Subgroups — C₅×#35265 #11957985

Order 1280.

1666 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2864	C₂	1	2	yes	maximal, complemented
2	C₅×#2859	C₂	1	2	yes	maximal, complemented
3	C₅×#2647	C₂	1	2	yes	maximal, complemented
4	C₅×#2647	C₂	1	2	yes	maximal, complemented
5	C₅×#2509	C₂	1	2	yes	maximal, complemented
6	C₅×#2508	C₂	1	2	yes	maximal, complemented
7	C₅×#2507	C₂	1	2	yes	maximal, complemented
8	C₅×#2506	C₂	1	2	yes	maximal, complemented
9	C₅×#2505	C₂	1	4	yes	maximal
10	C₅×#2473	C₂	1	4	yes	maximal
11	C₅×#2317	C₂	1	1	yes	maximal, characteristic
12	C₅×#2317	C₂	1	1	yes	maximal, characteristic
13	C₅×#2317	C₂	1	1	yes	maximal, characteristic
14	C₅×#2316	C₂	1	1	yes	maximal, characteristic
15	C₅×#2296	C₂	1	1	yes	maximal, characteristic
16	C₂×C₅×#389	C₂	1	1	yes	maximal, characteristic
17	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
18	C₂×C₅×#125	C₂²	1	4	yes	—
19	C₂×C₅×#117	C₂²	1	1	yes	characteristic
20	C₂×C₅×#117	C₂²	1	1	yes	characteristic
21	C₂×C₅×#117	C₂²	1	1	yes	characteristic
22	C₂×C₅×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₅×#400	C₂²	1	4	yes	—
28	C₅×#400	C₂²	1	4	yes	—
29	C₅×#400	C₂²	1	4	yes	—
30	C₅×#400	C₂²	1	4	yes	—
31	C₅×#398	C₂²	1	4	yes	—
32	C₅×#398	C₂²	1	4	yes	—
33	C₅×#398	C₂²	1	4	yes	—
34	C₅×#398	C₂²	1	4	yes	—
35	C₅×#389	C₂²	1	4	yes	—
36	C₅×#389	C₂²	1	4	yes	—
37	C₅×#389	C₂²	1	2	yes	—
38	C₅×#389	C₂²	1	2	yes	—
39	C₅×#389	C₂²	1	2	yes	—
40	C₅×#389	C₂²	1	2	yes	—
41	C₅×#389	C₂²	1	2	yes	—
42	C₅×#389	C₂²	1	2	yes	—
43	C₅×#389	C₂²	1	2	yes	—
44	C₅×#389	C₂²	1	2	yes	—
45	C₅×#389	C₂²	1	2	yes	—
46	C₅×#389	C₂²	1	2	yes	—
47	C₅×#389	C₂²	1	1	yes	characteristic
48	C₅×#389	C₂²	1	1	yes	characteristic
49	C₅×#389	C₂²	1	1	yes	characteristic
50	C₅×#389	C₂²	1	1	yes	characteristic
51	C₅×#389	C₂²	1	2	yes	—
52	C₅×#389	C₂²	1	2	yes	—
53	C₅×#389	C₂²	1	2	yes	—
54	C₅×#389	C₂²	1	2	yes	—
55	C₅×#387	C₂²	1	4	yes	—
56	C₅×#387	C₂²	1	4	yes	—
57	C₅×#387	C₂²	1	4	yes	—
58	C₅×#387	C₂²	1	4	yes	—
59	C₅×#385	C₂²	1	2	yes	—
60	C₅×#385	C₂²	1	4	yes	—
61	C₅×#385	C₂²	1	4	yes	—
62	C₅×#385	C₂²	1	4	yes	—
63	C₅×#385	C₂²	1	2	yes	—
64	C₅×#384	C₂²	1	2	yes	—
65	C₅×#384	C₂²	1	2	yes	—
66	C₅×#384	C₂²	1	2	yes	—
67	C₅×#382	C₂²	1	4	yes	—
68	C₅×#382	C₂²	1	2	yes	—
69	C₅×#382	C₂²	1	2	yes	—
70	C₅×#382	C₂²	1	2	yes	—
71	C₅×#380	C₂²	1	4	yes	—
72	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
73	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
74	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
75	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
76	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
77	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
78	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
79	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
80	C₅×#376	C₂²	1	2	yes	—
81	C₂×C₅×#95	C₂²	1	2	yes	—
82	C₄³×C₅	C₂²	1	2	yes	—
83	#35265	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₅	C₂³	1	2	yes	—
87	C₂³×C₄×C₅	C₂³	1	2	yes	—
88	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
89	C₅×#125	—	4	8	no	—
90	C₅×#117	—	2	4	no	—
91	C₅×#117	—	2	4	no	—
92	C₅×#117	—	2	4	no	—
93	C₅×#117	Q₈	1	8	yes	—
94	C₂×C₅×C₄⋊C₄	—	2	4	no	—
95	C₂×C₅×C₄⋊C₄	—	2	4	no	—
96	C₂×C₅×C₄⋊C₄	—	4	4	no	—
97	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
100	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
101	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
102	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
104	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
105	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
106	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
109	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
111	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
113	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
114	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
115	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
116	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
117	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
118	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
119	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
120	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
121	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
122	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
124	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
125	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂×C₄²×C₅	—	2	4	no	—
128	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅	C₂³	1	2	yes	—
130	C₂×C₄²×C₅	C₂³	1	2	yes	—
131	C₂×C₄²×C₅	C₂³	1	2	yes	—
132	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₅	C₂³	1	2	yes	—
136	C₂×C₄²×C₅	C₂³	1	2	yes	—
137	C₂×C₄²×C₅	C₂³	1	2	yes	—
138	C₂×C₄²×C₅	C₂³	1	2	yes	—
139	C₂×C₄²×C₅	C₂³	1	2	yes	—
140	C₅×#95	C₂³	1	4	yes	—
141	C₅×#95	C₂³	1	4	yes	—
142	C₅×#95	C₂³	1	2	yes	—
143	C₅×#95	C₂³	1	2	yes	—
144	C₅×#95	C₂³	1	2	yes	—
145	C₅×#95	C₂³	1	2	yes	—
146	C₅×#95	C₂³	1	2	yes	—
147	C₅×#95	C₂³	1	2	yes	—
148	C₅×#95	C₂³	1	2	yes	—
149	C₅×#95	C₂³	1	2	yes	—
150	C₅×#95	C₂³	1	4	yes	—
151	C₅×#95	C₂³	1	4	yes	—
152	C₅×#95	C₂³	1	4	yes	—
153	C₅×#95	C₂³	1	4	yes	—
154	C₅×#95	C₂³	1	4	yes	—
155	C₅×#95	C₂³	1	4	yes	—
156	C₅×#95	C₂³	1	4	yes	—
157	C₅×#95	C₂³	1	4	yes	—
158	#2864	C₂×C₅	1	2	yes	complemented
159	#2859	C₂×C₅	1	2	yes	complemented
160	#2647	C₂×C₅	1	2	yes	complemented
161	#2647	C₂×C₅	1	2	yes	complemented
162	#2509	C₂×C₅	1	2	yes	complemented
163	#2508	C₂×C₅	1	2	yes	complemented
164	#2507	C₂×C₅	1	2	yes	complemented
165	#2506	C₂×C₅	1	2	yes	complemented
166	#2505	C₂×C₅	1	4	yes	—
167	#2473	C₂×C₅	1	4	yes	—
168	#2317	C₂×C₅	1	1	yes	characteristic
169	#2317	C₂×C₅	1	1	yes	characteristic
170	#2317	C₂×C₅	1	1	yes	characteristic
171	#2316	C₂×C₅	1	1	yes	characteristic
172	#2296	C₂×C₅	1	1	yes	characteristic
173	C₂×#389	C₂×C₅	1	1	yes	characteristic
174	C₄×#117	C₂×C₅	1	1	yes	characteristic
175	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₅	—	2	2	no	—
177	C₂²×C₄×C₅	—	2	2	no	—
178	C₂²×C₄×C₅	—	2	4	no	—
179	C₂²×C₄×C₅	—	2	4	no	—
180	C₂²×C₄×C₅	—	2	2	no	—
181	C₂²×C₄×C₅	—	2	2	no	—
182	C₂²×C₄×C₅	—	4	2	no	—
183	C₂²×C₄×C₅	—	4	1	no	—
184	C₂²×C₄×C₅	—	4	1	no	—
185	C₂²×C₄×C₅	—	4	2	no	—
186	C₂²×C₄×C₅	—	4	2	no	—
187	C₂²×C₄×C₅	—	4	2	no	—
188	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
189	C₂²×C₄×C₅	#41	1	4	yes	—
190	C₂²×C₄×C₅	#41	1	4	yes	—
191	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
200	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
207	C₅×C₄⋊C₄	—	2	4	no	—
208	C₅×C₄⋊C₄	—	2	4	no	—
209	C₅×C₄⋊C₄	—	2	4	no	—
210	C₅×C₄⋊C₄	—	2	4	no	—
211	C₅×C₄⋊C₄	—	2	2	no	—
212	C₅×C₄⋊C₄	—	2	2	no	—
213	C₅×C₄⋊C₄	—	2	2	no	—
214	C₅×C₄⋊C₄	—	2	2	no	—
215	C₅×C₄⋊C₄	—	2	2	no	—
216	C₅×C₄⋊C₄	—	2	2	no	—
217	C₅×C₄⋊C₄	—	2	8	no	—
218	C₅×C₄⋊C₄	—	2	4	no	—
219	C₅×C₄⋊C₄	—	2	4	no	—
220	C₅×C₄⋊C₄	—	4	4	no	—
221	C₅×C₄⋊C₄	—	4	4	no	—
222	C₅×C₄⋊C₄	—	4	4	no	—
223	C₅×C₄⋊C₄	—	4	4	no	—
224	C₅×C₄⋊C₄	—	4	4	no	—
225	C₅×C₄⋊C₄	—	4	2	no	—
226	C₅×C₄⋊C₄	—	4	2	no	—
227	C₅×C₄⋊C₄	—	4	4	no	—
228	C₅×C₄⋊C₄	—	4	4	no	—
229	C₅×C₄⋊C₄	—	4	4	no	—
230	C₅×C₄⋊C₄	—	4	4	no	—
231	C₅×C₄⋊C₄	C₂×Q₈	1	4	yes	—
232	C₅×C₄⋊C₄	C₂×Q₈	1	4	yes	—
233	C₅×C₂²⋊C₄	—	2	4	no	—
234	C₅×C₂²⋊C₄	—	2	4	no	—
235	C₅×C₂²⋊C₄	—	2	4	no	—
236	C₅×C₂²⋊C₄	C₂×Q₈	1	8	yes	—
237	C₄²×C₅	—	2	2	no	—
238	C₄²×C₅	—	2	2	no	—
239	C₄²×C₅	—	2	2	no	—
240	C₄²×C₅	—	2	2	no	—
241	C₄²×C₅	—	2	4	no	—
242	C₄²×C₅	—	2	4	no	—
243	C₄²×C₅	—	2	4	no	—
244	C₄²×C₅	—	2	4	no	—
245	C₄²×C₅	—	2	4	no	—
246	C₄²×C₅	—	4	2	no	—
247	C₄²×C₅	—	4	2	no	—
248	C₄²×C₅	C₂×Q₈	1	8	yes	—
249	C₂×#125	C₂²×C₅	1	4	yes	—
250	C₂×#117	C₂²×C₅	1	1	yes	characteristic
251	C₂×#117	C₂²×C₅	1	1	yes	characteristic
252	C₂×#117	C₂²×C₅	1	1	yes	characteristic
253	C₂×#117	C₂²×C₅	1	1	yes	characteristic
254	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
255	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
256	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
257	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
258	#400	C₂²×C₅	1	4	yes	—
259	#400	C₂²×C₅	1	4	yes	—
260	#400	C₂²×C₅	1	4	yes	—
261	#400	C₂²×C₅	1	4	yes	—
262	#398	C₂²×C₅	1	4	yes	—
263	#398	C₂²×C₅	1	4	yes	—
264	#398	C₂²×C₅	1	4	yes	—
265	#398	C₂²×C₅	1	4	yes	—
266	#389	C₂²×C₅	1	2	yes	—
267	#389	C₂²×C₅	1	2	yes	—
268	#389	C₂²×C₅	1	2	yes	—
269	#389	C₂²×C₅	1	2	yes	—
270	#389	C₂²×C₅	1	2	yes	—
271	#389	C₂²×C₅	1	2	yes	—
272	#389	C₂²×C₅	1	2	yes	—
273	#389	C₂²×C₅	1	2	yes	—
274	#389	C₂²×C₅	1	2	yes	—
275	#389	C₂²×C₅	1	2	yes	—
276	#389	C₂²×C₅	1	1	yes	characteristic
277	#389	C₂²×C₅	1	2	yes	—
278	#389	C₂²×C₅	1	4	yes	—
279	#389	C₂²×C₅	1	4	yes	—
280	#389	C₂²×C₅	1	1	yes	characteristic
281	#389	C₂²×C₅	1	2	yes	—
282	#389	C₂²×C₅	1	1	yes	characteristic
283	#389	C₂²×C₅	1	1	yes	characteristic
284	#389	C₂²×C₅	1	2	yes	—
285	#389	C₂²×C₅	1	2	yes	—
286	#387	C₂²×C₅	1	4	yes	—
287	#387	C₂²×C₅	1	4	yes	—
288	#387	C₂²×C₅	1	4	yes	—
289	#387	C₂²×C₅	1	4	yes	—
290	#385	C₂²×C₅	1	4	yes	—
291	#385	C₂²×C₅	1	4	yes	—
292	#385	C₂²×C₅	1	4	yes	—
293	#385	C₂²×C₅	1	2	yes	—
294	#385	C₂²×C₅	1	2	yes	—
295	#384	C₂²×C₅	1	2	yes	—
296	#384	C₂²×C₅	1	2	yes	—
297	#384	C₂²×C₅	1	2	yes	—
298	#382	C₂²×C₅	1	2	yes	—
299	#382	C₂²×C₅	1	2	yes	—
300	#382	C₂²×C₅	1	2	yes	—
301	#382	C₂²×C₅	1	4	yes	—
302	#380	C₂²×C₅	1	4	yes	—
303	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
304	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
306	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
309	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
310	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
311	#376	C₂²×C₅	1	2	yes	—
312	C₂×#95	C₂²×C₅	1	2	yes	—
313	C₄³	C₂²×C₅	1	2	yes	—
314	C₂³×C₅	—	2	1	no	—
315	C₂³×C₅	—	2	1	no	—
316	C₂³×C₅	—	2	1	no	—
317	C₂³×C₅	—	2	1	no	—
318	C₂³×C₅	C₂²×Q₈	1	2	yes	—
319	C₂³×C₅	C₂×#41	1	2	yes	—
320	C₂³×C₅	C₂×#41	1	2	yes	—
321	C₂³×C₅	C₂⁵	1	1	yes	characteristic
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	2	1	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	2	1	no	—
327	C₂×C₄×C₅	—	2	1	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	2	1	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	2	2	no	—
333	C₂×C₄×C₅	—	2	2	no	—
334	C₂×C₄×C₅	—	2	2	no	—
335	C₂×C₄×C₅	—	2	2	no	—
336	C₂×C₄×C₅	—	2	2	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	2	4	no	—
339	C₂×C₄×C₅	—	2	4	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	2	no	—
343	C₂×C₄×C₅	—	2	4	no	—
344	C₂×C₄×C₅	—	2	4	no	—
345	C₂×C₄×C₅	—	2	1	no	—
346	C₂×C₄×C₅	—	2	2	no	—
347	C₂×C₄×C₅	—	2	2	no	—
348	C₂×C₄×C₅	—	2	2	no	—
349	C₂×C₄×C₅	—	2	1	no	—
350	C₂×C₄×C₅	—	2	2	no	—
351	C₂×C₄×C₅	—	2	2	no	—
352	C₂×C₄×C₅	—	2	2	no	—
353	C₂×C₄×C₅	—	2	2	no	—
354	C₂×C₄×C₅	—	2	2	no	—
355	C₂×C₄×C₅	—	2	2	no	—
356	C₂×C₄×C₅	—	2	4	no	—
357	C₂×C₄×C₅	—	2	1	no	—
358	C₂×C₄×C₅	—	2	4	no	—
359	C₂×C₄×C₅	—	2	1	no	—
360	C₂×C₄×C₅	—	4	4	no	—
361	C₂×C₄×C₅	—	4	4	no	—
362	C₂×C₄×C₅	—	4	8	no	—
363	C₂×C₄×C₅	—	8	4	no	—
364	C₂×C₄×C₅	#125	1	16	yes	complemented
365	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
366	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
367	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
368	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
369	C₂×C₄×C₅	C₂×#41	1	2	yes	—
370	C₂×C₄×C₅	C₂×#41	1	2	yes	—
371	C₂×C₄×C₅	C₂×#41	1	2	yes	—
372	C₂×C₄×C₅	C₂×#41	1	2	yes	—
373	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
374	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
375	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₅	1	2	yes	—
378	C₂³×C₄	C₂³×C₅	1	2	yes	—
379	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
380	#125	—	4	8	no	—
381	#117	—	2	4	no	—
382	#117	—	2	4	no	—
383	#117	—	2	4	no	—
384	#117	C₅×Q₈	1	8	yes	—
385	C₂×C₄⋊C₄	—	2	4	no	—
386	C₂×C₄⋊C₄	—	2	4	no	—
387	C₂×C₄⋊C₄	—	4	4	no	—
388	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
414	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
418	C₂×C₄²	—	2	4	no	—
419	C₂×C₄²	C₂³×C₅	1	2	yes	—
420	C₂×C₄²	C₂³×C₅	1	2	yes	—
421	C₂×C₄²	C₂³×C₅	1	2	yes	—
422	C₂×C₄²	C₂³×C₅	1	2	yes	—
423	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₅	1	2	yes	—
425	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₅	1	2	yes	—
429	C₂×C₄²	C₂³×C₅	1	2	yes	—
430	C₂×C₄²	C₂³×C₅	1	2	yes	—
431	#95	C₂³×C₅	1	4	yes	—
432	#95	C₂³×C₅	1	4	yes	—
433	#95	C₂³×C₅	1	2	yes	—
434	#95	C₂³×C₅	1	2	yes	—
435	#95	C₂³×C₅	1	2	yes	—
436	#95	C₂³×C₅	1	2	yes	—
437	#95	C₂³×C₅	1	2	yes	—
438	#95	C₂³×C₅	1	2	yes	—
439	#95	C₂³×C₅	1	2	yes	—
440	#95	C₂³×C₅	1	2	yes	—
441	#95	C₂³×C₅	1	4	yes	—
442	#95	C₂³×C₅	1	4	yes	—
443	#95	C₂³×C₅	1	4	yes	—
444	#95	C₂³×C₅	1	4	yes	—
445	#95	C₂³×C₅	1	4	yes	—
446	#95	C₂³×C₅	1	4	yes	—
447	#95	C₂³×C₅	1	4	yes	—
448	#95	C₂³×C₅	1	4	yes	—
449	C₂²×C₅	—	2	2	no	—
450	C₂²×C₅	—	2	2	no	—
451	C₂²×C₅	—	2	2	no	—
452	C₂²×C₅	—	2	2	no	—
453	C₂²×C₅	—	2	2	no	—
454	C₂²×C₅	—	2	2	no	—
455	C₂²×C₅	C₂×#125	1	4	yes	—
456	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
457	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
458	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
459	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
460	C₂²×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
461	C₂²×C₅	#585	1	1	yes	characteristic
462	C₂²×C₅	#585	1	1	yes	characteristic
463	C₄×C₅	—	2	2	no	—
464	C₄×C₅	—	2	2	no	—
465	C₄×C₅	—	2	4	no	—
466	C₄×C₅	—	2	4	no	—
467	C₄×C₅	—	4	2	no	—
468	C₄×C₅	—	4	2	no	—
469	C₄×C₅	—	4	4	no	—
470	C₄×C₅	—	4	2	no	—
471	C₄×C₅	—	4	2	no	—
472	C₄×C₅	—	4	2	no	—
473	C₄×C₅	—	4	1	no	—
474	C₄×C₅	—	4	4	no	—
475	C₄×C₅	—	4	2	no	—
476	C₄×C₅	—	4	1	no	—
477	C₄×C₅	C₂×#125	1	4	yes	—
478	C₄×C₅	C₂×#125	1	4	yes	—
479	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	1	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
493	C₂²×C₄	C₅×#41	1	4	yes	—
494	C₂²×C₄	C₅×#41	1	4	yes	—
495	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
500	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
501	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
509	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
510	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	8	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	4	4	no	—
525	C₄⋊C₄	—	4	4	no	—
526	C₄⋊C₄	—	4	4	no	—
527	C₄⋊C₄	—	4	4	no	—
528	C₄⋊C₄	—	4	4	no	—
529	C₄⋊C₄	—	4	4	no	—
530	C₄⋊C₄	—	4	4	no	—
531	C₄⋊C₄	—	4	4	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	C₂×C₅×Q₈	1	4	yes	—
536	C₄⋊C₄	C₂×C₅×Q₈	1	4	yes	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	C₂×C₅×Q₈	1	8	yes	—
541	C₄²	—	2	2	no	—
542	C₄²	—	2	2	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	2	no	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	4	no	—
547	C₄²	—	2	4	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	2	4	no	—
550	C₄²	—	4	2	no	—
551	C₄²	—	4	2	no	—
552	C₄²	C₂×C₅×Q₈	1	8	yes	—
553	C₂×C₅	—	2	4	no	—
554	C₂×C₅	C₂²×#125	1	1	yes	characteristic
555	C₂×C₅	#3460	1	1	yes	characteristic
556	C₂×C₅	#3461	1	1	yes	characteristic
557	C₂×C₅	#3494	1	1	yes	characteristic
558	C₂×C₅	#3495	1	1	yes	characteristic
559	C₂×C₅	#3503	1	1	yes	characteristic
560	C₂×C₅	#3504	1	1	yes	characteristic
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	C₂²×C₅×Q₈	1	2	yes	—
566	C₂³	C₂×C₅×#41	1	2	yes	—
567	C₂³	C₂×C₅×#41	1	2	yes	—
568	C₂³	C₂⁵×C₅	1	1	yes	characteristic
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	4	4	no	—
608	C₂×C₄	—	4	4	no	—
609	C₂×C₄	—	4	8	no	—
610	C₂×C₄	—	8	4	no	—
611	C₂×C₄	C₅×#125	1	16	yes	complemented
612	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
613	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
614	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
615	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
616	C₂×C₄	C₂×C₅×#41	1	2	yes	—
617	C₂×C₄	C₂×C₅×#41	1	2	yes	—
618	C₂×C₄	C₂×C₅×#41	1	2	yes	—
619	C₂×C₄	C₂×C₅×#41	1	2	yes	—
620	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
621	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
622	C₅	#35265	1	1	yes	characteristic, direct product factor, Sylow
623	C₂²	—	2	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	C₂×C₅×#125	1	4	yes	—
630	C₂²	C₂³×C₅×Q₈	1	1	yes	characteristic
631	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
632	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
633	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
634	C₂²	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
635	C₂²	C₅×#585	1	1	yes	characteristic
636	C₂²	C₅×#585	1	1	yes	characteristic
637	C₄	—	2	2	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	4	no	—
640	C₄	—	2	4	no	—
641	C₄	—	4	4	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	2	no	—
651	C₄	C₂×C₅×#125	1	4	yes	—
652	C₄	C₂×C₅×#125	1	4	yes	—
653	C₂	—	2	4	no	—
654	C₂	C₂²×C₅×#125	1	1	yes	characteristic
655	C₂	C₅×#3460	1	1	yes	characteristic
656	C₂	C₅×#3461	1	1	yes	characteristic
657	C₂	C₅×#3494	1	1	yes	characteristic
658	C₂	C₅×#3495	1	1	yes	characteristic
659	C₂	C₅×#3503	1	1	yes	characteristic
660	C₂	C₅×#3504	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2864	C₂	semidirect
2. C₅×#2859	C₂	semidirect
3. C₅×#2647	C₂	semidirect
4. C₅×#2647	C₂	semidirect
5. C₅×#2509	C₂	semidirect
6. C₅×#2508	C₂	semidirect
7. C₅×#2507	C₂	semidirect
8. C₅×#2506	C₂	semidirect
83. #35265	C₅	direct (N × H)
158. #2864	C₂×C₅	semidirect
159. #2859	C₂×C₅	semidirect
160. #2647	C₂×C₅	semidirect
161. #2647	C₂×C₅	semidirect
162. #2509	C₂×C₅	semidirect
163. #2508	C₂×C₅	semidirect
164. #2507	C₂×C₅	semidirect
165. #2506	C₂×C₅	semidirect
364. C₂×C₄×C₅	#125	semidirect
611. C₂×C₄	C₅×#125	semidirect
622. C₅	#35265	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2864	2
C₅×#2859	2
C₅×#2647	2
C₅×#2509	2
C₅×#2508	2
C₅×#2507	2
C₅×#2506	2
C₅×#2505	2
C₅×#2473	2
C₅×#2317	2
C₅×#2316	2
C₅×#2296	2
C₂×C₅×#389	2
C₄×C₅×#117	2
#35265	5

Subgroups — C5×#35265 #11957985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35265 #11957985