Subgroups — C₅×#9287 #11932007

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3227	C₂	1	1	yes	maximal, characteristic
2	C₅×#2932	C₂	1	1	yes	maximal, characteristic
3	C₅×#1993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1517	C₂	1	1	yes	maximal, characteristic
6	C₅×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	—	2	1	no	—
9	C₅×#532	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#498	C₄	1	2	yes	complemented
12	C₅×#486	C₄	1	2	yes	complemented
13	C₅×#481	—	2	2	no	—
14	C₅×#476	—	2	2	no	—
15	C₅×#435	—	2	2	no	—
16	C₅×#435	—	2	2	no	—
17	C₅×#431	C₂²	1	1	yes	characteristic
18	C₅×#431	C₂²	1	1	yes	characteristic
19	C₅×#423	C₂²	1	1	yes	characteristic
20	C₅×#408	—	2	1	no	—
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
22	C₅×#382	C₂²	1	1	yes	characteristic
23	C₅×#340	—	2	1	no	—
24	C₅×#340	—	2	1	no	—
25	C₅×#333	—	2	2	no	—
26	C₅×#327	—	2	2	no	—
27	#9287	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×Q₈	—	2	1	no	—
30	C₅×#131	—	4	2	no	—
31	C₅×#131	—	4	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₅×C₄⋊Q₈	—	2	1	no	—
38	C₅×C₄⋊Q₈	—	2	1	no	—
39	C₅×C₄⋊Q₈	—	2	1	no	—
40	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₅×#127	C₂×C₄	1	1	yes	characteristic
43	C₅×#124	—	2	1	no	—
44	C₅×#121	—	2	1	no	—
45	C₅×#121	D₄	1	2	yes	—
46	C₄×C₅×D₄	—	2	1	no	—
47	C₄×C₅×D₄	D₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×#106	—	2	4	no	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₄⋊C₈	—	2	1	no	—
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×C₄⋊C₈	D₄	1	2	yes	—
57	C₅×#102	—	2	4	no	—
58	C₅×C₂²⋊C₈	—	2	4	no	—
59	C₅×C₂²⋊C₈	—	2	1	no	—
60	C₅×C₂²⋊C₈	—	2	1	no	—
61	C₅×#97	—	2	1	no	—
62	C₅×#97	—	2	1	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
65	C₅×#95	—	2	1	no	—
66	C₅×#95	—	2	1	no	—
67	#3227	C₂×C₅	1	1	yes	characteristic
68	#2932	C₂×C₅	1	1	yes	characteristic
69	#1993	C₂×C₅	1	1	yes	characteristic, complemented
70	#1548	C₂×C₅	1	1	yes	characteristic, complemented
71	#1517	C₂×C₅	1	1	yes	characteristic
72	#1286	C₂×C₅	1	1	yes	characteristic, complemented
73	#1275	C₂×C₅	1	1	yes	characteristic, complemented
74	C₅×#41	—	4	1	no	—
75	C₅×#41	—	4	1	no	—
76	C₅×#41	—	4	1	no	—
77	C₅×#41	—	4	1	no	—
78	C₂×C₅×Q₈	—	2	1	no	—
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂×C₅×Q₈	—	8	2	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	2	1	no	—
84	C₂×C₅×D₄	—	2	1	no	—
85	C₂×C₅×D₄	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	4	1	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	#41	1	1	yes	characteristic
92	C₅×M₁₆	—	4	4	no	—
93	C₅×M₁₆	—	4	1	no	—
94	C₅×M₁₆	—	4	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	4	4	no	—
103	C₂×C₅×C₈	—	4	1	no	—
104	C₂×C₅×C₈	SD₁₆	1	2	yes	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	#41	1	1	yes	characteristic
119	C₂×#131	—	2	1	no	—
120	#532	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
122	#498	C₄×C₅	1	2	yes	complemented
123	#486	C₄×C₅	1	2	yes	complemented
124	#481	—	2	2	no	—
125	#476	—	2	2	no	—
126	#435	—	2	2	no	—
127	#435	—	2	2	no	—
128	#431	C₂²×C₅	1	1	yes	characteristic
129	#431	C₂²×C₅	1	1	yes	characteristic
130	#423	C₂²×C₅	1	1	yes	characteristic
131	#408	—	2	1	no	—
132	C₂×#97	C₂²×C₅	1	1	yes	characteristic
133	#382	C₂²×C₅	1	1	yes	characteristic
134	#340	—	2	1	no	—
135	#340	—	2	1	no	—
136	#333	—	2	2	no	—
137	#327	—	2	2	no	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	#121	1	1	yes	characteristic
140	C₅×Q₈	—	4	1	no	—
141	C₅×Q₈	—	4	1	no	—
142	C₅×Q₈	—	8	2	no	—
143	C₅×Q₈	—	8	2	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	4	2	no	—
146	C₅×D₄	—	4	2	no	—
147	C₅×D₄	—	4	1	no	—
148	C₅×D₄	—	4	1	no	—
149	C₅×D₄	—	8	1	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	8	2	no	—
157	C₂×C₄×C₅	—	8	1	no	—
158	C₂×C₄×C₅	—	8	1	no	—
159	C₂×C₄×C₅	—	8	1	no	—
160	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	#120	1	1	yes	characteristic
164	C₂×C₄×C₅	#121	1	1	yes	characteristic
165	C₂×C₄×C₅	#123	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₅	#135	1	1	yes	characteristic
168	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₅	#137	1	1	yes	characteristic
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	—	4	2	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	—	4	1	no	—
174	C₅×C₈	—	8	1	no	—
175	C₂×#41	—	2	1	no	—
176	C₂²×Q₈	—	2	1	no	—
177	#131	—	4	2	no	—
178	#131	—	4	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
188	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
189	#127	C₂×C₄×C₅	1	1	yes	characteristic
190	#124	—	2	1	no	—
191	#121	—	2	1	no	—
192	#121	C₅×D₄	1	2	yes	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₅×D₄	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	#106	—	2	4	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	C₅×D₄	1	2	yes	—
204	#102	—	2	4	no	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	—	4	2	no	—
218	C₂²×C₅	—	4	2	no	—
219	C₂²×C₅	#386	1	1	yes	characteristic
220	C₂²×C₅	#461	1	1	yes	characteristic
221	C₂²×C₅	#463	1	1	yes	characteristic
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	—	16	1	no	—
229	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
230	C₄×C₅	#440	1	1	yes	characteristic
231	C₄×C₅	#450	1	1	yes	characteristic
232	C₄×C₅	#452	1	1	yes	characteristic
233	#41	—	4	1	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₅×C₂²⋊C₄	1	2	yes	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₅×#41	1	1	yes	characteristic
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	C₅×SD₁₆	1	2	yes	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
275	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
277	C₄²	C₅×#41	1	1	yes	characteristic
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	8	2	no	—
281	C₂×C₅	#1877	1	1	yes	characteristic
282	C₂×C₅	#1903	1	1	yes	characteristic
283	C₂×C₅	#1909	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	C₅×#121	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	8	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
308	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
309	C₂×C₄	C₅×#120	1	1	yes	characteristic
310	C₂×C₄	C₅×#121	1	1	yes	characteristic
311	C₂×C₄	C₅×#123	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
313	C₂×C₄	C₅×#135	1	1	yes	characteristic
314	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₄	C₅×#137	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	4	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	1	no	—
320	C₈	—	8	1	no	—
321	C₅	#9287	1	1	yes	characteristic, direct product factor, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₅×#386	1	1	yes	characteristic
328	C₂²	C₅×#461	1	1	yes	characteristic
329	C₂²	C₅×#463	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	16	1	no	—
337	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
338	C₄	C₅×#440	1	1	yes	characteristic
339	C₄	C₅×#450	1	1	yes	characteristic
340	C₄	C₅×#452	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	2	no	—
344	C₂	C₅×#1877	1	1	yes	characteristic
345	C₂	C₅×#1903	1	1	yes	characteristic
346	C₂	C₅×#1909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1993	C₂	semidirect
4. C₅×#1548	C₂	semidirect
6. C₅×#1286	C₂	semidirect
7. C₅×#1275	C₂	semidirect
11. C₅×#498	C₄	semidirect
12. C₅×#486	C₄	semidirect
27. #9287	C₅	direct (N × H)
69. #1993	C₂×C₅	semidirect
70. #1548	C₂×C₅	semidirect
72. #1286	C₂×C₅	semidirect
73. #1275	C₂×C₅	semidirect
122. #498	C₄×C₅	semidirect
123. #486	C₄×C₅	semidirect
321. C₅	#9287	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×#3227	2
C₅×#2932	2
C₅×#1993	2
C₅×#1548	2
C₅×#1517	2
C₅×#1286	2
C₅×#1275	2
#9287	5

Subgroups — C5×#9287 #11932007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9287 #11932007