Subgroups — C₅×#65824 #13106645

Order 1600.

800 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4075	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4031	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4030	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4030	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4029	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#4028	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#4027	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×D₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#4024	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
17	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₅×C₂²⋊D₅	—	2	4	no	—
21	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
22	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
23	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
24	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
26	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	C₅×#709	—	2	4	no	—
30	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
36	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
37	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
40	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
41	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
42	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
43	C₅×#685	C₂²	1	1	yes	characteristic, complemented
44	C₅×#685	C₂²	1	1	yes	characteristic, complemented
45	C₅×#685	C₂²	1	1	yes	characteristic, complemented
46	C₅×#685	C₂²	1	1	yes	characteristic, complemented
47	C₅×#685	C₂²	1	1	yes	characteristic, complemented
48	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
49	C₅×#683	C₂²	1	1	yes	characteristic, complemented
50	C₅×#683	C₂²	1	1	yes	characteristic, complemented
51	C₅×#683	C₂²	1	1	yes	characteristic, complemented
52	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
53	C₅×#546	—	5	1	no	maximal
54	#65824	C₅	1	1	yes	maximal, characteristic, direct product factor
55	C₂³×C₅²	—	2	2	no	—
56	C₂³×C₅²	—	2	2	no	—
57	C₂³×C₅²	—	4	1	no	—
58	C₂³×C₅²	C₂³	1	1	yes	characteristic
59	C₂³×C₅²	C₂³	1	1	yes	characteristic
60	C₂³×C₅²	C₂³	1	1	yes	characteristic
61	C₂²×C₅×D₅	—	2	2	no	—
62	C₂²×C₅×D₅	—	2	2	no	—
63	C₂²×C₅×D₅	—	4	1	no	—
64	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
65	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
66	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₅²	—	2	2	no	—
68	C₂×C₄×C₅²	—	4	1	no	—
69	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
73	C₅×C₂²⋊D₅	—	2	2	no	—
74	C₅×C₂²⋊D₅	—	2	2	no	—
75	C₅×C₂²⋊D₅	—	2	2	no	—
76	C₅×C₂²⋊D₅	—	2	2	no	—
77	C₅×C₂²⋊D₅	—	2	2	no	—
78	C₅×C₂²⋊D₅	—	4	1	no	—
79	C₅×C₂²⋊D₅	D₄	1	4	yes	complemented
80	C₂×C₅×Dic₅	—	2	2	no	—
81	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
82	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
83	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
84	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
85	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
86	C₅×D₂₀	—	2	2	no	—
87	C₅×D₂₀	—	4	1	no	—
88	C₄×C₅×D₅	—	2	2	no	—
89	C₄×C₅×D₅	—	2	2	no	—
90	C₄×C₅×D₅	—	4	1	no	—
91	C₅×Dic₁₀	—	2	2	no	—
92	C₂×C₅×#41	—	5	1	no	—
93	C₂²×C₅×D₄	—	5	1	no	—
94	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
96	C₅×#124	—	5	1	no	—
97	C₅×#123	—	5	1	no	—
98	C₅×#123	—	5	1	no	—
99	C₅×C₂²⋊Q₈	—	5	1	no	—
100	C₅×#121	—	5	1	no	—
101	C₅×#121	—	5	1	no	—
102	C₅×#121	—	5	1	no	—
103	C₅×#120	—	5	1	no	—
104	C₅×#120	—	5	1	no	—
105	C₄×C₅×D₄	—	5	1	no	—
106	C₄×C₅×D₄	—	5	1	no	—
107	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
108	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
109	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
110	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
111	#4099	C₂×C₅	1	1	yes	characteristic, complemented
112	#4076	C₂×C₅	1	1	yes	characteristic, complemented
113	#4075	C₂×C₅	1	1	yes	characteristic, complemented
114	C₄×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic, complemented
115	#4031	C₂×C₅	1	1	yes	characteristic, complemented
116	#4030	C₂×C₅	1	1	yes	characteristic, complemented
117	#4030	C₂×C₅	1	1	yes	characteristic, complemented
118	#4029	C₂×C₅	1	1	yes	characteristic, complemented
119	#4028	C₂×C₅	1	1	yes	characteristic, complemented
120	#4027	C₂×C₅	1	1	yes	characteristic, complemented
121	D₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
122	#4024	C₂×C₅	1	1	yes	characteristic, complemented
123	C₂²×C₅²	—	2	1	no	—
124	C₂²×C₅²	—	2	1	no	—
125	C₂²×C₅²	—	2	1	no	—
126	C₂²×C₅²	—	2	1	no	—
127	C₂²×C₅²	—	2	1	no	—
128	C₂²×C₅²	—	2	1	no	—
129	C₂²×C₅²	—	2	1	no	—
130	C₂²×C₅²	—	4	4	no	—
131	C₂²×C₅²	C₂×D₄	1	2	yes	—
132	C₂²×C₅²	#41	1	2	yes	complemented
133	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
134	C₂×C₅×D₅	—	2	1	no	—
135	C₂×C₅×D₅	—	2	1	no	—
136	C₂×C₅×D₅	—	2	1	no	—
137	C₂×C₅×D₅	—	2	1	no	—
138	C₂×C₅×D₅	—	2	1	no	—
139	C₂×C₅×D₅	—	2	1	no	—
140	C₂×C₅×D₅	—	2	1	no	—
141	C₂×C₅×D₅	—	2	1	no	—
142	C₂×C₅×D₅	—	4	4	no	—
143	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
144	C₄×C₅²	—	2	1	no	—
145	C₄×C₅²	—	2	1	no	—
146	C₄×C₅²	—	2	1	no	—
147	C₄×C₅²	—	2	1	no	—
148	C₅×Dic₅	—	2	1	no	—
149	C₅×Dic₅	—	2	1	no	—
150	C₅×Dic₅	—	2	1	no	—
151	C₅×Dic₅	—	2	1	no	—
152	C₅×Dic₅	C₂×D₄	1	2	yes	—
153	C₂⁴×C₅	—	2	2	no	—
154	C₂⁴×C₅	—	5	1	no	—
155	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
156	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
157	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
158	C₅×#41	—	10	4	no	—
159	C₂×C₅×Q₈	—	5	1	no	—
160	C₂×C₅×D₄	—	5	1	no	—
161	C₂×C₅×D₄	—	5	1	no	—
162	C₂×C₅×D₄	—	5	1	no	—
163	C₂×C₅×D₄	—	5	1	no	—
164	C₂×C₅×D₄	—	5	1	no	—
165	C₂×C₅×D₄	—	5	1	no	—
166	C₂×C₅×D₄	—	5	1	no	—
167	C₂×C₅×D₄	—	5	1	no	—
168	C₂×C₅×D₄	—	5	1	no	—
169	C₂×C₅×D₄	—	10	4	no	—
170	C₂²×C₄×C₅	—	2	2	no	—
171	C₂²×C₄×C₅	—	2	2	no	—
172	C₂²×C₄×C₅	—	5	1	no	—
173	C₂²×C₄×C₅	—	5	1	no	—
174	C₂²×C₄×C₅	—	5	1	no	—
175	C₂²×C₄×C₅	—	5	1	no	—
176	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic, complemented
177	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic, complemented
178	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
179	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
180	C₂×C₂²⋊D₅	—	2	4	no	—
181	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
184	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
185	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
186	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic, complemented
187	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic, complemented
188	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
189	#709	—	2	4	no	—
190	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
191	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
193	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
194	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
195	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
196	C₅×C₄⋊C₄	—	5	1	no	—
197	C₅×C₄⋊C₄	—	5	1	no	—
198	C₅×C₄⋊C₄	—	5	1	no	—
199	C₅×C₄⋊C₄	—	5	1	no	—
200	C₅×C₂²⋊C₄	—	2	2	no	—
201	C₅×C₂²⋊C₄	—	2	2	no	—
202	C₅×C₂²⋊C₄	—	2	2	no	—
203	C₅×C₂²⋊C₄	—	2	2	no	—
204	C₅×C₂²⋊C₄	—	5	1	no	—
205	C₅×C₂²⋊C₄	—	5	1	no	—
206	C₅×C₂²⋊C₄	—	5	1	no	—
207	C₅×C₂²⋊C₄	—	5	1	no	—
208	C₅×C₂²⋊C₄	—	5	1	no	—
209	C₅×C₂²⋊C₄	—	5	1	no	—
210	C₅×C₂²⋊C₄	—	5	1	no	—
211	C₅×C₂²⋊C₄	—	5	1	no	—
212	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
213	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
214	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
215	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
216	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
217	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
218	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
219	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
220	C₄²×C₅	—	5	1	no	—
221	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
222	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
223	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
224	#685	C₂²×C₅	1	1	yes	characteristic, complemented
225	#685	C₂²×C₅	1	1	yes	characteristic, complemented
226	#685	C₂²×C₅	1	1	yes	characteristic, complemented
227	#685	C₂²×C₅	1	1	yes	characteristic, complemented
228	#685	C₂²×C₅	1	1	yes	characteristic, complemented
229	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
230	#683	C₂²×C₅	1	1	yes	characteristic, complemented
231	#683	C₂²×C₅	1	1	yes	characteristic, complemented
232	#683	C₂²×C₅	1	1	yes	characteristic, complemented
233	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
234	#546	—	5	1	no	Sylow
235	C₂×C₅²	—	2	2	no	—
236	C₂×C₅²	—	2	2	no	—
237	C₂×C₅²	—	4	1	no	—
238	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₅²	C₂×#41	1	1	yes	characteristic
240	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
241	C₅×D₅	—	2	2	no	—
242	C₅×D₅	—	4	1	no	—
243	C₅×D₅	—	4	1	no	—
244	C₂³×C₅	—	2	8	no	—
245	C₂³×C₅	—	2	2	no	—
246	C₂³×C₅	—	2	2	no	—
247	C₂³×C₅	—	2	2	no	—
248	C₂³×C₅	—	2	2	no	—
249	C₂³×C₅	—	2	2	no	—
250	C₂³×C₅	—	2	2	no	—
251	C₂³×C₅	—	2	2	no	—
252	C₂³×C₅	—	4	4	no	—
253	C₂³×C₅	—	4	1	no	—
254	C₂³×C₅	—	4	1	no	—
255	C₂³×C₅	—	4	4	no	—
256	C₂³×C₅	—	5	1	no	—
257	C₂³×C₅	—	5	1	no	—
258	C₂³×C₅	—	5	1	no	—
259	C₂³×C₅	—	10	2	no	—
260	C₂³×C₅	—	10	2	no	—
261	C₂³×C₅	—	20	1	no	—
262	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
263	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
264	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
265	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
266	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
267	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
268	C₂²×D₅	—	2	2	no	—
269	C₂²×D₅	—	2	2	no	—
270	C₂²×D₅	—	4	1	no	—
271	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
272	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
273	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
274	C₅×Q₈	—	10	2	no	—
275	C₅×D₄	—	5	4	no	—
276	C₅×D₄	—	10	2	no	—
277	C₅×D₄	—	10	2	no	—
278	C₅×D₄	—	10	2	no	—
279	C₅×D₄	—	10	2	no	—
280	C₅×D₄	—	10	2	no	—
281	C₅×D₄	—	10	2	no	—
282	C₅×D₄	—	20	1	no	—
283	C₅×D₄	—	20	1	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	2	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	2	no	—
290	C₂×C₄×C₅	—	4	4	no	—
291	C₂×C₄×C₅	—	4	1	no	—
292	C₂×C₄×C₅	—	4	1	no	—
293	C₂×C₄×C₅	—	4	4	no	—
294	C₂×C₄×C₅	—	5	1	no	—
295	C₂×C₄×C₅	—	5	1	no	—
296	C₂×C₄×C₅	—	5	1	no	—
297	C₂×C₄×C₅	—	5	1	no	—
298	C₂×C₄×C₅	—	5	1	no	—
299	C₂×C₄×C₅	—	10	2	no	—
300	C₂×C₄×C₅	—	10	2	no	—
301	C₂×C₄×C₅	—	10	2	no	—
302	C₂×C₄×C₅	—	20	1	no	—
303	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
304	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
305	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
306	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
307	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
308	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
309	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
311	C₂²⋊D₅	—	2	2	no	—
312	C₂²⋊D₅	—	2	2	no	—
313	C₂²⋊D₅	—	2	2	no	—
314	C₂²⋊D₅	—	2	2	no	—
315	C₂²⋊D₅	—	2	2	no	—
316	C₂²⋊D₅	—	4	1	no	—
317	C₂²⋊D₅	C₅×D₄	1	4	yes	complemented
318	C₂×Dic₅	—	2	2	no	—
319	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
320	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
321	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
322	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
323	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
324	D₂₀	—	2	2	no	—
325	D₂₀	—	4	1	no	—
326	C₄×D₅	—	2	2	no	—
327	C₄×D₅	—	2	2	no	—
328	C₄×D₅	—	4	1	no	—
329	Dic₁₀	—	2	2	no	—
330	C₂×#41	—	5	1	no	—
331	C₂²×D₄	—	5	1	no	—
332	#124	—	5	1	no	—
333	#123	—	5	1	no	—
334	#123	—	5	1	no	—
335	C₂²⋊Q₈	—	5	1	no	—
336	#121	—	5	1	no	—
337	#121	—	5	1	no	—
338	#121	—	5	1	no	—
339	#120	—	5	1	no	—
340	#120	—	5	1	no	—
341	C₄×D₄	—	5	1	no	—
342	C₄×D₄	—	5	1	no	—
343	C₂×C₂²⋊C₄	—	5	1	no	—
344	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
345	C₅²	#546	1	1	yes	characteristic, complemented, Sylow
346	C₂²×C₅	—	2	4	no	—
347	C₂²×C₅	—	2	4	no	—
348	C₂²×C₅	—	2	1	no	—
349	C₂²×C₅	—	2	1	no	—
350	C₂²×C₅	—	2	1	no	—
351	C₂²×C₅	—	2	1	no	—
352	C₂²×C₅	—	2	1	no	—
353	C₂²×C₅	—	2	1	no	—
354	C₂²×C₅	—	2	1	no	—
355	C₂²×C₅	—	2	1	no	—
356	C₂²×C₅	—	2	4	no	—
357	C₂²×C₅	—	2	1	no	—
358	C₂²×C₅	—	2	1	no	—
359	C₂²×C₅	—	2	1	no	—
360	C₂²×C₅	—	2	1	no	—
361	C₂²×C₅	—	2	1	no	—
362	C₂²×C₅	—	2	1	no	—
363	C₂²×C₅	—	2	4	no	—
364	C₂²×C₅	—	2	4	no	—
365	C₂²×C₅	—	2	2	no	—
366	C₂²×C₅	—	4	2	no	—
367	C₂²×C₅	—	4	2	no	—
368	C₂²×C₅	—	4	2	no	—
369	C₂²×C₅	—	4	2	no	—
370	C₂²×C₅	—	4	4	no	—
371	C₂²×C₅	—	4	4	no	—
372	C₂²×C₅	—	4	16	no	—
373	C₂²×C₅	—	5	2	no	—
374	C₂²×C₅	—	10	1	no	—
375	C₂²×C₅	—	10	1	no	—
376	C₂²×C₅	—	10	1	no	—
377	C₂²×C₅	—	10	1	no	—
378	C₂²×C₅	—	10	1	no	—
379	C₂²×C₅	—	10	1	no	—
380	C₂²×C₅	—	10	1	no	—
381	C₂²×C₅	—	10	1	no	—
382	C₂²×C₅	—	20	4	no	—
383	C₂²×C₅	#709	1	2	yes	complemented
384	C₂²×C₅	D₄×D₅	1	2	yes	—
385	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
386	C₂²×C₅	C₅×#41	1	2	yes	complemented
387	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
388	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
389	C₂×D₅	—	2	1	no	—
390	C₂×D₅	—	2	1	no	—
391	C₂×D₅	—	2	1	no	—
392	C₂×D₅	—	2	1	no	—
393	C₂×D₅	—	2	1	no	—
394	C₂×D₅	—	2	1	no	—
395	C₂×D₅	—	2	1	no	—
396	C₂×D₅	—	2	1	no	—
397	C₂×D₅	—	4	4	no	—
398	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
399	C₄×C₅	—	2	1	no	—
400	C₄×C₅	—	2	1	no	—
401	C₄×C₅	—	2	1	no	—
402	C₄×C₅	—	2	1	no	—
403	C₄×C₅	—	2	1	no	—
404	C₄×C₅	—	2	1	no	—
405	C₄×C₅	—	2	1	no	—
406	C₄×C₅	—	2	1	no	—
407	C₄×C₅	—	4	2	no	—
408	C₄×C₅	—	4	2	no	—
409	C₄×C₅	—	4	2	no	—
410	C₄×C₅	—	4	2	no	—
411	C₄×C₅	—	5	2	no	—
412	C₄×C₅	—	10	1	no	—
413	C₄×C₅	—	10	1	no	—
414	C₄×C₅	—	10	1	no	—
415	C₄×C₅	—	10	1	no	—
416	Dic₅	—	2	1	no	—
417	Dic₅	—	2	1	no	—
418	Dic₅	—	2	1	no	—
419	Dic₅	—	2	1	no	—
420	Dic₅	C₂×C₅×D₄	1	2	yes	—
421	C₂⁴	—	5	1	no	—
422	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
423	#41	—	10	4	no	—
424	C₂×Q₈	—	5	1	no	—
425	C₂×D₄	—	5	1	no	—
426	C₂×D₄	—	5	1	no	—
427	C₂×D₄	—	5	1	no	—
428	C₂×D₄	—	5	1	no	—
429	C₂×D₄	—	5	1	no	—
430	C₂×D₄	—	5	1	no	—
431	C₂×D₄	—	5	1	no	—
432	C₂×D₄	—	5	1	no	—
433	C₂×D₄	—	5	1	no	—
434	C₂×D₄	—	10	4	no	—
435	C₂²×C₄	—	5	1	no	—
436	C₂²×C₄	—	5	1	no	—
437	C₂²×C₄	—	5	1	no	—
438	C₂²×C₄	—	5	1	no	—
439	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
440	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
441	C₄⋊C₄	—	5	1	no	—
442	C₄⋊C₄	—	5	1	no	—
443	C₄⋊C₄	—	5	1	no	—
444	C₄⋊C₄	—	5	1	no	—
445	C₂²⋊C₄	—	5	1	no	—
446	C₂²⋊C₄	—	5	1	no	—
447	C₂²⋊C₄	—	5	1	no	—
448	C₂²⋊C₄	—	5	1	no	—
449	C₂²⋊C₄	—	5	1	no	—
450	C₂²⋊C₄	—	5	1	no	—
451	C₂²⋊C₄	—	5	1	no	—
452	C₂²⋊C₄	—	5	1	no	—
453	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
454	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
455	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
456	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
457	C₄²	—	5	1	no	—
458	C₂×C₅	—	2	8	no	—
459	C₂×C₅	—	2	2	no	—
460	C₂×C₅	—	2	2	no	—
461	C₂×C₅	—	2	2	no	—
462	C₂×C₅	—	2	2	no	—
463	C₂×C₅	—	2	2	no	—
464	C₂×C₅	—	2	2	no	—
465	C₂×C₅	—	2	2	no	—
466	C₂×C₅	—	4	4	no	—
467	C₂×C₅	—	4	4	no	—
468	C₂×C₅	—	4	1	no	—
469	C₂×C₅	—	4	1	no	—
470	C₂×C₅	—	10	2	no	—
471	C₂×C₅	—	20	1	no	—
472	C₂×C₅	—	20	1	no	—
473	C₂×C₅	C₂×#709	1	1	yes	characteristic
474	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
475	C₂×C₅	#4144	1	1	yes	characteristic
476	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
477	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
478	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
479	D₅	—	2	2	no	—
480	D₅	—	4	1	no	—
481	D₅	—	4	1	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	5	1	no	—
486	C₂³	—	5	1	no	—
487	C₂³	—	5	1	no	—
488	C₂³	—	10	2	no	—
489	C₂³	—	10	2	no	—
490	C₂³	—	20	1	no	—
491	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
492	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
493	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
494	Q₈	—	10	2	no	—
495	D₄	—	5	4	no	—
496	D₄	—	10	2	no	—
497	D₄	—	10	2	no	—
498	D₄	—	10	2	no	—
499	D₄	—	10	2	no	—
500	D₄	—	10	2	no	—
501	D₄	—	10	2	no	—
502	D₄	—	20	1	no	—
503	D₄	—	20	1	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	5	1	no	—
507	C₂×C₄	—	5	1	no	—
508	C₂×C₄	—	5	1	no	—
509	C₂×C₄	—	5	1	no	—
510	C₂×C₄	—	5	1	no	—
511	C₂×C₄	—	10	2	no	—
512	C₂×C₄	—	10	2	no	—
513	C₂×C₄	—	10	2	no	—
514	C₂×C₄	—	20	1	no	—
515	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
516	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
517	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
518	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
519	C₅	—	2	2	no	—
520	C₅	#65824	1	1	yes	characteristic, direct product factor
521	C₅	C₅×#546	1	1	yes	characteristic, complemented
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	5	2	no	—
531	C₂²	—	10	1	no	—
532	C₂²	—	10	1	no	—
533	C₂²	—	10	1	no	—
534	C₂²	—	10	1	no	—
535	C₂²	—	10	1	no	—
536	C₂²	—	10	1	no	—
537	C₂²	—	10	1	no	—
538	C₂²	—	10	1	no	—
539	C₂²	—	20	4	no	—
540	C₂²	C₅×#709	1	2	yes	complemented
541	C₂²	C₅×D₄×D₅	1	2	yes	—
542	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
543	C₄	—	2	1	no	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	5	2	no	—
548	C₄	—	10	1	no	—
549	C₄	—	10	1	no	—
550	C₄	—	10	1	no	—
551	C₄	—	10	1	no	—
552	C₂	—	2	2	no	—
553	C₂	—	2	2	no	—
554	C₂	—	4	1	no	—
555	C₂	—	10	2	no	—
556	C₂	—	20	1	no	—
557	C₂	—	20	1	no	—
558	C₂	C₂×C₅×#709	1	1	yes	characteristic
559	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
560	C₂	C₅×#4144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₅×#546
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
4. C₅×#4099	C₂	semidirect
5. C₅×#4076	C₂	semidirect
6. C₅×#4075	C₂	semidirect
7. C₄×C₅×C₂²⋊D₅	C₂	semidirect
8. C₅×#4031	C₂	semidirect
9. C₅×#4030	C₂	semidirect
10. C₅×#4030	C₂	semidirect
11. C₅×#4029	C₂	semidirect
12. C₅×#4028	C₂	semidirect
13. C₅×#4027	C₂	semidirect
14. C₅×D₅×C₂²⋊C₄	C₂	semidirect
15. C₅×#4024	C₂	semidirect
18. C₂²×C₄×C₅²	C₂²	semidirect
19. C₂²×C₄×C₅²	C₂²	semidirect
26. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
27. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
30. C₂×C₅×D₂₀	C₂²	semidirect
31. C₂×C₅×D₂₀	C₂²	semidirect
32. C₂×C₄×C₅×D₅	C₂²	semidirect
33. C₂×C₄×C₅×D₅	C₂²	semidirect
34. C₂×C₄×C₅×D₅	C₂²	semidirect
35. C₂×C₅×Dic₁₀	C₂²	semidirect
38. C₅²×C₂²⋊C₄	C₂²	semidirect
39. C₅²×C₂²⋊C₄	C₂²	semidirect
41. C₅×C₂²⋊Dic₅	C₂²	semidirect
42. C₅×C₂²⋊Dic₅	C₂²	semidirect
43. C₅×#685	C₂²	semidirect
44. C₅×#685	C₂²	semidirect
45. C₅×#685	C₂²	semidirect
46. C₅×#685	C₂²	semidirect
47. C₅×#685	C₂²	semidirect
48. C₅×C₄⋊Dic₅	C₂²	semidirect
49. C₅×#683	C₂²	semidirect
50. C₅×#683	C₂²	semidirect
51. C₅×#683	C₂²	semidirect
52. C₄×C₅×Dic₅	C₂²	semidirect
54. #65824	C₅	direct (N × H)
79. C₅×C₂²⋊D₅	D₄	semidirect
94. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
95. C₂×#709	C₂×C₅	semidirect
109. C₂×C₅×C₂²⋊C₄	D₅	semidirect
110. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
111. #4099	C₂×C₅	semidirect
112. #4076	C₂×C₅	semidirect
113. #4075	C₂×C₅	semidirect
114. C₄×C₂²⋊D₅	C₂×C₅	semidirect
115. #4031	C₂×C₅	semidirect
116. #4030	C₂×C₅	semidirect
117. #4030	C₂×C₅	semidirect
118. #4029	C₂×C₅	semidirect
119. #4028	C₂×C₅	semidirect
120. #4027	C₂×C₅	semidirect
121. D₅×C₂²⋊C₄	C₂×C₅	semidirect
122. #4024	C₂×C₅	semidirect
132. C₂²×C₅²	#41	semidirect
176. C₂²×C₄×C₅	C₂×D₅	semidirect
177. C₂²×C₄×C₅	C₂×D₅	semidirect
178. C₂²×C₄×C₅	C₂²×C₅	semidirect
179. C₂²×C₄×C₅	C₂²×C₅	semidirect
186. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
187. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
190. C₂×D₂₀	C₂²×C₅	semidirect
191. C₂×D₂₀	C₂²×C₅	semidirect
192. C₂×C₄×D₅	C₂²×C₅	semidirect
193. C₂×C₄×D₅	C₂²×C₅	semidirect
194. C₂×C₄×D₅	C₂²×C₅	semidirect
195. C₂×Dic₁₀	C₂²×C₅	semidirect
214. C₅×C₂²⋊C₄	C₂×D₅	semidirect
215. C₅×C₂²⋊C₄	C₂×D₅	semidirect
218. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
219. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
222. C₂²⋊Dic₅	C₂²×C₅	semidirect
223. C₂²⋊Dic₅	C₂²×C₅	semidirect
224. #685	C₂²×C₅	semidirect
225. #685	C₂²×C₅	semidirect
226. #685	C₂²×C₅	semidirect
227. #685	C₂²×C₅	semidirect
228. #685	C₂²×C₅	semidirect
229. C₄⋊Dic₅	C₂²×C₅	semidirect
230. #683	C₂²×C₅	semidirect
231. #683	C₂²×C₅	semidirect
232. #683	C₂²×C₅	semidirect
233. C₄×Dic₅	C₂²×C₅	semidirect
317. C₂²⋊D₅	C₅×D₄	semidirect
344. C₂×C₂²⋊C₄	C₅×D₅	semidirect
345. C₅²	#546	semidirect
383. C₂²×C₅	#709	semidirect
386. C₂²×C₅	C₅×#41	semidirect
439. C₂²×C₄	C₂×C₅×D₅	semidirect
440. C₂²×C₄	C₂×C₅×D₅	semidirect
455. C₂²⋊C₄	C₂×C₅×D₅	semidirect
456. C₂²⋊C₄	C₂×C₅×D₅	semidirect
520. C₅	#65824	direct (N × H)
521. C₅	C₅×#546	semidirect
540. C₂²	C₅×#709	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₂²⋊D₅	2
C₂×C₅×#709	2
C₂×C₅²×C₂²⋊C₄	2
C₅×#4099	2
C₅×#4076	2
C₅×#4075	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4031	2
C₅×#4030	2
C₅×#4029	2
C₅×#4028	2
C₅×#4027	2
C₅×D₅×C₂²⋊C₄	2
C₅×#4024	2
C₅×#546	5
#65824	5

Subgroups — C5×#65824 #13106645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65824 #13106645