Subgroups — C₅×#33661 #11956381

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2235	C₂	1	2	yes	maximal, complemented
4	C₅×#2226	C₂	1	2	yes	maximal, complemented
5	C₅×#2208	C₂	1	2	yes	maximal, complemented
6	C₅×#2187	C₂	1	2	yes	maximal
7	C₄×C₅×SD₃₂	C₂	1	4	yes	maximal, complemented
8	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₅×#533	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅×#494	C₂²	1	1	yes	characteristic, complemented
15	C₅×#487	—	2	1	no	—
16	C₅×#467	—	2	1	no	—
17	C₅×#467	C₂²	1	2	yes	—
18	C₄×C₅×Q₁₆	—	2	1	no	—
19	C₄×C₅×Q₁₆	C₂²	1	2	yes	complemented
20	C₄×C₅×D₈	—	2	1	no	—
21	C₄×C₅×D₈	—	2	1	no	—
22	C₄×C₅×D₈	C₂²	1	2	yes	—
23	C₄×C₅×D₈	C₂²	1	2	yes	—
24	C₂×C₅×#107	C₂²	1	1	yes	characteristic
25	C₂×C₅×#102	—	2	1	no	—
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#367	C₂²	1	4	yes	complemented
28	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₅×#358	C₂²	1	4	yes	complemented
30	C₅×#357	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
33	#33661	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×D₄	—	2	1	no	—
35	C₂³×C₄×C₅	—	4	1	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
38	C₂×C₅×D₈	—	2	4	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×#125	—	4	1	no	—
46	C₅×C₂²⋊Q₈	—	2	2	no	—
47	C₅×C₂²⋊Q₈	—	4	1	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₄×C₅×Q₈	—	4	1	no	—
50	C₄×C₅×D₄	—	2	2	no	—
51	C₄×C₅×D₄	—	2	2	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₄×C₅×D₄	—	4	1	no	—
54	C₅×#117	—	4	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
57	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
58	C₅×SD₃₂	—	2	8	no	—
59	C₂×C₅×C₁₆	C₂³	1	4	yes	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#103	—	2	2	no	—
65	C₅×#103	—	2	2	no	—
66	C₅×#102	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	#3170	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
74	#2235	C₂×C₅	1	2	yes	complemented
75	#2226	C₂×C₅	1	2	yes	complemented
76	#2208	C₂×C₅	1	2	yes	complemented
77	#2187	C₂×C₅	1	2	yes	—
78	C₄×SD₃₂	C₂×C₅	1	4	yes	complemented
79	#2162	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	4	1	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×D₄	—	2	4	no	—
84	C₂×C₅×D₄	—	2	2	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂²×C₄×C₅	—	4	1	no	—
88	C₂²×C₄×C₅	—	4	2	no	—
89	C₂²×C₄×C₅	—	4	2	no	—
90	C₂²×C₄×C₅	—	8	4	no	—
91	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×Q₁₆	—	2	4	no	—
95	C₅×Q₁₆	—	4	2	no	—
96	C₅×D₈	—	2	4	no	—
97	C₅×D₈	—	2	2	no	—
98	C₅×D₈	#41	1	4	yes	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₂²⋊C₄	—	4	2	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₄²×C₅	—	4	2	no	—
117	C₄²×C₅	D₈	1	1	yes	characteristic
118	C₄²×C₅	D₈	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₅×C₁₆	—	2	4	no	—
122	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
123	#533	—	2	1	no	—
124	C₂×C₄×D₄	—	2	1	no	—
125	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
126	#500	C₂²×C₅	1	1	yes	characteristic, complemented
127	#494	C₂²×C₅	1	1	yes	characteristic, complemented
128	#487	—	2	1	no	—
129	#467	—	2	1	no	—
130	#467	C₂²×C₅	1	2	yes	—
131	C₄×Q₁₆	—	2	1	no	—
132	C₄×Q₁₆	C₂²×C₅	1	2	yes	complemented
133	C₄×D₈	—	2	1	no	—
134	C₄×D₈	—	2	1	no	—
135	C₄×D₈	C₂²×C₅	1	2	yes	—
136	C₄×D₈	C₂²×C₅	1	2	yes	—
137	C₂×#107	C₂²×C₅	1	1	yes	characteristic
138	C₂×#102	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	#367	C₂²×C₅	1	4	yes	complemented
141	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
142	#358	C₂²×C₅	1	4	yes	complemented
143	#357	C₂²×C₅	1	4	yes	—
144	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
145	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
146	C₂³×C₅	—	4	4	no	—
147	C₂³×C₅	—	4	2	no	—
148	C₂³×C₅	—	8	2	no	—
149	C₂³×C₅	—	8	2	no	—
150	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
151	C₅×Q₈	—	4	2	no	—
152	C₅×Q₈	—	4	2	no	—
153	C₅×Q₈	—	8	2	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	4	4	no	—
156	C₅×D₄	—	4	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	8	2	no	—
164	C₂×C₄×C₅	—	8	4	no	—
165	C₂×C₄×C₅	—	8	4	no	—
166	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
167	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	C₂×#41	1	2	yes	—
177	C₂²×D₄	—	2	1	no	—
178	C₂³×C₄	—	4	1	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	—	2	1	no	—
183	C₂×D₈	C₂³×C₅	1	2	yes	—
184	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	#125	—	4	1	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	—	4	1	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	4	1	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	4	1	no	—
196	C₄×D₄	—	4	1	no	—
197	#117	—	4	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	4	1	no	—
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	SD₃₂	—	2	8	no	—
202	C₂×C₁₆	C₂³×C₅	1	4	yes	—
203	#107	—	2	1	no	—
204	#107	—	2	1	no	—
205	#107	C₂³×C₅	1	2	yes	—
206	#107	C₂³×C₅	1	2	yes	—
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	C₅×D₄	1	1	yes	characteristic
212	C₄×C₈	C₅×D₄	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	4	4	no	—
218	C₂²×C₅	—	8	8	no	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
222	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	8	2	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	C₂×SD₃₂	1	2	yes	—
230	C₄×C₅	C₂×SD₃₂	1	2	yes	—
231	C₄×C₅	#525	1	1	yes	characteristic
232	C₄×C₅	#575	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	8	4	no	—
244	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
245	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	4	2	no	—
249	D₈	—	2	4	no	—
250	D₈	—	2	2	no	—
251	D₈	C₅×#41	1	4	yes	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₅×D₈	1	1	yes	characteristic
271	C₄²	C₅×D₈	1	1	yes	characteristic
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	8	4	no	—
277	C₂×C₅	#3100	1	1	yes	characteristic
278	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
279	C₂×C₅	#3414	1	1	yes	characteristic
280	C₂³	—	4	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	C₅×SD₃₂	1	4	yes	complemented
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₂×C₅×#41	1	2	yes	—
311	C₅	#33661	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	8	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
319	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂×C₅×SD₃₂	1	2	yes	—
327	C₄	C₂×C₅×SD₃₂	1	2	yes	—
328	C₄	C₅×#525	1	1	yes	characteristic
329	C₄	C₅×#575	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₅×#3100	1	1	yes	characteristic
333	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
334	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₅×#3170	C₂	semidirect
3. C₅×#2235	C₂	semidirect
4. C₅×#2226	C₂	semidirect
5. C₅×#2208	C₂	semidirect
7. C₄×C₅×SD₃₂	C₂	semidirect
8. C₅×#2162	C₂	semidirect
13. C₅×#500	C₂²	semidirect
14. C₅×#494	C₂²	semidirect
19. C₄×C₅×Q₁₆	C₂²	semidirect
27. C₅×#367	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
32. C₄×C₅×C₁₆	C₂²	semidirect
33. #33661	C₅	direct (N × H)
72. #3170	C₂×C₅	semidirect
74. #2235	C₂×C₅	semidirect
75. #2226	C₂×C₅	semidirect
76. #2208	C₂×C₅	semidirect
78. C₄×SD₃₂	C₂×C₅	semidirect
79. #2162	C₂×C₅	semidirect
126. #500	C₂²×C₅	semidirect
127. #494	C₂²×C₅	semidirect
132. C₄×Q₁₆	C₂²×C₅	semidirect
140. #367	C₂²×C₅	semidirect
141. C₄⋊C₁₆	C₂²×C₅	semidirect
142. #358	C₂²×C₅	semidirect
145. C₄×C₁₆	C₂²×C₅	semidirect
166. C₂×C₄×C₅	SD₃₂	semidirect
300. C₂×C₄	C₅×SD₃₂	semidirect
311. C₅	#33661	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3170	2
C₂×C₄×C₅×D₈	2
C₅×#2235	2
C₅×#2226	2
C₅×#2208	2
C₅×#2187	2
C₄×C₅×SD₃₂	2
C₅×#2162	2
#33661	5

Subgroups — C5×#33661 #11956381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33661 #11956381