Subgroups — C₅×#21430 #11944150

Order 1280.

858 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₅×#1923	C₂	1	4	yes	maximal
3	C₅×#1916	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
19	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₅×#376	C₄	1	1	yes	characteristic
21	C₅×#376	C₄	1	1	yes	characteristic
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#336	C₂²	1	4	yes	—
25	C₅×#336	C₂²	1	4	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	#21430	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₅×#117	—	2	4	no	—
34	C₅×#117	C₈	1	8	yes	complemented
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	—	2	4	no	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	2	yes	—
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	8	no	—
53	C₅×C₂²⋊C₈	—	2	8	no	—
54	C₄×C₅×C₈	—	2	8	no	—
55	C₄×C₅×C₈	—	2	4	no	—
56	C₄×C₅×C₈	—	2	4	no	—
57	C₅×#95	C₂×C₄	1	2	yes	—
58	C₅×#95	C₂×C₄	1	2	yes	—
59	C₅×#95	C₂×C₄	1	2	yes	—
60	C₅×#95	C₂×C₄	1	2	yes	—
61	C₄×#117	C₂×C₅	1	1	yes	characteristic
62	#1923	C₂×C₅	1	4	yes	—
63	#1916	C₂×C₅	1	4	yes	complemented
64	C₄×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
65	C₄×C₂²⋊C₈	C₂×C₅	1	2	yes	—
66	#1744	C₂×C₅	1	1	yes	characteristic, complemented
67	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	4	no	—
91	C₂×C₅×C₈	#41	1	8	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
94	C₅×C₂²⋊C₄	—	2	4	no	—
95	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	4	no	—
100	C₄²×C₅	—	2	4	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	4	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₄²×C₅	C₂×C₈	1	4	yes	—
105	C₄²×C₅	C₂×C₈	1	4	yes	—
106	C₄²×C₅	M₁₆	1	4	yes	—
107	C₄²×C₅	#41	1	4	yes	—
108	C₂×#117	C₄×C₅	1	1	yes	characteristic
109	C₂×#117	C₄×C₅	1	1	yes	characteristic
110	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
112	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
113	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
116	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
117	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
118	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
119	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
120	#376	C₄×C₅	1	1	yes	characteristic
121	#376	C₄×C₅	1	1	yes	characteristic
122	C₄³	C₂²×C₅	1	1	yes	characteristic
123	C₄³	C₂²×C₅	1	1	yes	characteristic
124	#336	C₂²×C₅	1	4	yes	—
125	#336	C₂²×C₅	1	4	yes	—
126	#336	C₂²×C₅	1	4	yes	—
127	C₂³×C₅	—	2	1	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	C₂²×C₈	1	2	yes	—
132	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	4	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	#117	1	2	yes	—
152	C₂×C₄×C₅	#117	1	2	yes	—
153	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
154	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
155	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
158	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₅	#131	1	4	yes	—
160	C₂×C₄×C₅	C₂×#41	1	2	yes	—
161	C₂×C₄×C₅	C₂×#41	1	4	yes	—
162	C₅×C₈	—	2	8	no	—
163	C₅×C₈	—	4	4	no	—
164	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	4	yes	—
168	C₂²×C₈	C₂³×C₅	1	4	yes	—
169	#117	—	2	4	no	—
170	#117	C₅×C₈	1	8	yes	complemented
171	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₅	1	2	yes	—
186	C₂×C₄²	C₂³×C₅	1	2	yes	—
187	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
188	C₄⋊C₈	—	2	8	no	—
189	C₂²⋊C₈	—	2	8	no	—
190	C₄×C₈	—	2	8	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	#95	C₂×C₄×C₅	1	2	yes	—
194	#95	C₂×C₄×C₅	1	2	yes	—
195	#95	C₂×C₄×C₅	1	2	yes	—
196	#95	C₂×C₄×C₅	1	2	yes	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	4	1	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	C₂×#117	1	1	yes	characteristic
203	C₂²×C₅	C₄×#41	1	2	yes	—
204	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
205	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
206	C₂²×C₅	C₂×#131	1	2	yes	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	4	no	—
209	C₄×C₅	—	2	4	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	4	1	no	—
212	C₄×C₅	—	4	2	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	#431	1	4	yes	—
215	C₄×C₅	#433	1	4	yes	—
216	C₄×C₅	#529	1	4	yes	—
217	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₅×#41	1	8	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂×C₅×C₈	1	4	yes	—
254	C₄²	C₂×C₅×C₈	1	4	yes	—
255	C₄²	C₅×M₁₆	1	4	yes	—
256	C₄²	C₅×#41	1	4	yes	—
257	C₂×C₅	—	4	2	no	—
258	C₂×C₅	#2298	1	1	yes	characteristic
259	C₂×C₅	C₂×#431	1	1	yes	characteristic
260	C₂×C₅	C₂×#433	1	1	yes	characteristic
261	C₂×C₅	C₈×#41	1	2	yes	—
262	C₂×C₅	#2965	1	2	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂²×C₅×C₈	1	2	yes	—
268	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₅×#117	1	2	yes	—
288	C₂×C₄	C₅×#117	1	2	yes	—
289	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
290	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
291	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
292	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
293	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
294	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
295	C₂×C₄	C₅×#131	1	4	yes	—
296	C₂×C₄	C₂×C₅×#41	1	2	yes	—
297	C₂×C₄	C₂×C₅×#41	1	4	yes	—
298	C₈	—	2	8	no	—
299	C₈	—	4	4	no	—
300	C₅	#21430	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂×C₅×#117	1	1	yes	characteristic
307	C₂²	C₄×C₅×#41	1	2	yes	—
308	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
309	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
310	C₂²	C₂×C₅×#131	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	C₅×#431	1	4	yes	—
319	C₄	C₅×#433	1	4	yes	—
320	C₄	C₅×#529	1	4	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₅×#2298	1	1	yes	characteristic
323	C₂	C₂×C₅×#431	1	1	yes	characteristic
324	C₂	C₂×C₅×#433	1	1	yes	characteristic
325	C₂	C₅×C₈×#41	1	2	yes	—
326	C₂	C₅×#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1916	C₂	semidirect
4. C₄×C₅×C₄⋊C₈	C₂	semidirect
6. C₅×#1744	C₂	semidirect
7. C₄²×C₅×C₈	C₂	semidirect
27. #21430	C₅	direct (N × H)
34. C₅×#117	C₈	semidirect
63. #1916	C₂×C₅	semidirect
64. C₄×C₄⋊C₈	C₂×C₅	semidirect
66. #1744	C₂×C₅	semidirect
67. C₄²×C₈	C₂×C₅	semidirect
170. #117	C₅×C₈	semidirect
300. C₅	#21430	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#117	2
C₅×#1923	2
C₅×#1916	2
C₄×C₅×C₄⋊C₈	2
C₄×C₅×C₂²⋊C₈	2
C₅×#1744	2
C₄²×C₅×C₈	2
#21430	5

Subgroups — C5×#21430 #11944150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21430 #11944150