Subgroups — C₅×#10618096 #72258476

Order 3200.

3994 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#4144	C₂	1	2	yes	maximal, complemented
2	C₂×C₅²×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#66139	C₂	1	2	yes	maximal, complemented
4	C₅×D₄×C₂²⋊D₅	C₂	1	8	yes	maximal, complemented
5	C₅×#66051	C₂	1	4	yes	maximal, complemented
6	C₅×#66050	C₂	1	2	yes	maximal, complemented
7	C₅×#66049	C₂	1	4	yes	maximal, complemented
8	C₅×D₅×#127	C₂	1	2	yes	maximal, complemented
9	C₅×#66052	C₂	1	4	yes	maximal, complemented
10	C₅×#65811	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅²×D₄	C₂²	1	2	yes	complemented
13	C₂²×C₅²×D₄	C₂²	1	4	yes	complemented
14	C₂²×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
15	C₅×#4144	—	2	16	no	—
16	C₂×C₅×#711	C₂²	1	4	yes	complemented
17	C₂×C₅×#711	C₂²	1	4	yes	complemented
18	C₂×C₅×D₄×D₅	C₂²	1	4	yes	complemented
19	C₂×C₅×D₄×D₅	C₂²	1	4	yes	complemented
20	C₂×C₅×D₄×D₅	C₂²	1	8	yes	complemented
21	C₂×C₅×#709	C₂²	1	1	yes	characteristic
22	C₂×C₅×#709	C₂²	1	2	yes	complemented
23	C₅²×#127	C₂²	1	4	yes	complemented
24	C₅²×#127	C₂²	1	2	yes	complemented
25	C₅²×#127	C₂²	1	2	yes	complemented
26	C₂×C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
27	C₅×#4099	C₂²	1	8	yes	complemented
28	C₅×#4086	C₂²	1	8	yes	complemented
29	C₅×#4086	C₂²	1	4	yes	complemented
30	C₅×#4085	C₂²	1	16	yes	complemented
31	C₅×#4084	C₂²	1	8	yes	complemented
32	C₅×#4084	C₂²	1	8	yes	complemented
33	C₅×#4083	C₂²	1	16	yes	complemented
34	C₅×#4082	C₂²	1	4	yes	complemented
35	C₅×D₄×Dic₅	C₂²	1	8	yes	complemented
36	C₅×#4076	C₂²	1	4	yes	complemented
37	C₄×C₅×C₂²⋊D₅	C₂²	1	4	yes	complemented
38	C₅×#4072	C₂²	1	1	yes	characteristic, complemented
39	C₅×#4021	C₂²	1	2	yes	complemented
40	C₅×#4020	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₅×D₂₀	C₂²	1	4	yes	complemented
42	C₄×C₅×D₂₀	C₂²	1	2	yes	complemented
43	C₅×#4018	C₂²	1	2	yes	complemented
44	C₄²×C₅×D₅	C₂²	1	2	yes	complemented
45	C₅×#4015	C₂²	1	1	yes	characteristic, complemented
46	C₄×C₅×Dic₁₀	C₂²	1	2	yes	complemented
47	C₅×#3498	—	5	1	no	maximal
48	#10618096	C₅	1	1	yes	maximal, characteristic, direct product factor
49	C₂⁴×C₅²	C₂³	1	4	yes	—
50	C₂³×C₅×D₅	C₂³	1	8	yes	—
51	C₂×C₅²×D₄	—	2	8	no	—
52	C₂×C₅²×D₄	—	4	8	no	—
53	C₂×C₅²×D₄	C₂³	1	8	yes	—
54	C₂×C₅²×D₄	C₂³	1	4	yes	—
55	C₂×C₅²×D₄	C₂³	1	4	yes	—
56	C₂×C₅²×D₄	C₂³	1	8	yes	—
57	C₂²×C₄×C₅²	C₂³	1	2	yes	—
58	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₂²⋊D₅	—	2	32	no	—
60	C₂×C₅×C₂²⋊D₅	C₂³	1	4	yes	—
61	C₂×C₅×C₂²⋊D₅	C₂³	1	8	yes	—
62	C₂×C₅×C₂²⋊D₅	C₂³	1	8	yes	—
63	C₂×C₅×C₂²⋊D₅	C₂³	1	16	yes	—
64	C₂×C₅×C₂²⋊D₅	C₂³	1	2	yes	—
65	C₂²×C₅×Dic₅	C₂³	1	8	yes	—
66	C₅×#711	—	2	16	no	—
67	C₅×#711	—	2	16	no	—
68	C₅×D₄×D₅	—	2	16	no	—
69	C₅×D₄×D₅	—	2	16	no	—
70	C₅×D₄×D₅	—	4	16	no	—
71	C₅×#709	—	2	8	no	—
72	C₅×#709	D₄	1	8	yes	complemented
73	C₂×C₅×D₂₀	C₂³	1	4	yes	—
74	C₂×C₅×D₂₀	C₂³	1	2	yes	—
75	C₂×C₅×D₂₀	C₂³	1	1	yes	characteristic
76	C₂×C₄×C₅×D₅	C₂³	1	4	yes	—
77	C₂×C₄×C₅×D₅	C₂³	1	4	yes	—
78	C₂×C₄×C₅×D₅	C₂³	1	2	yes	—
79	C₂×C₅×Dic₁₀	C₂³	1	2	yes	—
80	C₂×C₅×Dic₁₀	C₂³	1	1	yes	characteristic
81	C₄²×C₅²	C₂³	1	2	yes	—
82	C₄²×C₅²	C₂³	1	1	yes	characteristic
83	C₄²×C₅²	C₂³	1	1	yes	characteristic
84	C₅×C₂²⋊Dic₅	C₂³	1	2	yes	—
85	C₅×C₂²⋊Dic₅	C₂³	1	16	yes	—
86	C₅×#685	C₂³	1	4	yes	—
87	C₅×#685	C₂³	1	8	yes	—
88	C₅×C₄⋊Dic₅	C₂³	1	2	yes	—
89	C₅×C₄⋊Dic₅	C₂³	1	4	yes	—
90	C₅×#683	C₂³	1	4	yes	—
91	C₄×C₅×Dic₅	C₂³	1	2	yes	—
92	C₄×C₅×Dic₅	C₂³	1	4	yes	—
93	C₂×C₅×H(5,𝔽₂)	—	5	2	no	—
94	C₂×#4144	C₂×C₅	1	2	yes	complemented
95	C₅×#550	—	5	4	no	—
96	C₅×D₄²	—	5	8	no	—
97	C₅×D₄²	—	5	4	no	—
98	C₅×#535	—	5	2	no	—
99	C₅×#535	—	5	4	no	—
100	C₅×#532	—	5	2	no	—
101	C₅×#532	—	5	1	no	—
102	C₂×C₅×#127	—	2	2	no	—
103	C₂×C₅×#127	—	5	2	no	—
104	C₂×C₅×#127	D₅	1	1	yes	characteristic, complemented
105	C₂×C₅×#127	C₂×C₅	1	1	yes	characteristic, complemented
106	C₄×C₅×#41	—	5	1	no	—
107	#66139	C₂×C₅	1	2	yes	complemented
108	D₄×C₂²⋊D₅	C₂×C₅	1	8	yes	complemented
109	#66052	C₂×C₅	1	4	yes	complemented
110	#66051	C₂×C₅	1	4	yes	complemented
111	#66050	C₂×C₅	1	2	yes	complemented
112	#66049	C₂×C₅	1	4	yes	complemented
113	D₅×#127	C₂×C₅	1	2	yes	complemented
114	#65811	C₂×C₅	1	1	yes	characteristic, complemented
115	C₄×#709	C₂×C₅	1	1	yes	characteristic, complemented
116	C₂³×C₅²	—	2	8	no	—
117	C₂³×C₅²	—	4	8	no	—
118	C₂³×C₅²	C₂⁴	1	8	yes	—
119	C₂³×C₅²	C₂⁴	1	1	yes	characteristic
120	C₂²×C₅×D₅	—	2	16	no	—
121	C₂²×C₅×D₅	—	4	16	no	—
122	C₂²×C₅×D₅	C₂⁴	1	4	yes	—
123	C₂²×C₅×D₅	C₂⁴	1	2	yes	—
124	C₅²×D₄	—	2	8	no	—
125	C₅²×D₄	—	2	8	no	—
126	C₅²×D₄	—	2	16	no	—
127	C₅²×D₄	—	4	8	no	—
128	C₂×C₄×C₅²	—	2	4	no	—
129	C₂×C₄×C₅²	C₂×D₄	1	4	yes	complemented
130	C₂×C₄×C₅²	C₂⁴	1	2	yes	—
131	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
133	C₂×C₄×C₅²	C₂⁴	1	2	yes	—
134	C₅×C₂²⋊D₅	—	2	8	no	—
135	C₅×C₂²⋊D₅	—	2	16	no	—
136	C₅×C₂²⋊D₅	—	2	16	no	—
137	C₅×C₂²⋊D₅	—	4	16	no	—
138	C₅×C₂²⋊D₅	C₂×D₄	1	8	yes	complemented
139	C₂×C₅×Dic₅	—	2	16	no	—
140	C₂×C₅×Dic₅	C₂⁴	1	4	yes	—
141	C₂×C₅×Dic₅	C₂⁴	1	4	yes	—
142	C₂×C₅×Dic₅	C₂⁴	1	2	yes	—
143	C₅×D₂₀	—	2	4	no	—
144	C₅×D₂₀	—	2	8	no	—
145	C₅×D₂₀	C₂×D₄	1	4	yes	complemented
146	C₄×C₅×D₅	—	2	8	no	—
147	C₄×C₅×D₅	—	2	8	no	—
148	C₄×C₅×D₅	C₂×D₄	1	8	yes	complemented
149	C₅×Dic₁₀	—	2	4	no	—
150	C₅×Dic₁₀	C₂×D₄	1	4	yes	complemented
151	C₅×H(5,𝔽₂)	—	10	16	no	—
152	C₂×C₅×#41	—	5	4	no	—
153	C₂×C₅×#41	—	5	4	no	—
154	C₂×C₅×#41	—	5	1	no	—
155	C₂×C₅×#41	—	5	2	no	—
156	C₂²×C₅×D₄	—	2	4	no	—
157	C₂²×C₅×D₄	—	2	8	no	—
158	C₂²×C₅×D₄	—	5	8	no	—
159	C₂²×C₅×D₄	—	5	4	no	—
160	C₂²×C₅×D₄	—	5	4	no	—
161	C₂²×C₅×D₄	—	5	8	no	—
162	C₂²×C₅×D₄	C₂×D₅	1	2	yes	complemented
163	C₂²×C₅×D₄	C₂×D₅	1	4	yes	complemented
164	C₂²×C₅×D₄	C₂²×C₅	1	2	yes	complemented
165	C₂²×C₅×D₄	C₂²×C₅	1	4	yes	complemented
166	C₂²×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
167	#4144	—	2	16	no	—
168	C₂×#711	C₂²×C₅	1	4	yes	complemented
169	C₂×#711	C₂²×C₅	1	4	yes	complemented
170	C₂×D₄×D₅	C₂²×C₅	1	4	yes	complemented
171	C₂×D₄×D₅	C₂²×C₅	1	4	yes	complemented
172	C₂×D₄×D₅	C₂²×C₅	1	8	yes	complemented
173	C₂×#709	C₂²×C₅	1	1	yes	characteristic
174	C₂×#709	C₂²×C₅	1	2	yes	complemented
175	C₅×C₄⋊Q₈	—	5	1	no	—
176	C₅×#127	—	2	4	no	—
177	C₅×#127	—	2	4	no	—
178	C₅×#127	—	2	8	no	—
179	C₅×#127	—	5	4	no	—
180	C₅×#127	—	5	8	no	—
181	C₅×#127	—	5	1	no	—
182	C₅×#127	C₂×D₅	1	2	yes	complemented
183	C₅×#127	C₂×D₅	1	2	yes	complemented
184	C₅×#127	C₂×D₅	1	4	yes	complemented
185	C₅×#127	C₂²×C₅	1	2	yes	complemented
186	C₅×#127	C₂²×C₅	1	2	yes	complemented
187	C₅×#127	C₂²×C₅	1	4	yes	complemented
188	C₅×#124	—	5	4	no	—
189	C₅×#124	—	5	2	no	—
190	C₅×#121	—	5	8	no	—
191	C₅×#121	—	5	8	no	—
192	C₅×#121	—	5	16	no	—
193	C₅×#121	—	5	4	no	—
194	C₅×#120	—	5	8	no	—
195	C₅×#120	—	5	16	no	—
196	C₄×C₅×Q₈	—	5	2	no	—
197	C₄×C₅×D₄	—	5	8	no	—
198	C₄×C₅×D₄	—	5	4	no	—
199	C₄×C₅×D₄	—	5	2	no	—
200	C₄×C₅×D₄	—	5	4	no	—
201	C₅×#117	—	5	1	no	—
202	C₅×#117	—	5	2	no	—
203	C₂×C₄²×C₅	—	2	2	no	—
204	C₂×C₄²×C₅	—	5	2	no	—
205	C₂×C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
206	C₂×C₄²×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
207	#4099	C₂²×C₅	1	8	yes	complemented
208	#4086	C₂²×C₅	1	4	yes	complemented
209	#4086	C₂²×C₅	1	8	yes	complemented
210	#4085	C₂²×C₅	1	16	yes	complemented
211	#4084	C₂²×C₅	1	8	yes	complemented
212	#4084	C₂²×C₅	1	8	yes	complemented
213	#4083	C₂²×C₅	1	16	yes	complemented
214	#4082	C₂²×C₅	1	4	yes	complemented
215	D₄×Dic₅	C₂²×C₅	1	8	yes	complemented
216	#4076	C₂²×C₅	1	4	yes	complemented
217	C₄×C₂²⋊D₅	C₂²×C₅	1	4	yes	complemented
218	#4072	C₂²×C₅	1	1	yes	characteristic, complemented
219	#4021	C₂²×C₅	1	2	yes	complemented
220	#4020	C₂²×C₅	1	1	yes	characteristic, complemented
221	C₄×D₂₀	C₂²×C₅	1	2	yes	complemented
222	C₄×D₂₀	C₂²×C₅	1	4	yes	complemented
223	#4018	C₂²×C₅	1	2	yes	complemented
224	C₄²×D₅	C₂²×C₅	1	2	yes	complemented
225	#4015	C₂²×C₅	1	1	yes	characteristic, complemented
226	C₄×Dic₁₀	C₂²×C₅	1	2	yes	complemented
227	#3498	—	5	1	no	Sylow
228	C₂²×C₅²	—	2	2	no	—
229	C₂²×C₅²	—	2	8	no	—
230	C₂²×C₅²	—	2	16	no	—
231	C₂²×C₅²	—	8	8	no	—
232	C₂²×C₅²	C₂²×D₄	1	2	yes	—
233	C₂²×C₅²	C₂⁵	1	1	yes	characteristic
234	C₂×C₅×D₅	—	2	4	no	—
235	C₂×C₅×D₅	—	2	4	no	—
236	C₂×C₅×D₅	—	2	8	no	—
237	C₂×C₅×D₅	—	8	16	no	—
238	C₂×C₅×D₅	C₂²×D₄	1	4	yes	—
239	C₄×C₅²	—	2	2	no	—
240	C₄×C₅²	C₂²×D₄	1	2	yes	—
241	C₄×C₅²	C₂²×D₄	1	2	yes	—
242	C₄×C₅²	H(5,𝔽₂)	1	4	yes	complemented
243	C₅×Dic₅	—	2	4	no	—
244	C₅×Dic₅	—	2	4	no	—
245	C₅×Dic₅	C₂²×D₄	1	4	yes	—
246	C₂⁴×C₅	—	2	8	no	—
247	C₂⁴×C₅	—	5	8	no	—
248	C₂⁴×C₅	C₂²×D₅	1	4	yes	—
249	C₂⁴×C₅	C₂³×C₅	1	4	yes	—
250	C₂³×D₅	C₂³×C₅	1	8	yes	—
251	C₅×#41	—	5	8	no	—
252	C₅×#41	—	10	16	no	—
253	C₅×#41	—	10	16	no	—
254	C₅×#41	—	10	8	no	—
255	C₂×C₅×Q₈	—	5	1	no	—
256	C₂×C₅×Q₈	—	5	2	no	—
257	C₂×C₅×D₄	—	2	8	no	—
258	C₂×C₅×D₄	—	2	8	no	—
259	C₂×C₅×D₄	—	2	16	no	—
260	C₂×C₅×D₄	—	2	8	no	—
261	C₂×C₅×D₄	—	2	8	no	—
262	C₂×C₅×D₄	—	2	16	no	—
263	C₂×C₅×D₄	—	4	16	no	—
264	C₂×C₅×D₄	—	4	8	no	—
265	C₂×C₅×D₄	—	4	8	no	—
266	C₂×C₅×D₄	—	4	32	no	—
267	C₂×C₅×D₄	—	5	2	no	—
268	C₂×C₅×D₄	—	5	8	no	—
269	C₂×C₅×D₄	—	5	4	no	—
270	C₂×C₅×D₄	—	5	8	no	—
271	C₂×C₅×D₄	—	5	16	no	—
272	C₂×C₅×D₄	—	5	1	no	—
273	C₂×C₅×D₄	—	5	4	no	—
274	C₂×C₅×D₄	—	5	2	no	—
275	C₂×C₅×D₄	—	10	16	no	—
276	C₂×C₅×D₄	—	10	16	no	—
277	C₂×C₅×D₄	—	10	32	no	—
278	C₂×C₅×D₄	—	20	16	no	—
279	C₂×C₅×D₄	C₂²×D₅	1	8	yes	—
280	C₂×C₅×D₄	C₂²×D₅	1	4	yes	—
281	C₂×C₅×D₄	C₂²×D₅	1	4	yes	—
282	C₂×C₅×D₄	C₂²×D₅	1	8	yes	—
283	C₂×C₅×D₄	C₂³×C₅	1	4	yes	—
284	C₂×C₅×D₄	C₂³×C₅	1	4	yes	—
285	C₂×C₅×D₄	C₂³×C₅	1	8	yes	—
286	C₂×C₅×D₄	C₂³×C₅	1	8	yes	—
287	C₂²×C₄×C₅	—	2	4	no	—
288	C₂²×C₄×C₅	—	2	2	no	—
289	C₂²×C₄×C₅	—	5	2	no	—
290	C₂²×C₄×C₅	—	5	8	no	—
291	C₂²×C₄×C₅	—	5	4	no	—
292	C₂²×C₄×C₅	—	5	4	no	—
293	C₂²×C₄×C₅	C₂²×D₅	1	2	yes	—
294	C₂²×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
295	C₂²×C₄×C₅	C₂³×C₅	1	2	yes	—
296	C₂²×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
297	C₂×C₂²⋊D₅	—	2	32	no	—
298	C₂×C₂²⋊D₅	C₂³×C₅	1	4	yes	—
299	C₂×C₂²⋊D₅	C₂³×C₅	1	8	yes	—
300	C₂×C₂²⋊D₅	C₂³×C₅	1	8	yes	—
301	C₂×C₂²⋊D₅	C₂³×C₅	1	16	yes	—
302	C₂×C₂²⋊D₅	C₂³×C₅	1	2	yes	—
303	C₂²×Dic₅	C₂³×C₅	1	8	yes	—
304	#711	—	2	16	no	—
305	#711	—	2	16	no	—
306	D₄×D₅	—	2	16	no	—
307	D₄×D₅	—	2	16	no	—
308	D₄×D₅	—	4	16	no	—
309	#709	—	2	8	no	—
310	#709	C₅×D₄	1	8	yes	complemented
311	C₂×D₂₀	C₂³×C₅	1	2	yes	—
312	C₂×D₂₀	C₂³×C₅	1	4	yes	—
313	C₂×D₂₀	C₂³×C₅	1	1	yes	characteristic
314	C₂×C₄×D₅	C₂³×C₅	1	4	yes	—
315	C₂×C₄×D₅	C₂³×C₅	1	4	yes	—
316	C₂×C₄×D₅	C₂³×C₅	1	2	yes	—
317	C₂×Dic₁₀	C₂³×C₅	1	2	yes	—
318	C₂×Dic₁₀	C₂³×C₅	1	1	yes	characteristic
319	C₅×C₄⋊C₄	—	5	4	no	—
320	C₅×C₄⋊C₄	—	5	2	no	—
321	C₅×C₄⋊C₄	—	5	4	no	—
322	C₅×C₂²⋊C₄	—	5	2	no	—
323	C₅×C₂²⋊C₄	—	5	16	no	—
324	C₅×C₂²⋊C₄	—	5	4	no	—
325	C₅×C₂²⋊C₄	—	5	8	no	—
326	C₄²×C₅	—	2	2	no	—
327	C₄²×C₅	—	2	2	no	—
328	C₄²×C₅	—	2	4	no	—
329	C₄²×C₅	—	5	2	no	—
330	C₄²×C₅	—	5	4	no	—
331	C₄²×C₅	C₂²×D₅	1	1	yes	characteristic
332	C₄²×C₅	C₂²×D₅	1	1	yes	characteristic
333	C₄²×C₅	C₂²×D₅	1	2	yes	—
334	C₄²×C₅	C₂³×C₅	1	2	yes	—
335	C₄²×C₅	C₂³×C₅	1	1	yes	characteristic
336	C₄²×C₅	C₂³×C₅	1	1	yes	characteristic
337	C₂²⋊Dic₅	C₂³×C₅	1	2	yes	—
338	C₂²⋊Dic₅	C₂³×C₅	1	16	yes	—
339	#685	C₂³×C₅	1	4	yes	—
340	#685	C₂³×C₅	1	8	yes	—
341	C₄⋊Dic₅	C₂³×C₅	1	2	yes	—
342	C₄⋊Dic₅	C₂³×C₅	1	4	yes	—
343	#683	C₂³×C₅	1	4	yes	—
344	C₄×Dic₅	C₂³×C₅	1	4	yes	—
345	C₄×Dic₅	C₂³×C₅	1	2	yes	—
346	C₂×H(5,𝔽₂)	—	5	2	no	—
347	#550	—	5	4	no	—
348	D₄²	—	5	8	no	—
349	D₄²	—	5	4	no	—
350	#535	—	5	2	no	—
351	#535	—	5	4	no	—
352	#532	—	5	2	no	—
353	#532	—	5	1	no	—
354	C₂×#127	—	5	2	no	—
355	C₂×#127	C₅×D₅	1	1	yes	characteristic, complemented
356	C₄×#41	—	5	1	no	—
357	C₂×C₅²	—	2	2	no	—
358	C₂×C₅²	—	4	8	no	—
359	C₂×C₅²	C₂³×D₄	1	1	yes	characteristic
360	C₂×C₅²	C₂×H(5,𝔽₂)	1	2	yes	—
361	C₅×D₅	—	2	4	no	—
362	C₅×D₅	—	4	4	no	—
363	C₂³×C₅	—	2	8	no	—
364	C₂³×C₅	—	2	8	no	—
365	C₂³×C₅	—	2	2	no	—
366	C₂³×C₅	—	2	16	no	—
367	C₂³×C₅	—	4	16	no	—
368	C₂³×C₅	—	4	8	no	—
369	C₂³×C₅	—	4	8	no	—
370	C₂³×C₅	—	4	32	no	—
371	C₂³×C₅	—	5	2	no	—
372	C₂³×C₅	—	5	4	no	—
373	C₂³×C₅	—	10	16	no	—
374	C₂³×C₅	—	20	16	no	—
375	C₂³×C₅	C₂³×D₅	1	8	yes	—
376	C₂³×C₅	C₂³×D₅	1	1	yes	characteristic
377	C₂³×C₅	C₂⁴×C₅	1	8	yes	—
378	C₂³×C₅	C₂⁴×C₅	1	1	yes	characteristic
379	C₂²×D₅	—	2	16	no	—
380	C₂²×D₅	—	4	16	no	—
381	C₂²×D₅	C₂⁴×C₅	1	4	yes	—
382	C₂²×D₅	C₂⁴×C₅	1	2	yes	—
383	C₅×Q₈	—	5	4	no	—
384	C₅×Q₈	—	10	4	no	—
385	C₅×D₄	—	2	8	no	—
386	C₅×D₄	—	2	8	no	—
387	C₅×D₄	—	2	16	no	—
388	C₅×D₄	—	2	8	no	—
389	C₅×D₄	—	2	8	no	—
390	C₅×D₄	—	2	16	no	—
391	C₅×D₄	—	4	16	no	—
392	C₅×D₄	—	4	16	no	—
393	C₅×D₄	—	4	32	no	—
394	C₅×D₄	—	4	8	no	—
395	C₅×D₄	—	4	8	no	—
396	C₅×D₄	—	4	32	no	—
397	C₅×D₄	—	5	8	no	—
398	C₅×D₄	—	5	4	no	—
399	C₅×D₄	—	10	16	no	—
400	C₅×D₄	—	10	16	no	—
401	C₅×D₄	—	10	8	no	—
402	C₅×D₄	—	10	4	no	—
403	C₅×D₄	—	10	8	no	—
404	C₅×D₄	—	20	16	no	—
405	C₂×C₄×C₅	—	2	16	no	—
406	C₂×C₄×C₅	—	2	4	no	—
407	C₂×C₄×C₅	—	2	4	no	—
408	C₂×C₄×C₅	—	2	8	no	—
409	C₂×C₄×C₅	—	2	4	no	—
410	C₂×C₄×C₅	—	2	2	no	—
411	C₂×C₄×C₅	—	2	2	no	—
412	C₂×C₄×C₅	—	2	4	no	—
413	C₂×C₄×C₅	—	5	2	no	—
414	C₂×C₄×C₅	—	5	8	no	—
415	C₂×C₄×C₅	—	5	4	no	—
416	C₂×C₄×C₅	—	5	4	no	—
417	C₂×C₄×C₅	—	10	16	no	—
418	C₂×C₄×C₅	—	10	8	no	—
419	C₂×C₄×C₅	—	10	8	no	—
420	C₂×C₄×C₅	D₄×D₅	1	4	yes	complemented
421	C₂×C₄×C₅	C₂×C₅×D₄	1	4	yes	complemented
422	C₂×C₄×C₅	C₂³×D₅	1	2	yes	—
423	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
424	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
425	C₂×C₄×C₅	C₂³×D₅	1	2	yes	—
426	C₂×C₄×C₅	C₂⁴×C₅	1	2	yes	—
427	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
428	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
429	C₂×C₄×C₅	C₂⁴×C₅	1	2	yes	—
430	C₂²⋊D₅	—	2	8	no	—
431	C₂²⋊D₅	—	2	16	no	—
432	C₂²⋊D₅	—	2	16	no	—
433	C₂²⋊D₅	—	4	16	no	—
434	C₂²⋊D₅	C₂×C₅×D₄	1	8	yes	complemented
435	C₂×Dic₅	—	2	16	no	—
436	C₂×Dic₅	C₂⁴×C₅	1	4	yes	—
437	C₂×Dic₅	C₂⁴×C₅	1	4	yes	—
438	C₂×Dic₅	C₂⁴×C₅	1	2	yes	—
439	D₂₀	—	2	4	no	—
440	D₂₀	—	2	8	no	—
441	D₂₀	C₂×C₅×D₄	1	4	yes	complemented
442	C₄×D₅	—	2	8	no	—
443	C₄×D₅	—	2	8	no	—
444	C₄×D₅	C₂×C₅×D₄	1	8	yes	complemented
445	Dic₁₀	—	2	4	no	—
446	Dic₁₀	C₂×C₅×D₄	1	4	yes	complemented
447	H(5,𝔽₂)	—	10	16	no	—
448	C₂×#41	—	5	4	no	—
449	C₂×#41	—	5	4	no	—
450	C₂×#41	—	5	1	no	—
451	C₂×#41	—	5	2	no	—
452	C₂²×D₄	—	5	8	no	—
453	C₂²×D₄	—	5	4	no	—
454	C₂²×D₄	—	5	4	no	—
455	C₂²×D₄	—	5	8	no	—
456	C₂²×D₄	C₂×C₅×D₅	1	2	yes	complemented
457	C₂²×D₄	C₂×C₅×D₅	1	4	yes	complemented
458	C₄⋊Q₈	—	5	1	no	—
459	#127	—	5	4	no	—
460	#127	—	5	8	no	—
461	#127	—	5	1	no	—
462	#127	C₂×C₅×D₅	1	2	yes	complemented
463	#127	C₂×C₅×D₅	1	2	yes	complemented
464	#127	C₂×C₅×D₅	1	4	yes	complemented
465	#124	—	5	4	no	—
466	#124	—	5	2	no	—
467	#121	—	5	8	no	—
468	#121	—	5	8	no	—
469	#121	—	5	16	no	—
470	#121	—	5	4	no	—
471	#120	—	5	8	no	—
472	#120	—	5	16	no	—
473	C₄×Q₈	—	5	2	no	—
474	C₄×D₄	—	5	8	no	—
475	C₄×D₄	—	5	4	no	—
476	C₄×D₄	—	5	2	no	—
477	C₄×D₄	—	5	4	no	—
478	#117	—	5	1	no	—
479	#117	—	5	2	no	—
480	C₂×C₄²	—	5	2	no	—
481	C₂×C₄²	C₂×C₅×D₅	1	1	yes	characteristic, complemented
482	C₅²	#3498	1	1	yes	characteristic, complemented, Sylow
483	C₂²×C₅	—	2	8	no	—
484	C₂²×C₅	—	2	2	no	—
485	C₂²×C₅	—	2	2	no	—
486	C₂²×C₅	—	2	8	no	—
487	C₂²×C₅	—	2	16	no	—
488	C₂²×C₅	—	2	8	no	—
489	C₂²×C₅	—	2	16	no	—
490	C₂²×C₅	—	2	4	no	—
491	C₂²×C₅	—	2	2	no	—
492	C₂²×C₅	—	4	16	no	—
493	C₂²×C₅	—	4	32	no	—
494	C₂²×C₅	—	5	4	no	—
495	C₂²×C₅	—	8	8	no	—
496	C₂²×C₅	—	8	8	no	—
497	C₂²×C₅	—	8	32	no	—
498	C₂²×C₅	—	10	4	no	—
499	C₂²×C₅	—	10	4	no	—
500	C₂²×C₅	—	10	8	no	—
501	C₂²×C₅	—	40	16	no	—
502	C₂²×C₅	C₂×D₄×D₅	1	2	yes	—
503	C₂²×C₅	C₂²×C₅×D₄	1	2	yes	—
504	C₂²×C₅	C₂⁴×D₅	1	1	yes	characteristic
505	C₂²×C₅	C₂⁵×C₅	1	1	yes	characteristic
506	C₂×D₅	—	2	4	no	—
507	C₂×D₅	—	2	4	no	—
508	C₂×D₅	—	2	8	no	—
509	C₂×D₅	—	8	16	no	—
510	C₂×D₅	C₂²×C₅×D₄	1	4	yes	—
511	C₄×C₅	—	2	8	no	—
512	C₄×C₅	—	2	2	no	—
513	C₄×C₅	—	2	2	no	—
514	C₄×C₅	—	2	4	no	—
515	C₄×C₅	—	2	4	no	—
516	C₄×C₅	—	2	8	no	—
517	C₄×C₅	—	5	4	no	—
518	C₄×C₅	—	10	4	no	—
519	C₄×C₅	—	10	4	no	—
520	C₄×C₅	C₂×D₄×D₅	1	2	yes	—
521	C₄×C₅	C₂×D₄×D₅	1	2	yes	—
522	C₄×C₅	#4144	1	4	yes	complemented
523	C₄×C₅	C₂²×C₅×D₄	1	2	yes	—
524	C₄×C₅	C₂²×C₅×D₄	1	2	yes	—
525	C₄×C₅	C₅×H(5,𝔽₂)	1	4	yes	complemented
526	Dic₅	—	2	4	no	—
527	Dic₅	—	2	4	no	—
528	Dic₅	C₂²×C₅×D₄	1	4	yes	—
529	C₂⁴	—	5	8	no	—
530	C₂⁴	C₂²×C₅×D₅	1	4	yes	—
531	#41	—	5	8	no	—
532	#41	—	10	16	no	—
533	#41	—	10	16	no	—
534	#41	—	10	8	no	—
535	C₂×Q₈	—	5	1	no	—
536	C₂×Q₈	—	5	2	no	—
537	C₂×D₄	—	2	8	no	—
538	C₂×D₄	—	4	8	no	—
539	C₂×D₄	—	5	2	no	—
540	C₂×D₄	—	5	8	no	—
541	C₂×D₄	—	5	4	no	—
542	C₂×D₄	—	5	8	no	—
543	C₂×D₄	—	5	16	no	—
544	C₂×D₄	—	5	1	no	—
545	C₂×D₄	—	5	4	no	—
546	C₂×D₄	—	5	2	no	—
547	C₂×D₄	—	10	16	no	—
548	C₂×D₄	—	10	16	no	—
549	C₂×D₄	—	10	32	no	—
550	C₂×D₄	—	20	16	no	—
551	C₂×D₄	C₂²×C₅×D₅	1	8	yes	—
552	C₂×D₄	C₂²×C₅×D₅	1	4	yes	—
553	C₂×D₄	C₂²×C₅×D₅	1	4	yes	—
554	C₂×D₄	C₂²×C₅×D₅	1	8	yes	—
555	C₂²×C₄	—	5	2	no	—
556	C₂²×C₄	—	5	8	no	—
557	C₂²×C₄	—	5	4	no	—
558	C₂²×C₄	—	5	4	no	—
559	C₂²×C₄	C₂²×C₅×D₅	1	2	yes	—
560	C₂²×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
561	C₄⋊C₄	—	5	4	no	—
562	C₄⋊C₄	—	5	2	no	—
563	C₄⋊C₄	—	5	4	no	—
564	C₂²⋊C₄	—	5	2	no	—
565	C₂²⋊C₄	—	5	16	no	—
566	C₂²⋊C₄	—	5	4	no	—
567	C₂²⋊C₄	—	5	8	no	—
568	C₄²	—	5	2	no	—
569	C₄²	—	5	4	no	—
570	C₄²	C₂²×C₅×D₅	1	1	yes	characteristic
571	C₄²	C₂²×C₅×D₅	1	1	yes	characteristic
572	C₄²	C₂²×C₅×D₅	1	2	yes	—
573	C₂×C₅	—	2	8	no	—
574	C₂×C₅	—	2	2	no	—
575	C₂×C₅	—	2	2	no	—
576	C₂×C₅	—	2	2	no	—
577	C₂×C₅	—	2	4	no	—
578	C₂×C₅	—	4	8	no	—
579	C₂×C₅	—	4	8	no	—
580	C₂×C₅	—	8	16	no	—
581	C₂×C₅	—	10	4	no	—
582	C₂×C₅	—	20	4	no	—
583	C₂×C₅	C₂²×D₄×D₅	1	1	yes	characteristic
584	C₂×C₅	C₂×#4144	1	2	yes	—
585	C₂×C₅	C₂³×C₅×D₄	1	1	yes	characteristic
586	C₂×C₅	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
587	D₅	—	2	4	no	—
588	D₅	—	4	4	no	—
589	C₂³	—	2	8	no	—
590	C₂³	—	4	8	no	—
591	C₂³	—	5	2	no	—
592	C₂³	—	5	4	no	—
593	C₂³	—	10	16	no	—
594	C₂³	—	20	16	no	—
595	C₂³	C₂³×C₅×D₅	1	8	yes	—
596	C₂³	C₂³×C₅×D₅	1	1	yes	characteristic
597	Q₈	—	5	4	no	—
598	Q₈	—	10	4	no	—
599	D₄	—	2	8	no	—
600	D₄	—	2	8	no	—
601	D₄	—	2	16	no	—
602	D₄	—	4	8	no	—
603	D₄	—	5	8	no	—
604	D₄	—	5	4	no	—
605	D₄	—	10	16	no	—
606	D₄	—	10	16	no	—
607	D₄	—	10	8	no	—
608	D₄	—	10	4	no	—
609	D₄	—	10	8	no	—
610	D₄	—	20	16	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	5	2	no	—
613	C₂×C₄	—	5	8	no	—
614	C₂×C₄	—	5	4	no	—
615	C₂×C₄	—	5	4	no	—
616	C₂×C₄	—	10	16	no	—
617	C₂×C₄	—	10	8	no	—
618	C₂×C₄	—	10	8	no	—
619	C₂×C₄	C₅×D₄×D₅	1	4	yes	complemented
620	C₂×C₄	C₂³×C₅×D₅	1	2	yes	—
621	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
622	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
623	C₂×C₄	C₂³×C₅×D₅	1	2	yes	—
624	C₅	—	2	2	no	—
625	C₅	#10618096	1	1	yes	characteristic, direct product factor
626	C₅	C₅×#3498	1	1	yes	characteristic, complemented
627	C₂²	—	2	2	no	—
628	C₂²	—	2	8	no	—
629	C₂²	—	2	16	no	—
630	C₂²	—	5	4	no	—
631	C₂²	—	8	8	no	—
632	C₂²	—	10	4	no	—
633	C₂²	—	10	4	no	—
634	C₂²	—	10	8	no	—
635	C₂²	—	40	16	no	—
636	C₂²	C₂×C₅×D₄×D₅	1	2	yes	—
637	C₂²	C₂⁴×C₅×D₅	1	1	yes	characteristic
638	C₄	—	2	2	no	—
639	C₄	—	5	4	no	—
640	C₄	—	10	4	no	—
641	C₄	—	10	4	no	—
642	C₄	C₂×C₅×D₄×D₅	1	2	yes	—
643	C₄	C₂×C₅×D₄×D₅	1	2	yes	—
644	C₄	C₅×#4144	1	4	yes	complemented
645	C₂	—	2	2	no	—
646	C₂	—	4	8	no	—
647	C₂	—	10	4	no	—
648	C₂	—	20	4	no	—
649	C₂	C₂²×C₅×D₄×D₅	1	1	yes	characteristic
650	C₂	C₂×C₅×#4144	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂⁴×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₅
Fitting	C₂×C₅²×#127	C₂
socle	C₂²×C₅²	C₂⁵
hypercenter	C₂×C₅×#127	D₅
hypocenter	C₅	C₅×#3498
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#4144	C₂	semidirect
2. C₂×C₅²×#127	C₂	semidirect
3. C₅×#66139	C₂	semidirect
4. C₅×D₄×C₂²⋊D₅	C₂	semidirect
5. C₅×#66051	C₂	semidirect
6. C₅×#66050	C₂	semidirect
7. C₅×#66049	C₂	semidirect
8. C₅×D₅×#127	C₂	semidirect
9. C₅×#66052	C₂	semidirect
10. C₅×#65811	C₂	semidirect
11. C₄×C₅×#709	C₂	semidirect
12. C₂²×C₅²×D₄	C₂²	semidirect
13. C₂²×C₅²×D₄	C₂²	semidirect
14. C₂²×C₅×C₂²⋊D₅	C₂²	semidirect
16. C₂×C₅×#711	C₂²	semidirect
17. C₂×C₅×#711	C₂²	semidirect
18. C₂×C₅×D₄×D₅	C₂²	semidirect
19. C₂×C₅×D₄×D₅	C₂²	semidirect
20. C₂×C₅×D₄×D₅	C₂²	semidirect
22. C₂×C₅×#709	C₂²	semidirect
23. C₅²×#127	C₂²	semidirect
24. C₅²×#127	C₂²	semidirect
25. C₅²×#127	C₂²	semidirect
26. C₂×C₄²×C₅²	C₂²	semidirect
27. C₅×#4099	C₂²	semidirect
28. C₅×#4086	C₂²	semidirect
29. C₅×#4086	C₂²	semidirect
30. C₅×#4085	C₂²	semidirect
31. C₅×#4084	C₂²	semidirect
32. C₅×#4084	C₂²	semidirect
33. C₅×#4083	C₂²	semidirect
34. C₅×#4082	C₂²	semidirect
35. C₅×D₄×Dic₅	C₂²	semidirect
36. C₅×#4076	C₂²	semidirect
37. C₄×C₅×C₂²⋊D₅	C₂²	semidirect
38. C₅×#4072	C₂²	semidirect
39. C₅×#4021	C₂²	semidirect
40. C₅×#4020	C₂²	semidirect
41. C₄×C₅×D₂₀	C₂²	semidirect
42. C₄×C₅×D₂₀	C₂²	semidirect
43. C₅×#4018	C₂²	semidirect
44. C₄²×C₅×D₅	C₂²	semidirect
45. C₅×#4015	C₂²	semidirect
46. C₄×C₅×Dic₁₀	C₂²	semidirect
48. #10618096	C₅	direct (N × H)
72. C₅×#709	D₄	semidirect
94. C₂×#4144	C₂×C₅	semidirect
104. C₂×C₅×#127	D₅	semidirect
105. C₂×C₅×#127	C₂×C₅	semidirect
107. #66139	C₂×C₅	semidirect
108. D₄×C₂²⋊D₅	C₂×C₅	semidirect
109. #66052	C₂×C₅	semidirect
110. #66051	C₂×C₅	semidirect
111. #66050	C₂×C₅	semidirect
112. #66049	C₂×C₅	semidirect
113. D₅×#127	C₂×C₅	semidirect
114. #65811	C₂×C₅	semidirect
115. C₄×#709	C₂×C₅	semidirect
129. C₂×C₄×C₅²	C₂×D₄	semidirect
138. C₅×C₂²⋊D₅	C₂×D₄	semidirect
145. C₅×D₂₀	C₂×D₄	semidirect
148. C₄×C₅×D₅	C₂×D₄	semidirect
150. C₅×Dic₁₀	C₂×D₄	semidirect
162. C₂²×C₅×D₄	C₂×D₅	semidirect
163. C₂²×C₅×D₄	C₂×D₅	semidirect
164. C₂²×C₅×D₄	C₂²×C₅	semidirect
165. C₂²×C₅×D₄	C₂²×C₅	semidirect
166. C₂²×C₂²⋊D₅	C₂²×C₅	semidirect
168. C₂×#711	C₂²×C₅	semidirect
169. C₂×#711	C₂²×C₅	semidirect
170. C₂×D₄×D₅	C₂²×C₅	semidirect
171. C₂×D₄×D₅	C₂²×C₅	semidirect
172. C₂×D₄×D₅	C₂²×C₅	semidirect
174. C₂×#709	C₂²×C₅	semidirect
182. C₅×#127	C₂×D₅	semidirect
183. C₅×#127	C₂×D₅	semidirect
184. C₅×#127	C₂×D₅	semidirect
185. C₅×#127	C₂²×C₅	semidirect
186. C₅×#127	C₂²×C₅	semidirect
187. C₅×#127	C₂²×C₅	semidirect
205. C₂×C₄²×C₅	C₂×D₅	semidirect
206. C₂×C₄²×C₅	C₂²×C₅	semidirect
207. #4099	C₂²×C₅	semidirect
208. #4086	C₂²×C₅	semidirect
209. #4086	C₂²×C₅	semidirect
210. #4085	C₂²×C₅	semidirect
211. #4084	C₂²×C₅	semidirect
212. #4084	C₂²×C₅	semidirect
213. #4083	C₂²×C₅	semidirect
214. #4082	C₂²×C₅	semidirect
215. D₄×Dic₅	C₂²×C₅	semidirect
216. #4076	C₂²×C₅	semidirect
217. C₄×C₂²⋊D₅	C₂²×C₅	semidirect
218. #4072	C₂²×C₅	semidirect
219. #4021	C₂²×C₅	semidirect
220. #4020	C₂²×C₅	semidirect
221. C₄×D₂₀	C₂²×C₅	semidirect
222. C₄×D₂₀	C₂²×C₅	semidirect
223. #4018	C₂²×C₅	semidirect
224. C₄²×D₅	C₂²×C₅	semidirect
225. #4015	C₂²×C₅	semidirect
226. C₄×Dic₁₀	C₂²×C₅	semidirect
242. C₄×C₅²	H(5,𝔽₂)	semidirect
310. #709	C₅×D₄	semidirect
355. C₂×#127	C₅×D₅	semidirect
420. C₂×C₄×C₅	D₄×D₅	semidirect
421. C₂×C₄×C₅	C₂×C₅×D₄	semidirect
434. C₂²⋊D₅	C₂×C₅×D₄	semidirect
441. D₂₀	C₂×C₅×D₄	semidirect
444. C₄×D₅	C₂×C₅×D₄	semidirect
446. Dic₁₀	C₂×C₅×D₄	semidirect
456. C₂²×D₄	C₂×C₅×D₅	semidirect
457. C₂²×D₄	C₂×C₅×D₅	semidirect
462. #127	C₂×C₅×D₅	semidirect
463. #127	C₂×C₅×D₅	semidirect
464. #127	C₂×C₅×D₅	semidirect
481. C₂×C₄²	C₂×C₅×D₅	semidirect
482. C₅²	#3498	semidirect
522. C₄×C₅	#4144	semidirect
525. C₄×C₅	C₅×H(5,𝔽₂)	semidirect
619. C₂×C₄	C₅×D₄×D₅	semidirect
625. C₅	#10618096	direct (N × H)
626. C₅	C₅×#3498	semidirect
644. C₄	C₅×#4144	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#4144	2
C₂×C₅²×#127	2
C₅×#66139	2
C₅×D₄×C₂²⋊D₅	2
C₅×#66051	2
C₅×#66050	2
C₅×#66049	2
C₅×D₅×#127	2
C₅×#66052	2
C₅×#65811	2
C₄×C₅×#709	2
C₅×#3498	5
#10618096	5

Subgroups — C5×#10618096 #72258476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10618096 #72258476