Subgroups — C₅×#21145 #11943865

Order 1280.

1010 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#424	C₂	1	2	yes	maximal
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#2285	C₂	1	1	yes	maximal, characteristic
5	C₅×#1940	C₂	1	4	yes	maximal, complemented
6	C₅×#1939	C₂	1	4	yes	maximal, complemented
7	C₅×#1729	C₂	1	2	yes	maximal
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₅×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#424	—	2	8	no	—
14	C₂×C₅×C₄⋊C₈	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
18	C₄×C₅×M₁₆	—	2	4	no	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₅×#384	C₄	1	2	yes	—
22	C₅×#382	C₄	1	2	yes	—
23	C₅×#380	C₄	1	2	yes	—
24	C₅×#380	C₄	1	2	yes	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	8	yes	—
28	#21145	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	D₄	1	4	yes	—
37	C₂×C₅×M₁₆	Q₈	1	4	yes	—
38	C₂²×C₅×C₈	—	2	4	no	—
39	C₂²×C₅×C₈	—	2	4	no	—
40	C₂²×C₅×C₈	—	2	4	no	—
41	C₂²×C₅×C₈	C₂³	1	4	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₅×#117	—	2	4	no	—
45	C₅×#117	—	2	4	no	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	D₄	1	4	yes	—
53	C₂×C₄²×C₅	Q₈	1	4	yes	—
54	C₂×C₄²×C₅	C₂³	1	4	yes	—
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	16	no	—
58	C₅×C₄⋊C₈	—	2	4	no	—
59	C₅×C₄⋊C₈	—	2	4	no	—
60	C₅×#97	—	2	4	no	—
61	C₄×C₅×C₈	—	2	4	no	—
62	C₅×#95	C₂×C₄	1	4	yes	—
63	C₅×#95	C₂×C₄	1	4	yes	—
64	C₂×#424	C₂×C₅	1	2	yes	—
65	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
66	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
67	#2285	C₂×C₅	1	1	yes	characteristic
68	#1940	C₂×C₅	1	4	yes	complemented
69	#1939	C₂×C₅	1	4	yes	complemented
70	#1729	C₂×C₅	1	2	yes	—
71	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
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	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₅	#41	1	2	yes	—
87	C₂²×C₄×C₅	#41	1	2	yes	—
88	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₅×M₁₆	—	2	8	no	—
90	C₅×M₁₆	—	2	8	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	4	8	no	—
99	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
100	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
101	C₅×C₄⋊C₄	—	2	8	no	—
102	C₅×C₄⋊C₄	—	2	8	no	—
103	C₅×C₄⋊C₄	—	4	4	no	—
104	C₅×C₂²⋊C₄	—	2	8	no	—
105	C₄²×C₅	—	2	4	no	—
106	C₄²×C₅	—	2	4	no	—
107	C₄²×C₅	C₄⋊C₄	1	4	yes	—
108	C₄²×C₅	C₄⋊C₄	1	4	yes	—
109	C₄²×C₅	C₂×D₄	1	2	yes	—
110	C₄²×C₅	C₂×D₄	1	2	yes	—
111	C₄²×C₅	C₂×Q₈	1	2	yes	—
112	C₄²×C₅	C₂×Q₈	1	2	yes	—
113	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
114	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
115	C₂³×C₈	C₂²×C₅	1	2	yes	—
116	C₂×#117	C₂²×C₅	1	2	yes	—
117	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
118	#424	—	2	8	no	—
119	C₂×C₄⋊C₈	—	2	4	no	—
120	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
123	C₄×M₁₆	—	2	4	no	—
124	C₂×#97	C₂²×C₅	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
126	#384	C₄×C₅	1	2	yes	—
127	#382	C₄×C₅	1	2	yes	—
128	#380	C₄×C₅	1	2	yes	—
129	#380	C₄×C₅	1	2	yes	—
130	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
131	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
132	#336	C₂²×C₅	1	8	yes	—
133	C₂³×C₅	—	2	2	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂³×C₅	C₄×D₄	1	2	yes	—
137	C₂³×C₅	C₄×Q₈	1	2	yes	—
138	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
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	8	no	—
143	C₂×C₄×C₅	—	2	8	no	—
144	C₂×C₄×C₅	—	2	8	no	—
145	C₂×C₄×C₅	—	2	8	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
154	C₂×C₄×C₅	#121	1	4	yes	—
155	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₄×C₅	#125	1	4	yes	—
157	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
158	C₂×C₄×C₅	#131	1	4	yes	—
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
165	C₅×C₈	—	2	4	no	—
166	C₅×C₈	—	2	4	no	—
167	C₅×C₈	—	4	4	no	—
168	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₅×D₄	1	4	yes	—
176	C₂×M₁₆	C₅×Q₈	1	4	yes	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	C₂³×C₅	1	4	yes	—
181	C₂²×C₈	C₂³×C₅	1	2	yes	—
182	C₂²×C₈	C₂³×C₅	1	2	yes	—
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₅×D₄	1	4	yes	—
192	C₂×C₄²	C₅×Q₈	1	4	yes	—
193	C₂×C₄²	C₂³×C₅	1	4	yes	—
194	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	16	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	4	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	4	no	—
201	#95	C₂×C₄×C₅	1	4	yes	—
202	#95	C₂×C₄×C₅	1	4	yes	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	—	2	4	no	—
207	C₂²×C₅	#389	1	4	yes	—
208	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
211	C₂²×C₅	C₂×#131	1	2	yes	—
212	C₂²×C₅	#568	1	2	yes	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	4	8	no	—
217	C₄×C₅	#389	1	4	yes	—
218	C₄×C₅	#424	1	4	yes	complemented
219	C₄×C₅	C₂×#121	1	1	yes	characteristic
220	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
221	C₄×C₅	C₂×#125	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
223	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
227	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
229	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
236	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
237	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
238	C₂²×C₄	C₅×#41	1	2	yes	—
239	C₂²×C₄	C₅×#41	1	2	yes	—
240	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
241	M₁₆	—	2	8	no	—
242	M₁₆	—	2	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	8	no	—
251	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
252	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	C₅×C₄⋊C₄	1	4	yes	—
260	C₄²	C₅×C₄⋊C₄	1	4	yes	—
261	C₄²	C₂×C₅×D₄	1	2	yes	—
262	C₄²	C₂×C₅×D₄	1	2	yes	—
263	C₄²	C₂×C₅×Q₈	1	2	yes	—
264	C₄²	C₂×C₅×Q₈	1	2	yes	—
265	C₂×C₅	—	2	4	no	—
266	C₂×C₅	C₂×#389	1	1	yes	characteristic
267	C₂×C₅	C₂×#424	1	1	yes	characteristic
268	C₂×C₅	#2905	1	1	yes	characteristic
269	C₂×C₅	#2935	1	2	yes	—
270	C₂×C₅	#2962	1	2	yes	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
274	C₂³	C₄×C₅×D₄	1	2	yes	—
275	C₂³	C₄×C₅×Q₈	1	2	yes	—
276	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
286	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
290	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
291	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
292	C₂×C₄	C₅×#121	1	4	yes	—
293	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
294	C₂×C₄	C₅×#125	1	4	yes	—
295	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
296	C₂×C₄	C₅×#131	1	4	yes	—
297	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	2	4	no	—
305	C₈	—	4	4	no	—
306	C₅	#21145	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	C₅×#389	1	4	yes	—
312	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
313	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
314	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
315	C₂²	C₂×C₅×#131	1	2	yes	—
316	C₂²	C₅×#568	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	8	no	—
321	C₄	C₅×#389	1	4	yes	—
322	C₄	C₅×#424	1	4	yes	complemented
323	C₄	C₂×C₅×#121	1	1	yes	characteristic
324	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
325	C₄	C₂×C₅×#125	1	1	yes	characteristic
326	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
327	C₂	—	2	4	no	—
328	C₂	C₂×C₅×#389	1	1	yes	characteristic
329	C₂	C₂×C₅×#424	1	1	yes	characteristic
330	C₂	C₅×#2905	1	1	yes	characteristic
331	C₂	C₅×#2935	1	2	yes	—
332	C₂	C₅×#2962	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₅×#1940	C₂	semidirect
6. C₅×#1939	C₂	semidirect
28. #21145	C₅	direct (N × H)
68. #1940	C₂×C₅	semidirect
69. #1939	C₂×C₅	semidirect
218. C₄×C₅	#424	semidirect
306. C₅	#21145	direct (N × H)
322. C₄	C₅×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#424	2
C₂²×C₅×C₄⋊C₈	2
C₂×C₄×C₅×M₁₆	2
C₅×#2285	2
C₅×#1940	2
C₅×#1939	2
C₅×#1729	2
#21145	5

Subgroups — C5×#21145 #11943865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21145 #11943865