Subgroups — C₅×#33798 #11956518

Order 1280.

468 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	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#499	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×#2236	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2233	C₂	1	1	yes	maximal, characteristic
8	C₅×#2232	C₂	1	1	yes	maximal, characteristic
9	C₅×#2229	C₂	1	2	yes	maximal, complemented
10	C₅×#2228	C₂	1	2	yes	maximal, complemented
11	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#529	—	2	1	no	—
13	C₂×C₅×#125	—	2	1	no	—
14	C₅×#499	—	2	1	no	—
15	C₅×#499	—	2	1	no	—
16	C₅×#499	C₂²	1	2	yes	—
17	C₅×#499	C₂²	1	2	yes	—
18	C₅×#499	C₂²	1	1	yes	characteristic
19	C₅×#495	C₂²	1	1	yes	characteristic
20	C₅×#488	—	2	1	no	—
21	C₅×#467	C₂²	1	1	yes	characteristic
22	C₅×#466	C₂²	1	1	yes	characteristic
23	C₅×#465	—	2	1	no	—
24	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₅×SD₁₆	—	2	1	no	—
26	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
27	C₂×C₅×#107	C₂²	1	1	yes	characteristic
28	C₂×C₅×#107	C₂²	1	1	yes	characteristic
29	C₂×C₅×#106	—	2	1	no	—
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	2	yes	—
36	C₅×#358	C₂²	1	2	yes	—
37	C₅×#358	C₂²	1	2	yes	—
38	C₅×#357	C₂²	1	2	yes	—
39	C₅×#357	C₂²	1	2	yes	—
40	C₅×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₅×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄×C₅×C₁₆	C₂²	1	2	yes	—
43	#33798	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×#126	—	4	1	no	—
50	C₅×#125	—	2	1	no	—
51	C₅×#125	—	2	2	no	—
52	C₅×#125	—	2	2	no	—
53	C₅×#125	—	4	1	no	—
54	C₅×#125	—	4	1	no	—
55	C₅×#124	—	2	1	no	—
56	C₅×#123	—	4	1	no	—
57	C₅×C₂²⋊Q₈	—	2	1	no	—
58	C₅×#121	—	2	1	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×D₄	—	2	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₅×#117	—	4	1	no	—
63	C₂×C₅×C₄⋊C₄	—	2	1	no	—
64	C₂×C₅×C₄⋊C₄	—	2	1	no	—
65	C₂×C₅×C₄⋊C₄	—	4	1	no	—
66	C₂×C₅×C₄⋊C₄	—	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	C₂×#499	C₂×C₅	1	1	yes	characteristic, complemented
89	#2253	C₂×C₅	1	2	yes	complemented
90	#2242	C₂×C₅	1	2	yes	complemented
91	#2236	C₂×C₅	1	1	yes	characteristic, complemented
92	#2235	C₂×C₅	1	1	yes	characteristic, complemented
93	#2233	C₂×C₅	1	1	yes	characteristic
94	#2232	C₂×C₅	1	1	yes	characteristic
95	#2229	C₂×C₅	1	2	yes	complemented
96	#2228	C₂×C₅	1	2	yes	complemented
97	#2162	C₂×C₅	1	1	yes	characteristic, complemented
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×D₄	—	2	1	no	—
100	C₂×C₅×D₄	—	4	1	no	—
101	C₂²×C₄×C₅	—	4	1	no	—
102	C₂²×C₄×C₅	—	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	2	no	—
108	C₅×SD₁₆	—	4	2	no	—
109	C₅×D₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₄⋊C₄	—	4	1	no	—
121	C₅×C₄⋊C₄	—	4	1	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₅×C₄⋊C₄	—	4	2	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×C₄⋊C₄	—	4	1	no	—
129	C₅×C₄⋊C₄	—	4	1	no	—
130	C₅×C₂²⋊C₄	—	4	1	no	—
131	C₅×C₂²⋊C₄	—	4	1	no	—
132	C₅×C₂²⋊C₄	—	4	1	no	—
133	C₅×C₂²⋊C₄	—	4	1	no	—
134	C₅×C₂²⋊C₄	—	4	1	no	—
135	C₄²×C₅	—	4	1	no	—
136	C₄²×C₅	D₈	1	1	yes	characteristic
137	C₄²×C₅	D₈	1	1	yes	characteristic
138	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
140	C₅×C₁₆	—	2	2	no	—
141	C₅×C₁₆	—	2	2	no	—
142	#529	—	2	1	no	—
143	C₂×#125	—	2	1	no	—
144	#499	—	2	1	no	—
145	#499	—	2	1	no	—
146	#499	C₂²×C₅	1	2	yes	—
147	#499	C₂²×C₅	1	2	yes	—
148	#499	C₂²×C₅	1	1	yes	characteristic
149	#495	C₂²×C₅	1	1	yes	characteristic
150	#488	—	2	1	no	—
151	#467	C₂²×C₅	1	1	yes	characteristic
152	#466	C₂²×C₅	1	1	yes	characteristic
153	#465	—	2	1	no	—
154	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
155	C₄×SD₁₆	—	2	1	no	—
156	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
157	C₂×#107	C₂²×C₅	1	1	yes	characteristic
158	C₂×#107	C₂²×C₅	1	1	yes	characteristic
159	C₂×#106	—	2	1	no	—
160	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
161	#367	C₂²×C₅	1	2	yes	—
162	#367	C₂²×C₅	1	2	yes	—
163	#366	C₂²×C₅	1	2	yes	—
164	#366	C₂²×C₅	1	2	yes	—
165	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
166	#358	C₂²×C₅	1	2	yes	—
167	#358	C₂²×C₅	1	2	yes	—
168	#357	C₂²×C₅	1	2	yes	—
169	#357	C₂²×C₅	1	2	yes	—
170	C₂²⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
171	C₂²⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
172	C₄×C₁₆	C₂²×C₅	1	2	yes	—
173	C₂³×C₅	—	4	1	no	—
174	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
175	C₅×Q₈	—	4	1	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₅×D₄	—	4	1	no	—
178	C₅×D₄	—	4	1	no	—
179	C₅×D₄	—	8	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	—	8	2	no	—
190	C₂×C₄×C₅	—	8	2	no	—
191	C₂×C₄×C₅	—	8	2	no	—
192	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₅×C₈	—	2	1	no	—
199	C₅×C₈	—	2	1	no	—
200	C₅×C₈	#137	1	2	yes	—
201	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
202	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
203	C₂×SD₁₆	—	2	1	no	—
204	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
205	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
207	#126	—	4	1	no	—
208	#125	—	2	2	no	—
209	#125	—	2	2	no	—
210	#125	—	2	1	no	—
211	#125	—	4	1	no	—
212	#125	—	4	1	no	—
213	#124	—	2	1	no	—
214	#123	—	4	1	no	—
215	C₂²⋊Q₈	—	2	1	no	—
216	#121	—	2	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	4	1	no	—
220	#117	—	4	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	4	1	no	—
224	C₂×C₄⋊C₄	—	4	1	no	—
225	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₅	1	2	yes	—
227	C₂×C₁₆	C₂³×C₅	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₅	1	2	yes	—
231	#107	C₂³×C₅	1	2	yes	—
232	#107	C₂³×C₅	1	1	yes	characteristic
233	#107	C₂³×C₅	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	#106	—	2	2	no	—
236	#106	—	2	1	no	—
237	#103	—	2	1	no	—
238	#103	—	2	1	no	—
239	#102	—	2	1	no	—
240	#102	—	2	1	no	—
241	C₄×C₈	C₅×D₄	1	1	yes	characteristic
242	C₄×C₈	C₅×D₄	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	2	1	no	—
247	C₂²×C₅	—	8	1	no	—
248	C₂²×C₅	—	8	1	no	—
249	C₂²×C₅	—	8	1	no	—
250	C₂²×C₅	#508	1	2	yes	—
251	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	8	2	no	—
258	C₄×C₅	—	8	1	no	—
259	C₄×C₅	—	8	1	no	—
260	C₄×C₅	—	8	2	no	—
261	C₄×C₅	—	8	1	no	—
262	C₄×C₅	#536	1	1	yes	characteristic
263	C₄×C₅	C₂×#137	1	1	yes	characteristic
264	C₂×Q₈	—	2	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	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	2	no	—
274	SD₁₆	—	4	2	no	—
275	D₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
279	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
281	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₅×D₈	1	1	yes	characteristic
303	C₄²	C₅×D₈	1	1	yes	characteristic
304	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
305	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	2	2	no	—
308	C₂×C₅	—	2	2	no	—
309	C₂×C₅	—	16	1	no	—
310	C₂×C₅	#3226	1	1	yes	characteristic
311	C₂×C₅	C₂×#508	1	1	yes	characteristic
312	C₂×C₅	#3414	1	1	yes	characteristic
313	C₂³	—	4	1	no	—
314	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
315	Q₈	—	4	1	no	—
316	Q₈	—	4	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
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	C₂²×C₅×D₄	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	C₅×#137	1	2	yes	—
341	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
342	C₅	#33798	1	1	yes	characteristic, direct product factor, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	8	1	no	—
348	C₂²	C₅×#508	1	2	yes	—
349	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	1	no	—
357	C₄	—	8	1	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	1	no	—
360	C₄	C₅×#536	1	1	yes	characteristic
361	C₄	C₂×C₅×#137	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	16	1	no	—
364	C₂	C₅×#3226	1	1	yes	characteristic
365	C₂	C₂×C₅×#508	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×#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. C₂×C₅×#499	C₂	semidirect
3. C₅×#2253	C₂	semidirect
4. C₅×#2242	C₂	semidirect
5. C₅×#2236	C₂	semidirect
6. C₅×#2235	C₂	semidirect
9. C₅×#2229	C₂	semidirect
10. C₅×#2228	C₂	semidirect
11. C₅×#2162	C₂	semidirect
43. #33798	C₅	direct (N × H)
88. C₂×#499	C₂×C₅	semidirect
89. #2253	C₂×C₅	semidirect
90. #2242	C₂×C₅	semidirect
91. #2236	C₂×C₅	semidirect
92. #2235	C₂×C₅	semidirect
95. #2229	C₂×C₅	semidirect
96. #2228	C₂×C₅	semidirect
97. #2162	C₂×C₅	semidirect
342. C₅	#33798	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
C₂×C₅×#499	2
C₅×#2253	2
C₅×#2242	2
C₅×#2236	2
C₅×#2235	2
C₅×#2233	2
C₅×#2232	2
C₅×#2229	2
C₅×#2228	2
C₅×#2162	2
#33798	5

Subgroups — C5×#33798 #11956518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33798 #11956518