Subgroups — C₅×#8601 #11931321

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₅×#1564	C₂	1	2	yes	maximal, complemented
3	C₅×#1458	C₂	1	2	yes	maximal
4	C₅×#1275	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₅×#117	C₂²	1	1	yes	characteristic
8	C₅×#431	C₂²	1	4	yes	—
9	C₅×#405	—	2	2	no	—
10	C₅×#405	—	2	2	no	—
11	C₂×C₅×#97	C₄	1	1	yes	characteristic
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#335	—	2	2	no	—
16	C₅×#334	—	2	2	no	—
17	C₅×#324	—	2	2	no	—
18	C₅×#323	—	2	2	no	—
19	#8601	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂²×C₅×C₈	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	C₈	1	4	yes	complemented
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₅×#117	—	2	2	no	—
33	C₅×#117	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
36	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×C₂²⋊C₈	—	2	4	no	—
42	C₅×#97	—	2	2	no	—
43	C₅×#97	—	2	2	no	—
44	C₄×C₅×C₈	—	2	1	no	—
45	C₄×C₅×C₈	—	2	1	no	—
46	C₄×C₅×C₈	—	2	4	no	—
47	C₄×C₅×C₈	D₄	1	2	yes	—
48	C₄×C₅×C₈	D₄	1	2	yes	—
49	C₂×#405	C₂×C₅	1	1	yes	characteristic
50	#1564	C₂×C₅	1	2	yes	complemented
51	#1458	C₂×C₅	1	2	yes	—
52	#1275	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅	M₁₆	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₅×M₁₆	—	2	8	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	2	1	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	—	2	1	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	4	4	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
80	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
81	C₂×C₅×C₈	M₁₆	1	2	yes	—
82	C₂×C₅×C₈	M₁₆	1	2	yes	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₂²⋊C₄	—	2	2	no	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₄²×C₅	—	2	1	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	C₂×D₄	1	2	yes	—
90	C₄²×C₅	#41	1	2	yes	—
91	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
92	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
93	C₂×#117	C₂²×C₅	1	1	yes	characteristic
94	#431	C₂²×C₅	1	4	yes	—
95	#405	—	2	2	no	—
96	#405	—	2	2	no	—
97	C₂×#97	C₄×C₅	1	1	yes	characteristic
98	C₂×#97	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	#335	—	2	2	no	—
102	#334	—	2	2	no	—
103	#324	—	2	2	no	—
104	#323	—	2	2	no	—
105	C₂³×C₅	—	2	1	no	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	4	1	no	—
108	C₂³×C₅	—	4	1	no	—
109	C₂³×C₅	C₄⋊C₈	1	2	yes	—
110	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	4	1	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	—	4	1	no	—
123	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#117	1	1	yes	characteristic
126	C₂×C₄×C₅	#121	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
132	C₅×C₈	—	2	2	no	—
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	—	2	2	no	—
135	C₅×C₈	—	8	4	no	—
136	C₅×C₈	C₂²⋊C₈	1	2	yes	—
137	C₅×C₈	C₂²⋊C₈	1	2	yes	—
138	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₅×C₈	1	4	yes	complemented
146	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
150	#117	—	2	2	no	—
151	#117	—	2	2	no	—
152	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₅×Q₈	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
158	C₄⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	2	4	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	C₅×D₄	1	2	yes	—
166	C₄×C₈	C₅×D₄	1	2	yes	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	#380	1	1	yes	characteristic
175	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
176	C₂²×C₅	#423	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	4	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₄×C₅	#407	1	1	yes	characteristic
184	C₄×C₅	#431	1	1	yes	characteristic
185	C₄×C₅	#432	1	1	yes	characteristic
186	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
195	C₂²×C₄	C₅×M₁₆	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
199	M₁₆	—	2	8	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
210	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
213	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
214	C₂×C₈	C₅×M₁₆	1	2	yes	—
215	C₂×C₈	C₅×M₁₆	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	C₂×C₅×D₄	1	2	yes	—
223	C₄²	C₅×#41	1	2	yes	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	#1797	1	1	yes	characteristic
228	C₂×C₅	#1812	1	1	yes	characteristic
229	C₂×C₅	#1813	1	1	yes	characteristic
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	4	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	C₅×C₄⋊C₈	1	2	yes	—
235	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₅×#117	1	1	yes	characteristic
251	C₂×C₄	C₅×#121	1	2	yes	—
252	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	8	4	no	—
261	C₈	C₅×C₂²⋊C₈	1	2	yes	—
262	C₈	C₅×C₂²⋊C₈	1	2	yes	—
263	C₅	#8601	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₅×#380	1	1	yes	characteristic
272	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
273	C₂²	C₅×#423	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
280	C₄	C₅×#407	1	1	yes	characteristic
281	C₄	C₅×#431	1	1	yes	characteristic
282	C₄	C₅×#432	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	C₅×#1797	1	1	yes	characteristic
287	C₂	C₅×#1812	1	1	yes	characteristic
288	C₂	C₅×#1813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1564	C₂	semidirect
4. C₅×#1275	C₂	semidirect
19. #8601	C₅	direct (N × H)
27. C₂²×C₅×C₈	C₈	semidirect
50. #1564	C₂×C₅	semidirect
52. #1275	C₂×C₅	semidirect
145. C₂²×C₈	C₅×C₈	semidirect
263. C₅	#8601	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×#405	2
C₅×#1564	2
C₅×#1458	2
C₅×#1275	2
#8601	5

Subgroups — C5×#8601 #11931321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8601 #11931321