Subgroups — C₅×#33799 #11956519

Order 1280.

466 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2253	C₂	1	2	yes	maximal, complemented
4	C₅×#2242	C₂	1	2	yes	maximal, complemented
5	C₅×#2237	C₂	1	2	yes	maximal, complemented
6	C₅×#2234	C₂	1	2	yes	maximal
7	C₅×#2227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2226	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2225	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2224	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#533	—	2	1	no	—
13	C₅×#529	—	2	1	no	—
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#500	C₂²	1	1	yes	characteristic
16	C₅×#499	—	2	1	no	—
17	C₅×#499	C₂²	1	2	yes	—
18	C₅×#499	C₂²	1	1	yes	characteristic
19	C₅×#498	—	2	1	no	—
20	C₅×#495	C₂²	1	1	yes	characteristic
21	C₅×#488	—	2	1	no	—
22	C₅×#467	C₂²	1	1	yes	characteristic
23	C₅×#466	C₂²	1	1	yes	characteristic
24	C₅×#465	—	2	1	no	—
25	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₅×SD₁₆	—	2	1	no	—
27	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
28	C₅×#427	—	2	1	no	—
29	C₅×#427	C₂²	1	2	yes	—
30	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
31	C₅×#367	C₂²	1	2	yes	—
32	C₅×#367	C₂²	1	2	yes	—
33	C₅×#366	C₂²	1	2	yes	—
34	C₅×#366	C₂²	1	2	yes	—
35	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₅×#358	C₂²	1	2	yes	—
38	C₅×#358	C₂²	1	2	yes	—
39	C₅×#357	C₂²	1	2	yes	—
40	C₅×#357	C₂²	1	2	yes	—
41	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
42	C₄×C₅×C₁₆	C₂²	1	2	yes	—
43	#33799	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₅×SD₁₆	—	2	1	no	—
46	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×#126	—	4	1	no	—
52	C₅×#125	—	2	1	no	—
53	C₅×#125	—	2	2	no	—
54	C₅×#124	—	2	1	no	—
55	C₅×#123	—	4	1	no	—
56	C₅×C₂²⋊Q₈	—	2	1	no	—
57	C₅×C₂²⋊Q₈	—	4	1	no	—
58	C₅×C₂²⋊Q₈	—	4	1	no	—
59	C₅×#121	—	2	1	no	—
60	C₄×C₅×Q₈	—	2	1	no	—
61	C₄×C₅×Q₈	—	4	1	no	—
62	C₄×C₅×Q₈	—	4	1	no	—
63	C₄×C₅×D₄	—	2	1	no	—
64	C₄×C₅×D₄	—	4	1	no	—
65	C₅×#117	—	2	2	no	—
66	C₅×#117	—	4	1	no	—
67	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
68	C₂×C₅×C₁₆	C₂³	1	2	yes	—
69	C₂×C₅×C₁₆	C₂³	1	2	yes	—
70	C₅×#107	—	2	1	no	—
71	C₅×#107	—	2	1	no	—
72	C₅×#107	C₂³	1	2	yes	—
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#107	C₂³	1	1	yes	characteristic
75	C₅×#107	C₂³	1	1	yes	characteristic
76	C₅×#106	—	2	2	no	—
77	C₅×#106	—	2	2	no	—
78	C₅×#106	—	2	1	no	—
79	C₅×#103	—	2	1	no	—
80	C₅×#103	—	2	1	no	—
81	C₅×#102	—	2	1	no	—
82	C₅×#102	—	2	1	no	—
83	C₄×C₅×C₈	D₄	1	1	yes	characteristic
84	C₄×C₅×C₈	D₄	1	1	yes	characteristic
85	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
87	#3168	C₂×C₅	1	1	yes	characteristic
88	#3160	C₂×C₅	1	1	yes	characteristic, complemented
89	#2253	C₂×C₅	1	2	yes	complemented
90	#2242	C₂×C₅	1	2	yes	complemented
91	#2237	C₂×C₅	1	2	yes	complemented
92	#2234	C₂×C₅	1	2	yes	—
93	#2227	C₂×C₅	1	1	yes	characteristic, complemented
94	#2226	C₂×C₅	1	1	yes	characteristic, complemented
95	#2225	C₂×C₅	1	1	yes	characteristic, complemented
96	#2224	C₂×C₅	1	1	yes	characteristic, complemented
97	#2162	C₂×C₅	1	1	yes	characteristic, complemented
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	4	1	no	—
100	C₂×C₅×Q₈	—	4	1	no	—
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	4	1	no	—
103	C₂²×C₄×C₅	—	4	1	no	—
104	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
107	C₅×Q₁₆	—	2	1	no	—
108	C₅×Q₁₆	—	2	1	no	—
109	C₅×SD₁₆	—	4	2	no	—
110	C₅×D₈	—	2	1	no	—
111	C₅×D₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	4	1	no	—
124	C₅×C₄⋊C₄	—	4	1	no	—
125	C₅×C₄⋊C₄	—	4	1	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	C₅×C₄⋊C₄	—	4	2	no	—
128	C₅×C₄⋊C₄	—	4	2	no	—
129	C₅×C₄⋊C₄	—	4	1	no	—
130	C₅×C₄⋊C₄	—	4	1	no	—
131	C₅×C₂²⋊C₄	—	4	2	no	—
132	C₅×C₂²⋊C₄	—	4	1	no	—
133	C₅×C₂²⋊C₄	—	4	1	no	—
134	C₅×C₂²⋊C₄	—	4	1	no	—
135	C₅×C₂²⋊C₄	—	4	1	no	—
136	C₅×C₂²⋊C₄	—	4	1	no	—
137	C₄²×C₅	—	4	2	no	—
138	C₄²×C₅	—	4	1	no	—
139	C₄²×C₅	D₈	1	1	yes	characteristic
140	C₄²×C₅	D₈	1	1	yes	characteristic
141	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
143	C₅×C₁₆	—	2	2	no	—
144	C₅×C₁₆	—	2	2	no	—
145	#533	—	2	1	no	—
146	#529	—	2	1	no	—
147	#500	C₂²×C₅	1	1	yes	characteristic
148	#500	C₂²×C₅	1	1	yes	characteristic
149	#499	—	2	1	no	—
150	#499	C₂²×C₅	1	2	yes	—
151	#499	C₂²×C₅	1	1	yes	characteristic
152	#498	—	2	1	no	—
153	#495	C₂²×C₅	1	1	yes	characteristic
154	#488	—	2	1	no	—
155	#467	C₂²×C₅	1	1	yes	characteristic
156	#466	C₂²×C₅	1	1	yes	characteristic
157	#465	—	2	1	no	—
158	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
159	C₄×SD₁₆	—	2	1	no	—
160	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
161	#427	—	2	1	no	—
162	#427	C₂²×C₅	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
164	#367	C₂²×C₅	1	2	yes	—
165	#367	C₂²×C₅	1	2	yes	—
166	#366	C₂²×C₅	1	2	yes	—
167	#366	C₂²×C₅	1	2	yes	—
168	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
169	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
170	#358	C₂²×C₅	1	2	yes	—
171	#358	C₂²×C₅	1	2	yes	—
172	#357	C₂²×C₅	1	2	yes	—
173	#357	C₂²×C₅	1	2	yes	—
174	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
175	C₄×C₁₆	C₂²×C₅	1	2	yes	—
176	C₂³×C₅	—	4	1	no	—
177	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
178	C₅×Q₈	—	4	1	no	—
179	C₅×Q₈	—	4	1	no	—
180	C₅×Q₈	—	8	2	no	—
181	C₅×Q₈	—	8	2	no	—
182	C₅×D₄	—	4	1	no	—
183	C₅×D₄	—	4	1	no	—
184	C₅×D₄	—	8	2	no	—
185	C₂×C₄×C₅	—	2	1	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	1	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	—	4	2	no	—
193	C₂×C₄×C₅	—	4	1	no	—
194	C₂×C₄×C₅	—	8	2	no	—
195	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
201	C₅×C₈	—	2	1	no	—
202	C₅×C₈	—	2	1	no	—
203	C₅×C₈	#137	1	2	yes	—
204	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
205	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
206	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
209	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#126	—	4	1	no	—
214	#125	—	2	1	no	—
215	#125	—	2	2	no	—
216	#124	—	2	1	no	—
217	#123	—	4	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	#121	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	—	4	1	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	4	1	no	—
227	#117	—	2	2	no	—
228	#117	—	4	1	no	—
229	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₅	1	2	yes	—
231	C₂×C₁₆	C₂³×C₅	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	C₂³×C₅	1	2	yes	—
235	#107	C₂³×C₅	1	2	yes	—
236	#107	C₂³×C₅	1	1	yes	characteristic
237	#107	C₂³×C₅	1	1	yes	characteristic
238	#106	—	2	2	no	—
239	#106	—	2	2	no	—
240	#106	—	2	1	no	—
241	#103	—	2	1	no	—
242	#103	—	2	1	no	—
243	#102	—	2	1	no	—
244	#102	—	2	1	no	—
245	C₄×C₈	C₅×D₄	1	1	yes	characteristic
246	C₄×C₈	C₅×D₄	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	8	1	no	—
253	C₂²×C₅	—	8	1	no	—
254	C₂²×C₅	—	8	1	no	—
255	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	1	no	—
260	C₄×C₅	—	8	1	no	—
261	C₄×C₅	—	8	2	no	—
262	C₄×C₅	—	8	1	no	—
263	C₄×C₅	—	8	2	no	—
264	C₄×C₅	—	8	1	no	—
265	C₄×C₅	#508	1	2	yes	—
266	C₄×C₅	#536	1	1	yes	characteristic
267	C₄×C₅	C₂×#137	1	1	yes	characteristic
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
275	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	2	1	no	—
279	SD₁₆	—	4	2	no	—
280	D₈	—	2	1	no	—
281	D₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	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	C₂⁴×C₅	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₄²	—	4	2	no	—
308	C₄²	—	4	1	no	—
309	C₄²	C₅×D₈	1	1	yes	characteristic
310	C₄²	C₅×D₈	1	1	yes	characteristic
311	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
312	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
313	C₁₆	—	2	2	no	—
314	C₁₆	—	2	2	no	—
315	C₂×C₅	—	4	1	no	—
316	C₂×C₅	—	16	1	no	—
317	C₂×C₅	#3226	1	1	yes	characteristic
318	C₂×C₅	C₂×#508	1	1	yes	characteristic
319	C₂×C₅	#3414	1	1	yes	characteristic
320	C₂³	—	4	1	no	—
321	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
322	Q₈	—	4	1	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	4	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	8	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
340	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
341	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
342	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
343	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	C₅×#137	1	2	yes	—
348	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
349	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
350	C₅	#33799	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	1	no	—
367	C₄	C₅×#508	1	2	yes	—
368	C₄	C₅×#536	1	1	yes	characteristic
369	C₄	C₂×C₅×#137	1	1	yes	characteristic
370	C₂	—	4	1	no	—
371	C₂	—	16	1	no	—
372	C₂	C₅×#3226	1	1	yes	characteristic
373	C₂	C₂×C₅×#508	1	1	yes	characteristic
374	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×#466	C₂²
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
2. C₅×#3160	C₂	semidirect
3. C₅×#2253	C₂	semidirect
4. C₅×#2242	C₂	semidirect
5. C₅×#2237	C₂	semidirect
7. C₅×#2227	C₂	semidirect
8. C₅×#2226	C₂	semidirect
9. C₅×#2225	C₂	semidirect
10. C₅×#2224	C₂	semidirect
11. C₅×#2162	C₂	semidirect
43. #33799	C₅	direct (N × H)
88. #3160	C₂×C₅	semidirect
89. #2253	C₂×C₅	semidirect
90. #2242	C₂×C₅	semidirect
91. #2237	C₂×C₅	semidirect
93. #2227	C₂×C₅	semidirect
94. #2226	C₂×C₅	semidirect
95. #2225	C₂×C₅	semidirect
96. #2224	C₂×C₅	semidirect
97. #2162	C₂×C₅	semidirect
350. C₅	#33799	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×#3168	2
C₅×#3160	2
C₅×#2253	2
C₅×#2242	2
C₅×#2237	2
C₅×#2234	2
C₅×#2227	2
C₅×#2226	2
C₅×#2225	2
C₅×#2224	2
C₅×#2162	2
#33799	5

Subgroups — C5×#33799 #11956519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33799 #11956519