Subgroups — C₅×#10836 #11933556

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×H(3,ℤ/4)	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂×C₅×#117	C₂²	1	2	yes	—
7	C₂×C₅×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#423	—	2	2	no	—
10	C₅×#423	—	2	1	no	—
11	C₅×#423	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
13	C₅×#384	—	2	1	no	—
14	C₅×#382	—	2	1	no	—
15	C₅×#380	—	2	2	no	—
16	C₄×C₅×C₄⋊C₄	—	2	2	no	—
17	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	complemented
18	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
19	C₅×#376	C₄	1	2	yes	complemented
20	C₄³×C₅	C₄	1	2	yes	complemented
21	C₅×H(3,ℤ/4)	—	2	8	no	—
22	#10836	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₅	D₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₅×#117	—	2	2	no	—
31	C₅×#117	—	2	2	no	—
32	C₅×#117	—	2	4	no	—
33	C₅×#117	—	2	4	no	—
34	C₅×#117	D₄	1	4	yes	—
35	C₅×#117	Q₈	1	4	yes	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₄²×C₅	—	2	4	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	D₄	1	2	yes	—
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	C₅×#95	—	2	4	no	—
56	C₅×#95	—	2	4	no	—
57	C₂×#423	C₂×C₅	1	1	yes	characteristic
58	#2285	C₂×C₅	1	1	yes	characteristic
59	C₄×#117	C₂×C₅	1	1	yes	characteristic
60	C₂×H(3,ℤ/4)	C₂×C₅	1	4	yes	—
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₅	#41	1	2	yes	—
80	C₂²×C₄×C₅	#41	1	2	yes	—
81	C₅×M₁₆	—	4	4	no	—
82	C₅×M₁₆	—	4	2	no	—
83	C₅×M₁₆	—	4	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	4	8	no	—
89	C₅×C₄⋊C₄	—	4	2	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₅×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	8	no	—
96	C₄²×C₅	—	4	8	no	—
97	C₄²×C₅	C₄²	1	2	yes	—
98	C₄²×C₅	C₄²	1	2	yes	—
99	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₄⋊C₄	1	2	yes	—
105	C₄²×C₅	C₄⋊C₄	1	2	yes	—
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	8	yes	complemented
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂×#117	C₂²×C₅	1	2	yes	—
110	C₂×#117	C₂²×C₅	1	2	yes	—
111	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
112	#423	—	2	2	no	—
113	#423	—	2	1	no	—
114	#423	—	2	1	no	—
115	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
116	#384	—	2	1	no	—
117	#382	—	2	1	no	—
118	#380	—	2	2	no	—
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₄⋊C₄	C₄×C₅	1	4	yes	complemented
121	C₄×C₂²⋊C₄	—	2	2	no	—
122	#376	C₄×C₅	1	2	yes	complemented
123	C₄³	C₄×C₅	1	2	yes	complemented
124	H(3,ℤ/4)	—	2	8	no	—
125	C₂³×C₅	—	4	2	no	—
126	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅	#121	1	2	yes	—
130	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	8	no	—
139	C₂×C₄×C₅	—	4	8	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	#95	1	2	yes	—
143	C₂×C₄×C₅	#95	1	2	yes	—
144	C₂×C₄×C₅	#95	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#117	1	2	yes	—
151	C₂×C₄×C₅	C₄×D₄	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	C₄×Q₈	1	2	yes	—
155	C₂×C₄×C₅	#121	1	2	yes	—
156	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
157	C₂×C₄×C₅	#124	1	1	yes	characteristic
158	C₂×C₄×C₅	#124	1	1	yes	characteristic
159	C₂×C₄×C₅	#125	1	2	yes	—
160	C₂×C₄×C₅	#127	1	1	yes	characteristic
161	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
162	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
163	C₅×C₈	—	4	4	no	—
164	C₂³×C₄	C₅×D₄	1	2	yes	—
165	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
171	#117	—	2	2	no	—
172	#117	—	2	2	no	—
173	#117	—	2	4	no	—
174	#117	—	2	4	no	—
175	#117	C₅×D₄	1	4	yes	—
176	#117	C₅×Q₈	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
185	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
189	C₂×C₄²	C₅×D₄	1	2	yes	—
190	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
191	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
192	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
193	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	4	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	4	2	no	—
202	C₂²×C₅	—	4	2	no	—
203	C₂²×C₅	C₂×#95	1	1	yes	characteristic
204	C₂²×C₅	#380	1	2	yes	—
205	C₂²×C₅	#389	1	2	yes	—
206	C₂²×C₅	#390	1	1	yes	characteristic
207	C₂²×C₅	#391	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	4	8	no	—
212	C₄×C₅	—	8	4	no	—
213	C₄×C₅	H(3,ℤ/4)	1	4	yes	complemented
214	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
215	C₄×C₅	#382	1	1	yes	characteristic
216	C₄×C₅	#384	1	1	yes	characteristic
217	C₄×C₅	#389	1	2	yes	—
218	C₄×C₅	#390	1	1	yes	characteristic
219	C₄×C₅	#391	1	1	yes	characteristic
220	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
225	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
226	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
228	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
229	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
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	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
236	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
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	M₁₆	—	4	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	8	no	—
255	C₄²	—	4	8	no	—
256	C₄²	C₄²×C₅	1	2	yes	—
257	C₄²	C₄²×C₅	1	2	yes	—
258	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
259	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
260	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
263	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₄²	C₅×C₄⋊C₄	1	2	yes	—
265	C₄²	C₅×C₄⋊C₄	1	2	yes	—
266	C₄²	C₅×C₄⋊C₄	1	8	yes	complemented
267	C₂×C₅	—	2	2	no	—
268	C₂×C₅	—	2	1	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
272	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
277	C₂³	C₅×#121	1	2	yes	—
278	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₅×#95	1	2	yes	—
290	C₂×C₄	C₅×#95	1	2	yes	—
291	C₂×C₄	C₅×#95	1	2	yes	—
292	C₂×C₄	C₅×#95	1	2	yes	—
293	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
294	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
296	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₅×#117	1	2	yes	—
299	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
300	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
301	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
302	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
303	C₂×C₄	C₅×#121	1	2	yes	—
304	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
305	C₂×C₄	C₅×#124	1	1	yes	characteristic
306	C₂×C₄	C₅×#124	1	1	yes	characteristic
307	C₂×C₄	C₅×#125	1	2	yes	—
308	C₂×C₄	C₅×#127	1	1	yes	characteristic
309	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
310	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₅	#10836	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₂×C₅×#95	1	1	yes	characteristic
319	C₂²	C₅×#380	1	2	yes	—
320	C₂²	C₅×#389	1	2	yes	—
321	C₂²	C₅×#390	1	1	yes	characteristic
322	C₂²	C₅×#391	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	8	4	no	—
328	C₄	C₅×H(3,ℤ/4)	1	4	yes	complemented
329	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
330	C₄	C₅×#382	1	1	yes	characteristic
331	C₄	C₅×#384	1	1	yes	characteristic
332	C₄	C₅×#389	1	2	yes	—
333	C₄	C₅×#390	1	1	yes	characteristic
334	C₄	C₅×#391	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
340	C₂	C₂×C₅×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₅×C₄⋊C₈	C₄	semidirect
17. C₄×C₅×C₄⋊C₄	C₄	semidirect
19. C₅×#376	C₄	semidirect
20. C₄³×C₅	C₄	semidirect
22. #10836	C₅	direct (N × H)
107. C₄²×C₅	C₄⋊C₄	semidirect
115. C₂×C₄⋊C₈	C₄×C₅	semidirect
120. C₄×C₄⋊C₄	C₄×C₅	semidirect
122. #376	C₄×C₅	semidirect
123. C₄³	C₄×C₅	semidirect
213. C₄×C₅	H(3,ℤ/4)	semidirect
266. C₄²	C₅×C₄⋊C₄	semidirect
312. C₅	#10836	direct (N × H)
328. C₄	C₅×H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
C₅×#2285	2
C₄×C₅×#117	2
C₂×C₅×H(3,ℤ/4)	2
#10836	5

Subgroups — C5×#10836 #11933556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10836 #11933556