Subgroups — C₅×#33805 #11956525

Order 1280.

496 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×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2255	C₂	1	2	yes	maximal, complemented
4	C₅×#2243	C₂	1	2	yes	maximal, complemented
5	C₅×#2236	C₂	1	2	yes	maximal, complemented
6	C₅×#2233	C₂	1	2	yes	maximal
7	C₅×#2225	C₂	1	2	yes	maximal, complemented
8	C₅×#2224	C₂	1	2	yes	maximal, complemented
9	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₅×#529	—	2	1	no	—
12	C₅×#500	—	2	1	no	—
13	C₅×#500	—	2	1	no	—
14	C₅×#500	C₂²	1	2	yes	—
15	C₅×#500	C₂²	1	2	yes	—
16	C₅×#499	C₂²	1	1	yes	characteristic
17	C₅×#495	C₂²	1	1	yes	characteristic
18	C₅×#488	—	2	1	no	—
19	C₅×#467	C₂²	1	1	yes	characteristic
20	C₅×#466	C₂²	1	1	yes	characteristic
21	C₅×#465	—	2	1	no	—
22	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
23	C₄×C₅×SD₁₆	—	2	1	no	—
24	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
25	C₂×C₅×#107	—	2	1	no	—
26	C₂×C₅×#107	C₂²	1	2	yes	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#367	C₂²	1	4	yes	—
29	C₅×#366	C₂²	1	4	yes	—
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅×#358	C₂²	1	4	yes	—
32	C₅×#357	C₂²	1	4	yes	—
33	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₅×#346	C₂²	1	2	yes	—
35	#33805	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	—	4	1	no	—
37	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₅×SD₁₆	—	2	1	no	—
39	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	—	2	2	no	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	4	1	no	—
45	C₅×C₄⋊Q₈	—	4	1	no	—
46	C₅×#126	—	4	1	no	—
47	C₅×#125	—	2	1	no	—
48	C₅×#124	—	2	1	no	—
49	C₅×#123	—	4	1	no	—
50	C₅×C₂²⋊Q₈	—	2	1	no	—
51	C₅×#121	—	2	1	no	—
52	C₄×C₅×Q₈	—	2	1	no	—
53	C₄×C₅×D₄	—	2	1	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₅×#117	—	4	1	no	—
56	C₂×C₅×C₄⋊C₄	—	2	2	no	—
57	C₂×C₅×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂³	1	4	yes	—
60	C₅×#107	—	2	2	no	—
61	C₅×#107	—	2	2	no	—
62	C₅×#107	—	2	2	no	—
63	C₅×#107	C₂³	1	4	yes	—
64	C₅×#107	C₂³	1	1	yes	characteristic
65	C₅×#107	C₂³	1	1	yes	characteristic
66	C₅×#106	—	2	1	no	—
67	C₅×#103	—	2	1	no	—
68	C₅×#103	—	2	1	no	—
69	C₅×#102	—	2	1	no	—
70	C₅×#102	—	2	1	no	—
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	D₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3168	C₂×C₅	1	1	yes	characteristic
76	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
77	#2255	C₂×C₅	1	2	yes	complemented
78	#2243	C₂×C₅	1	2	yes	complemented
79	#2236	C₂×C₅	1	2	yes	complemented
80	#2233	C₂×C₅	1	2	yes	—
81	#2225	C₂×C₅	1	2	yes	complemented
82	#2224	C₂×C₅	1	2	yes	complemented
83	#2163	C₂×C₅	1	1	yes	characteristic, complemented
84	C₂×C₅×Q₈	—	2	1	no	—
85	C₂×C₅×Q₈	—	4	2	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	8	4	no	—
88	C₂×C₅×D₄	—	2	1	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×SD₁₆	—	4	2	no	—
97	C₅×D₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₂²⋊C₄	—	4	1	no	—
115	C₅×C₂²⋊C₄	—	4	1	no	—
116	C₅×C₂²⋊C₄	—	4	1	no	—
117	C₅×C₂²⋊C₄	—	4	1	no	—
118	C₅×C₂²⋊C₄	—	4	1	no	—
119	C₄²×C₅	—	4	1	no	—
120	C₄²×C₅	D₈	1	1	yes	characteristic
121	C₄²×C₅	D₈	1	1	yes	characteristic
122	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
124	C₅×C₁₆	—	2	4	no	—
125	C₂×C₄⋊Q₈	—	2	1	no	—
126	#529	—	2	1	no	—
127	#500	—	2	1	no	—
128	#500	—	2	1	no	—
129	#500	C₂²×C₅	1	2	yes	—
130	#500	C₂²×C₅	1	2	yes	—
131	#499	C₂²×C₅	1	1	yes	characteristic
132	#495	C₂²×C₅	1	1	yes	characteristic
133	#488	—	2	1	no	—
134	#467	C₂²×C₅	1	1	yes	characteristic
135	#466	C₂²×C₅	1	1	yes	characteristic
136	#465	—	2	1	no	—
137	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
138	C₄×SD₁₆	—	2	1	no	—
139	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
140	C₂×#107	—	2	1	no	—
141	C₂×#107	C₂²×C₅	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
143	#367	C₂²×C₅	1	4	yes	—
144	#366	C₂²×C₅	1	4	yes	—
145	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
146	#358	C₂²×C₅	1	4	yes	—
147	#357	C₂²×C₅	1	4	yes	—
148	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
149	#346	C₂²×C₅	1	2	yes	—
150	C₂³×C₅	—	4	1	no	—
151	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
152	C₅×Q₈	—	4	1	no	—
153	C₅×Q₈	—	4	1	no	—
154	C₅×Q₈	—	8	4	no	—
155	C₅×Q₈	—	8	4	no	—
156	C₅×D₄	—	4	1	no	—
157	C₅×D₄	—	4	1	no	—
158	C₅×D₄	—	8	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	4	no	—
167	C₂×C₄×C₅	—	8	4	no	—
168	C₂×C₄×C₅	—	8	2	no	—
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	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	#137	1	2	yes	—
178	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
179	C₂²×Q₈	—	4	1	no	—
180	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
181	C₂×SD₁₆	—	2	1	no	—
182	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	4	1	no	—
188	C₄⋊Q₈	—	4	1	no	—
189	#126	—	4	1	no	—
190	#125	—	2	1	no	—
191	#124	—	2	1	no	—
192	#123	—	4	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	#121	—	2	1	no	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	4	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
202	C₂×C₁₆	C₂³×C₅	1	4	yes	—
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	C₂³×C₅	1	4	yes	—
207	#107	C₂³×C₅	1	1	yes	characteristic
208	#107	C₂³×C₅	1	1	yes	characteristic
209	#106	—	2	1	no	—
210	#103	—	2	1	no	—
211	#103	—	2	1	no	—
212	#102	—	2	1	no	—
213	#102	—	2	1	no	—
214	C₄×C₈	C₅×D₄	1	1	yes	characteristic
215	C₄×C₈	C₅×D₄	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	8	1	no	—
221	C₂²×C₅	—	8	1	no	—
222	C₂²×C₅	—	8	1	no	—
223	C₂²×C₅	#510	1	2	yes	—
224	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	8	1	no	—
230	C₄×C₅	—	8	4	no	—
231	C₄×C₅	—	8	1	no	—
232	C₄×C₅	—	8	1	no	—
233	C₄×C₅	#509	1	2	yes	—
234	C₄×C₅	#536	1	1	yes	characteristic
235	C₄×C₅	C₂×#137	1	1	yes	characteristic
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	8	4	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	1	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	2	no	—
248	SD₁₆	—	4	2	no	—
249	D₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₅×D₈	1	1	yes	characteristic
273	C₄²	C₅×D₈	1	1	yes	characteristic
274	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
276	C₁₆	—	2	4	no	—
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	16	1	no	—
279	C₂×C₅	#3226	1	1	yes	characteristic
280	C₂×C₅	C₂×#509	1	1	yes	characteristic
281	C₂×C₅	C₂×#510	1	1	yes	characteristic
282	C₂³	—	4	1	no	—
283	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	8	4	no	—
287	Q₈	—	8	4	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	1	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	2	no	—
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	C₅×#137	1	2	yes	—
310	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
311	C₅	#33805	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₅×#510	1	2	yes	—
318	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	4	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	C₅×#509	1	2	yes	—
328	C₄	C₅×#536	1	1	yes	characteristic
329	C₄	C₂×C₅×#137	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₅×#3226	1	1	yes	characteristic
333	C₂	C₂×C₅×#509	1	1	yes	characteristic
334	C₂	C₂×C₅×#510	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×#500	C₂	semidirect
3. C₅×#2255	C₂	semidirect
4. C₅×#2243	C₂	semidirect
5. C₅×#2236	C₂	semidirect
7. C₅×#2225	C₂	semidirect
8. C₅×#2224	C₂	semidirect
9. C₅×#2163	C₂	semidirect
35. #33805	C₅	direct (N × H)
76. C₂×#500	C₂×C₅	semidirect
77. #2255	C₂×C₅	semidirect
78. #2243	C₂×C₅	semidirect
79. #2236	C₂×C₅	semidirect
81. #2225	C₂×C₅	semidirect
82. #2224	C₂×C₅	semidirect
83. #2163	C₂×C₅	semidirect
311. C₅	#33805	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×#500	2
C₅×#2255	2
C₅×#2243	2
C₅×#2236	2
C₅×#2233	2
C₅×#2225	2
C₅×#2224	2
C₅×#2163	2
#33805	5

Subgroups — C5×#33805 #11956525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33805 #11956525