	2	2	no	—
315	C₂²×C₅	—	4	4	no	—
316	C₂²×C₅	—	4	4	no	—
317	C₂²×C₅	—	4	4	no	—
318	C₂²×C₅	—	8	2	no	—
319	C₂²×C₅	—	8	2	no	—
320	C₂²×C₅	—	8	2	no	—
321	C₂²×C₅	—	8	2	no	—
322	C₂²×C₅	C₄×D₈	1	4	yes	complemented
323	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
324	C₂²×C₅	D₄²	1	1	yes	characteristic
325	C₂²×C₅	#547	1	1	yes	characteristic
326	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
327	C₂²×C₅	C₂×#135	1	1	yes	characteristic
328	C₂²×C₅	#576	1	1	yes	characteristic
329	C₂²×C₅	#578	1	1	yes	characteristic
330	C₄×C₅	—	2	2	no	—
331	C₄×C₅	—	2	2	no	—
332	C₄×C₅	—	2	2	no	—
333	C₄×C₅	—	4	1	no	—
334	C₄×C₅	—	4	4	no	—
335	C₄×C₅	—	4	4	no	—
336	C₄×C₅	—	4	1	no	—
337	C₄×C₅	—	8	2	no	—
338	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
339	C₄×C₅	#520	1	1	yes	characteristic
340	C₄×C₅	#548	1	1	yes	characteristic
341	C₄×C₅	#553	1	1	yes	characteristic
342	C₂⁴	—	2	2	no	—
343	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	4	4	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	8	2	no	—
365	C₂²×C₄	—	8	2	no	—
366	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
367	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
368	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
369	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
370	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
371	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
372	C₂²×C₄	C₅×#41	1	2	yes	—
373	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
374	D₈	—	2	8	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	8	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
385	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
386	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
387	C₂×C₈	C₅×#41	1	2	yes	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	4	no	—
393	C₄⋊C₄	—	4	4	no	—
394	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
395	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	2	8	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	8	2	no	—
401	C₂²⋊C₄	C₅×D₈	1	4	yes	complemented
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	4	4	no	—
407	C₂×C₅	—	2	4	no	—
408	C₂×C₅	—	8	4	no	—
409	C₂×C₅	#2361	1	1	yes	characteristic
410	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
411	C₂×C₅	#2951	1	1	yes	characteristic
412	C₂×C₅	D₄×D₈	1	1	yes	characteristic
413	C₂×C₅	#3285	1	1	yes	characteristic
414	C₂×C₅	#3334	1	1	yes	characteristic
415	C₂×C₅	#3339	1	1	yes	characteristic
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	C₄×C₅×D₄	1	2	yes	—
428	C₂³	C₂×C₅×D₈	1	2	yes	—
429	C₂³	C₅×#135	1	2	yes	—
430	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
431	D₄	—	2	4	no	—
432	D₄	—	2	4	no	—
433	D₄	—	4	16	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
464	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
465	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
466	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
468	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
469	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
470	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
471	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
472	C₈	—	2	2	no	—
473	C₈	—	4	1	no	—
474	C₈	—	4	1	no	—
475	C₈	C₄×C₅×D₄	1	4	yes	complemented
476	C₅	#23025	1	1	yes	characteristic, direct product factor, Sylow
477	C₂²	—	2	2	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	8	2	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	C₄×C₅×D₈	1	4	yes	complemented
491	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
492	C₂²	C₅×D₄²	1	1	yes	characteristic
493	C₂²	C₅×#547	1	1	yes	characteristic
494	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
495	C₂²	C₂×C₅×#135	1	1	yes	characteristic
496	C₂²	C₅×#576	1	1	yes	characteristic
497	C₂²	C₅×#578	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	4	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	4	no	—
505	C₄	—	8	2	no	—
506	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
507	C₄	C₅×#520	1	1	yes	characteristic
508	C₄	C₅×#548	1	1	yes	characteristic
509	C₄	C₅×#553	1	1	yes	characteristic
510	C₂	—	2	4	no	—
511	C₂	—	8	4	no	—
512	C₂	C₅×#2361	1	1	yes	characteristic
513	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
514	C₂	C₅×#2951	1	1	yes	characteristic
515	C₂	C₅×D₄×D₈	1	1	yes	characteristic
516	C₂	C₅×#3285	1	1	yes	characteristic
517	C₂	C₅×#3334	1	1	yes	characteristic
518	C₂	C₅×#3339	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	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×D₈	C₂	semidirect
3. C₂²×C₅×#107	C₂	semidirect
4. C₅×#2361	C₂	semidirect
5. C₅×#1970	C₂	semidirect
6. C₅×#1942	C₂	semidirect
7. C₅×#1925	C₂	semidirect
8. C₅×#1874	C₂	semidirect
9. C₅×#1862	C₂	semidirect
10. C₄×C₅×#107	C₂	semidirect
11. C₅×C₈×C₂²⋊C₄	C₂	semidirect
18. C₅×#466	C₄	semidirect
42. #23025	C₅	direct (N × H)
50. C₂×C₅×D₈	C₂×C₄	semidirect
53. C₂²×C₅×C₈	C₂×C₄	semidirect
59. C₅×#121	C₂×C₄	semidirect
81. C₅×#107	C₂×C₄	semidirect
82. C₅×#107	D₄	semidirect
85. C₅×#102	C₂×C₄	semidirect
94. C₂×C₄×D₈	C₂×C₅	semidirect
95. C₂²×#107	C₂×C₅	semidirect
96. #2361	C₂×C₅	semidirect
97. #1970	C₂×C₅	semidirect
98. #1942	C₂×C₅	semidirect
99. #1925	C₂×C₅	semidirect
100. #1874	C₂×C₅	semidirect
101. #1862	C₂×C₅	semidirect
102. C₄×#107	C₂×C₅	semidirect
103. C₈×C₂²⋊C₄	C₂×C₅	semidirect
163. C₅×C₂²⋊C₄	D₈	semidirect
175. #466	C₄×C₅	semidirect
258. C₅×C₈	C₄×D₄	semidirect
266. C₂×D₈	C₂×C₄×C₅	semidirect
269. C₂²×C₈	C₂×C₄×C₅	semidirect
275. #121	C₂×C₄×C₅	semidirect
297. #107	C₂×C₄×C₅	semidirect
298. #107	C₅×D₄	semidirect
301. #102	C₂×C₄×C₅	semidirect
322. C₂²×C₅	C₄×D₈	semidirect
401. C₂²⋊C₄	C₅×D₈	semidirect
475. C₈	C₄×C₅×D₄	semidirect
476. C₅	#23025	direct (N × H)
490. C₂²	C₄×C₅×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#466	2
C₂×C₄×C₅×D₈	2
C₂²×C₅×#107	2
C₅×#2361	2
C₅×#1970	2
C₅×#1942	2
C₅×#1925	2
C₅×#1874	2
C₅×#1862	2
C₄×C₅×#107	2
C₅×C₈×C₂²⋊C₄	2
#23025	5

Subgroups — C5×#23025 #11945745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23025 #11945745