Subgroups — C₅×#19289 #11942009

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1698	C₂	1	4	yes	maximal, complemented
5	C₅×#1696	C₂	1	2	yes	maximal, complemented
6	C₅×#1688	C₂	1	2	yes	maximal, complemented
7	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1551	C₂	1	2	yes	maximal, complemented
9	C₅×#532	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#500	C₂²	1	1	yes	characteristic
12	C₅×#493	C₂²	1	1	yes	characteristic
13	C₅×#486	C₂²	1	2	yes	—
14	C₅×#480	—	2	2	no	—
15	C₅×#477	C₂²	1	4	yes	—
16	C₅×#466	—	2	1	no	—
17	C₅×#466	—	2	1	no	—
18	C₅×#460	C₂²	1	4	yes	—
19	C₅×#448	—	2	2	no	—
20	C₅×#446	—	2	1	no	—
21	C₅×#446	—	2	1	no	—
22	C₅×#446	C₂²	1	2	yes	—
23	C₅×#446	C₂²	1	2	yes	—
24	C₅×#440	—	2	4	no	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₄×C₅×D₈	—	2	1	no	—
27	C₅×#432	C₂²	1	2	yes	—
28	C₂×C₅×C₄⋊C₈	—	2	1	no	—
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₅×#97	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#333	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19289	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
39	C₂×C₅×SD₁₆	—	2	4	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂²×C₅×C₈	—	2	1	no	—
43	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₅×#127	C₂³	1	2	yes	—
47	C₅×#124	—	2	2	no	—
48	C₅×#121	—	2	2	no	—
49	C₅×#121	—	2	2	no	—
50	C₄×C₅×Q₈	—	2	1	no	—
51	C₄×C₅×Q₈	—	2	1	no	—
52	C₄×C₅×Q₈	D₄	1	2	yes	—
53	C₄×C₅×Q₈	C₂³	1	2	yes	—
54	C₄×C₅×D₄	—	2	2	no	—
55	C₄×C₅×D₄	—	2	2	no	—
56	C₂×C₅×C₄⋊C₄	—	2	1	no	—
57	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅	—	2	1	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	—	2	4	no	—
63	C₅×C₄⋊C₈	—	2	2	no	—
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	—	2	1	no	—
66	C₅×C₄⋊C₈	C₂³	1	2	yes	—
67	C₅×C₄⋊C₈	C₂³	1	4	yes	—
68	C₅×#103	—	2	4	no	—
69	C₅×#102	—	2	2	no	—
70	C₅×#102	—	2	2	no	—
71	C₅×#102	—	2	4	no	—
72	C₅×C₂²⋊C₈	—	2	2	no	—
73	C₅×#97	—	2	2	no	—
74	C₅×#97	—	2	2	no	—
75	C₅×#97	—	2	2	no	—
76	C₄×C₅×C₈	D₄	1	1	yes	characteristic
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	#3169	C₂×C₅	1	1	yes	characteristic
81	#3112	C₂×C₅	1	2	yes	complemented
82	C₂×#446	C₂×C₅	1	1	yes	characteristic, complemented
83	#1698	C₂×C₅	1	4	yes	complemented
84	#1696	C₂×C₅	1	2	yes	complemented
85	#1688	C₂×C₅	1	2	yes	complemented
86	#1568	C₂×C₅	1	1	yes	characteristic, complemented
87	#1551	C₂×C₅	1	2	yes	complemented
88	C₅×#41	—	8	4	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	4	2	no	—
92	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	4	2	no	—
97	C₂²×C₄×C₅	—	2	1	no	—
98	C₂²×C₄×C₅	—	2	1	no	—
99	C₂²×C₄×C₅	—	4	2	no	—
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	#41	1	1	yes	characteristic
103	C₅×SD₁₆	—	4	8	no	—
104	C₅×D₈	—	4	2	no	—
105	C₅×D₈	—	4	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	4	no	—
109	C₂×C₅×C₈	—	4	1	no	—
110	C₂×C₅×C₈	—	4	2	no	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
121	C₅×C₂²⋊C₄	—	4	2	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅	#41	1	1	yes	characteristic
128	C₄²×C₅	C₂⁴	1	1	yes	characteristic
129	#532	C₂²×C₅	1	2	yes	—
130	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
131	#500	C₂²×C₅	1	1	yes	characteristic
132	#493	C₂²×C₅	1	1	yes	characteristic
133	#486	C₂²×C₅	1	2	yes	—
134	#480	—	2	2	no	—
135	#477	C₂²×C₅	1	4	yes	—
136	#466	—	2	1	no	—
137	#466	—	2	1	no	—
138	#460	C₂²×C₅	1	4	yes	—
139	#448	—	2	2	no	—
140	#446	—	2	1	no	—
141	#446	—	2	1	no	—
142	#446	C₂²×C₅	1	2	yes	—
143	#446	C₂²×C₅	1	2	yes	—
144	#440	—	2	4	no	—
145	C₄×D₈	—	2	1	no	—
146	C₄×D₈	—	2	1	no	—
147	#432	C₂²×C₅	1	2	yes	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
150	C₂×#97	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
152	#335	C₂²×C₅	1	2	yes	—
153	#334	C₂²×C₅	1	2	yes	—
154	#333	C₂²×C₅	1	4	yes	—
155	#331	C₂²×C₅	1	4	yes	—
156	C₂³×C₅	—	4	2	no	—
157	C₂³×C₅	#121	1	1	yes	characteristic
158	C₅×Q₈	—	2	4	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	8	2	no	—
161	C₅×D₄	—	4	2	no	—
162	C₅×D₄	—	4	2	no	—
163	C₅×D₄	—	4	4	no	—
164	C₅×D₄	—	8	2	no	—
165	C₅×D₄	—	8	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	8	4	no	—
176	C₂×C₄×C₅	#121	1	1	yes	characteristic
177	C₂×C₄×C₅	#121	1	1	yes	characteristic
178	C₂×C₄×C₅	#121	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	—	2	1	no	—
185	C₅×C₈	—	4	2	no	—
186	C₅×C₈	—	4	4	no	—
187	C₅×C₈	#137	1	2	yes	—
188	C₂×#41	—	2	2	no	—
189	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
195	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
196	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
197	#127	C₂³×C₅	1	2	yes	—
198	#124	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×Q₈	C₅×D₄	1	2	yes	—
204	C₄×Q₈	C₂³×C₅	1	2	yes	—
205	C₄×D₄	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	C₂³×C₅	1	2	yes	—
218	C₄⋊C₈	C₂³×C₅	1	4	yes	—
219	#103	—	2	4	no	—
220	#102	—	2	2	no	—
221	#102	—	2	2	no	—
222	#102	—	2	4	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	#97	—	2	2	no	—
225	#97	—	2	2	no	—
226	#97	—	2	2	no	—
227	C₄×C₈	C₅×D₄	1	1	yes	characteristic
228	C₄×C₈	C₅×D₄	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	8	2	no	—
235	C₂²×C₅	—	8	2	no	—
236	C₂²×C₅	#472	1	2	yes	—
237	C₂²×C₅	C₂×#121	1	1	yes	characteristic
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	4	2	no	—
244	C₄×C₅	—	8	2	no	—
245	C₄×C₅	C₂×#137	1	1	yes	characteristic
246	C₄×C₅	#575	1	1	yes	characteristic
247	C₄×C₅	#575	1	2	yes	—
248	#41	—	8	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
262	C₂²×C₄	C₅×#41	1	1	yes	characteristic
263	SD₁₆	—	4	8	no	—
264	D₈	—	4	2	no	—
265	D₈	—	4	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
286	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
287	C₄²	C₅×#41	1	1	yes	characteristic
288	C₄²	C₂⁴×C₅	1	1	yes	characteristic
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	16	2	no	—
291	C₂×C₅	#3037	1	1	yes	characteristic
292	C₂×C₅	#3054	1	1	yes	characteristic
293	C₂×C₅	C₂×#472	1	1	yes	characteristic
294	C₂³	—	4	2	no	—
295	C₂³	C₅×#121	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	4	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	C₅×#121	1	1	yes	characteristic
315	C₂×C₄	C₅×#121	1	1	yes	characteristic
316	C₂×C₄	C₅×#121	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	4	2	no	—
324	C₈	—	4	4	no	—
325	C₈	C₅×#137	1	2	yes	—
326	C₅	#19289	1	1	yes	characteristic, direct product factor, Sylow
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	C₅×#472	1	2	yes	—
333	C₂²	C₂×C₅×#121	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	C₂×C₅×#137	1	1	yes	characteristic
342	C₄	C₅×#575	1	1	yes	characteristic
343	C₄	C₅×#575	1	2	yes	—
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₅×#3037	1	1	yes	characteristic
347	C₂	C₅×#3054	1	1	yes	characteristic
348	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3169	C₂
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
2. C₅×#3112	C₂	semidirect
3. C₂×C₅×#446	C₂	semidirect
4. C₅×#1698	C₂	semidirect
5. C₅×#1696	C₂	semidirect
6. C₅×#1688	C₂	semidirect
7. C₅×#1568	C₂	semidirect
8. C₅×#1551	C₂	semidirect
36. #19289	C₅	direct (N × H)
81. #3112	C₂×C₅	semidirect
82. C₂×#446	C₂×C₅	semidirect
83. #1698	C₂×C₅	semidirect
84. #1696	C₂×C₅	semidirect
85. #1688	C₂×C₅	semidirect
86. #1568	C₂×C₅	semidirect
87. #1551	C₂×C₅	semidirect
326. C₅	#19289	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×#3169	2
C₅×#3112	2
C₂×C₅×#446	2
C₅×#1698	2
C₅×#1696	2
C₅×#1688	2
C₅×#1568	2
C₅×#1551	2
#19289	5

Subgroups — C5×#19289 #11942009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19289 #11942009