Subgroups — C₅×#8829 #11931549

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#335	C₂	1	2	yes	maximal, complemented
5	C₅×#1277	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	—	2	1	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	—	2	4	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₄	1	2	yes	—
19	C₅×#335	—	2	4	no	—
20	#8829	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	—	4	2	no	—
29	C₂²×C₅×C₈	—	4	2	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	—	2	2	no	—
33	C₂×C₄²×C₅	—	2	2	no	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	D₄	1	4	yes	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	4	2	no	—
45	C₅×C₂²⋊C₈	C₈	1	4	yes	complemented
46	C₄×C₅×C₈	—	2	2	no	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
49	#1797	C₂×C₅	1	1	yes	characteristic
50	C₄×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
51	C₂×#335	C₂×C₅	1	2	yes	complemented
52	#1277	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅	M₁₆	1	2	yes	—
68	C₂²×C₄×C₅	D₈	1	2	yes	—
69	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	4	8	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	C₂×C₈	1	4	yes	complemented
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	C₂×D₄	1	2	yes	—
94	C₄²×C₅	#41	1	2	yes	—
95	C₂³×C₈	—	2	1	no	—
96	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄⋊C₈	—	2	4	no	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
102	C₂×C₂²⋊C₈	—	2	1	no	—
103	C₂×C₂²⋊C₈	C₄×C₅	1	1	yes	characteristic
104	C₂×C₂²⋊C₈	C₄×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
106	#336	—	2	1	no	—
107	#336	C₄×C₅	1	2	yes	—
108	#335	—	2	4	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	—	2	1	no	—
113	C₂³×C₅	C₄⋊C₈	1	2	yes	—
114	C₂³×C₅	#107	1	2	yes	—
115	C₂³×C₅	#108	1	2	yes	—
116	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
134	C₂×C₄×C₅	#107	1	2	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	#120	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅	#123	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	#131	1	2	yes	—
142	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₅	#137	1	1	yes	characteristic
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	—	4	2	no	—
148	C₅×C₈	—	4	4	no	—
149	C₅×C₈	—	8	1	no	—
150	C₅×C₈	—	8	1	no	—
151	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	—	4	2	no	—
160	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
166	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	C₅×D₄	1	4	yes	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	4	2	no	—
175	C₂²⋊C₈	C₅×C₈	1	4	yes	complemented
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	#335	1	4	yes	complemented
185	C₂²×C₅	#385	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
187	C₂²×C₅	#424	1	1	yes	characteristic
188	C₂²×C₅	C₂×#107	1	1	yes	characteristic
189	C₂²×C₅	#428	1	1	yes	characteristic
190	C₂²×C₅	C₂×#108	1	1	yes	characteristic
191	C₂²×C₅	#430	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	C₈×D₄	1	2	yes	—
197	C₄×C₅	#435	1	2	yes	—
198	C₄×C₅	#447	1	1	yes	characteristic
199	C₄×C₅	#451	1	1	yes	characteristic
200	C₄×C₅	#480	1	1	yes	characteristic
201	C₄×C₅	#484	1	1	yes	characteristic
202	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
216	C₂²×C₄	C₅×M₁₆	1	2	yes	—
217	C₂²×C₄	C₅×D₈	1	2	yes	—
218	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₂×C₅×C₈	1	4	yes	complemented
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	C₂×C₅×D₄	1	2	yes	—
243	C₄²	C₅×#41	1	2	yes	—
244	C₂×C₅	—	2	4	no	—
245	C₂×C₅	C₂×#335	1	1	yes	characteristic
246	C₂×C₅	#1565	1	1	yes	characteristic
247	C₂×C₅	#1571	1	1	yes	characteristic
248	C₂×C₅	#1573	1	1	yes	characteristic
249	C₂×C₅	#1850	1	1	yes	characteristic
250	C₂×C₅	#1852	1	1	yes	characteristic
251	C₂×C₅	#1855	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₅×C₄⋊C₈	1	2	yes	—
257	C₂³	C₅×#107	1	2	yes	—
258	C₂³	C₅×#108	1	2	yes	—
259	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
277	C₂×C₄	C₅×#107	1	2	yes	—
278	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
279	C₂×C₄	C₅×#120	1	1	yes	characteristic
280	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₅×#123	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
284	C₂×C₄	C₅×#131	1	2	yes	—
285	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
287	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
288	C₂×C₄	C₅×#137	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	4	no	—
292	C₈	—	8	1	no	—
293	C₈	—	8	1	no	—
294	C₅	#8829	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	C₅×#335	1	4	yes	complemented
302	C₂²	C₅×#385	1	1	yes	characteristic
303	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
304	C₂²	C₅×#424	1	1	yes	characteristic
305	C₂²	C₂×C₅×#107	1	1	yes	characteristic
306	C₂²	C₅×#428	1	1	yes	characteristic
307	C₂²	C₂×C₅×#108	1	1	yes	characteristic
308	C₂²	C₅×#430	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	C₅×C₈×D₄	1	2	yes	—
314	C₄	C₅×#435	1	2	yes	—
315	C₄	C₅×#447	1	1	yes	characteristic
316	C₄	C₅×#451	1	1	yes	characteristic
317	C₄	C₅×#480	1	1	yes	characteristic
318	C₄	C₅×#484	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×C₅×#335	1	1	yes	characteristic
321	C₂	C₅×#1565	1	1	yes	characteristic
322	C₂	C₅×#1571	1	1	yes	characteristic
323	C₂	C₅×#1573	1	1	yes	characteristic
324	C₂	C₅×#1850	1	1	yes	characteristic
325	C₂	C₅×#1852	1	1	yes	characteristic
326	C₂	C₅×#1855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#335	C₂	semidirect
5. C₅×#1277	C₂	semidirect
20. #8829	C₅	direct (N × H)
45. C₅×C₂²⋊C₈	C₈	semidirect
51. C₂×#335	C₂×C₅	semidirect
52. #1277	C₂×C₅	semidirect
88. C₂×C₅×C₈	C₂×C₈	semidirect
175. C₂²⋊C₈	C₅×C₈	semidirect
184. C₂²×C₅	#335	semidirect
237. C₂×C₈	C₂×C₅×C₈	semidirect
294. C₅	#8829	direct (N × H)
301. C₂²	C₅×#335	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₅×#1797	2
C₄×C₅×C₂²⋊C₈	2
C₂×C₅×#335	2
C₅×#1277	2
#8829	5

Subgroups — C5×#8829 #11931549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8829 #11931549