Subgroups — C₅×#9829 #11932549

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#492	C₂	1	1	yes	maximal, characteristic
2	C₅×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1957	C₂	1	1	yes	maximal, characteristic
5	C₅×#1957	C₂	1	1	yes	maximal, characteristic
6	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×SD₁₆	—	2	2	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#127	C₂²	1	1	yes	characteristic
11	C₅×#492	—	2	2	no	—
12	C₅×#492	C₄	1	4	yes	complemented
13	C₂×C₅×#102	—	2	2	no	—
14	C₂×C₅×#102	—	2	2	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#384	C₂²	1	1	yes	characteristic
19	C₅×#384	C₂²	1	1	yes	characteristic
20	C₅×#340	—	2	2	no	—
21	C₅×#340	—	2	2	no	—
22	C₅×#340	—	2	2	no	—
23	C₅×#322	—	2	2	no	—
24	#9829	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×C₅×D₄	—	2	1	no	—
27	C₂²×C₅×D₄	—	2	1	no	—
28	C₂²×C₅×D₄	—	2	1	no	—
29	C₂×C₅×SD₁₆	—	2	8	no	—
30	C₂×C₅×SD₁₆	—	4	4	no	—
31	C₂×C₅×SD₁₆	—	4	4	no	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₅×C₄⋊Q₈	—	2	1	no	—
36	C₅×C₄⋊Q₈	—	2	1	no	—
37	C₅×C₄⋊Q₈	—	2	1	no	—
38	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₅×#127	—	2	1	no	—
40	C₅×#127	—	2	1	no	—
41	C₅×#127	—	2	1	no	—
42	C₅×#127	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
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×#102	—	4	4	no	—
53	C₅×#102	—	4	4	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	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	D₄	1	2	yes	—
60	C₂×#492	C₂×C₅	1	1	yes	characteristic
61	#1990	C₂×C₅	1	1	yes	characteristic, complemented
62	#1990	C₂×C₅	1	1	yes	characteristic, complemented
63	#1957	C₂×C₅	1	1	yes	characteristic
64	#1957	C₂×C₅	1	1	yes	characteristic
65	#1834	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂×#322	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂⁴×C₅	—	4	1	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	—	4	1	no	—
70	C₂×C₅×Q₈	—	4	2	no	—
71	C₂×C₅×Q₈	—	4	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₂×C₅×D₄	—	4	1	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂×C₅×D₄	—	4	1	no	—
78	C₂×C₅×D₄	—	4	2	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂×C₅×D₄	—	4	2	no	—
82	C₂×C₅×D₄	—	4	2	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₂²×C₄×C₅	—	4	1	no	—
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×SD₁₆	—	4	16	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
99	C₂×C₅×C₈	SD₁₆	1	4	yes	complemented
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	4	1	no	—
103	C₅×C₄⋊C₄	—	4	1	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	#41	1	1	yes	characteristic
113	C₄²×C₅	#41	1	1	yes	characteristic
114	C₂²×SD₁₆	—	2	2	no	—
115	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
116	C₂×#127	C₂²×C₅	1	1	yes	characteristic
117	#492	—	2	2	no	—
118	#492	C₄×C₅	1	4	yes	complemented
119	C₂×#102	—	2	2	no	—
120	C₂×#102	—	2	2	no	—
121	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
124	#384	C₂²×C₅	1	1	yes	characteristic
125	#384	C₂²×C₅	1	1	yes	characteristic
126	#340	—	2	2	no	—
127	#340	—	2	2	no	—
128	#340	—	2	2	no	—
129	#322	—	2	2	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	—	8	1	no	—
132	C₂³×C₅	—	8	1	no	—
133	C₂³×C₅	—	8	1	no	—
134	C₂³×C₅	—	8	1	no	—
135	C₂³×C₅	—	8	1	no	—
136	C₂³×C₅	—	8	1	no	—
137	C₂³×C₅	#127	1	1	yes	characteristic
138	C₅×Q₈	—	4	4	no	—
139	C₅×Q₈	—	8	2	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	4	4	no	—
143	C₅×D₄	—	8	2	no	—
144	C₅×D₄	—	8	2	no	—
145	C₅×D₄	—	8	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	8	1	no	—
148	C₂×C₄×C₅	—	8	1	no	—
149	C₂×C₄×C₅	—	8	1	no	—
150	C₂×C₄×C₅	—	8	1	no	—
151	C₂×C₄×C₅	—	8	1	no	—
152	C₂×C₄×C₅	—	8	1	no	—
153	C₂×C₄×C₅	—	8	1	no	—
154	C₂×C₄×C₅	—	8	1	no	—
155	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	#121	1	1	yes	characteristic
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	#124	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
162	C₂×C₄×C₅	#135	1	2	yes	—
163	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
165	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₅	#137	1	2	yes	—
167	C₅×C₈	—	2	4	no	—
168	C₅×C₈	—	2	4	no	—
169	C₅×C₈	—	4	2	no	—
170	C₂²×Q₈	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂²×D₄	—	2	1	no	—
174	C₂×SD₁₆	—	2	8	no	—
175	C₂×SD₁₆	—	4	4	no	—
176	C₂×SD₁₆	—	4	4	no	—
177	C₂²×C₈	C₅×D₄	1	2	yes	—
178	C₂²×C₈	C₅×D₄	1	2	yes	—
179	C₂²×C₈	C₅×D₄	1	2	yes	—
180	C₄⋊Q₈	—	2	1	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
184	#127	—	2	1	no	—
185	#127	—	2	1	no	—
186	#127	—	2	1	no	—
187	#127	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	#102	—	4	4	no	—
198	#102	—	4	4	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
204	C₄×C₈	C₅×D₄	1	2	yes	—
205	C₂²×C₅	—	8	2	no	—
206	C₂²×C₅	—	8	2	no	—
207	C₂²×C₅	—	8	2	no	—
208	C₂²×C₅	—	16	1	no	—
209	C₂²×C₅	—	16	1	no	—
210	C₂²×C₅	—	16	1	no	—
211	C₂²×C₅	—	16	1	no	—
212	C₂²×C₅	#390	1	1	yes	characteristic
213	C₂²×C₅	#492	1	1	yes	characteristic
214	C₂²×C₅	#495	1	1	yes	characteristic
215	C₂²×C₅	#496	1	1	yes	characteristic
216	C₂²×C₅	#496	1	1	yes	characteristic
217	C₂²×C₅	#496	1	1	yes	characteristic
218	C₂²×C₅	#497	1	1	yes	characteristic
219	C₄×C₅	—	8	2	no	—
220	C₄×C₅	—	16	1	no	—
221	C₄×C₅	—	16	1	no	—
222	C₄×C₅	#418	1	2	yes	—
223	C₄×C₅	C₄×SD₁₆	1	2	yes	—
224	C₄×C₅	#440	1	2	yes	—
225	C₄×C₅	#465	1	2	yes	—
226	C₄×C₅	#468	1	2	yes	—
227	C₄×C₅	#490	1	2	yes	—
228	C₂⁴	—	4	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂²×C₄	—	4	1	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	C₂×C₅×D₄	1	1	yes	characteristic
251	SD₁₆	—	4	16	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
260	C₂×C₈	C₅×SD₁₆	1	4	yes	complemented
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₄²	C₅×#41	1	1	yes	characteristic
274	C₄²	C₅×#41	1	1	yes	characteristic
275	C₂×C₅	—	16	2	no	—
276	C₂×C₅	#1714	1	1	yes	characteristic
277	C₂×C₅	#1715	1	1	yes	characteristic
278	C₂×C₅	#1717	1	1	yes	characteristic
279	C₂×C₅	#1719	1	1	yes	characteristic
280	C₂×C₅	#1961	1	1	yes	characteristic
281	C₂×C₅	#1968	1	1	yes	characteristic
282	C₂×C₅	#1973	1	1	yes	characteristic
283	C₂³	—	4	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	C₅×#127	1	1	yes	characteristic
291	Q₈	—	4	4	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	4	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
310	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
311	C₂×C₄	C₅×#121	1	1	yes	characteristic
312	C₂×C₄	C₅×#121	1	1	yes	characteristic
313	C₂×C₄	C₅×#124	1	1	yes	characteristic
314	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
315	C₂×C₄	C₅×#135	1	2	yes	—
316	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
317	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
318	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
319	C₂×C₄	C₅×#137	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	2	4	no	—
322	C₈	—	4	2	no	—
323	C₅	#9829	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	1	no	—
328	C₂²	—	16	1	no	—
329	C₂²	—	16	1	no	—
330	C₂²	—	16	1	no	—
331	C₂²	C₅×#390	1	1	yes	characteristic
332	C₂²	C₅×#492	1	1	yes	characteristic
333	C₂²	C₅×#495	1	1	yes	characteristic
334	C₂²	C₅×#496	1	1	yes	characteristic
335	C₂²	C₅×#496	1	1	yes	characteristic
336	C₂²	C₅×#496	1	1	yes	characteristic
337	C₂²	C₅×#497	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	16	1	no	—
340	C₄	—	16	1	no	—
341	C₄	C₅×#418	1	2	yes	—
342	C₄	C₄×C₅×SD₁₆	1	2	yes	—
343	C₄	C₅×#440	1	2	yes	—
344	C₄	C₅×#465	1	2	yes	—
345	C₄	C₅×#468	1	2	yes	—
346	C₄	C₅×#490	1	2	yes	—
347	C₂	—	16	2	no	—
348	C₂	C₅×#1714	1	1	yes	characteristic
349	C₂	C₅×#1715	1	1	yes	characteristic
350	C₂	C₅×#1717	1	1	yes	characteristic
351	C₂	C₅×#1719	1	1	yes	characteristic
352	C₂	C₅×#1961	1	1	yes	characteristic
353	C₂	C₅×#1968	1	1	yes	characteristic
354	C₂	C₅×#1973	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1990	C₂	semidirect
3. C₅×#1990	C₂	semidirect
6. C₅×#1834	C₂	semidirect
7. C₂×C₅×#322	C₂	semidirect
12. C₅×#492	C₄	semidirect
24. #9829	C₅	direct (N × H)
61. #1990	C₂×C₅	semidirect
62. #1990	C₂×C₅	semidirect
65. #1834	C₂×C₅	semidirect
66. C₂×#322	C₂×C₅	semidirect
99. C₂×C₅×C₈	SD₁₆	semidirect
118. #492	C₄×C₅	semidirect
260. C₂×C₈	C₅×SD₁₆	semidirect
323. C₅	#9829	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×#492	2
C₅×#1990	2
C₅×#1957	2
C₅×#1834	2
C₂×C₅×#322	2
#9829	5

Subgroups — C5×#9829 #11932549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9829 #11932549