Subgroups — C₅×#8727 #11931447

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 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	#8727	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	4	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	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	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₅	—	2	4	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₂²⋊C₈	—	4	2	no	—
44	C₅×C₂²⋊C₈	—	4	2	no	—
45	C₅×#95	—	2	2	no	—
46	C₅×#95	—	2	2	no	—
47	#2285	C₂×C₅	1	1	yes	characteristic
48	#1797	C₂×C₅	1	1	yes	characteristic
49	#1797	C₂×C₅	1	1	yes	characteristic
50	#1296	C₂×C₅	1	2	yes	complemented
51	#1296	C₂×C₅	1	2	yes	complemented
52	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	2	no	—
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	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	—	4	1	no	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×C₈	—	4	2	no	—
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	C₅×C₄⋊C₄	—	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	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	4	4	no	—
90	C₄²×C₅	#41	1	4	yes	—
91	C₂×#117	—	2	1	no	—
92	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
95	C₂×C₂²⋊C₈	—	2	1	no	—
96	C₂×C₂²⋊C₈	—	2	1	no	—
97	#384	C₂²×C₅	1	1	yes	characteristic
98	#384	C₂²×C₅	1	1	yes	characteristic
99	#382	—	2	1	no	—
100	#380	—	2	1	no	—
101	#380	C₄×C₅	1	2	yes	—
102	C₄×C₄⋊C₄	—	2	1	no	—
103	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
104	#336	—	2	1	no	—
105	#336	—	2	1	no	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	2	1	no	—
108	C₂³×C₅	—	2	1	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	#103	1	2	yes	—
113	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	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	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	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	#103	1	2	yes	—
136	C₂×C₄×C₅	#117	1	2	yes	—
137	C₂×C₄×C₅	#123	1	2	yes	—
138	C₂×C₄×C₅	#123	1	2	yes	—
139	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	#135	1	2	yes	—
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₅	#137	1	1	yes	characteristic
145	C₅×C₈	—	8	2	no	—
146	C₅×C₈	—	8	2	no	—
147	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	#117	—	4	4	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₂²⋊C₈	—	4	2	no	—
169	C₂²⋊C₈	—	4	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	#381	1	1	yes	characteristic
180	C₂²×C₅	#413	1	1	yes	characteristic
181	C₂²×C₅	#413	1	1	yes	characteristic
182	C₂²×C₅	C₂×#103	1	1	yes	characteristic
183	C₂²×C₅	#416	1	1	yes	characteristic
184	C₂²×C₅	#417	1	1	yes	characteristic
185	C₂²×C₅	#418	1	1	yes	characteristic
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	—	4	1	no	—
190	C₄×C₅	—	8	2	no	—
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	#481	1	2	yes	—
193	C₄×C₅	#482	1	2	yes	—
194	C₄×C₅	#484	1	2	yes	—
195	C₄×C₅	#485	1	2	yes	—
196	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
212	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₅×#41	1	4	yes	—
235	C₂×C₅	—	4	2	no	—
236	C₂×C₅	#1542	1	1	yes	characteristic
237	C₂×C₅	#1550	1	1	yes	characteristic
238	C₂×C₅	#1553	1	1	yes	characteristic
239	C₂×C₅	#1557	1	1	yes	characteristic
240	C₂×C₅	#1822	1	1	yes	characteristic
241	C₂×C₅	#1824	1	1	yes	characteristic
242	C₂×C₅	#1825	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₅×#103	1	2	yes	—
250	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
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	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₅×#103	1	2	yes	—
273	C₂×C₄	C₅×#117	1	2	yes	—
274	C₂×C₄	C₅×#123	1	2	yes	—
275	C₂×C₄	C₅×#123	1	2	yes	—
276	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
277	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
278	C₂×C₄	C₅×#135	1	2	yes	—
279	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
280	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂×C₄	C₅×#137	1	1	yes	characteristic
282	C₈	—	8	2	no	—
283	C₈	—	8	2	no	—
284	C₅	#8727	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₅×#381	1	1	yes	characteristic
293	C₂²	C₅×#413	1	1	yes	characteristic
294	C₂²	C₅×#413	1	1	yes	characteristic
295	C₂²	C₂×C₅×#103	1	1	yes	characteristic
296	C₂²	C₅×#416	1	1	yes	characteristic
297	C₂²	C₅×#417	1	1	yes	characteristic
298	C₂²	C₅×#418	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₅×#481	1	2	yes	—
306	C₄	C₅×#482	1	2	yes	—
307	C₄	C₅×#484	1	2	yes	—
308	C₄	C₅×#485	1	2	yes	—
309	C₂	—	4	2	no	—
310	C₂	C₅×#1542	1	1	yes	characteristic
311	C₂	C₅×#1550	1	1	yes	characteristic
312	C₂	C₅×#1553	1	1	yes	characteristic
313	C₂	C₅×#1557	1	1	yes	characteristic
314	C₂	C₅×#1822	1	1	yes	characteristic
315	C₂	C₅×#1824	1	1	yes	characteristic
316	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. #8727	C₅	direct (N × H)
50. #1296	C₂×C₅	semidirect
51. #1296	C₂×C₅	semidirect
284. C₅	#8727	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
#8727	5

Subgroups — C5×#8727 #11931447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8727 #11931447