Subgroups — C₅×#8662 #11931382

Order 1280.

450 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1486	C₂	1	2	yes	maximal
5	C₅×#1301	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	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×#436	C₄	1	2	yes	—
11	C₅×#436	C₄	1	2	yes	—
12	C₅×#431	C₂²	1	2	yes	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#108	—	2	1	no	—
15	C₅×#423	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×D₄⋊C₈	—	2	4	no	—
22	#8662	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	4	2	no	—
29	C₂×C₅×M₁₆	—	4	2	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×D₄	—	2	1	no	—
34	C₄×C₅×D₄	—	2	1	no	—
35	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
36	C₄×C₅×D₄	D₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×#108	—	4	2	no	—
41	C₅×#108	—	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	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	2	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₅×C₂²⋊C₈	—	2	2	no	—
50	C₄×C₅×C₈	—	2	2	no	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₂×#436	C₂×C₅	1	1	yes	characteristic
56	#1848	C₂×C₅	1	1	yes	characteristic, complemented
57	#1564	C₂×C₅	1	1	yes	characteristic, complemented
58	#1486	C₂×C₅	1	2	yes	—
59	#1301	C₂×C₅	1	2	yes	complemented
60	C₂⁴×C₅	—	2	1	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	4	2	no	—
63	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	4	1	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₅×M₁₆	—	4	4	no	—
75	C₅×M₁₆	—	4	4	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	D₈	1	2	yes	—
90	C₂×C₅×C₈	SD₁₆	1	2	yes	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅	#41	1	1	yes	characteristic
99	C₂²×M₁₆	—	2	1	no	—
100	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
101	#436	—	2	1	no	—
102	#436	—	2	1	no	—
103	#436	C₄×C₅	1	2	yes	—
104	#436	C₄×C₅	1	2	yes	—
105	#431	C₂²×C₅	1	2	yes	—
106	C₂×#108	—	2	1	no	—
107	C₂×#108	—	2	1	no	—
108	#423	C₂²×C₅	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₂²⋊C₈	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#335	—	2	1	no	—
113	#334	—	2	1	no	—
114	D₄⋊C₈	—	2	4	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	—	4	2	no	—
117	C₂³×C₅	—	4	2	no	—
118	C₂³×C₅	—	4	1	no	—
119	C₂³×C₅	—	4	1	no	—
120	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	4	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	#117	1	1	yes	characteristic
135	C₂×C₄×C₅	#121	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	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	#137	1	1	yes	characteristic
142	C₅×C₈	—	2	1	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	4	2	no	—
145	C₅×C₈	—	8	2	no	—
146	C₅×C₈	—	8	2	no	—
147	C₅×C₈	#102	1	2	yes	—
148	C₅×C₈	#102	1	2	yes	—
149	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic
150	C₂³×C₄	—	2	1	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	4	2	no	—
155	C₂×M₁₆	—	4	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
162	C₄×D₄	C₅×D₄	1	2	yes	—
163	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	2	1	no	—
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	#108	—	4	2	no	—
167	#108	—	4	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₄×C₈	C₅×D₄	1	1	yes	characteristic
180	C₄×C₈	C₅×D₄	1	1	yes	characteristic
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	8	2	no	—
187	C₂²×C₅	—	8	2	no	—
188	C₂²×C₅	#380	1	1	yes	characteristic
189	C₂²×C₅	#474	1	1	yes	characteristic
190	C₂²×C₅	#476	1	1	yes	characteristic
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	C₂×#102	1	1	yes	characteristic
196	C₄×C₅	#417	1	1	yes	characteristic
197	C₄×C₅	#430	1	2	yes	—
198	C₄×C₅	#459	1	1	yes	characteristic
199	C₄×C₅	#461	1	1	yes	characteristic
200	C₂⁴	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₅×#41	1	1	yes	characteristic
214	M₁₆	—	4	4	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
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	C₅×C₄⋊C₄	1	2	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
237	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
238	C₄²	C₅×#41	1	1	yes	characteristic
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	8	2	no	—
242	C₂×C₅	#1802	1	1	yes	characteristic
243	C₂×C₅	#1811	1	1	yes	characteristic
244	C₂×C₅	#1814	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	1	no	—
250	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
251	D₄	—	2	4	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₅×#117	1	1	yes	characteristic
265	C₂×C₄	C₅×#121	1	1	yes	characteristic
266	C₂×C₄	C₅×#121	1	1	yes	characteristic
267	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₂×C₄	C₅×#137	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	8	2	no	—
276	C₈	—	8	2	no	—
277	C₈	C₅×#102	1	2	yes	—
278	C₈	C₅×#102	1	2	yes	—
279	C₅	#8662	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₅×#380	1	1	yes	characteristic
288	C₂²	C₅×#474	1	1	yes	characteristic
289	C₂²	C₅×#476	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	2	no	—
294	C₄	C₂×C₅×#102	1	1	yes	characteristic
295	C₄	C₅×#417	1	1	yes	characteristic
296	C₄	C₅×#430	1	2	yes	—
297	C₄	C₅×#459	1	1	yes	characteristic
298	C₄	C₅×#461	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅×#1802	1	1	yes	characteristic
303	C₂	C₅×#1811	1	1	yes	characteristic
304	C₂	C₅×#1814	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	C₂²×C₅×D₄	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. C₅×#1848	C₂	semidirect
3. C₅×#1564	C₂	semidirect
5. C₅×#1301	C₂	semidirect
22. #8662	C₅	direct (N × H)
56. #1848	C₂×C₅	semidirect
57. #1564	C₂×C₅	semidirect
59. #1301	C₂×C₅	semidirect
279. C₅	#8662	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×#1848	2
C₅×#1564	2
C₅×#1486	2
C₅×#1301	2
#8662	5

Subgroups — C5×#8662 #11931382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8662 #11931382