Subgroups — C₂×C₅×#1811 #11943386

Order 1280.

1438 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#407	C₂	1	2	yes	maximal
4	C₅×#1811	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#341	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	—	2	8	no	—
8	C₂²×C₅×M₁₆	C₄	1	4	yes	—
9	C₂²×C₅×M₁₆	C₄	1	4	yes	—
10	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#108	—	2	8	no	—
15	C₂×C₅×#108	C₂²	1	8	yes	complemented
16	C₅×#407	—	2	4	no	—
17	C₅×#407	C₂²	1	8	yes	—
18	C₂×C₅×C₂²⋊C₈	—	2	4	no	—
19	C₅×#341	C₂²	1	8	yes	complemented
20	C₂×#1811	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₅	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	—	2	4	no	—
23	C₂³×C₄×C₅	—	2	4	no	—
24	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	D₄	1	2	yes	—
27	C₂³×C₄×C₅	D₄	1	2	yes	—
28	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₅	Q₈	1	2	yes	—
30	C₂³×C₄×C₅	C₂³	1	2	yes	—
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×M₁₆	—	2	4	no	—
33	C₂×C₅×M₁₆	—	2	16	no	—
34	C₂×C₅×M₁₆	—	2	16	no	—
35	C₂×C₅×M₁₆	—	2	16	no	—
36	C₂×C₅×M₁₆	—	2	8	no	—
37	C₂×C₅×M₁₆	—	2	8	no	—
38	C₂×C₅×M₁₆	C₂×C₄	1	16	yes	—
39	C₂×C₅×M₁₆	C₂³	1	8	yes	—
40	C₂²×C₅×C₈	—	2	4	no	—
41	C₂²×C₅×C₈	—	2	4	no	—
42	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₅×C₈	D₄	1	8	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	2	yes	—
49	C₅×#108	—	2	16	no	—
50	C₅×C₂²⋊C₈	—	2	16	no	—
51	C₂³×M₁₆	C₂×C₅	1	1	yes	characteristic
52	C₂²×#108	C₂×C₅	1	2	yes	complemented
53	C₂×#407	C₂×C₅	1	2	yes	—
54	#1811	C₂×C₅	1	8	yes	direct product factor
55	C₂×#341	C₂×C₅	1	2	yes	complemented
56	C₂⁴×C₅	—	2	4	no	—
57	C₂⁴×C₅	—	4	4	no	—
58	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
59	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
60	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂⁴×C₅	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	8	no	—
68	C₂²×C₄×C₅	—	2	8	no	—
69	C₂²×C₄×C₅	—	2	8	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
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	4	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	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	#41	1	4	yes	—
86	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₅×M₁₆	—	2	32	no	—
88	C₅×M₁₆	—	2	8	no	—
89	C₅×M₁₆	—	2	8	no	—
90	C₅×M₁₆	—	4	8	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	8	no	—
95	C₂×C₅×C₈	—	4	8	no	—
96	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
97	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
98	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
101	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
102	C₂²×M₁₆	—	2	8	no	—
103	C₂²×M₁₆	C₄×C₅	1	4	yes	—
104	C₂²×M₁₆	C₄×C₅	1	4	yes	—
105	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
106	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
107	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×#108	—	2	8	no	—
110	C₂×#108	C₂²×C₅	1	8	yes	complemented
111	#407	—	2	4	no	—
112	#407	C₂²×C₅	1	8	yes	—
113	C₂×C₂²⋊C₈	—	2	4	no	—
114	#341	C₂²×C₅	1	8	yes	complemented
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	2	4	no	—
119	C₂³×C₅	—	2	8	no	—
120	C₂³×C₅	—	4	4	no	—
121	C₂³×C₅	—	4	4	no	—
122	C₂³×C₅	—	4	16	no	—
123	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
124	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₅	C₂²⋊Q₈	1	4	yes	—
127	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	2	8	no	—
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	#117	1	2	yes	—
142	C₂×C₄×C₅	#117	1	2	yes	—
143	C₂×C₄×C₅	#121	1	4	yes	—
144	C₂×C₄×C₅	#121	1	4	yes	—
145	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
146	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×#41	1	2	yes	—
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	—	4	8	no	—
152	C₅×C₈	C₂×C₂²⋊C₄	1	8	yes	—
153	C₂⁵	C₅×Q₈	1	1	yes	characteristic
154	C₂³×C₄	—	2	4	no	—
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
157	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
158	C₂³×C₄	C₅×D₄	1	2	yes	—
159	C₂³×C₄	C₅×D₄	1	2	yes	—
160	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
161	C₂³×C₄	C₅×Q₈	1	2	yes	—
162	C₂³×C₄	C₂³×C₅	1	2	yes	—
163	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	16	no	—
166	C₂×M₁₆	—	2	16	no	—
167	C₂×M₁₆	—	2	16	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	C₂×C₄×C₅	1	16	yes	—
171	C₂×M₁₆	C₂³×C₅	1	8	yes	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
175	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
176	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
177	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
178	C₂²×C₈	C₅×D₄	1	8	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₂²×C₈	C₂³×C₅	1	2	yes	—
181	#108	—	2	16	no	—
182	C₂²⋊C₈	—	2	16	no	—
183	C₂²×C₅	—	2	8	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	8	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	4	8	no	—
189	C₂²×C₅	—	4	8	no	—
190	C₂²×C₅	—	4	8	no	—
191	C₂²×C₅	#380	1	2	yes	—
192	C₂²×C₅	#380	1	2	yes	—
193	C₂²×C₅	#430	1	4	yes	—
194	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
196	C₄×C₅	—	2	4	no	—
197	C₄×C₅	#380	1	2	yes	—
198	C₄×C₅	#380	1	2	yes	—
199	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₅	C₂×#117	1	1	yes	characteristic
201	C₄×C₅	C₂×#121	1	2	yes	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	4	no	—
204	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
205	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
206	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
207	C₂⁴	C₂×C₅×Q₈	1	2	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
219	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
220	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
221	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
226	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
227	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
230	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
231	C₂²×C₄	C₅×#41	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
233	M₁₆	—	2	32	no	—
234	M₁₆	—	2	8	no	—
235	M₁₆	—	2	8	no	—
236	M₁₆	—	4	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	4	8	no	—
242	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
243	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
244	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
245	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
246	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	4	4	no	—
250	C₂×C₅	#1811	1	4	yes	direct product factor
251	C₂×C₅	C₂×#380	1	1	yes	characteristic
252	C₂×C₅	C₂×#430	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	8	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	16	no	—
261	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
262	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
263	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂³	C₅×C₂²⋊Q₈	1	4	yes	—
265	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
275	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
276	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
277	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₅×#117	1	2	yes	—
280	C₂×C₄	C₅×#117	1	2	yes	—
281	C₂×C₄	C₅×#121	1	4	yes	—
282	C₂×C₄	C₅×#121	1	4	yes	—
283	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
284	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
286	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#41	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	8	no	—
290	C₈	C₂×C₅×C₂²⋊C₄	1	8	yes	—
291	C₅	C₂×#1811	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	C₅×#380	1	2	yes	—
301	C₂²	C₅×#380	1	2	yes	—
302	C₂²	C₅×#430	1	4	yes	—
303	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	C₅×#380	1	2	yes	—
307	C₄	C₅×#380	1	2	yes	—
308	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
309	C₄	C₂×C₅×#117	1	1	yes	characteristic
310	C₄	C₂×C₅×#121	1	2	yes	—
311	C₂	—	2	2	no	—
312	C₂	—	2	2	no	—
313	C₂	—	4	4	no	—
314	C₂	C₅×#1811	1	4	yes	direct product factor
315	C₂	C₂×C₅×#380	1	1	yes	characteristic
316	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁵×C₅	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#108	C₂	semidirect
4. C₅×#1811	C₂	direct (N × H)
5. C₂×C₅×#341	C₂	semidirect
15. C₂×C₅×#108	C₂²	semidirect
19. C₅×#341	C₂²	semidirect
20. C₂×#1811	C₅	direct (N × H)
52. C₂²×#108	C₂×C₅	semidirect
54. #1811	C₂×C₅	direct (N × H)
55. C₂×#341	C₂×C₅	semidirect
110. C₂×#108	C₂²×C₅	semidirect
114. #341	C₂²×C₅	semidirect
250. C₂×C₅	#1811	direct (N × H)
291. C₅	C₂×#1811	direct (N × H)
314. C₂	C₅×#1811	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×M₁₆	2
C₂²×C₅×#108	2
C₂×C₅×#407	2
C₅×#1811	2
C₂×C₅×#341	2
C₂×#1811	5

Subgroups — C2×C5×#1811 #11943386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1811 #11943386