Subgroups — C₂×C₅×#2211 #11956298

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2211	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
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×SD₁₆	—	2	1	no	—
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#124	—	2	1	no	—
14	C₂×C₄×C₅×Q₈	—	2	1	no	—
15	C₂×C₅×Q₃₂	—	2	4	no	—
16	C₂×C₅×Q₃₂	C₂²	1	4	yes	complemented
17	C₂×C₅×SD₃₂	—	2	4	no	—
18	C₂×C₅×SD₃₂	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	—
23	C₄×C₅×Q₁₆	—	2	2	no	—
24	C₄×C₅×Q₁₆	C₂²	1	4	yes	complemented
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#103	—	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	—
31	C₂×#2211	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	—	2	1	no	—
33	C₂²×C₅×Q₈	—	2	1	no	—
34	C₂²×C₅×D₄	—	2	1	no	—
35	C₂×C₅×Q₁₆	—	2	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	—	2	2	no	—
40	C₂×C₅×Q₁₆	D₄	1	4	yes	—
41	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₅×SD₁₆	—	2	4	no	—
44	C₂×C₅×SD₁₆	—	4	4	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₂×C₅×D₈	—	2	2	no	—
48	C₂×C₅×D₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×#124	—	2	4	no	—
52	C₅×#124	—	4	2	no	—
53	C₄×C₅×Q₈	—	2	4	no	—
54	C₄×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	—	4	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
58	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
59	C₂×C₄²×C₅	—	4	1	no	—
60	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
61	C₅×Q₃₂	—	2	8	no	—
62	C₅×SD₃₂	—	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×#103	—	2	4	no	—
70	C₄×C₅×C₈	D₄	1	2	yes	—
71	C₄×C₅×C₈	C₂³	1	2	yes	—
72	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic
74	C₂×#495	C₂×C₅	1	1	yes	characteristic
75	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
76	#2211	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
80	C₂⁴×C₅	—	4	1	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	4	2	no	—
88	C₂×C₅×Q₈	—	4	2	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	—	4	2	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂²×C₄×C₅	—	4	1	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	1	no	—
96	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₅×Q₁₆	—	2	4	no	—
100	C₅×Q₁₆	—	2	2	no	—
101	C₅×Q₁₆	—	2	2	no	—
102	C₅×Q₁₆	—	2	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×Q₁₆	C₂×D₄	1	4	yes	—
105	C₅×SD₁₆	—	4	8	no	—
106	C₅×D₈	—	2	2	no	—
107	C₅×D₈	—	2	2	no	—
108	C₅×D₈	—	4	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₅×C₈	#41	1	2	yes	—
114	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	4	1	no	—
117	C₅×C₄⋊C₄	—	4	4	no	—
118	C₅×C₂²⋊C₄	—	4	4	no	—
119	C₅×C₂²⋊C₄	—	4	4	no	—
120	C₄²×C₅	—	4	4	no	—
121	C₄²×C₅	D₈	1	2	yes	—
122	C₄²×C₅	C₂×D₄	1	2	yes	—
123	C₅×C₁₆	—	2	2	no	—
124	C₅×C₁₆	—	2	2	no	—
125	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
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	C₂×#124	—	2	1	no	—
130	C₂×C₄×Q₈	—	2	1	no	—
131	C₂×Q₃₂	—	2	4	no	—
132	C₂×Q₃₂	C₂²×C₅	1	4	yes	complemented
133	C₂×SD₃₂	—	2	4	no	—
134	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
135	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
136	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
137	#495	—	2	2	no	—
138	#495	C₂²×C₅	1	4	yes	—
139	C₄×Q₁₆	—	2	2	no	—
140	C₄×Q₁₆	C₂²×C₅	1	4	yes	complemented
141	C₂×#107	C₂²×C₅	1	1	yes	characteristic
142	C₂×#103	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
144	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
145	#358	C₂²×C₅	1	4	yes	complemented
146	#357	C₂²×C₅	1	4	yes	—
147	C₂³×C₅	—	4	2	no	—
148	C₂³×C₅	—	8	2	no	—
149	C₂³×C₅	—	8	2	no	—
150	C₂³×C₅	—	8	2	no	—
151	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
152	C₅×Q₈	—	2	4	no	—
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	—	4	4	no	—
155	C₅×Q₈	—	4	2	no	—
156	C₅×Q₈	—	4	2	no	—
157	C₅×Q₈	—	8	2	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	4	2	no	—
160	C₅×D₄	—	8	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	4	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	8	2	no	—
168	C₂×C₄×C₅	—	8	2	no	—
169	C₂×C₄×C₅	#121	1	2	yes	—
170	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
175	C₅×C₈	—	2	2	no	—
176	C₅×C₈	#121	1	2	yes	—
177	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
178	C₅×C₈	C₂×#41	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	C₅×D₄	1	4	yes	—
188	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
189	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×SD₁₆	—	4	4	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	C₂³×C₅	1	2	yes	—
196	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
198	#124	—	2	4	no	—
199	#124	—	4	2	no	—
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	—	4	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	C₂×C₂²⋊C₄	—	4	1	no	—
206	C₂×C₄²	—	4	1	no	—
207	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
208	Q₃₂	—	2	8	no	—
209	SD₃₂	—	2	8	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	C₂³×C₅	1	2	yes	—
213	C₂×C₁₆	C₂³×C₅	1	2	yes	—
214	#107	—	2	1	no	—
215	#107	C₂³×C₅	1	2	yes	—
216	#103	—	2	4	no	—
217	C₄×C₈	C₅×D₄	1	2	yes	—
218	C₄×C₈	C₂³×C₅	1	2	yes	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	—	8	2	no	—
222	C₂²×C₅	—	16	4	no	—
223	C₂²×C₅	#459	1	2	yes	—
224	C₂²×C₅	#508	1	2	yes	—
225	C₂²×C₅	#510	1	2	yes	—
226	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	—	8	2	no	—
231	C₄×C₅	—	8	2	no	—
232	C₄×C₅	#459	1	2	yes	—
233	C₄×C₅	C₂×#121	1	1	yes	characteristic
234	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂⁴	—	4	1	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
252	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	C₂×C₅×D₄	1	4	yes	—
260	SD₁₆	—	4	8	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	2	no	—
263	D₈	—	4	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
266	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
268	C₂×C₈	C₅×#41	1	2	yes	—
269	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₂²⋊C₄	—	4	4	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₄²	—	4	4	no	—
276	C₄²	C₅×D₈	1	2	yes	—
277	C₄²	C₂×C₅×D₄	1	2	yes	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	2	no	—
280	C₂×C₅	—	16	2	no	—
281	C₂×C₅	#2211	1	4	yes	direct product factor
282	C₂×C₅	C₂×#459	1	1	yes	characteristic
283	C₂×C₅	C₂×#508	1	1	yes	characteristic
284	C₂×C₅	C₂×#510	1	1	yes	characteristic
285	C₂³	—	4	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₅×#121	1	2	yes	—
308	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
309	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
310	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
311	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
312	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	C₅×#121	1	2	yes	—
315	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
316	C₈	C₂×C₅×#41	1	1	yes	characteristic
317	C₅	C₂×#2211	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	4	no	—
322	C₂²	C₅×#459	1	2	yes	—
323	C₂²	C₅×#508	1	2	yes	—
324	C₂²	C₅×#510	1	2	yes	—
325	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	C₅×#459	1	2	yes	—
332	C₄	C₂×C₅×#121	1	1	yes	characteristic
333	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	—	16	2	no	—
335	C₂	C₅×#2211	1	4	yes	direct product factor
336	C₂	C₂×C₅×#459	1	1	yes	characteristic
337	C₂	C₂×C₅×#508	1	1	yes	characteristic
338	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
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×Q₃₂	C₂	semidirect
4. C₂×C₄×C₅×Q₁₆	C₂	semidirect
5. C₅×#2211	C₂	direct (N × H)
6. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
7. C₂×C₅×#358	C₂	semidirect
16. C₂×C₅×Q₃₂	C₂²	semidirect
24. C₄×C₅×Q₁₆	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
31. C₂×#2211	C₅	direct (N × H)
72. C₂²×Q₃₂	C₂×C₅	semidirect
75. C₂×C₄×Q₁₆	C₂×C₅	semidirect
76. #2211	C₂×C₅	direct (N × H)
77. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
78. C₂×#358	C₂×C₅	semidirect
132. C₂×Q₃₂	C₂²×C₅	semidirect
140. C₄×Q₁₆	C₂²×C₅	semidirect
144. C₄⋊C₁₆	C₂²×C₅	semidirect
145. #358	C₂²×C₅	semidirect
281. C₂×C₅	#2211	direct (N × H)
317. C₅	C₂×#2211	direct (N × H)
335. C₂	C₅×#2211	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×Q₃₂	2
C₂²×C₅×SD₃₂	2
C₂×C₅×#495	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2211	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×C₅×#357	2
C₂×#2211	5

Subgroups — C2×C5×#2211 #11956298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2211 #11956298