Subgroups — C₂×C₅×#1271 #11930279

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#389	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
4	C₅×#1271	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×C₈	—	2	1	no	—
6	C₂³×C₅×C₈	—	2	1	no	—
7	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
8	C₂²×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#389	—	2	1	no	—
12	C₅×#389	—	2	1	no	—
13	C₅×#389	C₄	1	2	yes	—
14	C₅×#389	C₂²	1	2	yes	—
15	C₂×C₅×#95	—	2	1	no	—
16	C₅×#336	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₂²	1	2	yes	—
19	C₅×#336	C₂²	1	2	yes	—
20	C₂×#1271	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	1	no	—
22	C₂³×C₄×C₅	—	2	1	no	—
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	4	2	no	—
31	C₂²×C₅×C₈	—	4	2	no	—
32	C₂²×C₅×C₈	—	4	2	no	—
33	C₂²×C₅×C₈	—	4	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	4	2	no	—
40	C₂×C₅×C₄⋊C₄	—	4	2	no	—
41	C₂×C₅×C₄⋊C₄	—	4	2	no	—
42	C₂×C₅×C₄⋊C₄	C₈	1	4	yes	complemented
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	—	4	2	no	—
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	C₂×#389	C₂×C₅	1	1	yes	characteristic
53	C₂×#336	C₂×C₅	1	1	yes	characteristic
54	C₂×#336	C₂×C₅	1	1	yes	characteristic
55	#1271	C₂×C₅	1	4	yes	direct product factor
56	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	—	4	2	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
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×C₈	1	2	yes	—
75	C₂²×C₄×C₅	M₁₆	1	2	yes	—
76	C₂²×C₄×C₅	D₈	1	2	yes	—
77	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	4	8	no	—
86	C₂×C₅×C₈	—	4	8	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₂×C₅×C₈	—	4	2	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	complemented
99	C₄²×C₅	—	4	1	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₂³×C₈	—	2	1	no	—
104	C₂³×C₈	—	2	1	no	—
105	C₂²×C₄⋊C₄	—	2	1	no	—
106	C₂²×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
107	C₂²×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
108	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
109	#389	—	2	1	no	—
110	#389	—	2	1	no	—
111	#389	C₄×C₅	1	2	yes	—
112	#389	C₂²×C₅	1	2	yes	—
113	C₂×#95	—	2	1	no	—
114	#336	—	2	1	no	—
115	#336	—	2	1	no	—
116	#336	C₂²×C₅	1	2	yes	—
117	#336	C₂²×C₅	1	2	yes	—
118	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
119	C₂³×C₅	#99	1	2	yes	—
120	C₂³×C₅	#101	1	2	yes	—
121	C₂³×C₅	#102	1	2	yes	—
122	C₂³×C₅	#103	1	2	yes	—
123	C₂³×C₅	#104	1	2	yes	—
124	C₂³×C₅	#104	1	2	yes	—
125	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
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	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	4	8	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	8	4	no	—
152	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
153	C₂×C₄×C₅	#102	1	2	yes	—
154	C₂×C₄×C₅	#103	1	2	yes	—
155	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
161	C₅×C₈	—	4	4	no	—
162	C₅×C₈	—	4	4	no	—
163	C₂³×C₄	—	2	1	no	—
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
167	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	4	2	no	—
173	C₂²×C₈	—	4	2	no	—
174	C₂²×C₈	—	4	2	no	—
175	C₂²×C₈	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	4	2	no	—
182	C₂×C₄⋊C₄	—	4	2	no	—
183	C₂×C₄⋊C₄	—	4	2	no	—
184	C₂×C₄⋊C₄	C₅×C₈	1	4	yes	complemented
185	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
187	C₂×C₄²	—	4	2	no	—
188	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
190	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂²×C₅	#324	1	4	yes	—
195	C₂²×C₅	D₄⋊C₈	1	4	yes	—
196	C₂²×C₅	Q₈⋊C₈	1	4	yes	—
197	C₂²×C₅	#327	1	4	yes	—
198	C₂²×C₅	#328	1	4	yes	—
199	C₂²×C₅	#330	1	4	yes	—
200	C₂²×C₅	#332	1	4	yes	—
201	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
202	C₂²×C₅	C₂×#99	1	1	yes	characteristic
203	C₂²×C₅	C₂×#101	1	1	yes	characteristic
204	C₂²×C₅	C₂×#102	1	1	yes	characteristic
205	C₂²×C₅	C₂×#103	1	1	yes	characteristic
206	C₂²×C₅	C₂×#104	1	1	yes	characteristic
207	C₂²×C₅	C₂×#104	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	8	2	no	—
219	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
238	C₂²×C₄	C₅×M₁₆	1	2	yes	—
239	C₂²×C₄	C₅×D₈	1	2	yes	—
240	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
241	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
242	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂×C₈	—	4	8	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	complemented
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	1	no	—
266	C₂×C₅	#1271	1	8	yes	direct product factor
267	C₂×C₅	C₂×#324	1	1	yes	characteristic
268	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
269	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
270	C₂×C₅	C₂×#327	1	1	yes	characteristic
271	C₂×C₅	C₂×#328	1	1	yes	characteristic
272	C₂×C₅	C₂×#330	1	1	yes	characteristic
273	C₂×C₅	C₂×#332	1	1	yes	characteristic
274	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
275	C₂³	C₅×#99	1	2	yes	—
276	C₂³	C₅×#101	1	2	yes	—
277	C₂³	C₅×#102	1	2	yes	—
278	C₂³	C₅×#103	1	2	yes	—
279	C₂³	C₅×#104	1	2	yes	—
280	C₂³	C₅×#104	1	2	yes	—
281	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
309	C₂×C₄	C₅×#102	1	2	yes	—
310	C₂×C₄	C₅×#103	1	2	yes	—
311	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
313	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
314	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
317	C₈	—	4	4	no	—
318	C₈	—	4	4	no	—
319	C₅	C₂×#1271	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	C₅×#324	1	4	yes	—
321	C₂²	C₅×D₄⋊C₈	1	4	yes	—
322	C₂²	C₅×Q₈⋊C₈	1	4	yes	—
323	C₂²	C₅×#327	1	4	yes	—
324	C₂²	C₅×#328	1	4	yes	—
325	C₂²	C₅×#330	1	4	yes	—
326	C₂²	C₅×#332	1	4	yes	—
327	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
328	C₂²	C₂×C₅×#99	1	1	yes	characteristic
329	C₂²	C₂×C₅×#101	1	1	yes	characteristic
330	C₂²	C₂×C₅×#102	1	1	yes	characteristic
331	C₂²	C₂×C₅×#103	1	1	yes	characteristic
332	C₂²	C₂×C₅×#104	1	1	yes	characteristic
333	C₂²	C₂×C₅×#104	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₂	C₅×#1271	1	8	yes	direct product factor
346	C₂	C₂×C₅×#324	1	1	yes	characteristic
347	C₂	C₂×C₅×D₄⋊C₈	1	1	yes	characteristic
348	C₂	C₂×C₅×Q₈⋊C₈	1	1	yes	characteristic
349	C₂	C₂×C₅×#327	1	1	yes	characteristic
350	C₂	C₂×C₅×#328	1	1	yes	characteristic
351	C₂	C₂×C₅×#330	1	1	yes	characteristic
352	C₂	C₂×C₅×#332	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1271	C₂	direct (N × H)
20. C₂×#1271	C₅	direct (N × H)
42. C₂×C₅×C₄⋊C₄	C₈	semidirect
55. #1271	C₂×C₅	direct (N × H)
98. C₅×C₄⋊C₄	C₂×C₈	semidirect
184. C₂×C₄⋊C₄	C₅×C₈	semidirect
261. C₄⋊C₄	C₂×C₅×C₈	semidirect
266. C₂×C₅	#1271	direct (N × H)
319. C₅	C₂×#1271	direct (N × H)
345. C₂	C₅×#1271	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×#389	2
C₂×C₅×#336	2
C₅×#1271	2
C₂×#1271	5

Subgroups — C2×C5×#1271 #11930279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1271 #11930279