Subgroups — C₅×#8652 #11931372

Order 1280.

510 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	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1802	C₂	1	2	yes	maximal
4	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	—	2	1	no	—
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	2	no	—
9	C₅×#436	C₄	1	4	yes	complemented
10	C₂×C₅×#107	—	2	1	no	—
11	C₂×C₅×#107	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#102	—	2	2	no	—
15	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#384	C₂²	1	2	yes	—
18	C₅×#340	—	2	2	no	—
19	C₅×#334	—	2	2	no	—
20	#8652	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	—	2	1	no	—
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	4	2	no	—
25	C₂×C₅×M₁₆	—	4	2	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₄×C₅×D₄	Q₈	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×#107	—	4	2	no	—
41	C₅×#107	—	4	2	no	—
42	C₅×C₄⋊C₈	—	2	1	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₅×#102	—	4	4	no	—
47	C₅×C₂²⋊C₈	—	2	4	no	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₂×#436	C₂×C₅	1	1	yes	characteristic
51	#1845	C₂×C₅	1	1	yes	characteristic, complemented
52	#1802	C₂×C₅	1	2	yes	—
53	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
54	#1296	C₂×C₅	1	2	yes	complemented
55	C₂⁴×C₅	—	2	1	no	—
56	C₂×C₅×D₄	—	2	1	no	—
57	C₂×C₅×D₄	—	2	2	no	—
58	C₂×C₅×D₄	—	2	2	no	—
59	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
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	—	4	1	no	—
66	C₂²×C₄×C₅	—	4	1	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₅×M₁₆	—	4	8	no	—
72	C₂×C₅×C₈	—	2	1	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	—	4	1	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	D₈	1	2	yes	—
83	C₂×C₅×C₈	SD₁₆	1	2	yes	—
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	8	2	no	—
90	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	—	2	2	no	—
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₄²×C₅	#41	1	1	yes	characteristic
97	C₂²×M₁₆	—	2	1	no	—
98	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
99	#436	—	2	2	no	—
100	#436	C₄×C₅	1	4	yes	complemented
101	C₂×#107	—	2	1	no	—
102	C₂×#107	—	2	1	no	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×#102	—	2	2	no	—
106	C₂×C₂²⋊C₈	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
108	#384	C₂²×C₅	1	2	yes	—
109	#340	—	2	2	no	—
110	#334	—	2	2	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₂³×C₅	—	4	1	no	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	—	4	1	no	—
117	C₂³×C₅	—	4	1	no	—
118	C₂³×C₅	#121	1	1	yes	characteristic
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	4	2	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	2	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	—	8	2	no	—
131	C₂×C₄×C₅	—	8	2	no	—
132	C₂×C₄×C₅	—	8	2	no	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	#117	1	1	yes	characteristic
136	C₂×C₄×C₅	#121	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₅	#137	1	1	yes	characteristic
144	C₂×C₄×C₅	#137	1	2	yes	—
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	8	2	no	—
148	C₅×C₈	#102	1	4	yes	complemented
149	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
150	C₂³×C₄	—	2	1	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	4	2	no	—
153	C₂×M₁₆	—	4	2	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
161	C₄×D₄	C₅×Q₈	1	2	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
168	#107	—	4	2	no	—
169	#107	—	4	2	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
174	#102	—	4	4	no	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
177	C₄×C₈	C₅×D₄	1	2	yes	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	8	1	no	—
182	C₂²×C₅	—	8	1	no	—
183	C₂²×C₅	—	8	1	no	—
184	C₂²×C₅	—	8	1	no	—
185	C₂²×C₅	#380	1	1	yes	characteristic
186	C₂²×C₅	#459	1	1	yes	characteristic
187	C₂²×C₅	#461	1	1	yes	characteristic
188	C₂²×C₅	#469	1	1	yes	characteristic
189	C₂²×C₅	#470	1	1	yes	characteristic
190	C₂²×C₅	#472	1	1	yes	characteristic
191	C₂²×C₅	#473	1	1	yes	characteristic
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	—	16	2	no	—
196	C₄×C₅	C₂×#102	1	1	yes	characteristic
197	C₄×C₅	#417	1	1	yes	characteristic
198	C₄×C₅	#428	1	2	yes	—
199	C₄×C₅	#468	1	2	yes	—
200	C₄×C₅	#474	1	1	yes	characteristic
201	C₄×C₅	#476	1	1	yes	characteristic
202	C₂⁴	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
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	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₅×#41	1	1	yes	characteristic
218	M₁₆	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂×C₈	C₅×D₈	1	2	yes	—
230	C₂×C₈	C₅×SD₁₆	1	2	yes	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	8	2	no	—
237	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
242	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
243	C₄²	C₅×#41	1	1	yes	characteristic
244	C₂×C₅	—	8	2	no	—
245	C₂×C₅	#1687	1	1	yes	characteristic
246	C₂×C₅	#1689	1	1	yes	characteristic
247	C₂×C₅	#1693	1	1	yes	characteristic
248	C₂×C₅	#1695	1	1	yes	characteristic
249	C₂×C₅	#1802	1	1	yes	characteristic
250	C₂×C₅	#1809	1	1	yes	characteristic
251	C₂×C₅	#1816	1	1	yes	characteristic
252	C₂³	—	2	2	no	—
253	C₂³	—	4	1	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	C₅×#121	1	1	yes	characteristic
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	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	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₅×#117	1	1	yes	characteristic
277	C₂×C₄	C₅×#121	1	1	yes	characteristic
278	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
282	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄	C₅×#137	1	1	yes	characteristic
285	C₂×C₄	C₅×#137	1	2	yes	—
286	C₈	—	2	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	8	2	no	—
289	C₈	C₅×#102	1	4	yes	complemented
290	C₅	#8652	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	8	1	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	C₅×#380	1	1	yes	characteristic
299	C₂²	C₅×#459	1	1	yes	characteristic
300	C₂²	C₅×#461	1	1	yes	characteristic
301	C₂²	C₅×#469	1	1	yes	characteristic
302	C₂²	C₅×#470	1	1	yes	characteristic
303	C₂²	C₅×#472	1	1	yes	characteristic
304	C₂²	C₅×#473	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	16	2	no	—
309	C₄	C₂×C₅×#102	1	1	yes	characteristic
310	C₄	C₅×#417	1	1	yes	characteristic
311	C₄	C₅×#428	1	2	yes	—
312	C₄	C₅×#468	1	2	yes	—
313	C₄	C₅×#474	1	1	yes	characteristic
314	C₄	C₅×#476	1	1	yes	characteristic
315	C₂	—	8	2	no	—
316	C₂	C₅×#1687	1	1	yes	characteristic
317	C₂	C₅×#1689	1	1	yes	characteristic
318	C₂	C₅×#1693	1	1	yes	characteristic
319	C₂	C₅×#1695	1	1	yes	characteristic
320	C₂	C₅×#1802	1	1	yes	characteristic
321	C₂	C₅×#1809	1	1	yes	characteristic
322	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1845	C₂	semidirect
4. C₂×C₅×#334	C₂	semidirect
5. C₅×#1296	C₂	semidirect
9. C₅×#436	C₄	semidirect
20. #8652	C₅	direct (N × H)
51. #1845	C₂×C₅	semidirect
53. C₂×#334	C₂×C₅	semidirect
54. #1296	C₂×C₅	semidirect
100. #436	C₄×C₅	semidirect
148. C₅×C₈	#102	semidirect
289. C₈	C₅×#102	semidirect
290. C₅	#8652	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×#436	2
C₅×#1845	2
C₅×#1802	2
C₂×C₅×#334	2
C₅×#1296	2
#8652	5

Subgroups — C5×#8652 #11931372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8652 #11931372