Subgroups — C₅×#8726 #11931446

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2285	C₂	1	1	yes	maximal, characteristic
2	C₅×#1797	C₂	1	1	yes	maximal, characteristic
3	C₅×#1797	C₂	1	1	yes	maximal, characteristic
4	C₅×#1296	C₂	1	2	yes	maximal, complemented
5	C₅×#1296	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#117	—	2	1	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
11	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₅×#384	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	1	yes	characteristic
14	C₅×#382	—	2	1	no	—
15	C₅×#380	—	2	1	no	—
16	C₅×#380	C₄	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	—	2	1	no	—
18	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
19	C₅×#336	—	2	1	no	—
20	C₅×#336	—	2	1	no	—
21	#8726	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₅×#117	—	4	2	no	—
28	C₅×#117	—	4	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₅	—	2	4	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₂²⋊C₈	—	4	2	no	—
45	C₅×C₂²⋊C₈	—	4	2	no	—
46	C₅×#95	—	2	2	no	—
47	C₅×#95	—	2	2	no	—
48	#2285	C₂×C₅	1	1	yes	characteristic
49	#1797	C₂×C₅	1	1	yes	characteristic
50	#1797	C₂×C₅	1	1	yes	characteristic
51	#1296	C₂×C₅	1	2	yes	complemented
52	#1296	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	D₈	1	2	yes	—
69	C₂²×C₄×C₅	SD₁₆	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×D₄	1	1	yes	characteristic
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₄⋊C₄	—	4	1	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₄⋊C₄	—	4	1	no	—
88	C₅×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	4	4	no	—
93	C₄²×C₅	#41	1	4	yes	—
94	C₂×#117	—	2	1	no	—
95	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
97	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
98	C₂×C₂²⋊C₈	—	2	1	no	—
99	C₂×C₂²⋊C₈	—	2	1	no	—
100	#384	C₂²×C₅	1	1	yes	characteristic
101	#384	C₂²×C₅	1	1	yes	characteristic
102	#382	—	2	1	no	—
103	#380	—	2	1	no	—
104	#380	C₄×C₅	1	2	yes	—
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
107	#336	—	2	1	no	—
108	#336	—	2	1	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	—	2	1	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	#102	1	2	yes	—
116	C₂³×C₅	C₂×C₂²⋊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	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	#102	1	2	yes	—
139	C₂×C₄×C₅	#117	1	2	yes	—
140	C₂×C₄×C₅	#123	1	2	yes	—
141	C₂×C₄×C₅	#123	1	2	yes	—
142	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅	#135	1	2	yes	—
145	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₅	#137	1	2	yes	—
147	C₂×C₄×C₅	#137	1	1	yes	characteristic
148	C₅×C₈	—	8	2	no	—
149	C₅×C₈	—	8	2	no	—
150	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	#117	—	4	2	no	—
156	#117	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
168	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	4	2	no	—
173	C₂²⋊C₈	—	4	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	1	no	—
181	C₂²×C₅	—	4	1	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	#381	1	1	yes	characteristic
184	C₂²×C₅	#413	1	1	yes	characteristic
185	C₂²×C₅	#413	1	1	yes	characteristic
186	C₂²×C₅	C₂×#102	1	1	yes	characteristic
187	C₂²×C₅	#416	1	1	yes	characteristic
188	C₂²×C₅	#417	1	1	yes	characteristic
189	C₂²×C₅	#419	1	1	yes	characteristic
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	4	1	no	—
193	C₄×C₅	—	4	1	no	—
194	C₄×C₅	—	8	2	no	—
195	C₄×C₅	—	8	2	no	—
196	C₄×C₅	#480	1	2	yes	—
197	C₄×C₅	#482	1	2	yes	—
198	C₄×C₅	#483	1	2	yes	—
199	C₄×C₅	#485	1	2	yes	—
200	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₅×D₈	1	2	yes	—
216	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	4	4	no	—
240	C₄²	C₅×#41	1	4	yes	—
241	C₂×C₅	—	4	2	no	—
242	C₂×C₅	#1540	1	1	yes	characteristic
243	C₂×C₅	#1547	1	1	yes	characteristic
244	C₂×C₅	#1554	1	1	yes	characteristic
245	C₂×C₅	#1559	1	1	yes	characteristic
246	C₂×C₅	#1822	1	1	yes	characteristic
247	C₂×C₅	#1823	1	1	yes	characteristic
248	C₂×C₅	#1825	1	1	yes	characteristic
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	C₅×#102	1	2	yes	—
256	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₅×#102	1	2	yes	—
279	C₂×C₄	C₅×#117	1	2	yes	—
280	C₂×C₄	C₅×#123	1	2	yes	—
281	C₂×C₄	C₅×#123	1	2	yes	—
282	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
284	C₂×C₄	C₅×#135	1	2	yes	—
285	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₂×C₄	C₅×#137	1	2	yes	—
287	C₂×C₄	C₅×#137	1	1	yes	characteristic
288	C₈	—	8	2	no	—
289	C₈	—	8	2	no	—
290	C₅	#8726	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₅×#381	1	1	yes	characteristic
299	C₂²	C₅×#413	1	1	yes	characteristic
300	C₂²	C₅×#413	1	1	yes	characteristic
301	C₂²	C₂×C₅×#102	1	1	yes	characteristic
302	C₂²	C₅×#416	1	1	yes	characteristic
303	C₂²	C₅×#417	1	1	yes	characteristic
304	C₂²	C₅×#419	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₅×#480	1	2	yes	—
312	C₄	C₅×#482	1	2	yes	—
313	C₄	C₅×#483	1	2	yes	—
314	C₄	C₅×#485	1	2	yes	—
315	C₂	—	4	2	no	—
316	C₂	C₅×#1540	1	1	yes	characteristic
317	C₂	C₅×#1547	1	1	yes	characteristic
318	C₂	C₅×#1554	1	1	yes	characteristic
319	C₂	C₅×#1559	1	1	yes	characteristic
320	C₂	C₅×#1822	1	1	yes	characteristic
321	C₂	C₅×#1823	1	1	yes	characteristic
322	C₂	C₅×#1825	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1296	C₂	semidirect
5. C₅×#1296	C₂	semidirect
21. #8726	C₅	direct (N × H)
51. #1296	C₂×C₅	semidirect
52. #1296	C₂×C₅	semidirect
290. C₅	#8726	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×#2285	2
C₅×#1797	2
C₅×#1296	2
#8726	5

Subgroups — C5×#8726 #11931446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8726 #11931446