Subgroups — C₂×C₅×#2210 #11956295

Order 1280.

786 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2210	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×SD₁₆	—	2	1	no	—
11	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#124	—	2	1	no	—
14	C₂×C₄×C₅×D₄	—	2	1	no	—
15	C₂×C₅×SD₃₂	—	2	4	no	—
16	C₂×C₅×SD₃₂	C₂²	1	4	yes	complemented
17	C₂×C₅×D₁₆	—	2	4	no	—
18	C₂×C₅×D₁₆	C₂²	1	4	yes	—
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#495	—	2	2	no	—
22	C₅×#495	C₂²	1	4	yes	complemented
23	C₄×C₅×D₈	—	2	2	no	—
24	C₄×C₅×D₈	C₂²	1	4	yes	complemented
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#102	—	2	1	no	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
29	C₅×#358	C₂²	1	4	yes	complemented
30	C₅×#357	C₂²	1	4	yes	complemented
31	C₂×#2210	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	—	2	1	no	—
33	C₂²×C₅×D₄	—	2	1	no	—
34	C₂²×C₅×D₄	—	2	1	no	—
35	C₂³×C₄×C₅	—	4	1	no	—
36	C₂×C₅×Q₁₆	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₅×SD₁₆	—	2	4	no	—
41	C₂×C₅×SD₁₆	—	4	4	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	—	2	2	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	—	2	2	no	—
47	C₂×C₅×D₈	D₄	1	4	yes	—
48	C₂×C₅×D₈	C₂³	1	2	yes	—
49	C₂×C₅×D₈	C₂³	1	2	yes	—
50	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₅×#124	—	2	4	no	—
53	C₅×#124	—	4	2	no	—
54	C₄×C₅×D₄	—	2	4	no	—
55	C₄×C₅×D₄	—	4	2	no	—
56	C₂×C₅×C₄⋊C₄	—	2	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
58	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
59	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
60	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
61	C₅×SD₃₂	—	2	8	no	—
62	C₅×D₁₆	—	2	8	no	—
63	C₂×C₅×C₁₆	—	2	2	no	—
64	C₂×C₅×C₁₆	—	2	2	no	—
65	C₂×C₅×C₁₆	C₂³	1	2	yes	—
66	C₂×C₅×C₁₆	C₂³	1	2	yes	—
67	C₅×#107	—	2	1	no	—
68	C₅×#107	C₂³	1	2	yes	—
69	C₅×#102	—	2	4	no	—
70	C₄×C₅×C₈	D₄	1	2	yes	—
71	C₄×C₅×C₈	C₂³	1	2	yes	—
72	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic
74	C₂×#495	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
76	#2210	C₂×C₅	1	8	yes	direct product factor
77	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×#358	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂×#357	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	4	1	no	—
81	C₂⁴×C₅	—	4	1	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂×C₅×Q₈	—	4	2	no	—
86	C₂×C₅×D₄	—	2	4	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂×C₅×D₄	—	4	1	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	2	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	1	no	—
96	C₂²×C₄×C₅	—	4	4	no	—
97	C₂²×C₄×C₅	—	8	4	no	—
98	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₅×Q₁₆	—	2	2	no	—
102	C₅×Q₁₆	—	2	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×SD₁₆	—	4	8	no	—
105	C₅×D₈	—	2	4	no	—
106	C₅×D₈	—	2	2	no	—
107	C₅×D₈	—	2	2	no	—
108	C₅×D₈	—	2	2	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×D₈	C₂×D₄	1	4	yes	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₅×C₈	#41	1	2	yes	—
116	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₅×C₂²⋊C₄	—	4	4	no	—
120	C₅×C₂²⋊C₄	—	4	4	no	—
121	C₅×C₂²⋊C₄	—	4	4	no	—
122	C₄²×C₅	D₈	1	2	yes	—
123	C₄²×C₅	C₂×D₄	1	2	yes	—
124	C₅×C₁₆	—	2	2	no	—
125	C₅×C₁₆	—	2	2	no	—
126	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
127	C₂²×SD₁₆	—	2	1	no	—
128	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
129	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
130	C₂×#124	—	2	1	no	—
131	C₂×C₄×D₄	—	2	1	no	—
132	C₂×SD₃₂	—	2	4	no	—
133	C₂×SD₃₂	C₂²×C₅	1	4	yes	complemented
134	C₂×D₁₆	—	2	4	no	—
135	C₂×D₁₆	C₂²×C₅	1	4	yes	—
136	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
138	#495	—	2	2	no	—
139	#495	C₂²×C₅	1	4	yes	complemented
140	C₄×D₈	—	2	2	no	—
141	C₄×D₈	C₂²×C₅	1	4	yes	complemented
142	C₂×#107	C₂²×C₅	1	1	yes	characteristic
143	C₂×#102	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
145	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
146	#358	C₂²×C₅	1	4	yes	complemented
147	#357	C₂²×C₅	1	4	yes	complemented
148	C₂³×C₅	—	4	2	no	—
149	C₂³×C₅	—	4	4	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	8	2	no	—
152	C₂³×C₅	—	8	2	no	—
153	C₂³×C₅	—	8	2	no	—
154	C₂³×C₅	—	8	2	no	—
155	C₂³×C₅	—	8	2	no	—
156	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
157	C₅×Q₈	—	4	2	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	8	2	no	—
160	C₅×D₄	—	2	4	no	—
161	C₅×D₄	—	4	2	no	—
162	C₅×D₄	—	4	2	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	4	no	—
165	C₅×D₄	—	8	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	8	8	no	—
171	C₂×C₄×C₅	—	8	2	no	—
172	C₂×C₄×C₅	—	8	2	no	—
173	C₂×C₄×C₅	#121	1	2	yes	—
174	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	#121	1	2	yes	—
181	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
182	C₅×C₈	C₂×#41	1	1	yes	characteristic
183	C₂²×Q₈	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂³×C₄	—	4	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×SD₁₆	—	4	4	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	2	no	—
198	C₂×D₈	C₅×D₄	1	4	yes	—
199	C₂×D₈	C₂³×C₅	1	2	yes	—
200	C₂×D₈	C₂³×C₅	1	2	yes	—
201	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
203	#124	—	2	4	no	—
204	#124	—	4	2	no	—
205	C₄×D₄	—	2	4	no	—
206	C₄×D₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	4	1	no	—
209	C₂×C₂²⋊C₄	—	4	1	no	—
210	C₂×C₂²⋊C₄	—	4	1	no	—
211	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
212	SD₃₂	—	2	8	no	—
213	D₁₆	—	2	8	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	C₂³×C₅	1	2	yes	—
217	C₂×C₁₆	C₂³×C₅	1	2	yes	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₅	1	2	yes	—
220	#102	—	2	4	no	—
221	C₄×C₈	C₅×D₄	1	2	yes	—
222	C₄×C₈	C₂³×C₅	1	2	yes	—
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	2	no	—
226	C₂²×C₅	—	8	8	no	—
227	C₂²×C₅	—	8	2	no	—
228	C₂²×C₅	—	8	2	no	—
229	C₂²×C₅	—	8	2	no	—
230	C₂²×C₅	—	16	4	no	—
231	C₂²×C₅	#459	1	2	yes	—
232	C₂²×C₅	#508	1	2	yes	—
233	C₂²×C₅	#509	1	2	yes	—
234	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	8	2	no	—
238	C₄×C₅	—	8	2	no	—
239	C₄×C₅	#459	1	2	yes	—
240	C₄×C₅	C₂×#121	1	1	yes	characteristic
241	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂⁴	—	4	1	no	—
243	C₂⁴	—	4	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	4	no	—
259	C₂²×C₄	—	8	4	no	—
260	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
261	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	4	2	no	—
266	SD₁₆	—	4	8	no	—
267	D₈	—	2	4	no	—
268	D₈	—	2	2	no	—
269	D₈	—	2	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	4	2	no	—
272	D₈	C₂×C₅×D₄	1	4	yes	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
275	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
277	C₂×C₈	C₅×#41	1	2	yes	—
278	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	C₅×D₈	1	2	yes	—
285	C₄²	C₂×C₅×D₄	1	2	yes	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	2	no	—
288	C₂×C₅	—	8	4	no	—
289	C₂×C₅	—	16	2	no	—
290	C₂×C₅	#2210	1	4	yes	direct product factor
291	C₂×C₅	C₂×#459	1	1	yes	characteristic
292	C₂×C₅	C₂×#508	1	1	yes	characteristic
293	C₂×C₅	C₂×#509	1	1	yes	characteristic
294	C₂³	—	4	4	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	2	no	—
300	C₂³	—	8	2	no	—
301	C₂³	—	8	2	no	—
302	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	2	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	8	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	C₅×#121	1	2	yes	—
320	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
321	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
322	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
323	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
324	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	C₅×#121	1	2	yes	—
327	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
328	C₈	C₂×C₅×#41	1	1	yes	characteristic
329	C₅	C₂×#2210	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	4	4	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	8	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	16	4	no	—
338	C₂²	C₅×#459	1	2	yes	—
339	C₂²	C₅×#508	1	2	yes	—
340	C₂²	C₅×#509	1	2	yes	—
341	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	C₅×#459	1	2	yes	—
347	C₄	C₂×C₅×#121	1	1	yes	characteristic
348	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
349	C₂	—	8	4	no	—
350	C₂	—	16	2	no	—
351	C₂	C₅×#2210	1	4	yes	direct product factor
352	C₂	C₂×C₅×#459	1	1	yes	characteristic
353	C₂	C₂×C₅×#508	1	1	yes	characteristic
354	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×SD₃₂	C₂	semidirect
3. C₂×C₅×#495	C₂	semidirect
4. C₂×C₄×C₅×D₈	C₂	semidirect
5. C₅×#2210	C₂	direct (N × H)
6. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
7. C₂×C₅×#358	C₂	semidirect
8. C₂×C₅×#357	C₂	semidirect
16. C₂×C₅×SD₃₂	C₂²	semidirect
22. C₅×#495	C₂²	semidirect
24. C₄×C₅×D₈	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
30. C₅×#357	C₂²	semidirect
31. C₂×#2210	C₅	direct (N × H)
72. C₂²×SD₃₂	C₂×C₅	semidirect
74. C₂×#495	C₂×C₅	semidirect
75. C₂×C₄×D₈	C₂×C₅	semidirect
76. #2210	C₂×C₅	direct (N × H)
77. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
78. C₂×#358	C₂×C₅	semidirect
79. C₂×#357	C₂×C₅	semidirect
133. C₂×SD₃₂	C₂²×C₅	semidirect
139. #495	C₂²×C₅	semidirect
141. C₄×D₈	C₂²×C₅	semidirect
145. C₄⋊C₁₆	C₂²×C₅	semidirect
146. #358	C₂²×C₅	semidirect
147. #357	C₂²×C₅	semidirect
290. C₂×C₅	#2210	direct (N × H)
329. C₅	C₂×#2210	direct (N × H)
351. C₂	C₅×#2210	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×SD₃₂	2
C₂²×C₅×D₁₆	2
C₂×C₅×#495	2
C₂×C₄×C₅×D₈	2
C₅×#2210	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×C₅×#357	2
C₂×#2210	5

Subgroups — C2×C5×#2210 #11956295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2210 #11956295