Subgroups — C₅×#20549 #11943269

Order 1280.

966 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1771	C₂	1	4	yes	maximal, complemented
5	C₅×#1767	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₅×#1753	C₂	1	4	yes	maximal
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#432	C₄	1	8	yes	complemented
12	C₅×#423	C₄	1	8	yes	complemented
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	4	yes	—
18	C₂×C₅×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
20	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
23	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₅×#376	C₄	1	2	yes	—
25	C₄³×C₅	C₂²	1	1	yes	characteristic
26	C₄³×C₅	C₂²	1	1	yes	characteristic
27	C₅×#336	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	4	yes	—
29	#20549	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×M₁₆	—	4	4	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
35	C₂²×C₅×C₈	C₂³	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	4	yes	—
37	C₅×#117	—	2	8	no	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	2	yes	—
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₅×#97	—	4	4	no	—
57	C₅×#97	C₂×C₄	1	8	yes	—
58	C₄×C₅×C₈	—	4	4	no	—
59	C₅×#95	C₂×C₄	1	4	yes	—
60	C₅×#95	C₂×C₄	1	4	yes	—
61	C₂×#432	C₂×C₅	1	1	yes	characteristic
62	C₂×#423	C₂×C₅	1	1	yes	characteristic
63	C₄×#117	C₂×C₅	1	1	yes	characteristic
64	#1771	C₂×C₅	1	4	yes	complemented
65	#1767	C₂×C₅	1	2	yes	complemented
66	C₄×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
67	#1753	C₂×C₅	1	4	yes	—
68	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₄²	1	4	yes	—
74	C₂²×C₄×C₅	C₄²	1	4	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₅×M₁₆	—	4	8	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
92	C₅×C₄⋊C₄	—	2	8	no	—
93	C₅×C₂²⋊C₄	—	2	8	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	2	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	—	4	2	no	—
101	C₄²×C₅	C₄²	1	4	yes	—
102	C₄²×C₅	C₄²	1	4	yes	—
103	C₄²×C₅	M₁₆	1	8	yes	complemented
104	C₄²×C₅	C₂²×C₄	1	2	yes	—
105	C₄²×C₅	C₂²×C₄	1	2	yes	—
106	C₄²×C₅	C₂²×C₄	1	2	yes	—
107	C₄²×C₅	C₂²×C₄	1	2	yes	—
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂×#117	C₄×C₅	1	2	yes	—
110	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
111	#432	C₄×C₅	1	8	yes	complemented
112	#423	C₄×C₅	1	8	yes	complemented
113	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
117	C₂×#97	C₂²×C₅	1	4	yes	—
118	C₂×#97	C₂²×C₅	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
120	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
121	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
122	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
123	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
124	#376	C₄×C₅	1	2	yes	—
125	C₄³	C₂²×C₅	1	1	yes	characteristic
126	C₄³	C₂²×C₅	1	1	yes	characteristic
127	#336	C₂²×C₅	1	4	yes	—
128	#336	C₂²×C₅	1	4	yes	—
129	C₂³×C₅	—	2	2	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	C₂×C₄²	1	2	yes	—
133	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	8	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
149	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
150	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
152	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
153	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
154	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
155	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
158	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
159	C₅×C₈	—	2	8	no	—
160	C₅×C₈	—	2	8	no	—
161	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	4	4	no	—
165	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
166	C₂²×C₈	C₂³×C₅	1	4	yes	—
167	C₂²×C₈	C₂³×C₅	1	4	yes	—
168	#117	—	2	8	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
179	C₂×C₄²	C₂³×C₅	1	2	yes	—
180	C₂×C₄²	C₂³×C₅	1	2	yes	—
181	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
183	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
184	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
185	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
186	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
187	#97	—	4	4	no	—
188	#97	C₂×C₄×C₅	1	8	yes	—
189	C₄×C₈	—	4	4	no	—
190	#95	C₂×C₄×C₅	1	4	yes	—
191	#95	C₂×C₄×C₅	1	4	yes	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	4	4	no	—
196	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
197	C₂²×C₅	#520	1	2	yes	—
198	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
199	C₂²×C₅	#568	1	2	yes	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	8	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	C₄×M₁₆	1	4	yes	—
207	C₄×C₅	#518	1	1	yes	characteristic
208	C₄×C₅	#519	1	1	yes	characteristic
209	C₄×C₅	#520	1	2	yes	—
210	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₄²×C₅	1	4	yes	—
216	C₂²×C₄	C₄²×C₅	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	M₁₆	—	4	8	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
233	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₄²×C₅	1	4	yes	—
244	C₄²	C₄²×C₅	1	4	yes	—
245	C₄²	C₅×M₁₆	1	8	yes	complemented
246	C₄²	C₂²×C₄×C₅	1	2	yes	—
247	C₄²	C₂²×C₄×C₅	1	2	yes	—
248	C₄²	C₂²×C₄×C₅	1	2	yes	—
249	C₄²	C₂²×C₄×C₅	1	2	yes	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	#2274	1	1	yes	characteristic
252	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
253	C₂×C₅	#2873	1	1	yes	characteristic
254	C₂×C₅	#2989	1	4	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄²×C₅	1	2	yes	—
259	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
278	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
279	C₂×C₄	C₂×C₅×M₁₆	1	8	yes	—
280	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
282	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
283	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
284	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
285	C₈	—	2	8	no	—
286	C₈	—	2	8	no	—
287	C₅	#20549	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
293	C₂²	C₅×#520	1	2	yes	—
294	C₂²	C₂²×C₅×M₁₆	1	2	yes	—
295	C₂²	C₅×#568	1	2	yes	—
296	C₄	—	2	8	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	C₄×C₅×M₁₆	1	4	yes	—
303	C₄	C₅×#518	1	1	yes	characteristic
304	C₄	C₅×#519	1	1	yes	characteristic
305	C₄	C₅×#520	1	2	yes	—
306	C₂	—	4	2	no	—
307	C₂	C₅×#2274	1	1	yes	characteristic
308	C₂	C₂×C₄×C₅×M₁₆	1	1	yes	characteristic
309	C₂	C₅×#2873	1	1	yes	characteristic
310	C₂	C₅×#2989	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1771	C₂	semidirect
5. C₅×#1767	C₂	semidirect
6. C₄×C₅×C₄⋊C₈	C₂	semidirect
11. C₅×#432	C₄	semidirect
12. C₅×#423	C₄	semidirect
29. #20549	C₅	direct (N × H)
64. #1771	C₂×C₅	semidirect
65. #1767	C₂×C₅	semidirect
66. C₄×C₄⋊C₈	C₂×C₅	semidirect
103. C₄²×C₅	M₁₆	semidirect
111. #432	C₄×C₅	semidirect
112. #423	C₄×C₅	semidirect
245. C₄²	C₅×M₁₆	semidirect
287. C₅	#20549	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×#432	2
C₂×C₅×#423	2
C₄×C₅×#117	2
C₅×#1771	2
C₅×#1767	2
C₄×C₅×C₄⋊C₈	2
C₅×#1753	2
#20549	5

Subgroups — C5×#20549 #11943269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20549 #11943269