Subgroups — C₅×#33637 #11956357

Order 1280.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
3	C₅×#2231	C₂	1	4	yes	maximal, complemented
4	C₅×#2230	C₂	1	4	yes	maximal, complemented
5	C₅×#2155	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2145	C₂	1	2	yes	maximal
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂×C₅×#131	—	2	1	no	—
10	C₅×#533	—	2	1	no	—
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#500	—	2	1	no	—
13	C₅×#499	—	2	1	no	—
14	C₅×#499	C₂²	1	2	yes	complemented
15	C₅×#498	C₂²	1	2	yes	complemented
16	C₅×#443	—	2	1	no	—
17	C₅×#443	—	2	1	no	—
18	C₅×#443	C₂²	1	2	yes	—
19	C₅×#443	C₂²	1	2	yes	—
20	C₂×C₅×#108	C₂²	1	1	yes	characteristic
21	C₅×#427	—	2	1	no	—
22	C₅×#427	C₂²	1	2	yes	—
23	C₂×C₅×#104	—	2	1	no	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#365	C₂²	1	8	yes	complemented
26	C₅×#364	C₂²	1	2	yes	complemented
27	C₅×#364	C₂²	1	2	yes	complemented
28	C₅×#360	C₂²	1	8	yes	—
29	#33637	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	—	2	1	no	—
33	C₅×#135	Q₈	1	2	yes	complemented
34	C₅×#135	C₂³	1	2	yes	—
35	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic, complemented
36	C₂×C₅×SD₁₆	—	2	1	no	—
37	C₂×C₅×D₈	Q₈	1	1	yes	characteristic, complemented
38	C₅×#131	—	2	2	no	—
39	C₅×#131	—	2	2	no	—
40	C₅×#131	—	4	1	no	—
41	C₅×#131	—	4	1	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂×C₅×M₁₆	—	4	1	no	—
44	C₂²×C₅×C₈	—	4	1	no	—
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×#125	—	2	2	no	—
49	C₅×C₂²⋊Q₈	—	4	2	no	—
50	C₄×C₅×Q₈	—	4	2	no	—
51	C₅×#117	—	2	2	no	—
52	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
53	C₅×M₃₂	—	2	2	no	—
54	C₅×M₃₂	C₂³	1	4	yes	—
55	C₂×C₅×C₁₆	—	2	2	no	—
56	C₅×#108	—	2	1	no	—
57	C₅×#108	C₂³	1	2	yes	—
58	C₅×#107	—	2	2	no	—
59	C₅×#107	—	2	2	no	—
60	C₅×#106	—	2	2	no	—
61	C₅×#106	C₂³	1	4	yes	—
62	C₅×#104	—	2	2	no	—
63	C₅×#104	—	2	2	no	—
64	C₄×C₅×C₈	D₄	1	1	yes	characteristic
65	C₄×C₅×C₈	D₄	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
68	#3160	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#443	C₂×C₅	1	1	yes	characteristic
70	#2231	C₂×C₅	1	4	yes	complemented
71	#2230	C₂×C₅	1	4	yes	complemented
72	#2155	C₂×C₅	1	2	yes	complemented
73	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
74	#2145	C₂×C₅	1	2	yes	—
75	C₅×#41	—	2	2	no	—
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	4	1	no	—
78	C₅×#41	—	4	1	no	—
79	C₂×C₅×Q₈	—	2	1	no	—
80	C₂×C₅×Q₈	—	4	2	no	—
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₅×Q₁₆	—	2	1	no	—
87	C₅×Q₁₆	C₂×Q₈	1	2	yes	—
88	C₅×SD₁₆	—	2	2	no	—
89	C₅×SD₁₆	—	2	2	no	—
90	C₅×D₈	—	2	1	no	—
91	C₅×D₈	C₂×Q₈	1	2	yes	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	4	1	no	—
94	C₅×M₁₆	—	4	2	no	—
95	C₅×M₁₆	—	4	2	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	—	4	2	no	—
98	C₂×C₅×C₈	—	4	2	no	—
99	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
100	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
102	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₅×C₈	#41	1	1	yes	characteristic
109	C₂×C₅×C₈	#41	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	4	4	no	—
113	C₅×C₄⋊C₄	—	4	2	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	—	4	2	no	—
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	C₅×C₁₆	—	2	4	no	—
120	C₂×#135	C₂²×C₅	1	1	yes	characteristic
121	C₂×#131	—	2	1	no	—
122	#533	—	2	1	no	—
123	C₂×M₃₂	C₂²×C₅	1	2	yes	—
124	#500	—	2	1	no	—
125	#499	—	2	1	no	—
126	#499	C₂²×C₅	1	2	yes	complemented
127	#498	C₂²×C₅	1	2	yes	complemented
128	#443	—	2	1	no	—
129	#443	—	2	1	no	—
130	#443	C₂²×C₅	1	2	yes	—
131	#443	C₂²×C₅	1	2	yes	—
132	C₂×#108	C₂²×C₅	1	1	yes	characteristic
133	#427	—	2	1	no	—
134	#427	C₂²×C₅	1	2	yes	—
135	C₂×#104	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
137	#365	C₂²×C₅	1	8	yes	complemented
138	#364	C₂²×C₅	1	2	yes	complemented
139	#364	C₂²×C₅	1	2	yes	complemented
140	#360	C₂²×C₅	1	8	yes	—
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₅×Q₈	—	2	2	no	—
144	C₅×Q₈	—	4	1	no	—
145	C₅×Q₈	—	8	4	no	—
146	C₅×D₄	—	2	2	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	4	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
165	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
166	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
167	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
168	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
169	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
170	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₅×C₈	C₂×#41	1	1	yes	characteristic
172	C₂×#41	—	2	1	no	—
173	#135	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	C₅×Q₈	1	2	yes	complemented
176	#135	C₂³×C₅	1	2	yes	—
177	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic, complemented
178	C₂×SD₁₆	—	2	1	no	—
179	C₂×D₈	C₅×Q₈	1	1	yes	characteristic, complemented
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	#131	—	4	1	no	—
183	#131	—	4	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	4	1	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	2	no	—
190	#125	—	2	2	no	—
191	C₂²⋊Q₈	—	4	2	no	—
192	C₄×Q₈	—	4	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	C₂³×C₅	1	4	yes	—
197	C₂×C₁₆	—	2	2	no	—
198	#108	—	2	1	no	—
199	#108	C₂³×C₅	1	2	yes	—
200	#107	—	2	2	no	—
201	#107	—	2	2	no	—
202	#106	—	2	2	no	—
203	#106	C₂³×C₅	1	4	yes	—
204	#104	—	2	2	no	—
205	#104	—	2	2	no	—
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
210	C₂²×C₅	—	4	1	no	—
211	C₂²×C₅	—	4	2	no	—
212	C₂²×C₅	—	8	2	no	—
213	C₂²×C₅	#476	1	1	yes	characteristic
214	C₂²×C₅	#476	1	1	yes	characteristic
215	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	4	1	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	—	8	4	no	—
220	C₄×C₅	#476	1	1	yes	characteristic
221	C₄×C₅	#476	1	1	yes	characteristic
222	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
223	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	Q₁₆	—	2	1	no	—
236	Q₁₆	C₂×C₅×Q₈	1	2	yes	—
237	SD₁₆	—	2	2	no	—
238	SD₁₆	—	2	2	no	—
239	D₈	—	2	1	no	—
240	D₈	C₂×C₅×Q₈	1	2	yes	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
249	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
250	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
251	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
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	C₂×C₅×Q₈	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
257	C₂×C₈	C₅×#41	1	1	yes	characteristic
258	C₂×C₈	C₅×#41	1	1	yes	characteristic
259	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
267	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
268	C₁₆	—	2	4	no	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	8	2	no	—
273	C₂×C₅	C₂×#476	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	4	1	no	—
278	Q₈	—	8	4	no	—
279	D₄	—	2	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
291	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
293	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
294	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
296	C₈	—	4	2	no	—
297	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
298	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
299	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
300	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
301	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
302	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
303	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
304	C₈	C₂×C₅×#41	1	1	yes	characteristic
305	C₅	#33637	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₅×#476	1	1	yes	characteristic
310	C₂²	C₅×#476	1	1	yes	characteristic
311	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	4	no	—
316	C₄	C₅×#476	1	1	yes	characteristic
317	C₄	C₅×#476	1	1	yes	characteristic
318	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
319	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₅×#476	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×#476
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅×#3160	C₂	semidirect
3. C₅×#2231	C₂	semidirect
4. C₅×#2230	C₂	semidirect
5. C₅×#2155	C₂	semidirect
6. C₂×C₅×#364	C₂	semidirect
14. C₅×#499	C₂²	semidirect
15. C₅×#498	C₂²	semidirect
25. C₅×#365	C₂²	semidirect
26. C₅×#364	C₂²	semidirect
27. C₅×#364	C₂²	semidirect
29. #33637	C₅	direct (N × H)
33. C₅×#135	Q₈	semidirect
35. C₂×C₅×Q₁₆	Q₈	semidirect
37. C₂×C₅×D₈	Q₈	semidirect
68. #3160	C₂×C₅	semidirect
70. #2231	C₂×C₅	semidirect
71. #2230	C₂×C₅	semidirect
72. #2155	C₂×C₅	semidirect
73. C₂×#364	C₂×C₅	semidirect
126. #499	C₂²×C₅	semidirect
127. #498	C₂²×C₅	semidirect
137. #365	C₂²×C₅	semidirect
138. #364	C₂²×C₅	semidirect
139. #364	C₂²×C₅	semidirect
175. #135	C₅×Q₈	semidirect
177. C₂×Q₁₆	C₅×Q₈	semidirect
179. C₂×D₈	C₅×Q₈	semidirect
305. C₅	#33637	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×#3160	2
C₂×C₅×#443	2
C₅×#2231	2
C₅×#2230	2
C₅×#2155	2
C₂×C₅×#364	2
C₅×#2145	2
#33637	5

Subgroups — C5×#33637 #11956357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33637 #11956357