Subgroups — C₅×#9822 #11932542

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#486	C₂	1	1	yes	maximal, characteristic
2	C₅×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1957	C₂	1	1	yes	maximal, characteristic
4	C₅×#1957	C₂	1	1	yes	maximal, characteristic
5	C₅×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#127	C₂²	1	1	yes	characteristic
10	C₅×#486	—	2	2	no	—
11	C₅×#486	C₄	1	4	yes	complemented
12	C₂×C₅×#103	—	2	2	no	—
13	C₂×C₅×#102	—	2	2	no	—
14	C₂×C₅×#102	—	2	2	no	—
15	C₂×C₅×#102	—	2	2	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#384	C₂²	1	1	yes	characteristic
20	C₅×#384	C₂²	1	1	yes	characteristic
21	C₅×#340	—	2	2	no	—
22	C₅×#340	—	2	2	no	—
23	C₅×C₈²	—	2	2	no	—
24	#9822	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	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₅×C₄⋊Q₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	—	2	1	no	—
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₅×#127	—	2	1	no	—
37	C₅×#127	—	2	1	no	—
38	C₅×#127	—	2	1	no	—
39	C₅×#127	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₄⋊C₄	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
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×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#103	—	4	4	no	—
50	C₅×#102	—	2	8	no	—
51	C₅×#102	—	4	4	no	—
52	C₅×#102	—	4	4	no	—
53	C₄×C₅×C₈	—	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	C₂×C₄	1	2	yes	—
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₂×#486	C₂×C₅	1	1	yes	characteristic
60	#1990	C₂×C₅	1	1	yes	characteristic, complemented
61	#1957	C₂×C₅	1	1	yes	characteristic
62	#1957	C₂×C₅	1	1	yes	characteristic
63	#1956	C₂×C₅	1	1	yes	characteristic, complemented
64	#1834	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂⁴×C₅	—	4	1	no	—
67	C₂×C₅×Q₈	—	2	2	no	—
68	C₂×C₅×Q₈	—	4	1	no	—
69	C₂×C₅×Q₈	—	4	2	no	—
70	C₂×C₅×Q₈	—	4	2	no	—
71	C₂×C₅×D₄	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂×C₅×D₄	—	4	1	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂×C₅×D₄	—	4	2	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	C₂²×C₄×C₅	—	4	1	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	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	C₂²⋊C₄	1	4	yes	—
96	C₂×C₅×C₈	D₈	1	4	yes	—
97	C₂×C₅×C₈	SD₁₆	1	4	yes	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	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	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	#41	1	1	yes	characteristic
111	C₄²×C₅	#41	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×#127	C₂²×C₅	1	1	yes	characteristic
114	#486	—	2	2	no	—
115	#486	C₄×C₅	1	4	yes	complemented
116	C₂×#103	—	2	2	no	—
117	C₂×#102	—	2	2	no	—
118	C₂×#102	—	2	2	no	—
119	C₂×#102	—	2	2	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
123	#384	C₂²×C₅	1	1	yes	characteristic
124	#384	C₂²×C₅	1	1	yes	characteristic
125	#340	—	2	2	no	—
126	#340	—	2	2	no	—
127	C₈²	—	2	2	no	—
128	C₂³×C₅	—	4	2	no	—
129	C₂³×C₅	—	8	1	no	—
130	C₂³×C₅	—	8	1	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	#127	1	1	yes	characteristic
136	C₅×Q₈	—	4	4	no	—
137	C₅×Q₈	—	8	2	no	—
138	C₅×D₄	—	4	4	no	—
139	C₅×D₄	—	4	4	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	8	2	no	—
142	C₅×D₄	—	8	2	no	—
143	C₅×D₄	—	8	2	no	—
144	C₂×C₄×C₅	—	4	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	—	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	#124	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₅	#135	1	2	yes	—
162	C₂×C₄×C₅	#135	1	2	yes	—
163	C₂×C₄×C₅	#135	1	2	yes	—
164	C₂×C₄×C₅	#135	1	2	yes	—
165	C₅×C₈	—	2	4	no	—
166	C₅×C₈	—	2	4	no	—
167	C₅×C₈	—	2	4	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×C₈	C₅×D₄	1	2	yes	—
173	C₂²×C₈	C₅×D₄	1	2	yes	—
174	C₂²×C₈	C₅×D₄	1	2	yes	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
179	#127	—	2	1	no	—
180	#127	—	2	1	no	—
181	#127	—	2	1	no	—
182	#127	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
192	#103	—	4	4	no	—
193	#102	—	2	8	no	—
194	#102	—	4	4	no	—
195	#102	—	4	4	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
201	C₄×C₈	C₅×D₄	1	2	yes	—
202	C₂²×C₅	—	8	2	no	—
203	C₂²×C₅	—	8	2	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	—	16	1	no	—
206	C₂²×C₅	—	16	1	no	—
207	C₂²×C₅	—	16	1	no	—
208	C₂²×C₅	—	16	1	no	—
209	C₂²×C₅	#390	1	1	yes	characteristic
210	C₂²×C₅	#492	1	1	yes	characteristic
211	C₂²×C₅	#493	1	1	yes	characteristic
212	C₂²×C₅	#495	1	1	yes	characteristic
213	C₂²×C₅	#495	1	1	yes	characteristic
214	C₂²×C₅	#495	1	1	yes	characteristic
215	C₂²×C₅	#495	1	1	yes	characteristic
216	C₄×C₅	—	8	2	no	—
217	C₄×C₅	—	16	1	no	—
218	C₄×C₅	—	16	1	no	—
219	C₄×C₅	#416	1	2	yes	—
220	C₄×C₅	C₄×D₈	1	2	yes	—
221	C₄×C₅	C₄×SD₁₆	1	2	yes	—
222	C₄×C₅	#465	1	2	yes	—
223	C₄×C₅	#466	1	2	yes	—
224	C₄×C₅	#488	1	2	yes	—
225	C₂⁴	—	4	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
255	C₂×C₈	C₅×D₈	1	4	yes	—
256	C₂×C₈	C₅×SD₁₆	1	4	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
268	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
269	C₄²	C₅×#41	1	1	yes	characteristic
270	C₄²	C₅×#41	1	1	yes	characteristic
271	C₂×C₅	—	16	2	no	—
272	C₂×C₅	#1706	1	1	yes	characteristic
273	C₂×C₅	#1708	1	1	yes	characteristic
274	C₂×C₅	#1709	1	1	yes	characteristic
275	C₂×C₅	#1711	1	1	yes	characteristic
276	C₂×C₅	#1958	1	1	yes	characteristic
277	C₂×C₅	#1968	1	1	yes	characteristic
278	C₂×C₅	#1970	1	1	yes	characteristic
279	C₂³	—	4	2	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	C₅×#127	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	2	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	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	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
306	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
307	C₂×C₄	C₅×#121	1	1	yes	characteristic
308	C₂×C₄	C₅×#121	1	1	yes	characteristic
309	C₂×C₄	C₅×#124	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
311	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
312	C₂×C₄	C₅×#135	1	2	yes	—
313	C₂×C₄	C₅×#135	1	2	yes	—
314	C₂×C₄	C₅×#135	1	2	yes	—
315	C₂×C₄	C₅×#135	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₅	#9822	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	C₅×#390	1	1	yes	characteristic
328	C₂²	C₅×#492	1	1	yes	characteristic
329	C₂²	C₅×#493	1	1	yes	characteristic
330	C₂²	C₅×#495	1	1	yes	characteristic
331	C₂²	C₅×#495	1	1	yes	characteristic
332	C₂²	C₅×#495	1	1	yes	characteristic
333	C₂²	C₅×#495	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	16	1	no	—
336	C₄	—	16	1	no	—
337	C₄	C₅×#416	1	2	yes	—
338	C₄	C₄×C₅×D₈	1	2	yes	—
339	C₄	C₄×C₅×SD₁₆	1	2	yes	—
340	C₄	C₅×#465	1	2	yes	—
341	C₄	C₅×#466	1	2	yes	—
342	C₄	C₅×#488	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	C₅×#1706	1	1	yes	characteristic
345	C₂	C₅×#1708	1	1	yes	characteristic
346	C₂	C₅×#1709	1	1	yes	characteristic
347	C₂	C₅×#1711	1	1	yes	characteristic
348	C₂	C₅×#1958	1	1	yes	characteristic
349	C₂	C₅×#1968	1	1	yes	characteristic
350	C₂	C₅×#1970	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
5. C₅×#1956	C₂	semidirect
6. C₅×#1834	C₂	semidirect
7. C₂×C₅×C₈²	C₂	semidirect
11. C₅×#486	C₄	semidirect
24. #9822	C₅	direct (N × H)
60. #1990	C₂×C₅	semidirect
63. #1956	C₂×C₅	semidirect
64. #1834	C₂×C₅	semidirect
65. C₂×C₈²	C₂×C₅	semidirect
115. #486	C₄×C₅	semidirect
319. C₅	#9822	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×#486	2
C₅×#1990	2
C₅×#1957	2
C₅×#1956	2
C₅×#1834	2
C₂×C₅×C₈²	2
#9822	5

Subgroups — C5×#9822 #11932542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9822 #11932542