Subgroups — C₅×#21367 #11944087

Order 1280.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#433	C₂	1	1	yes	maximal, characteristic
2	C₅×#2314	C₂	1	1	yes	maximal, characteristic
3	C₅×#2274	C₂	1	1	yes	maximal, characteristic
4	C₅×#1949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1854	C₂	1	1	yes	maximal, characteristic
9	C₅×#1854	C₂	1	1	yes	maximal, characteristic
10	C₅×#1853	C₂	1	1	yes	maximal, characteristic
11	C₅×#1852	C₂	1	1	yes	maximal, characteristic
12	C₅×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1776	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#117	C₂²	1	1	yes	characteristic
17	C₂×C₅×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₅×#433	C₄	1	8	yes	complemented
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#107	C₂²	1	1	yes	characteristic
23	C₂×C₅×#107	C₂²	1	1	yes	characteristic
24	C₂×C₅×#106	C₂²	1	1	yes	characteristic
25	C₂×C₅×#106	C₂²	1	1	yes	characteristic
26	C₂×C₅×#106	C₂²	1	1	yes	characteristic
27	C₂×C₅×#106	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₅×#97	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₅×#388	—	2	1	no	—
35	C₅×#388	—	2	1	no	—
36	C₅×#387	—	2	1	no	—
37	C₅×#387	—	2	1	no	—
38	C₅×#384	C₂²	1	1	yes	characteristic
39	C₅×#382	C₂²	1	1	yes	characteristic
40	C₅×#380	C₂²	1	1	yes	characteristic
41	C₅×#380	C₂²	1	1	yes	characteristic
42	C₄×C₅×C₄⋊C₄	—	2	1	no	—
43	C₄×C₅×C₄⋊C₄	—	2	1	no	—
44	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
46	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
51	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
52	C₅×#340	C₂²	1	1	yes	characteristic
53	C₅×#340	C₂²	1	1	yes	characteristic
54	C₅×#340	C₂²	1	1	yes	characteristic
55	C₅×#340	C₂²	1	1	yes	characteristic
56	C₅×#340	C₂²	1	1	yes	characteristic
57	C₅×#340	C₂²	1	1	yes	characteristic
58	C₅×#340	C₂²	1	1	yes	characteristic
59	C₅×#340	C₂²	1	1	yes	characteristic
60	#21367	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₄×C₅	—	2	1	no	—
62	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₅×#117	—	2	2	no	—
68	C₅×#117	—	2	2	no	—
69	C₅×#117	C₂×C₄	1	4	yes	—
70	C₅×#117	Q₈	1	4	yes	—
71	C₂×C₅×C₄⋊C₄	—	2	1	no	—
72	C₂×C₅×C₄⋊C₄	—	2	2	no	—
73	C₂×C₅×C₄⋊C₄	—	2	2	no	—
74	C₂×C₅×C₄⋊C₄	—	2	1	no	—
75	C₂×C₅×C₄⋊C₄	—	2	1	no	—
76	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
77	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
78	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
87	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
88	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
89	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
90	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₅	—	2	1	no	—
92	C₂×C₄²×C₅	—	2	1	no	—
93	C₂×C₄²×C₅	—	2	1	no	—
94	C₂×C₄²×C₅	—	2	1	no	—
95	C₂×C₄²×C₅	—	2	1	no	—
96	C₂×C₄²×C₅	—	2	1	no	—
97	C₂×C₄²×C₅	—	2	1	no	—
98	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
100	C₅×#107	—	2	2	no	—
101	C₅×#107	—	2	2	no	—
102	C₅×#107	—	2	2	no	—
103	C₅×#107	—	2	2	no	—
104	C₅×#106	—	2	2	no	—
105	C₅×#106	—	2	2	no	—
106	C₅×#106	—	2	2	no	—
107	C₅×#106	—	2	2	no	—
108	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
109	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
110	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
111	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
112	C₅×#97	C₂×C₄	1	4	yes	complemented
113	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
114	C₅×#95	—	2	1	no	—
115	C₅×#95	—	2	1	no	—
116	C₅×#95	—	2	1	no	—
117	C₅×#95	—	2	1	no	—
118	C₅×#95	—	2	1	no	—
119	C₅×#95	—	2	1	no	—
120	C₅×#95	—	2	1	no	—
121	C₅×#95	—	2	1	no	—
122	C₅×#95	—	2	1	no	—
123	C₅×#95	—	2	1	no	—
124	C₂×#433	C₂×C₅	1	1	yes	characteristic
125	#2314	C₂×C₅	1	1	yes	characteristic
126	#2274	C₂×C₅	1	1	yes	characteristic
127	#1949	C₂×C₅	1	1	yes	characteristic, complemented
128	#1949	C₂×C₅	1	1	yes	characteristic, complemented
129	#1948	C₂×C₅	1	1	yes	characteristic, complemented
130	#1947	C₂×C₅	1	1	yes	characteristic, complemented
131	#1854	C₂×C₅	1	1	yes	characteristic
132	#1854	C₂×C₅	1	1	yes	characteristic
133	#1853	C₂×C₅	1	1	yes	characteristic
134	#1852	C₂×C₅	1	1	yes	characteristic
135	#1776	C₂×C₅	1	1	yes	characteristic, complemented
136	#1776	C₂×C₅	1	1	yes	characteristic, complemented
137	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
138	C₄×#106	C₂×C₅	1	1	yes	characteristic, complemented
139	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₅	—	2	1	no	—
141	C₂²×C₄×C₅	—	2	1	no	—
142	C₂²×C₄×C₅	—	2	1	no	—
143	C₂²×C₄×C₅	—	2	1	no	—
144	C₂²×C₄×C₅	—	2	1	no	—
145	C₂²×C₄×C₅	—	2	1	no	—
146	C₂²×C₄×C₅	—	2	1	no	—
147	C₂²×C₄×C₅	—	2	1	no	—
148	C₂²×C₄×C₅	—	2	4	no	—
149	C₂²×C₄×C₅	—	2	1	no	—
150	C₂²×C₄×C₅	—	2	1	no	—
151	C₂²×C₄×C₅	—	2	1	no	—
152	C₂²×C₄×C₅	—	2	1	no	—
153	C₂²×C₄×C₅	—	8	1	no	—
154	C₂²×C₄×C₅	—	8	1	no	—
155	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
162	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
163	C₂×C₅×C₈	—	2	1	no	—
164	C₂×C₅×C₈	—	2	1	no	—
165	C₂×C₅×C₈	—	2	1	no	—
166	C₂×C₅×C₈	—	2	1	no	—
167	C₂×C₅×C₈	—	2	1	no	—
168	C₂×C₅×C₈	—	2	1	no	—
169	C₂×C₅×C₈	—	2	1	no	—
170	C₂×C₅×C₈	—	2	1	no	—
171	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
172	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
173	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
174	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
175	C₅×C₄⋊C₄	—	2	4	no	—
176	C₅×C₄⋊C₄	—	2	1	no	—
177	C₅×C₄⋊C₄	—	2	1	no	—
178	C₅×C₄⋊C₄	—	2	1	no	—
179	C₅×C₄⋊C₄	—	2	1	no	—
180	C₅×C₄⋊C₄	—	2	1	no	—
181	C₅×C₄⋊C₄	—	2	1	no	—
182	C₅×C₄⋊C₄	—	2	1	no	—
183	C₅×C₄⋊C₄	—	2	1	no	—
184	C₅×C₄⋊C₄	—	2	1	no	—
185	C₅×C₄⋊C₄	—	2	1	no	—
186	C₅×C₄⋊C₄	—	2	1	no	—
187	C₅×C₄⋊C₄	—	2	1	no	—
188	C₅×C₄⋊C₄	—	8	1	no	—
189	C₅×C₄⋊C₄	—	8	1	no	—
190	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
191	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
192	C₅×C₄⋊C₄	#41	1	2	yes	—
193	C₅×C₄⋊C₄	#41	1	2	yes	—
194	C₅×C₄⋊C₄	#41	1	2	yes	—
195	C₅×C₄⋊C₄	#41	1	2	yes	—
196	C₅×C₂²⋊C₄	—	2	4	no	—
197	C₅×C₂²⋊C₄	—	8	1	no	—
198	C₅×C₂²⋊C₄	—	8	1	no	—
199	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
200	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
201	C₄²×C₅	—	2	4	no	—
202	C₄²×C₅	—	8	1	no	—
203	C₄²×C₅	—	8	1	no	—
204	C₄²×C₅	—	8	1	no	—
205	C₄²×C₅	—	8	1	no	—
206	C₄²×C₅	—	8	1	no	—
207	C₄²×C₅	—	8	1	no	—
208	C₄²×C₅	C₂²×C₄	1	2	yes	—
209	C₄²×C₅	C₂²×C₄	1	2	yes	—
210	C₄²×C₅	C₂×Q₈	1	2	yes	—
211	C₄²×C₅	C₂×Q₈	1	2	yes	—
212	C₂×#117	C₂²×C₅	1	1	yes	characteristic
213	C₂×#117	C₂²×C₅	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
215	#433	C₄×C₅	1	8	yes	complemented
216	C₂×#107	C₂²×C₅	1	1	yes	characteristic
217	C₂×#107	C₂²×C₅	1	1	yes	characteristic
218	C₂×#107	C₂²×C₅	1	1	yes	characteristic
219	C₂×#107	C₂²×C₅	1	1	yes	characteristic
220	C₂×#106	C₂²×C₅	1	1	yes	characteristic
221	C₂×#106	C₂²×C₅	1	1	yes	characteristic
222	C₂×#106	C₂²×C₅	1	1	yes	characteristic
223	C₂×#106	C₂²×C₅	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
226	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
227	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
228	C₂×#97	C₂²×C₅	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
230	#388	—	2	1	no	—
231	#388	—	2	1	no	—
232	#387	—	2	1	no	—
233	#387	—	2	1	no	—
234	#384	C₂²×C₅	1	1	yes	characteristic
235	#382	C₂²×C₅	1	1	yes	characteristic
236	#380	C₂²×C₅	1	1	yes	characteristic
237	#380	C₂²×C₅	1	1	yes	characteristic
238	C₄×C₄⋊C₄	—	2	1	no	—
239	C₄×C₄⋊C₄	—	2	1	no	—
240	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
241	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
242	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
243	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
245	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
246	C₄×C₂²⋊C₄	—	2	1	no	—
247	C₄×C₂²⋊C₄	—	2	1	no	—
248	#340	C₂²×C₅	1	1	yes	characteristic
249	#340	C₂²×C₅	1	1	yes	characteristic
250	#340	C₂²×C₅	1	1	yes	characteristic
251	#340	C₂²×C₅	1	1	yes	characteristic
252	#340	C₂²×C₅	1	1	yes	characteristic
253	#340	C₂²×C₅	1	1	yes	characteristic
254	#340	C₂²×C₅	1	1	yes	characteristic
255	#340	C₂²×C₅	1	1	yes	characteristic
256	C₂³×C₅	—	2	1	no	—
257	C₂³×C₅	—	2	1	no	—
258	C₂³×C₅	—	2	1	no	—
259	C₂³×C₅	—	2	1	no	—
260	C₂³×C₅	—	2	1	no	—
261	C₂³×C₅	—	2	1	no	—
262	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
263	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₅	—	2	1	no	—
265	C₂×C₄×C₅	—	2	1	no	—
266	C₂×C₄×C₅	—	2	1	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	1	no	—
269	C₂×C₄×C₅	—	2	1	no	—
270	C₂×C₄×C₅	—	2	1	no	—
271	C₂×C₄×C₅	—	2	1	no	—
272	C₂×C₄×C₅	—	2	1	no	—
273	C₂×C₄×C₅	—	2	1	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	2	2	no	—
276	C₂×C₄×C₅	—	2	2	no	—
277	C₂×C₄×C₅	—	2	2	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	2	no	—
282	C₂×C₄×C₅	—	4	1	no	—
283	C₂×C₄×C₅	—	4	1	no	—
284	C₂×C₄×C₅	—	4	1	no	—
285	C₂×C₄×C₅	—	4	1	no	—
286	C₂×C₄×C₅	—	4	1	no	—
287	C₂×C₄×C₅	—	4	1	no	—
288	C₂×C₄×C₅	—	4	1	no	—
289	C₂×C₄×C₅	—	4	1	no	—
290	C₂×C₄×C₅	—	4	1	no	—
291	C₂×C₄×C₅	—	4	1	no	—
292	C₂×C₄×C₅	—	4	1	no	—
293	C₂×C₄×C₅	—	4	1	no	—
294	C₂×C₄×C₅	—	4	1	no	—
295	C₂×C₄×C₅	—	4	1	no	—
296	C₂×C₄×C₅	—	4	1	no	—
297	C₂×C₄×C₅	—	4	1	no	—
298	C₂×C₄×C₅	—	4	1	no	—
299	C₂×C₄×C₅	—	4	1	no	—
300	C₂×C₄×C₅	—	4	1	no	—
301	C₂×C₄×C₅	—	4	1	no	—
302	C₂×C₄×C₅	—	8	4	no	—
303	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
304	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
305	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
306	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
307	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
308	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
309	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
312	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
313	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
314	C₅×C₈	—	2	2	no	—
315	C₅×C₈	—	2	2	no	—
316	C₅×C₈	—	2	2	no	—
317	C₅×C₈	—	2	2	no	—
318	C₂³×C₄	—	2	1	no	—
319	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
324	#117	—	2	2	no	—
325	#117	—	2	2	no	—
326	#117	C₂×C₄×C₅	1	4	yes	—
327	#117	C₅×Q₈	1	4	yes	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
357	#107	—	2	2	no	—
358	#107	—	2	2	no	—
359	#107	—	2	2	no	—
360	#107	—	2	2	no	—
361	#106	—	2	2	no	—
362	#106	—	2	2	no	—
363	#106	—	2	2	no	—
364	#106	—	2	2	no	—
365	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
366	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
367	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
368	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
369	#97	C₂×C₄×C₅	1	4	yes	complemented
370	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
371	#95	—	2	1	no	—
372	#95	—	2	1	no	—
373	#95	—	2	1	no	—
374	#95	—	2	1	no	—
375	#95	—	2	1	no	—
376	#95	—	2	1	no	—
377	#95	—	2	1	no	—
378	#95	—	2	1	no	—
379	#95	—	2	1	no	—
380	#95	—	2	1	no	—
381	C₂²×C₅	—	2	2	no	—
382	C₂²×C₅	—	2	2	no	—
383	C₂²×C₅	—	2	2	no	—
384	C₂²×C₅	—	4	1	no	—
385	C₂²×C₅	—	4	1	no	—
386	C₂²×C₅	—	4	1	no	—
387	C₂²×C₅	—	4	1	no	—
388	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
389	C₂²×C₅	#525	1	1	yes	characteristic
390	C₂²×C₅	#525	1	1	yes	characteristic
391	C₂²×C₅	#576	1	1	yes	characteristic
392	C₂²×C₅	#577	1	1	yes	characteristic
393	C₂²×C₅	#577	1	1	yes	characteristic
394	C₂²×C₅	#578	1	1	yes	characteristic
395	C₄×C₅	—	2	2	no	—
396	C₄×C₅	—	2	2	no	—
397	C₄×C₅	—	2	2	no	—
398	C₄×C₅	—	2	2	no	—
399	C₄×C₅	—	2	2	no	—
400	C₄×C₅	—	4	2	no	—
401	C₄×C₅	—	4	2	no	—
402	C₄×C₅	—	4	2	no	—
403	C₄×C₅	—	4	2	no	—
404	C₄×C₅	—	8	1	no	—
405	C₄×C₅	—	8	1	no	—
406	C₄×C₅	—	8	1	no	—
407	C₄×C₅	—	8	1	no	—
408	C₄×C₅	C₄×#41	1	1	yes	characteristic
409	C₄×C₅	C₄×#41	1	1	yes	characteristic
410	C₄×C₅	#533	1	1	yes	characteristic
411	C₄×C₅	#533	1	1	yes	characteristic
412	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	8	1	no	—
427	C₂²×C₄	—	8	1	no	—
428	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
429	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
431	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
432	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
433	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
434	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
445	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
446	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
447	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	8	1	no	—
462	C₄⋊C₄	—	8	1	no	—
463	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
464	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
465	C₄⋊C₄	C₅×#41	1	2	yes	—
466	C₄⋊C₄	C₅×#41	1	2	yes	—
467	C₄⋊C₄	C₅×#41	1	2	yes	—
468	C₄⋊C₄	C₅×#41	1	2	yes	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	8	1	no	—
471	C₂²⋊C₄	—	8	1	no	—
472	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
473	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
474	C₄²	—	2	4	no	—
475	C₄²	—	8	1	no	—
476	C₄²	—	8	1	no	—
477	C₄²	—	8	1	no	—
478	C₄²	—	8	1	no	—
479	C₄²	—	8	1	no	—
480	C₄²	—	8	1	no	—
481	C₄²	C₂²×C₄×C₅	1	2	yes	—
482	C₄²	C₂²×C₄×C₅	1	2	yes	—
483	C₄²	C₂×C₅×Q₈	1	2	yes	—
484	C₄²	C₂×C₅×Q₈	1	2	yes	—
485	C₂×C₅	—	4	2	no	—
486	C₂×C₅	#2296	1	1	yes	characteristic
487	C₂×C₅	#2912	1	1	yes	characteristic
488	C₂×C₅	#2913	1	1	yes	characteristic
489	C₂×C₅	#3118	1	1	yes	characteristic
490	C₂×C₅	#3119	1	1	yes	characteristic
491	C₂×C₅	#3120	1	1	yes	characteristic
492	C₂×C₅	#3121	1	1	yes	characteristic
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
500	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	8	4	no	—
540	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
541	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
542	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
543	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
544	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
545	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
546	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
547	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
548	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
549	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
550	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
551	C₈	—	2	2	no	—
552	C₈	—	2	2	no	—
553	C₈	—	2	2	no	—
554	C₈	—	2	2	no	—
555	C₅	#21367	1	1	yes	characteristic, direct product factor, Sylow
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
564	C₂²	C₅×#525	1	1	yes	characteristic
565	C₂²	C₅×#525	1	1	yes	characteristic
566	C₂²	C₅×#576	1	1	yes	characteristic
567	C₂²	C₅×#577	1	1	yes	characteristic
568	C₂²	C₅×#577	1	1	yes	characteristic
569	C₂²	C₅×#578	1	1	yes	characteristic
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	8	1	no	—
580	C₄	—	8	1	no	—
581	C₄	—	8	1	no	—
582	C₄	—	8	1	no	—
583	C₄	C₄×C₅×#41	1	1	yes	characteristic
584	C₄	C₄×C₅×#41	1	1	yes	characteristic
585	C₄	C₅×#533	1	1	yes	characteristic
586	C₄	C₅×#533	1	1	yes	characteristic
587	C₂	—	4	2	no	—
588	C₂	C₅×#2296	1	1	yes	characteristic
589	C₂	C₅×#2912	1	1	yes	characteristic
590	C₂	C₅×#2913	1	1	yes	characteristic
591	C₂	C₅×#3118	1	1	yes	characteristic
592	C₂	C₅×#3119	1	1	yes	characteristic
593	C₂	C₅×#3120	1	1	yes	characteristic
594	C₂	C₅×#3121	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	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1949	C₂	semidirect
5. C₅×#1949	C₂	semidirect
6. C₅×#1948	C₂	semidirect
7. C₅×#1947	C₂	semidirect
12. C₅×#1776	C₂	semidirect
13. C₅×#1776	C₂	semidirect
14. C₄×C₅×#107	C₂	semidirect
15. C₄×C₅×#106	C₂	semidirect
19. C₅×#433	C₄	semidirect
60. #21367	C₅	direct (N × H)
108. C₅×C₄⋊C₈	C₂×C₄	semidirect
109. C₅×C₄⋊C₈	C₂×C₄	semidirect
112. C₅×#97	C₂×C₄	semidirect
113. C₄×C₅×C₈	C₂×C₄	semidirect
127. #1949	C₂×C₅	semidirect
128. #1949	C₂×C₅	semidirect
129. #1948	C₂×C₅	semidirect
130. #1947	C₂×C₅	semidirect
135. #1776	C₂×C₅	semidirect
136. #1776	C₂×C₅	semidirect
137. C₄×#107	C₂×C₅	semidirect
138. C₄×#106	C₂×C₅	semidirect
215. #433	C₄×C₅	semidirect
365. C₄⋊C₈	C₂×C₄×C₅	semidirect
366. C₄⋊C₈	C₂×C₄×C₅	semidirect
369. #97	C₂×C₄×C₅	semidirect
370. C₄×C₈	C₂×C₄×C₅	semidirect
555. C₅	#21367	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×#433	2
C₅×#2314	2
C₅×#2274	2
C₅×#1949	2
C₅×#1948	2
C₅×#1947	2
C₅×#1854	2
C₅×#1853	2
C₅×#1852	2
C₅×#1776	2
C₄×C₅×#107	2
C₄×C₅×#106	2
#21367	5

Subgroups — C5×#21367 #11944087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21367 #11944087