Subgroups — C₅×#33801 #11956521

Order 1280.

500 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2255	C₂	1	2	yes	maximal, complemented
4	C₅×#2244	C₂	1	2	yes	maximal, complemented
5	C₅×#2234	C₂	1	4	yes	maximal, complemented
6	C₅×#2226	C₂	1	2	yes	maximal, complemented
7	C₅×#2224	C₂	1	2	yes	maximal, complemented
8	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#533	—	2	1	no	—
10	C₅×#532	—	2	1	no	—
11	C₅×#500	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#499	—	2	1	no	—
15	C₅×#499	C₂²	1	2	yes	—
16	C₅×#498	—	2	1	no	—
17	C₅×#493	C₂²	1	1	yes	characteristic
18	C₅×#486	—	2	1	no	—
19	C₅×#466	—	2	1	no	—
20	C₅×#466	C₂²	1	2	yes	—
21	C₄×C₅×D₈	—	2	1	no	—
22	C₄×C₅×D₈	C₂²	1	2	yes	—
23	C₅×#427	—	2	1	no	—
24	C₅×#427	C₂²	1	2	yes	—
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₅×#367	C₂²	1	4	yes	—
27	C₅×#366	C₂²	1	4	yes	—
28	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₅×#357	C₂²	1	4	yes	—
30	C₅×#357	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₅×#346	C₂²	1	2	yes	—
33	#33801	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	4	1	no	—
35	C₂×C₅×D₈	—	2	1	no	—
36	C₂×C₅×D₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊Q₈	—	2	1	no	—
40	C₅×C₄⋊Q₈	—	2	1	no	—
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×#127	—	2	1	no	—
43	C₅×#125	—	2	2	no	—
44	C₅×#124	—	4	1	no	—
45	C₅×C₂²⋊Q₈	—	4	1	no	—
46	C₅×C₂²⋊Q₈	—	4	1	no	—
47	C₅×#121	—	2	2	no	—
48	C₅×#121	—	4	1	no	—
49	C₄×C₅×Q₈	—	4	1	no	—
50	C₄×C₅×Q₈	—	4	1	no	—
51	C₄×C₅×D₄	—	2	2	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₅×#117	—	2	2	no	—
54	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
55	C₂×C₅×C₁₆	C₂³	1	4	yes	—
56	C₅×#107	—	2	1	no	—
57	C₅×#107	—	2	1	no	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂³	1	2	yes	—
60	C₅×#107	C₂³	1	2	yes	—
61	C₅×#107	C₂³	1	2	yes	—
62	C₅×#106	—	2	2	no	—
63	C₅×#106	—	2	2	no	—
64	C₅×#102	—	2	2	no	—
65	C₅×#102	—	2	2	no	—
66	C₄×C₅×C₈	D₄	1	1	yes	characteristic
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	#3169	C₂×C₅	1	1	yes	characteristic
71	#3160	C₂×C₅	1	1	yes	characteristic, complemented
72	#2255	C₂×C₅	1	2	yes	complemented
73	#2244	C₂×C₅	1	2	yes	complemented
74	#2234	C₂×C₅	1	4	yes	complemented
75	#2226	C₂×C₅	1	2	yes	complemented
76	#2224	C₂×C₅	1	2	yes	complemented
77	#2163	C₂×C₅	1	1	yes	characteristic, complemented
78	C₅×#41	—	8	4	no	—
79	C₂×C₅×Q₈	—	4	1	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂×C₅×Q₈	—	4	1	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂²×C₄×C₅	—	4	2	no	—
87	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×D₈	—	2	4	no	—
91	C₅×D₈	—	4	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₅×C₂²⋊C₄	—	4	2	no	—
108	C₅×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₅	—	4	2	no	—
110	C₄²×C₅	D₈	1	1	yes	characteristic
111	C₄²×C₅	D₈	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
114	C₅×C₁₆	—	2	4	no	—
115	#533	—	2	1	no	—
116	#532	—	2	1	no	—
117	#500	C₂²×C₅	1	1	yes	characteristic
118	#500	C₂²×C₅	1	1	yes	characteristic
119	#500	C₂²×C₅	1	1	yes	characteristic
120	#499	—	2	1	no	—
121	#499	C₂²×C₅	1	2	yes	—
122	#498	—	2	1	no	—
123	#493	C₂²×C₅	1	1	yes	characteristic
124	#486	—	2	1	no	—
125	#466	—	2	1	no	—
126	#466	C₂²×C₅	1	2	yes	—
127	C₄×D₈	—	2	1	no	—
128	C₄×D₈	C₂²×C₅	1	2	yes	—
129	#427	—	2	1	no	—
130	#427	C₂²×C₅	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
132	#367	C₂²×C₅	1	4	yes	—
133	#366	C₂²×C₅	1	4	yes	—
134	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
135	#357	C₂²×C₅	1	4	yes	—
136	#357	C₂²×C₅	1	4	yes	—
137	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
138	#346	C₂²×C₅	1	2	yes	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
141	C₅×Q₈	—	8	2	no	—
142	C₅×Q₈	—	8	2	no	—
143	C₅×Q₈	—	8	2	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	4	2	no	—
146	C₅×D₄	—	8	2	no	—
147	C₅×D₄	—	8	2	no	—
148	C₅×D₄	—	8	4	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	8	4	no	—
157	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₅×C₈	—	2	1	no	—
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	#137	1	2	yes	—
166	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
167	C₂×#41	—	4	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	2	1	no	—
175	#127	—	2	1	no	—
176	#125	—	2	2	no	—
177	#124	—	4	1	no	—
178	C₂²⋊Q₈	—	4	1	no	—
179	C₂²⋊Q₈	—	4	1	no	—
180	#121	—	2	2	no	—
181	#121	—	4	1	no	—
182	C₄×Q₈	—	4	1	no	—
183	C₄×Q₈	—	4	1	no	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	—	4	1	no	—
186	#117	—	2	2	no	—
187	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
188	C₂×C₁₆	C₂³×C₅	1	4	yes	—
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	#107	—	2	1	no	—
192	#107	C₂³×C₅	1	2	yes	—
193	#107	C₂³×C₅	1	2	yes	—
194	#107	C₂³×C₅	1	2	yes	—
195	#106	—	2	2	no	—
196	#106	—	2	2	no	—
197	#102	—	2	2	no	—
198	#102	—	2	2	no	—
199	C₄×C₈	C₅×D₄	1	1	yes	characteristic
200	C₄×C₈	C₅×D₄	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	8	2	no	—
207	C₂²×C₅	—	8	2	no	—
208	C₂²×C₅	—	8	2	no	—
209	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	8	2	no	—
215	C₄×C₅	—	8	2	no	—
216	C₄×C₅	—	8	2	no	—
217	C₄×C₅	#509	1	2	yes	—
218	C₄×C₅	#536	1	1	yes	characteristic
219	C₄×C₅	C₂×#137	1	1	yes	characteristic
220	#41	—	8	4	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
230	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	D₈	—	2	4	no	—
233	D₈	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	C₅×D₈	1	1	yes	characteristic
253	C₄²	C₅×D₈	1	1	yes	characteristic
254	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
255	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
256	C₁₆	—	2	4	no	—
257	C₂×C₅	—	4	1	no	—
258	C₂×C₅	—	16	2	no	—
259	C₂×C₅	#3226	1	1	yes	characteristic
260	C₂×C₅	C₂×#509	1	1	yes	characteristic
261	C₂×C₅	#3414	1	1	yes	characteristic
262	C₂³	—	4	2	no	—
263	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
264	Q₈	—	8	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	C₅×#137	1	2	yes	—
289	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
290	C₅	#33801	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₅×#509	1	2	yes	—
306	C₄	C₅×#536	1	1	yes	characteristic
307	C₄	C₂×C₅×#137	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	16	2	no	—
310	C₂	C₅×#3226	1	1	yes	characteristic
311	C₂	C₂×C₅×#509	1	1	yes	characteristic
312	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×#3169	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
2. C₅×#3160	C₂	semidirect
3. C₅×#2255	C₂	semidirect
4. C₅×#2244	C₂	semidirect
5. C₅×#2234	C₂	semidirect
6. C₅×#2226	C₂	semidirect
7. C₅×#2224	C₂	semidirect
8. C₅×#2163	C₂	semidirect
33. #33801	C₅	direct (N × H)
71. #3160	C₂×C₅	semidirect
72. #2255	C₂×C₅	semidirect
73. #2244	C₂×C₅	semidirect
74. #2234	C₂×C₅	semidirect
75. #2226	C₂×C₅	semidirect
76. #2224	C₂×C₅	semidirect
77. #2163	C₂×C₅	semidirect
290. C₅	#33801	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×#3169	2
C₅×#3160	2
C₅×#2255	2
C₅×#2244	2
C₅×#2234	2
C₅×#2226	2
C₅×#2224	2
C₅×#2163	2
#33801	5

Subgroups — C5×#33801 #11956521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33801 #11956521