Subgroups — C₅×#12646 #11935366

Order 1280.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	C₅×#2148	C₂	1	1	yes	maximal, characteristic
3	C₅×#2145	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂×C₅×#131	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#443	—	2	1	no	—
13	C₅×#443	—	2	1	no	—
14	C₅×#443	C₄	1	2	yes	complemented
15	C₅×#443	C₄	1	2	yes	complemented
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₅×#427	C₂²	1	1	yes	characteristic
19	C₂×C₅×#104	—	2	1	no	—
20	C₅×#405	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#362	C₄	1	2	yes	complemented
23	C₅×#361	C₄	1	2	yes	complemented
24	C₅×#360	C₄	1	4	yes	complemented
25	C₅×#346	—	2	1	no	—
26	C₄×C₅×C₁₆	—	2	1	no	—
27	C₅×#339	—	2	1	no	—
28	#12646	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×#41	—	2	1	no	—
30	C₅×#135	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	C₂×C₄	1	2	yes	—
33	C₅×#135	C₂×C₄	1	2	yes	—
34	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×SD₁₆	—	2	1	no	—
36	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
37	C₅×#131	—	2	2	no	—
38	C₅×#131	—	2	2	no	—
39	C₅×#131	—	4	1	no	—
40	C₅×#131	—	4	1	no	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂×C₅×M₁₆	—	4	1	no	—
44	C₂²×C₅×C₈	—	4	1	no	—
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×#117	—	2	1	no	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₅×M₃₂	—	2	1	no	—
50	C₅×M₃₂	—	2	1	no	—
51	C₅×M₃₂	C₂×C₄	1	2	yes	—
52	C₅×M₃₂	C₂×C₄	1	2	yes	—
53	C₂×C₅×C₁₆	—	2	1	no	—
54	C₂×C₅×C₁₆	—	2	1	no	—
55	C₅×#108	—	2	1	no	—
56	C₅×#108	—	2	1	no	—
57	C₅×#108	C₂×C₄	1	2	yes	—
58	C₅×#108	C₂×C₄	1	2	yes	—
59	C₅×#107	—	2	1	no	—
60	C₅×#106	C₂×C₄	1	2	yes	—
61	C₅×#104	—	2	2	no	—
62	C₅×#104	—	2	2	no	—
63	C₅×#97	—	4	1	no	—
64	C₄×C₅×C₈	—	4	1	no	—
65	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₂×#443	C₂×C₅	1	1	yes	characteristic
70	#2148	C₂×C₅	1	1	yes	characteristic
71	#2145	C₂×C₅	1	1	yes	characteristic
72	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
73	#1772	C₂×C₅	1	1	yes	characteristic, complemented
74	#1389	C₂×C₅	1	1	yes	characteristic, complemented
75	#1388	C₂×C₅	1	1	yes	characteristic, complemented
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	2	2	no	—
78	C₅×#41	—	4	1	no	—
79	C₅×#41	—	4	1	no	—
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₅×Q₁₆	—	2	1	no	—
87	C₅×Q₁₆	C₄²	1	2	yes	—
88	C₅×SD₁₆	—	2	2	no	—
89	C₅×SD₁₆	—	2	2	no	—
90	C₅×D₈	—	2	1	no	—
91	C₅×D₈	C₄²	1	2	yes	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	2	no	—
94	C₅×M₁₆	—	4	1	no	—
95	C₅×M₁₆	—	4	1	no	—
96	C₅×M₁₆	—	4	2	no	—
97	C₅×M₁₆	—	4	2	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	—	4	2	no	—
101	C₂×C₅×C₈	—	4	2	no	—
102	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	D₈	1	1	yes	characteristic
105	C₂×C₅×C₈	D₈	1	1	yes	characteristic
106	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
107	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	#41	1	1	yes	characteristic
113	C₂×C₅×C₈	#41	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	4	1	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₄²×C₅	—	4	1	no	—
118	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
119	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
120	C₅×C₁₆	—	2	2	no	—
121	C₅×C₁₆	—	2	2	no	—
122	C₂×#135	C₂²×C₅	1	1	yes	characteristic
123	C₂×#131	—	2	1	no	—
124	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
125	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
126	#443	—	2	1	no	—
127	#443	—	2	1	no	—
128	#443	C₄×C₅	1	2	yes	complemented
129	#443	C₄×C₅	1	2	yes	complemented
130	C₂×#108	C₂²×C₅	1	1	yes	characteristic
131	C₂×#108	C₂²×C₅	1	1	yes	characteristic
132	#427	C₂²×C₅	1	1	yes	characteristic
133	C₂×#104	—	2	1	no	—
134	#405	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
136	#362	C₄×C₅	1	2	yes	complemented
137	#361	C₄×C₅	1	2	yes	complemented
138	#360	C₄×C₅	1	4	yes	complemented
139	#346	—	2	1	no	—
140	C₄×C₁₆	—	2	1	no	—
141	#339	—	2	1	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	#135	1	1	yes	characteristic
144	C₅×Q₈	—	2	2	no	—
145	C₅×Q₈	—	4	1	no	—
146	C₅×D₄	—	2	2	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	4	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
162	C₂×C₄×C₅	#135	1	1	yes	characteristic
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	—	4	2	no	—
165	C₅×C₈	#102	1	1	yes	characteristic
166	C₅×C₈	#102	1	1	yes	characteristic
167	C₅×C₈	#102	1	1	yes	characteristic
168	C₅×C₈	#102	1	1	yes	characteristic
169	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
170	C₅×C₈	#117	1	1	yes	characteristic
171	C₅×C₈	C₄×D₄	1	1	yes	characteristic
172	C₅×C₈	C₄×D₄	1	1	yes	characteristic
173	C₂×#41	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	—	2	1	no	—
176	#135	C₂×C₄×C₅	1	2	yes	—
177	#135	C₂×C₄×C₅	1	2	yes	—
178	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂×SD₁₆	—	2	1	no	—
180	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
181	#131	—	2	2	no	—
182	#131	—	2	2	no	—
183	#131	—	4	1	no	—
184	#131	—	4	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	4	1	no	—
188	C₂²×C₈	—	4	1	no	—
189	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	M₃₂	C₂×C₄×C₅	1	2	yes	—
196	M₃₂	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	#108	—	2	1	no	—
200	#108	—	2	1	no	—
201	#108	C₂×C₄×C₅	1	2	yes	—
202	#108	C₂×C₄×C₅	1	2	yes	—
203	#107	—	2	1	no	—
204	#106	C₂×C₄×C₅	1	2	yes	—
205	#104	—	2	2	no	—
206	#104	—	2	2	no	—
207	#97	—	4	1	no	—
208	C₄×C₈	—	4	1	no	—
209	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
210	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
211	C₄×C₈	C₅×D₄	1	1	yes	characteristic
212	C₄×C₈	C₅×D₄	1	1	yes	characteristic
213	C₂²×C₅	—	4	1	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	8	2	no	—
216	C₂²×C₅	#416	1	1	yes	characteristic
217	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
218	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
224	C₄×C₅	C₂×#102	1	1	yes	characteristic
225	C₄×C₅	C₄×D₈	1	1	yes	characteristic
226	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	Q₁₆	—	2	1	no	—
238	Q₁₆	C₄²×C₅	1	2	yes	—
239	SD₁₆	—	2	2	no	—
240	SD₁₆	—	2	2	no	—
241	D₈	—	2	1	no	—
242	D₈	C₄²×C₅	1	2	yes	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₈	C₅×D₈	1	1	yes	characteristic
256	C₂×C₈	C₅×D₈	1	1	yes	characteristic
257	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
258	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
259	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
261	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
263	C₂×C₈	C₅×#41	1	1	yes	characteristic
264	C₂×C₈	C₅×#41	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	2	no	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	—	8	2	no	—
277	C₂×C₅	C₄×#102	1	1	yes	characteristic
278	C₂³	—	4	1	no	—
279	C₂³	C₅×#135	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	4	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
295	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
298	C₂×C₄	C₅×#135	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	C₅×#102	1	1	yes	characteristic
302	C₈	C₅×#102	1	1	yes	characteristic
303	C₈	C₅×#102	1	1	yes	characteristic
304	C₈	C₅×#102	1	1	yes	characteristic
305	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
306	C₈	C₅×#117	1	1	yes	characteristic
307	C₈	C₄×C₅×D₄	1	1	yes	characteristic
308	C₈	C₄×C₅×D₄	1	1	yes	characteristic
309	C₅	#12646	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	C₅×#416	1	1	yes	characteristic
314	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
315	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
321	C₄	C₂×C₅×#102	1	1	yes	characteristic
322	C₄	C₄×C₅×D₈	1	1	yes	characteristic
323	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	2	no	—
328	C₂	C₄×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#102
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	C₄×#102
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₅×M₃₂	C₂	semidirect
5. C₅×#1772	C₂	semidirect
6. C₅×#1389	C₂	semidirect
7. C₅×#1388	C₂	semidirect
14. C₅×#443	C₄	semidirect
15. C₅×#443	C₄	semidirect
22. C₅×#362	C₄	semidirect
23. C₅×#361	C₄	semidirect
24. C₅×#360	C₄	semidirect
28. #12646	C₅	direct (N × H)
72. C₄×M₃₂	C₂×C₅	semidirect
73. #1772	C₂×C₅	semidirect
74. #1389	C₂×C₅	semidirect
75. #1388	C₂×C₅	semidirect
128. #443	C₄×C₅	semidirect
129. #443	C₄×C₅	semidirect
136. #362	C₄×C₅	semidirect
137. #361	C₄×C₅	semidirect
138. #360	C₄×C₅	semidirect
309. C₅	#12646	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×#443	2
C₅×#2148	2
C₅×#2145	2
C₄×C₅×M₃₂	2
C₅×#1772	2
C₅×#1389	2
C₅×#1388	2
#12646	5

Subgroups — C5×#12646 #11935366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12646 #11935366