Subgroups — C₅×#20800 #11943520

Order 1280.

1190 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#428	C₂	1	4	yes	maximal
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic
5	C₅×#1847	C₂	1	4	yes	maximal, complemented
6	C₅×#1845	C₂	1	2	yes	maximal, complemented
7	C₅×#1844	C₂	1	2	yes	maximal, complemented
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#428	—	2	16	no	—
14	C₂×C₅×#107	C₂²	1	8	yes	—
15	C₂×C₅×#106	C₂²	1	8	yes	—
16	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₅×#384	—	2	2	no	—
20	C₅×#384	C₂²	1	2	yes	—
21	C₅×#384	C₂²	1	2	yes	—
22	C₅×#384	C₂²	1	1	yes	characteristic
23	C₅×#384	C₂²	1	1	yes	characteristic
24	C₅×#382	C₂²	1	2	yes	—
25	C₅×#380	—	2	2	no	—
26	C₄×C₅×C₄⋊C₄	—	2	2	no	—
27	#20800	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	D₄	1	2	yes	—
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
32	C₂×C₅×M₁₆	D₄	1	8	yes	—
33	C₂×C₅×M₁₆	Q₈	1	8	yes	—
34	C₂²×C₅×C₈	C₂³	1	4	yes	—
35	C₅×#117	—	2	8	no	—
36	C₂×C₅×C₄⋊C₄	—	2	8	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	—	2	4	no	—
42	C₂×C₅×C₄⋊C₄	—	4	8	no	—
43	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅	D₄	1	2	yes	—
50	C₂×C₄²×C₅	Q₈	1	4	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#107	—	2	16	no	—
54	C₅×#106	—	2	16	no	—
55	C₅×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
57	C₅×#95	—	2	4	no	—
58	C₂×#428	C₂×C₅	1	4	yes	—
59	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
60	#2285	C₂×C₅	1	1	yes	characteristic
61	C₂×#384	C₂×C₅	1	1	yes	characteristic
62	#1847	C₂×C₅	1	4	yes	complemented
63	#1845	C₂×C₅	1	2	yes	complemented
64	#1844	C₂×C₅	1	2	yes	complemented
65	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	4	8	no	—
72	C₂²×C₄×C₅	—	4	4	no	—
73	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₅×M₁₆	C₄⋊C₄	1	32	yes	complemented
83	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
85	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
86	C₅×C₄⋊C₄	—	2	8	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	4	8	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₅×C₄⋊C₄	—	4	4	no	—
92	C₅×C₄⋊C₄	—	4	8	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₂²⋊C₄	—	4	4	no	—
95	C₄²×C₅	—	4	4	no	—
96	C₄²×C₅	C₄⋊C₄	1	8	yes	—
97	C₄²×C₅	C₂²×C₄	1	2	yes	—
98	C₄²×C₅	C₂²×C₄	1	2	yes	—
99	C₄²×C₅	C₂×Q₈	1	2	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
102	C₂×#117	C₂²×C₅	1	2	yes	—
103	C₂²×C₄⋊C₄	—	2	2	no	—
104	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
105	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
106	#428	—	2	16	no	—
107	C₂×#107	C₂²×C₅	1	8	yes	—
108	C₂×#106	C₂²×C₅	1	8	yes	—
109	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
110	C₂×#97	C₂²×C₅	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
112	#384	—	2	2	no	—
113	#384	C₂²×C₅	1	1	yes	characteristic
114	#384	C₂²×C₅	1	2	yes	—
115	#384	C₂²×C₅	1	1	yes	characteristic
116	#384	C₂²×C₅	1	2	yes	—
117	#382	C₂²×C₅	1	2	yes	—
118	#380	—	2	2	no	—
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
123	C₂³×C₅	#127	1	2	yes	—
124	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
125	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	16	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	8	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	8	8	no	—
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄×C₅	#127	1	2	yes	—
141	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
142	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
144	C₂×C₄×C₅	#137	1	4	yes	—
145	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
149	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	complemented
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	C₅×D₄	1	2	yes	—
152	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
154	C₂×M₁₆	C₅×D₄	1	8	yes	—
155	C₂×M₁₆	C₅×Q₈	1	8	yes	—
156	C₂²×C₈	C₂³×C₅	1	4	yes	—
157	#117	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	8	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	4	8	no	—
165	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
166	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
168	C₂×C₂²⋊C₄	—	2	2	no	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₄²	C₅×D₄	1	2	yes	—
172	C₂×C₄²	C₅×Q₈	1	4	yes	—
173	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
175	#107	—	2	16	no	—
176	#106	—	2	16	no	—
177	#97	C₂×C₄×C₅	1	8	yes	complemented
178	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
179	#95	—	2	4	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	4	no	—
183	C₂²×C₅	—	2	4	no	—
184	C₂²×C₅	#384	1	4	yes	—
185	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₅	C₂×#127	1	1	yes	characteristic
187	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
189	C₂²×C₅	C₂×#137	1	2	yes	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	—	8	4	no	—
193	C₄×C₅	—	8	2	no	—
194	C₄×C₅	#384	1	4	yes	—
195	C₄×C₅	#428	1	8	yes	complemented
196	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
197	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
198	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
207	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
208	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
212	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
213	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
215	M₁₆	C₅×C₄⋊C₄	1	32	yes	complemented
216	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
217	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
218	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₅×C₄⋊C₄	1	8	yes	—
230	C₄²	C₂²×C₄×C₅	1	2	yes	—
231	C₄²	C₂²×C₄×C₅	1	2	yes	—
232	C₄²	C₂×C₅×Q₈	1	2	yes	—
233	C₄²	C₂×C₅×Q₈	1	2	yes	—
234	C₂×C₅	—	2	4	no	—
235	C₂×C₅	C₂×#384	1	1	yes	characteristic
236	C₂×C₅	C₂×#428	1	2	yes	—
237	C₂×C₅	#3151	1	1	yes	characteristic
238	C₂×C₅	#3152	1	1	yes	characteristic
239	C₂×C₅	#3165	1	2	yes	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
243	C₂³	C₅×#127	1	2	yes	—
244	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
245	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	8	no	—
256	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
257	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
258	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
259	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
260	C₂×C₄	C₅×#127	1	2	yes	—
261	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
262	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
263	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
264	C₂×C₄	C₅×#137	1	4	yes	—
265	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
267	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
268	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
269	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	complemented
270	C₅	#20800	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₅×#384	1	4	yes	—
276	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
277	C₂²	C₂×C₅×#127	1	1	yes	characteristic
278	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
279	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
280	C₂²	C₂×C₅×#137	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	4	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	C₅×#384	1	4	yes	—
286	C₄	C₅×#428	1	8	yes	complemented
287	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
288	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₂×C₅×#384	1	1	yes	characteristic
291	C₂	C₂×C₅×#428	1	2	yes	—
292	C₂	C₅×#3151	1	1	yes	characteristic
293	C₂	C₅×#3152	1	1	yes	characteristic
294	C₂	C₅×#3165	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1847	C₂	semidirect
6. C₅×#1845	C₂	semidirect
7. C₅×#1844	C₂	semidirect
16. C₄×C₅×M₁₆	C₄	semidirect
27. #20800	C₅	direct (N × H)
55. C₅×#97	C₂×C₄	semidirect
56. C₄×C₅×C₈	C₂×C₄	semidirect
62. #1847	C₂×C₅	semidirect
63. #1845	C₂×C₅	semidirect
64. #1844	C₂×C₅	semidirect
82. C₅×M₁₆	C₄⋊C₄	semidirect
109. C₄×M₁₆	C₄×C₅	semidirect
149. C₅×C₈	C₂×C₄⋊C₄	semidirect
177. #97	C₂×C₄×C₅	semidirect
178. C₄×C₈	C₂×C₄×C₅	semidirect
195. C₄×C₅	#428	semidirect
215. M₁₆	C₅×C₄⋊C₄	semidirect
269. C₈	C₂×C₅×C₄⋊C₄	semidirect
270. C₅	#20800	direct (N × H)
286. C₄	C₅×#428	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#428	2
C₂×C₄×C₅×M₁₆	2
C₅×#2285	2
C₂×C₅×#384	2
C₅×#1847	2
C₅×#1845	2
C₅×#1844	2
#20800	5

Subgroups — C5×#20800 #11943520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20800 #11943520