Subgroups — C₅×#8609 #11931329

Order 1280.

550 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×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#1808	C₂	1	2	yes	maximal, complemented
3	C₅×#1799	C₂	1	2	yes	maximal
4	C₅×#1289	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×C₈×D₄	—	2	2	no	—
9	C₅×C₈×D₄	—	2	2	no	—
10	C₂×C₅×#107	—	2	2	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₅×#99	—	2	2	no	—
13	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
16	C₅×#380	C₂²	1	4	yes	—
17	C₅×#342	—	2	2	no	—
18	C₅×#340	—	2	2	no	—
19	#8609	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	1	no	—
25	C₂²×C₅×C₈	—	4	2	no	—
26	C₂²×C₅×C₈	—	4	2	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₄×C₅×D₄	—	2	2	no	—
34	C₄×C₅×D₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
38	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₅×C₂²⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₅×#107	—	4	4	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×#99	—	4	4	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₅×#95	—	2	4	no	—
49	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
50	#1808	C₂×C₅	1	2	yes	complemented
51	#1799	C₂×C₅	1	2	yes	—
52	#1289	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×D₄	—	2	2	no	—
56	C₂×C₅×D₄	—	2	2	no	—
57	C₂×C₅×D₄	D₈	1	2	yes	—
58	C₂×C₅×D₄	Q₁₆	1	2	yes	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	4	1	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	—	4	4	no	—
66	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	4	1	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₅×C₂²⋊C₄	—	2	2	no	—
90	C₅×C₂²⋊C₄	—	2	2	no	—
91	C₅×C₂²⋊C₄	—	4	4	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₂³×C₈	C₄×C₅	1	1	yes	characteristic, complemented
96	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
98	C₈×D₄	—	2	2	no	—
99	C₈×D₄	—	2	2	no	—
100	C₂×#107	—	2	2	no	—
101	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic, complemented
102	C₂×#99	—	2	2	no	—
103	C₂×C₂²⋊C₈	C₄×C₅	1	1	yes	characteristic, complemented
104	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic, complemented
106	#380	C₂²×C₅	1	4	yes	—
107	#342	—	2	2	no	—
108	#340	—	2	2	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	4	1	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₂³×C₅	—	4	1	no	—
115	C₂³×C₅	#107	1	2	yes	complemented
116	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₅	#121	1	2	yes	—
118	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
119	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
120	C₂³×C₅	#135	1	2	yes	—
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	2	4	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	8	4	no	—
139	C₂×C₄×C₅	—	8	4	no	—
140	C₂×C₄×C₅	—	8	4	no	—
141	C₂×C₄×C₅	#107	1	2	yes	complemented
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	#117	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	4	1	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	#99	1	2	yes	complemented
150	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
151	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	4	2	no	—
156	C₂²×C₈	—	4	2	no	—
157	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₈	C₅×D₄	1	2	yes	—
162	C₂²×C₈	C₅×D₄	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	4	no	—
168	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₅×Q₈	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
171	#107	—	4	4	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#99	—	4	4	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	#95	—	2	4	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	1	no	—
186	C₂²×C₅	—	4	2	no	—
187	C₂²×C₅	—	4	2	no	—
188	C₂²×C₅	—	4	2	no	—
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	#380	1	1	yes	characteristic
192	C₂²×C₅	C₂×#107	1	1	yes	characteristic
193	C₂²×C₅	#427	1	1	yes	characteristic
194	C₂²×C₅	#466	1	2	yes	—
195	C₂²×C₅	#467	1	2	yes	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	—	4	1	no	—
201	C₄×C₅	—	16	4	no	—
202	C₄×C₅	C₂×#99	1	1	yes	characteristic
203	C₄×C₅	#410	1	1	yes	characteristic
204	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
205	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₅×D₈	1	2	yes	—
209	C₂×D₄	C₅×Q₁₆	1	2	yes	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₅×#41	1	2	yes	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
236	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	#1799	1	1	yes	characteristic
252	C₂×C₅	#1808	1	1	yes	characteristic
253	C₂×C₅	#1812	1	1	yes	characteristic
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₅×#107	1	2	yes	complemented
261	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂³	C₅×#121	1	2	yes	—
263	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
264	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
265	C₂³	C₅×#135	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	C₅×#107	1	2	yes	complemented
287	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₅×#117	1	1	yes	characteristic
289	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
290	C₈	—	2	1	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₈	C₅×#99	1	2	yes	complemented
295	C₅	#8609	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₅×#380	1	1	yes	characteristic
309	C₂²	C₂×C₅×#107	1	1	yes	characteristic
310	C₂²	C₅×#427	1	1	yes	characteristic
311	C₂²	C₅×#466	1	2	yes	—
312	C₂²	C₅×#467	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	16	4	no	—
319	C₄	C₂×C₅×#99	1	1	yes	characteristic
320	C₄	C₅×#410	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₅×#1799	1	1	yes	characteristic
327	C₂	C₅×#1808	1	1	yes	characteristic
328	C₂	C₅×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1808	C₂	semidirect
4. C₅×#1289	C₂	semidirect
5. C₂³×C₅×C₈	C₄	semidirect
11. C₂×C₅×C₄⋊C₈	C₄	semidirect
13. C₂×C₅×C₂²⋊C₈	C₄	semidirect
15. C₂×C₄×C₅×C₈	C₄	semidirect
19. #8609	C₅	direct (N × H)
50. #1808	C₂×C₅	semidirect
52. #1289	C₂×C₅	semidirect
95. C₂³×C₈	C₄×C₅	semidirect
101. C₂×C₄⋊C₈	C₄×C₅	semidirect
103. C₂×C₂²⋊C₈	C₄×C₅	semidirect
105. C₂×C₄×C₈	C₄×C₅	semidirect
115. C₂³×C₅	#107	semidirect
141. C₂×C₄×C₅	#107	semidirect
149. C₅×C₈	#99	semidirect
260. C₂³	C₅×#107	semidirect
286. C₂×C₄	C₅×#107	semidirect
294. C₈	C₅×#99	semidirect
295. C₅	#8609	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×C₈×D₄	2
C₅×#1808	2
C₅×#1799	2
C₅×#1289	2
#8609	5

Subgroups — C5×#8609 #11931329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8609 #11931329