Subgroups — C₂×C₅×#4060 #13106913

Order 1600.

664 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#688	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#686	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4060	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₅×D₅	—	2	1	no	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
13	C₅×#713	—	2	2	no	—
14	C₅×#713	C₂²	1	4	yes	complemented
15	C₅×D₄×D₅	—	2	2	no	—
16	C₅×D₄×D₅	C₂²	1	4	yes	complemented
17	C₂×C₅×D₂₀	—	2	1	no	—
18	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₄×C₅×D₅	—	2	1	no	—
21	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
22	C₅²×SD₁₆	C₂²	1	4	yes	complemented
23	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
24	C₅×#688	C₂²	1	4	yes	complemented
25	C₅×#686	C₂²	1	4	yes	complemented
26	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
27	C₅×D₄₀	C₂²	1	4	yes	complemented
28	C₅×#676	C₂²	1	4	yes	complemented
29	C₂×C₅×AGL(1,ℤ/8)	—	5	1	no	maximal
30	C₂×#4060	C₅	1	1	yes	maximal, characteristic, direct product factor
31	C₂³×C₅²	—	2	1	no	—
32	C₂²×C₅×D₅	—	2	1	no	—
33	C₂²×C₅×D₅	—	2	1	no	—
34	C₂²×C₅×D₅	—	2	4	no	—
35	C₂²×C₅×D₅	—	4	4	no	—
36	C₂²×C₅×D₅	D₄	1	1	yes	characteristic
37	C₅²×Q₈	—	2	1	no	—
38	C₅²×Q₈	C₂³	1	2	yes	complemented
39	C₅²×D₄	—	2	1	no	—
40	C₅²×D₄	C₂³	1	2	yes	—
41	C₂×C₄×C₅²	—	2	1	no	—
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₅×C₂²⋊D₅	—	2	4	no	—
44	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
45	C₅×D₂₀	—	2	4	no	—
46	C₅×D₂₀	—	2	1	no	—
47	C₅×D₂₀	—	2	1	no	—
48	C₅×D₂₀	C₂³	1	2	yes	—
49	C₅×D₂₀	C₂³	1	2	yes	complemented
50	C₄×C₅×D₅	—	2	4	no	—
51	C₄×C₅×D₅	D₄	1	2	yes	—
52	C₄×C₅×D₅	C₂³	1	2	yes	—
53	C₅²×C₈	C₂³	1	2	yes	complemented
54	C₅×#168	C₂³	1	2	yes	complemented
55	C₂×C₅×#41	—	5	1	no	—
56	C₂²×C₅×D₄	—	5	1	no	—
57	C₂×#713	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×D₄×D₅	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×AGL(1,ℤ/8)	—	5	8	no	—
60	C₂×C₅×SD₁₆	—	2	2	no	—
61	C₂×C₅×SD₁₆	—	5	1	no	—
62	C₂×C₅×SD₁₆	D₅	1	1	yes	characteristic, complemented
63	C₂×C₅×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×D₈	—	5	1	no	—
65	C₂×C₅×D₈	—	5	1	no	—
66	C₂×C₅×M₁₆	—	5	1	no	—
67	C₂×#688	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×#686	C₂×C₅	1	1	yes	characteristic, complemented
69	#4060	C₂×C₅	1	8	yes	direct product factor
70	C₂×D₄₀	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#676	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂²×C₅²	—	2	2	no	—
73	C₂²×C₅²	—	4	2	no	—
74	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×D₅	—	2	2	no	—
76	C₂×C₅×D₅	—	2	2	no	—
77	C₂×C₅×D₅	—	2	2	no	—
78	C₂×C₅×D₅	—	4	8	no	—
79	C₂×C₅×D₅	—	4	2	no	—
80	C₂×C₅×D₅	—	4	2	no	—
81	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
82	C₄×C₅²	—	2	2	no	—
83	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
84	C₄×C₅²	C₂⁴	1	1	yes	characteristic
85	C₅×Dic₅	C₂×D₄	1	2	yes	—
86	C₂⁴×C₅	—	10	1	no	—
87	C₂³×D₅	—	2	1	no	—
88	C₅×#41	—	5	4	no	—
89	C₅×#41	—	10	2	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
92	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	5	4	no	—
95	C₂×C₅×D₄	—	5	1	no	—
96	C₂×C₅×D₄	—	5	1	no	—
97	C₂×C₅×D₄	—	10	1	no	—
98	C₂×C₅×D₄	—	10	2	no	—
99	C₂×C₅×D₄	—	10	1	no	—
100	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
101	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	5	1	no	—
103	C₂²×C₄×C₅	—	10	1	no	—
104	C₂×C₂²⋊D₅	—	2	1	no	—
105	#713	—	2	2	no	—
106	#713	C₂²×C₅	1	4	yes	complemented
107	D₄×D₅	—	2	2	no	—
108	D₄×D₅	C₂²×C₅	1	4	yes	complemented
109	C₂×D₂₀	—	2	1	no	—
110	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
111	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
112	C₂×C₄×D₅	—	2	1	no	—
113	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
114	C₅×SD₁₆	—	2	8	no	—
115	C₅×SD₁₆	—	5	4	no	—
116	C₅×SD₁₆	C₂×D₅	1	4	yes	complemented
117	C₅×SD₁₆	C₂²×C₅	1	4	yes	complemented
118	C₅×D₈	—	5	4	no	—
119	C₅×D₈	—	5	4	no	—
120	C₅×M₁₆	—	5	4	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	5	1	no	—
123	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
124	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
125	#688	C₂²×C₅	1	4	yes	complemented
126	#686	C₂²×C₅	1	4	yes	complemented
127	C₂×#168	C₂²×C₅	1	1	yes	characteristic, complemented
128	D₄₀	C₂²×C₅	1	4	yes	complemented
129	#676	C₂²×C₅	1	4	yes	complemented
130	C₂×AGL(1,ℤ/8)	—	5	1	no	Sylow
131	C₂×C₅²	—	4	2	no	—
132	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	complemented
133	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
134	C₅×D₅	—	2	2	no	—
135	C₅×D₅	—	4	2	no	—
136	C₅×D₅	—	4	2	no	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	—	2	1	no	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	—	5	1	no	—
141	C₂³×C₅	—	10	4	no	—
142	C₂³×C₅	—	10	1	no	—
143	C₂³×C₅	—	10	1	no	—
144	C₂³×C₅	—	20	4	no	—
145	C₂²×D₅	—	2	1	no	—
146	C₂²×D₅	—	2	1	no	—
147	C₂²×D₅	—	2	4	no	—
148	C₂²×D₅	—	4	4	no	—
149	C₂²×D₅	C₅×D₄	1	1	yes	characteristic
150	C₅×Q₈	—	2	1	no	—
151	C₅×Q₈	—	2	1	no	—
152	C₅×Q₈	—	2	4	no	—
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	C₂²×D₅	1	2	yes	complemented
155	C₅×Q₈	C₂³×C₅	1	2	yes	complemented
156	C₅×D₄	—	2	1	no	—
157	C₅×D₄	—	2	1	no	—
158	C₅×D₄	—	2	4	no	—
159	C₅×D₄	—	4	2	no	—
160	C₅×D₄	—	5	2	no	—
161	C₅×D₄	—	5	2	no	—
162	C₅×D₄	—	10	4	no	—
163	C₅×D₄	—	10	4	no	—
164	C₅×D₄	—	10	1	no	—
165	C₅×D₄	—	10	1	no	—
166	C₅×D₄	C₂²×D₅	1	2	yes	—
167	C₅×D₄	C₂³×C₅	1	2	yes	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	5	1	no	—
173	C₂×C₄×C₅	—	5	2	no	—
174	C₂×C₄×C₅	—	5	2	no	—
175	C₂×C₄×C₅	—	10	4	no	—
176	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
178	C₂²⋊D₅	—	2	4	no	—
179	C₂×Dic₅	C₅×D₄	1	1	yes	characteristic
180	D₂₀	—	2	4	no	—
181	D₂₀	—	2	1	no	—
182	D₂₀	—	2	1	no	—
183	D₂₀	C₂³×C₅	1	2	yes	—
184	D₂₀	C₂³×C₅	1	2	yes	complemented
185	C₄×D₅	—	2	4	no	—
186	C₄×D₅	C₅×D₄	1	2	yes	—
187	C₄×D₅	C₂³×C₅	1	2	yes	—
188	C₅×C₈	—	2	4	no	—
189	C₅×C₈	—	5	2	no	—
190	C₅×C₈	C₂²×D₅	1	2	yes	complemented
191	C₅×C₈	C₂³×C₅	1	2	yes	complemented
192	#168	C₂³×C₅	1	2	yes	complemented
193	C₂×#41	—	5	1	no	—
194	C₂²×D₄	—	5	1	no	—
195	AGL(1,ℤ/8)	—	5	8	no	—
196	C₂×SD₁₆	—	5	1	no	—
197	C₂×SD₁₆	C₅×D₅	1	1	yes	characteristic, complemented
198	C₂×D₈	—	5	1	no	—
199	C₂×D₈	—	5	1	no	—
200	C₂×M₁₆	—	5	1	no	—
201	C₅²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	4	4	no	—
206	C₂²×C₅	—	4	2	no	—
207	C₂²×C₅	—	4	2	no	—
208	C₂²×C₅	—	5	2	no	—
209	C₂²×C₅	—	8	4	no	—
210	C₂²×C₅	—	10	2	no	—
211	C₂²×C₅	—	10	2	no	—
212	C₂²×C₅	—	10	2	no	—
213	C₂²×C₅	—	20	8	no	—
214	C₂²×C₅	—	20	2	no	—
215	C₂²×C₅	—	20	2	no	—
216	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
217	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	4	8	no	—
222	C₂×D₅	—	4	2	no	—
223	C₂×D₅	—	4	2	no	—
224	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	—	5	2	no	—
231	C₄×C₅	D₄×D₅	1	1	yes	characteristic
232	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
233	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
234	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
235	Dic₅	C₂×C₅×D₄	1	2	yes	—
236	C₂⁴	—	10	1	no	—
237	#41	—	5	4	no	—
238	#41	—	10	2	no	—
239	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
240	C₂×D₄	—	5	4	no	—
241	C₂×D₄	—	5	1	no	—
242	C₂×D₄	—	5	1	no	—
243	C₂×D₄	—	10	1	no	—
244	C₂×D₄	—	10	2	no	—
245	C₂×D₄	—	10	1	no	—
246	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	10	1	no	—
249	SD₁₆	—	5	4	no	—
250	SD₁₆	C₂×C₅×D₅	1	4	yes	complemented
251	D₈	—	5	4	no	—
252	D₈	—	5	4	no	—
253	M₁₆	—	5	4	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
256	C₂×C₅	—	2	4	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	—	8	4	no	—
261	C₂×C₅	—	10	2	no	—
262	C₂×C₅	—	20	2	no	—
263	C₂×C₅	—	20	2	no	—
264	C₂×C₅	#4060	1	2	yes	direct product factor
265	C₂×C₅	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
266	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
267	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
268	D₅	—	2	2	no	—
269	D₅	—	4	2	no	—
270	D₅	—	4	2	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	5	1	no	—
273	C₂³	—	10	4	no	—
274	C₂³	—	10	1	no	—
275	C₂³	—	10	1	no	—
276	C₂³	—	20	4	no	—
277	Q₈	—	2	1	no	—
278	Q₈	C₂²×C₅×D₅	1	2	yes	complemented
279	D₄	—	2	1	no	—
280	D₄	—	5	2	no	—
281	D₄	—	5	2	no	—
282	D₄	—	10	4	no	—
283	D₄	—	10	4	no	—
284	D₄	—	10	1	no	—
285	D₄	—	10	1	no	—
286	D₄	C₂²×C₅×D₅	1	2	yes	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	10	4	no	—
292	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
293	C₈	—	5	2	no	—
294	C₈	C₂²×C₅×D₅	1	2	yes	complemented
295	C₅	—	2	2	no	—
296	C₅	C₂×#4060	1	1	yes	characteristic, direct product factor
297	C₅	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	5	2	no	—
301	C₂²	—	10	2	no	—
302	C₂²	—	10	2	no	—
303	C₂²	—	10	2	no	—
304	C₂²	—	20	8	no	—
305	C₂²	—	20	2	no	—
306	C₂²	—	20	2	no	—
307	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	5	2	no	—
310	C₄	C₅×D₄×D₅	1	1	yes	characteristic
311	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
312	C₂	—	4	2	no	—
313	C₂	—	10	2	no	—
314	C₂	—	20	2	no	—
315	C₂	—	20	2	no	—
316	C₂	C₅×#4060	1	2	yes	direct product factor
317	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×SD₁₆	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₅×SD₁₆	D₅
hypocenter	C₅	C₂×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#713	C₂	semidirect
2. C₂×C₅×D₄×D₅	C₂	semidirect
3. C₂×C₅²×SD₁₆	C₂	semidirect
4. C₂×C₅×#688	C₂	semidirect
5. C₂×C₅×#686	C₂	semidirect
6. C₅×#4060	C₂	direct (N × H)
7. C₂×C₅×D₄₀	C₂	semidirect
8. C₂×C₅×#676	C₂	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
14. C₅×#713	C₂²	semidirect
16. C₅×D₄×D₅	C₂²	semidirect
19. C₂×C₅×D₂₀	C₂²	semidirect
22. C₅²×SD₁₆	C₂²	semidirect
23. C₂×C₅²×C₈	C₂²	semidirect
24. C₅×#688	C₂²	semidirect
25. C₅×#686	C₂²	semidirect
26. C₂×C₅×#168	C₂²	semidirect
27. C₅×D₄₀	C₂²	semidirect
28. C₅×#676	C₂²	semidirect
30. C₂×#4060	C₅	direct (N × H)
38. C₅²×Q₈	C₂³	semidirect
49. C₅×D₂₀	C₂³	semidirect
53. C₅²×C₈	C₂³	semidirect
54. C₅×#168	C₂³	semidirect
57. C₂×#713	C₂×C₅	semidirect
58. C₂×D₄×D₅	C₂×C₅	semidirect
62. C₂×C₅×SD₁₆	D₅	semidirect
63. C₂×C₅×SD₁₆	C₂×C₅	semidirect
67. C₂×#688	C₂×C₅	semidirect
68. C₂×#686	C₂×C₅	semidirect
69. #4060	C₂×C₅	direct (N × H)
70. C₂×D₄₀	C₂×C₅	semidirect
71. C₂×#676	C₂×C₅	semidirect
91. C₂×C₅×Q₈	C₂×D₅	semidirect
92. C₂×C₅×Q₈	C₂²×C₅	semidirect
106. #713	C₂²×C₅	semidirect
108. D₄×D₅	C₂²×C₅	semidirect
111. C₂×D₂₀	C₂²×C₅	semidirect
116. C₅×SD₁₆	C₂×D₅	semidirect
117. C₅×SD₁₆	C₂²×C₅	semidirect
123. C₂×C₅×C₈	C₂×D₅	semidirect
124. C₂×C₅×C₈	C₂²×C₅	semidirect
125. #688	C₂²×C₅	semidirect
126. #686	C₂²×C₅	semidirect
127. C₂×#168	C₂²×C₅	semidirect
128. D₄₀	C₂²×C₅	semidirect
129. #676	C₂²×C₅	semidirect
132. C₂×C₅²	AGL(1,ℤ/8)	semidirect
154. C₅×Q₈	C₂²×D₅	semidirect
155. C₅×Q₈	C₂³×C₅	semidirect
184. D₂₀	C₂³×C₅	semidirect
190. C₅×C₈	C₂²×D₅	semidirect
191. C₅×C₈	C₂³×C₅	semidirect
192. #168	C₂³×C₅	semidirect
197. C₂×SD₁₆	C₅×D₅	semidirect
201. C₅²	C₂×AGL(1,ℤ/8)	semidirect
239. C₂×Q₈	C₂×C₅×D₅	semidirect
250. SD₁₆	C₂×C₅×D₅	semidirect
255. C₂×C₈	C₂×C₅×D₅	semidirect
264. C₂×C₅	#4060	direct (N × H)
265. C₂×C₅	C₅×AGL(1,ℤ/8)	semidirect
278. Q₈	C₂²×C₅×D₅	semidirect
294. C₈	C₂²×C₅×D₅	semidirect
296. C₅	C₂×#4060	direct (N × H)
297. C₅	C₂×C₅×AGL(1,ℤ/8)	semidirect
316. C₂	C₅×#4060	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×#713	2
C₂×C₅×D₄×D₅	2
C₂×C₅²×SD₁₆	2
C₂×C₅×#688	2
C₂×C₅×#686	2
C₅×#4060	2
C₂×C₅×D₄₀	2
C₂×C₅×#676	2
C₂×C₅×AGL(1,ℤ/8)	5
C₂×#4060	5

Subgroups — C2×C5×#4060 #13106913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4060 #13106913