Subgroups — C₅×#33581 #11956301

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	2	yes	maximal, complemented
2	C₅×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2211	C₂	1	4	yes	maximal, complemented
5	C₅×#2209	C₂	1	2	yes	maximal, complemented
6	C₅×#2207	C₂	1	2	yes	maximal, complemented
7	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2139	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#135	—	2	1	no	—
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
12	C₅×#532	—	2	1	no	—
13	C₂×C₄×C₅×Q₈	—	2	1	no	—
14	C₅×#510	—	2	8	no	—
15	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
16	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
17	C₂×C₅×M₃₂	C₂²	1	2	yes	—
18	C₅×#495	—	2	1	no	—
19	C₅×#495	C₂²	1	2	yes	—
20	C₅×#494	C₂²	1	1	yes	characteristic
21	C₅×#493	C₂²	1	1	yes	characteristic
22	C₅×#492	—	2	1	no	—
23	C₄×C₅×Q₁₆	—	2	1	no	—
24	C₄×C₅×Q₁₆	—	2	1	no	—
25	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
27	C₂×C₅×#107	C₂²	1	1	yes	characteristic
28	C₂×C₅×#103	—	2	1	no	—
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₅×#358	C₂²	1	4	yes	—
33	C₅×#357	C₂²	1	4	yes	—
34	#33581	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅×#41	—	2	2	no	—
36	C₂²×C₅×Q₈	—	2	1	no	—
37	C₅×#135	—	2	4	no	—
38	C₅×#135	—	2	4	no	—
39	C₅×#135	—	4	4	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×Q₁₆	—	2	1	no	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×Q₁₆	D₄	1	4	yes	complemented
44	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
46	C₂×C₅×SD₁₆	—	2	2	no	—
47	C₂×C₅×SD₁₆	—	2	2	no	—
48	C₂×C₅×D₈	—	2	1	no	—
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×C₄⋊Q₈	—	2	1	no	—
53	C₅×#127	—	2	1	no	—
54	C₅×#124	—	2	2	no	—
55	C₅×#121	—	4	1	no	—
56	C₅×#121	—	4	1	no	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₄×C₅×Q₈	—	2	2	no	—
59	C₄×C₅×Q₈	—	4	1	no	—
60	C₄×C₅×Q₈	—	4	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₄×C₅×D₄	—	4	1	no	—
63	C₂×C₅×C₄⋊C₄	—	2	1	no	—
64	C₂×C₅×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄²×C₅	—	4	1	no	—
66	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
67	C₅×Q₃₂	—	2	8	no	—
68	C₅×SD₃₂	—	2	8	no	—
69	C₅×M₃₂	—	2	4	no	—
70	C₂×C₅×C₁₆	C₂³	1	4	yes	—
71	C₅×#107	—	2	1	no	—
72	C₅×#107	C₂³	1	2	yes	—
73	C₅×#103	—	2	2	no	—
74	C₅×#103	—	2	2	no	—
75	C₄×C₅×C₈	D₄	1	1	yes	characteristic
76	C₄×C₅×C₈	D₄	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₂×#510	C₂×C₅	1	2	yes	complemented
80	#3147	C₂×C₅	1	1	yes	characteristic
81	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
82	#2211	C₂×C₅	1	4	yes	complemented
83	#2209	C₂×C₅	1	2	yes	complemented
84	#2207	C₂×C₅	1	2	yes	complemented
85	#2150	C₂×C₅	1	1	yes	characteristic, complemented
86	#2139	C₂×C₅	1	2	yes	complemented
87	C₅×#41	—	4	4	no	—
88	C₅×#41	—	4	4	no	—
89	C₂×C₅×Q₈	—	2	4	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	4	1	no	—
93	C₂×C₅×Q₈	—	4	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	4	2	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂×C₅×D₄	—	4	1	no	—
98	C₂²×C₄×C₅	—	4	1	no	—
99	C₂²×C₄×C₅	—	4	1	no	—
100	C₂²×C₄×C₅	—	4	2	no	—
101	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₅×Q₁₆	—	2	4	no	—
105	C₅×Q₁₆	—	2	2	no	—
106	C₅×Q₁₆	—	2	2	no	—
107	C₅×Q₁₆	—	2	2	no	—
108	C₅×Q₁₆	—	4	2	no	—
109	C₅×Q₁₆	C₂×D₄	1	4	yes	—
110	C₅×SD₁₆	—	4	4	no	—
111	C₅×SD₁₆	—	4	4	no	—
112	C₅×D₈	—	2	2	no	—
113	C₅×D₈	—	2	2	no	—
114	C₅×D₈	—	4	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₅×C₈	#41	1	2	yes	—
121	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	4	1	no	—
124	C₅×C₄⋊C₄	—	4	2	no	—
125	C₅×C₄⋊C₄	—	4	2	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	C₅×C₄⋊C₄	—	4	1	no	—
128	C₅×C₂²⋊C₄	—	4	2	no	—
129	C₅×C₂²⋊C₄	—	4	2	no	—
130	C₄²×C₅	—	4	2	no	—
131	C₄²×C₅	—	4	2	no	—
132	C₄²×C₅	D₈	1	1	yes	characteristic
133	C₄²×C₅	D₈	1	1	yes	characteristic
134	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	—	2	4	no	—
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₂²×C₅	1	2	yes	—
139	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
140	#532	—	2	1	no	—
141	C₂×C₄×Q₈	—	2	1	no	—
142	#510	—	2	8	no	—
143	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
144	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
145	C₂×M₃₂	C₂²×C₅	1	2	yes	—
146	#495	—	2	1	no	—
147	#495	C₂²×C₅	1	2	yes	—
148	#494	C₂²×C₅	1	1	yes	characteristic
149	#493	C₂²×C₅	1	1	yes	characteristic
150	#492	—	2	1	no	—
151	C₄×Q₁₆	—	2	1	no	—
152	C₄×Q₁₆	—	2	1	no	—
153	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
154	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
155	C₂×#107	C₂²×C₅	1	1	yes	characteristic
156	C₂×#103	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
158	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
159	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
160	#358	C₂²×C₅	1	4	yes	—
161	#357	C₂²×C₅	1	4	yes	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
164	C₅×Q₈	—	2	4	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	4	no	—
167	C₅×Q₈	—	4	2	no	—
168	C₅×Q₈	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	8	2	no	—
172	C₅×D₄	—	8	4	no	—
173	C₅×D₄	—	8	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	4	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	8	2	no	—
181	C₂×C₄×C₅	—	8	4	no	—
182	C₂×C₄×C₅	#121	1	2	yes	—
183	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
188	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	2	1	no	—
192	C₅×C₈	#121	1	2	yes	—
193	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
194	C₅×C₈	C₂×#41	1	1	yes	characteristic
195	C₂×#41	—	2	2	no	—
196	C₂²×Q₈	—	2	1	no	—
197	#135	—	2	4	no	—
198	#135	—	2	4	no	—
199	#135	—	4	4	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	C₅×D₄	1	4	yes	complemented
204	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
205	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	C₂³×C₅	1	2	yes	—
210	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	1	no	—
213	#127	—	2	1	no	—
214	#124	—	2	2	no	—
215	#121	—	4	1	no	—
216	#121	—	4	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	—	4	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	4	1	no	—
225	C₂×C₄²	—	4	1	no	—
226	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
227	Q₃₂	—	2	8	no	—
228	SD₃₂	—	2	8	no	—
229	M₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₅	1	4	yes	—
231	#107	—	2	1	no	—
232	#107	C₂³×C₅	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	C₄×C₈	C₅×D₄	1	1	yes	characteristic
236	C₄×C₈	C₅×D₄	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	—	8	2	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	—	8	2	no	—
244	C₂²×C₅	#459	1	2	yes	—
245	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	4	4	no	—
250	C₄×C₅	—	8	2	no	—
251	C₄×C₅	—	8	2	no	—
252	C₄×C₅	#459	1	2	yes	—
253	C₄×C₅	#510	1	2	yes	complemented
254	C₄×C₅	C₂×#121	1	1	yes	characteristic
255	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
256	#41	—	4	4	no	—
257	#41	—	4	4	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
271	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
273	Q₁₆	—	2	4	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	4	2	no	—
278	Q₁₆	C₂×C₅×D₄	1	4	yes	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	4	4	no	—
281	D₈	—	2	2	no	—
282	D₈	—	2	2	no	—
283	D₈	—	4	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
286	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
287	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
288	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
289	C₂×C₈	C₅×#41	1	2	yes	—
290	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	C₅×D₈	1	1	yes	characteristic
302	C₄²	C₅×D₈	1	1	yes	characteristic
303	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
304	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
305	C₁₆	—	2	4	no	—
306	C₂×C₅	—	2	2	no	—
307	C₂×C₅	—	16	2	no	—
308	C₂×C₅	C₂×#459	1	1	yes	characteristic
309	C₂×C₅	C₂×#510	1	1	yes	characteristic
310	C₂×C₅	#3414	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
313	Q₈	—	2	4	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	4	no	—
322	D₄	—	8	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	C₅×#121	1	2	yes	—
332	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
333	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
334	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
335	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
336	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
337	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
338	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	C₅×#121	1	2	yes	—
342	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
343	C₈	C₂×C₅×#41	1	1	yes	characteristic
344	C₅	#33581	1	1	yes	characteristic, direct product factor, Sylow
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	C₅×#459	1	2	yes	—
351	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	4	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₅×#459	1	2	yes	—
359	C₄	C₅×#510	1	2	yes	complemented
360	C₄	C₂×C₅×#121	1	1	yes	characteristic
361	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₂×C₅×#459	1	1	yes	characteristic
365	C₂	C₂×C₅×#510	1	1	yes	characteristic
366	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂×C₅×#510	C₂	semidirect
3. C₂×C₄×C₅×Q₁₆	C₂	semidirect
4. C₅×#2211	C₂	semidirect
5. C₅×#2209	C₂	semidirect
6. C₅×#2207	C₂	semidirect
7. C₅×#2150	C₂	semidirect
8. C₅×#2139	C₂	semidirect
34. #33581	C₅	direct (N × H)
43. C₂×C₅×Q₁₆	D₄	semidirect
79. C₂×#510	C₂×C₅	semidirect
81. C₂×C₄×Q₁₆	C₂×C₅	semidirect
82. #2211	C₂×C₅	semidirect
83. #2209	C₂×C₅	semidirect
84. #2207	C₂×C₅	semidirect
85. #2150	C₂×C₅	semidirect
86. #2139	C₂×C₅	semidirect
203. C₂×Q₁₆	C₅×D₄	semidirect
253. C₄×C₅	#510	semidirect
344. C₅	#33581	direct (N × H)
359. C₄	C₅×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#510	2
C₅×#3147	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2211	2
C₅×#2209	2
C₅×#2207	2
C₅×#2150	2
C₅×#2139	2
#33581	5

Subgroups — C5×#33581 #11956301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33581 #11956301