Subgroups — C₅×#9550 #11932270

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#100	C₂	1	1	yes	maximal, characteristic
2	C₅×#1565	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1478	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#100	C₄	1	1	yes	characteristic
13	C₂×C₅×#100	C₄	1	1	yes	characteristic
14	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#343	C₄	1	1	yes	characteristic
18	C₅×#343	C₄	1	1	yes	characteristic
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×D₄⋊C₈	—	2	1	no	—
22	C₅×D₄⋊C₈	—	2	1	no	—
23	C₅×D₄⋊C₈	—	2	1	no	—
24	C₅×D₄⋊C₈	—	2	1	no	—
25	C₅×#322	—	2	1	no	—
26	C₅×C₈²	—	2	1	no	—
27	#9550	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	—	2	1	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
47	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₅×#100	—	2	2	no	—
49	C₅×#100	C₈	1	4	yes	complemented
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×#97	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×#100	C₂×C₅	1	1	yes	characteristic
62	#1565	C₂×C₅	1	1	yes	characteristic, complemented
63	#1478	C₂×C₅	1	1	yes	characteristic
64	C₂×D₄⋊C₈	C₂×C₅	1	1	yes	characteristic
65	C₈×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
66	#1283	C₂×C₅	1	1	yes	characteristic, complemented
67	#1278	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	—	2	1	no	—
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	—	2	1	no	—
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	8	1	no	—
79	C₂²×C₄×C₅	—	8	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅	#41	1	1	yes	characteristic
83	C₅×M₁₆	—	2	1	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	1	no	—
86	C₅×M₁₆	—	2	1	no	—
87	C₅×M₁₆	—	2	1	no	—
88	C₅×M₁₆	C₂×C₈	1	2	yes	—
89	C₅×M₁₆	C₂×C₈	1	2	yes	—
90	C₅×M₁₆	M₁₆	1	2	yes	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	4	1	no	—
102	C₂×C₅×C₈	—	8	1	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	1	no	—
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅	#41	1	1	yes	characteristic
110	C₄²×C₅	#41	1	1	yes	characteristic
111	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
112	#431	C₂²×C₅	1	1	yes	characteristic
113	#431	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×#100	C₄×C₅	1	1	yes	characteristic
116	C₂×#100	C₄×C₅	1	1	yes	characteristic
117	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
120	#343	C₄×C₅	1	1	yes	characteristic
121	#343	C₄×C₅	1	1	yes	characteristic
122	#335	—	2	1	no	—
123	#334	—	2	1	no	—
124	D₄⋊C₈	—	2	1	no	—
125	D₄⋊C₈	—	2	1	no	—
126	D₄⋊C₈	—	2	1	no	—
127	D₄⋊C₈	—	2	1	no	—
128	#322	—	2	1	no	—
129	C₈²	—	2	1	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	4	1	no	—
132	C₂³×C₅	—	4	1	no	—
133	C₂³×C₅	—	8	1	no	—
134	C₂³×C₅	—	8	1	no	—
135	C₂³×C₅	#117	1	1	yes	characteristic
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	4	1	no	—
139	C₅×D₄	—	4	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	—	8	2	no	—
145	C₂×C₄×C₅	—	8	1	no	—
146	C₂×C₄×C₅	—	8	1	no	—
147	C₂×C₄×C₅	—	8	1	no	—
148	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#121	1	1	yes	characteristic
151	C₂×C₄×C₅	#123	1	1	yes	characteristic
152	C₂×C₄×C₅	#124	1	1	yes	characteristic
153	C₂×C₄×C₅	#126	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
156	C₂×C₄×C₅	#131	1	1	yes	characteristic
157	C₂×C₄×C₅	#131	1	1	yes	characteristic
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	—	2	2	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	4	1	no	—
163	C₅×C₈	—	4	1	no	—
164	C₅×C₈	—	4	1	no	—
165	C₅×C₈	—	4	1	no	—
166	C₅×C₈	—	8	1	no	—
167	C₅×C₈	—	8	1	no	—
168	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂³×C₄	—	2	1	no	—
170	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
171	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
172	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
173	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
187	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
188	#100	—	2	2	no	—
189	#100	C₅×C₈	1	4	yes	complemented
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	C₅×D₄	1	1	yes	characteristic
200	C₄×C₈	C₅×D₄	1	1	yes	characteristic
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	4	2	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	—	8	2	no	—
206	C₂²×C₅	—	8	1	no	—
207	C₂²×C₅	—	8	1	no	—
208	C₂²×C₅	#388	1	1	yes	characteristic
209	C₂²×C₅	#431	1	1	yes	characteristic
210	C₂²×C₅	#433	1	1	yes	characteristic
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	8	1	no	—
215	C₄×C₅	C₈×D₄	1	1	yes	characteristic
216	C₄×C₅	C₈×D₄	1	1	yes	characteristic
217	C₄×C₅	#435	1	1	yes	characteristic
218	C₄×C₅	#436	1	1	yes	characteristic
219	C₄×C₅	#471	1	1	yes	characteristic
220	C₄×C₅	#472	1	1	yes	characteristic
221	C₂⁴	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	8	1	no	—
232	C₂²×C₄	—	8	1	no	—
233	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₅×#41	1	1	yes	characteristic
235	C₂²×C₄	C₅×#41	1	1	yes	characteristic
236	M₁₆	—	2	1	no	—
237	M₁₆	—	2	1	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	2	1	no	—
240	M₁₆	—	2	1	no	—
241	M₁₆	C₂×C₅×C₈	1	2	yes	—
242	M₁₆	C₂×C₅×C₈	1	2	yes	—
243	M₁₆	C₅×M₁₆	1	2	yes	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	8	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
261	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
262	C₄²	C₅×#41	1	1	yes	characteristic
263	C₄²	C₅×#41	1	1	yes	characteristic
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	8	2	no	—
267	C₂×C₅	#1923	1	1	yes	characteristic
268	C₂×C₅	#1931	1	1	yes	characteristic
269	C₂×C₅	#1932	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	C₅×#117	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	1	no	—
279	D₄	—	4	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
289	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
290	C₂×C₄	C₅×#121	1	1	yes	characteristic
291	C₂×C₄	C₅×#123	1	1	yes	characteristic
292	C₂×C₄	C₅×#124	1	1	yes	characteristic
293	C₂×C₄	C₅×#126	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
296	C₂×C₄	C₅×#131	1	1	yes	characteristic
297	C₂×C₄	C₅×#131	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	2	2	no	—
302	C₈	—	4	1	no	—
303	C₈	—	4	1	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₅	#9550	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	C₅×#388	1	1	yes	characteristic
317	C₂²	C₅×#431	1	1	yes	characteristic
318	C₂²	C₅×#433	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	C₅×C₈×D₄	1	1	yes	characteristic
324	C₄	C₅×C₈×D₄	1	1	yes	characteristic
325	C₄	C₅×#435	1	1	yes	characteristic
326	C₄	C₅×#436	1	1	yes	characteristic
327	C₄	C₅×#471	1	1	yes	characteristic
328	C₄	C₅×#472	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₅×#1923	1	1	yes	characteristic
333	C₂	C₅×#1931	1	1	yes	characteristic
334	C₂	C₅×#1932	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#388
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1565	C₂	semidirect
5. C₅×C₈×M₁₆	C₂	semidirect
6. C₅×#1283	C₂	semidirect
7. C₅×#1278	C₂	semidirect
27. #9550	C₅	direct (N × H)
49. C₅×#100	C₈	semidirect
62. #1565	C₂×C₅	semidirect
65. C₈×M₁₆	C₂×C₅	semidirect
66. #1283	C₂×C₅	semidirect
67. #1278	C₂×C₅	semidirect
189. #100	C₅×C₈	semidirect
308. C₅	#9550	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×#100	2
C₅×#1565	2
C₅×#1478	2
C₂×C₅×D₄⋊C₈	2
C₅×C₈×M₁₆	2
C₅×#1283	2
C₅×#1278	2
#9550	5

Subgroups — C5×#9550 #11932270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9550 #11932270