Subgroups — C₅×#8670 #11931390

Order 1280.

408 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1489	C₂	1	2	yes	maximal
5	C₅×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₅×C₈×D₄	—	2	1	no	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	1	yes	characteristic
12	C₅×C₈×D₄	C₄	1	1	yes	characteristic
13	C₅×#431	—	2	1	no	—
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₅×#108	—	2	1	no	—
16	C₅×#423	C₂²	1	2	yes	—
17	C₄×C₅×M₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#341	—	2	1	no	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×Q₈⋊C₈	—	2	2	no	—
24	C₅×D₄⋊C₈	—	2	2	no	—
25	#8670	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	Q₈	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×D₄	D₄	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₅×#117	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#108	—	2	4	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₅×#97	—	2	2	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₈×#41	C₂×C₅	1	1	yes	characteristic
63	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
64	#1566	C₂×C₅	1	1	yes	characteristic, complemented
65	#1489	C₂×C₅	1	2	yes	—
66	#1299	C₂×C₅	1	2	yes	complemented
67	C₅×#41	—	2	2	no	—
68	C₅×#41	—	2	2	no	—
69	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₅×D₄	—	2	1	no	—
71	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	8	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
96	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₅×C₂²⋊C₄	—	2	1	no	—
99	C₅×C₂²⋊C₄	—	2	1	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅	#41	1	1	yes	characteristic
107	C₄×#41	C₂²×C₅	1	1	yes	characteristic
108	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
109	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
110	C₈×D₄	—	2	1	no	—
111	C₈×D₄	—	2	1	no	—
112	C₈×D₄	C₄×C₅	1	1	yes	characteristic
113	C₈×D₄	C₄×C₅	1	1	yes	characteristic
114	#431	—	2	1	no	—
115	#431	C₂²×C₅	1	1	yes	characteristic
116	C₂×#108	—	2	1	no	—
117	#423	C₂²×C₅	1	2	yes	—
118	C₄×M₁₆	C₂²×C₅	1	2	yes	—
119	C₂×C₄×C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#341	—	2	1	no	—
122	#334	—	2	1	no	—
123	#334	—	2	1	no	—
124	Q₈⋊C₈	—	2	2	no	—
125	D₄⋊C₈	—	2	2	no	—
126	C₂³×C₅	—	2	1	no	—
127	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
128	C₅×Q₈	—	2	1	no	—
129	C₅×Q₈	#108	1	2	yes	complemented
130	C₅×D₄	—	2	4	no	—
131	C₅×D₄	—	2	1	no	—
132	C₅×D₄	#108	1	2	yes	complemented
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	#117	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	#121	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	2	1	no	—
154	C₅×C₈	—	4	4	no	—
155	C₅×C₈	—	4	1	no	—
156	C₅×C₈	—	4	1	no	—
157	C₅×C₈	—	8	2	no	—
158	C₅×C₈	#104	1	2	yes	—
159	C₂×#41	C₅×Q₈	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
166	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×D₄	C₅×D₄	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	#117	C₅×Q₈	1	1	yes	characteristic
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
176	#108	—	2	4	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
182	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²⋊C₈	—	2	1	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	#97	—	2	2	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
192	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
193	C₄×C₈	C₅×D₄	1	1	yes	characteristic
194	C₄×C₈	C₅×D₄	1	1	yes	characteristic
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	4	1	no	—
200	C₂²×C₅	—	4	1	no	—
201	C₂²×C₅	#380	1	1	yes	characteristic
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	4	1	no	—
207	C₄×C₅	—	4	1	no	—
208	C₄×C₅	C₂×#104	1	1	yes	characteristic
209	C₄×C₅	#421	1	1	yes	characteristic
210	C₄×C₅	C₂×#108	1	1	yes	characteristic
211	C₄×C₅	C₂×#108	1	1	yes	characteristic
212	C₄×C₅	#471	1	2	yes	—
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₅×#41	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	8	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
251	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
252	C₄²	C₅×#41	1	1	yes	characteristic
253	C₂×C₅	—	4	1	no	—
254	C₂×C₅	—	4	2	no	—
255	C₂×C₅	#1806	1	1	yes	characteristic
256	C₂×C₅	#1810	1	1	yes	characteristic
257	C₂×C₅	#1815	1	1	yes	characteristic
258	C₂³	—	2	1	no	—
259	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
260	Q₈	—	2	1	no	—
261	Q₈	C₅×#108	1	2	yes	complemented
262	D₄	—	2	4	no	—
263	D₄	—	2	1	no	—
264	D₄	C₅×#108	1	2	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₅×#117	1	1	yes	characteristic
280	C₂×C₄	C₅×#121	1	1	yes	characteristic
281	C₂×C₄	C₅×#121	1	1	yes	characteristic
282	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	4	4	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	1	no	—
289	C₈	—	8	2	no	—
290	C₈	C₅×#104	1	2	yes	—
291	C₅	#8670	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₅×#380	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂×C₅×#104	1	1	yes	characteristic
306	C₄	C₅×#421	1	1	yes	characteristic
307	C₄	C₂×C₅×#108	1	1	yes	characteristic
308	C₄	C₂×C₅×#108	1	1	yes	characteristic
309	C₄	C₅×#471	1	2	yes	—
310	C₂	—	4	1	no	—
311	C₂	—	4	2	no	—
312	C₂	C₅×#1806	1	1	yes	characteristic
313	C₂	C₅×#1810	1	1	yes	characteristic
314	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂×C₅×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#108	C₂	semidirect
3. C₅×#1566	C₂	semidirect
5. C₅×#1299	C₂	semidirect
25. #8670	C₅	direct (N × H)
63. C₄×#108	C₂×C₅	semidirect
64. #1566	C₂×C₅	semidirect
66. #1299	C₂×C₅	semidirect
129. C₅×Q₈	#108	semidirect
132. C₅×D₄	#108	semidirect
261. Q₈	C₅×#108	semidirect
264. D₄	C₅×#108	semidirect
291. C₅	#8670	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×C₈×#41	2
C₄×C₅×#108	2
C₅×#1566	2
C₅×#1489	2
C₅×#1299	2
#8670	5

Subgroups — C5×#8670 #11931390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8670 #11931390