Subgroups — C₅×#11741 #11934461

Order 1280.

356 conjugacy classes of subgroups.

The subgroup lattice has 288 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×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1348	C₂	1	2	yes	maximal, complemented
5	C₅×#1347	C₂	1	2	yes	maximal
6	C₅×#529	—	2	1	no	—
7	C₅×#499	C₄	1	1	yes	characteristic, complemented
8	C₅×#495	C₄	1	1	yes	characteristic, complemented
9	C₅×#488	—	2	1	no	—
10	C₅×#467	C₂²	1	1	yes	characteristic
11	C₅×#466	C₂²	1	1	yes	characteristic
12	C₅×#465	—	2	1	no	—
13	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
14	C₄×C₅×SD₁₆	—	2	1	no	—
15	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
16	C₂×C₅×#106	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₄⋊C₄	—	2	1	no	—
19	C₅×C₄⋊C₁₆	—	2	1	no	—
20	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₄×C₅×C₁₆	—	2	1	no	—
22	C₅×#340	C₂²	1	2	yes	—
23	#11741	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×SD₁₆	—	2	1	no	—
26	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₅×#126	—	4	1	no	—
30	C₅×#125	—	2	1	no	—
31	C₅×#124	—	2	1	no	—
32	C₅×#123	—	4	1	no	—
33	C₅×C₂²⋊Q₈	—	2	1	no	—
34	C₅×#121	—	2	1	no	—
35	C₄×C₅×Q₈	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	4	1	no	—
38	C₅×#117	—	4	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₅	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₅×#107	C₂×C₄	1	1	yes	characteristic
44	C₅×#107	C₂×C₄	1	1	yes	characteristic
45	C₅×#106	—	2	4	no	—
46	C₅×#106	—	2	1	no	—
47	C₅×#103	—	2	1	no	—
48	C₅×#103	—	2	1	no	—
49	C₅×#102	—	2	1	no	—
50	C₅×#102	—	2	1	no	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₅×#95	—	4	1	no	—
56	#3168	C₂×C₅	1	1	yes	characteristic
57	#2162	C₂×C₅	1	1	yes	characteristic, complemented
58	C₄×#106	C₂×C₅	1	1	yes	characteristic, complemented
59	#1348	C₂×C₅	1	2	yes	complemented
60	#1347	C₂×C₅	1	2	yes	—
61	C₂×C₅×Q₈	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×Q₁₆	—	4	1	no	—
70	C₅×SD₁₆	—	4	2	no	—
71	C₅×D₈	—	4	1	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	2	4	no	—
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₂²⋊C₄	—	4	1	no	—
87	C₅×C₂²⋊C₄	—	4	1	no	—
88	C₅×C₂²⋊C₄	—	4	1	no	—
89	C₅×C₂²⋊C₄	—	4	1	no	—
90	C₅×C₂²⋊C₄	—	4	1	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	—	8	2	no	—
93	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅	SD₁₆	1	1	yes	characteristic
96	C₄²×C₅	SD₁₆	1	1	yes	characteristic
97	C₅×C₁₆	—	4	2	no	—
98	#529	—	2	1	no	—
99	#499	C₄×C₅	1	1	yes	characteristic, complemented
100	#495	C₄×C₅	1	1	yes	characteristic, complemented
101	#488	—	2	1	no	—
102	#467	C₂²×C₅	1	1	yes	characteristic
103	#466	C₂²×C₅	1	1	yes	characteristic
104	#465	—	2	1	no	—
105	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
106	C₄×SD₁₆	—	2	1	no	—
107	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
108	C₂×#106	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₁₆	—	2	1	no	—
112	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
113	C₄×C₁₆	—	2	1	no	—
114	#340	C₂²×C₅	1	2	yes	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
117	C₅×Q₈	—	4	1	no	—
118	C₅×Q₈	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	1	no	—
121	C₅×D₄	—	8	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	8	2	no	—
131	C₂×C₄×C₅	—	8	2	no	—
132	C₂×C₄×C₅	—	8	2	no	—
133	C₂×C₄×C₅	#102	1	1	yes	characteristic
134	C₂×C₄×C₅	#102	1	1	yes	characteristic
135	C₂×C₄×C₅	#102	1	1	yes	characteristic
136	C₂×C₄×C₅	#102	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	—	2	1	no	—
142	C₅×C₈	#104	1	2	yes	—
143	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂×SD₁₆	—	2	1	no	—
145	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
148	#126	—	4	1	no	—
149	#125	—	2	1	no	—
150	#124	—	2	1	no	—
151	#123	—	4	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	#121	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	4	1	no	—
157	#117	—	4	1	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄²	—	4	1	no	—
160	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₂×C₁₆	—	2	2	no	—
162	#107	C₂×C₄×C₅	1	1	yes	characteristic
163	#107	C₂×C₄×C₅	1	1	yes	characteristic
164	#106	—	2	4	no	—
165	#106	—	2	1	no	—
166	#103	—	2	1	no	—
167	#103	—	2	1	no	—
168	#102	—	2	1	no	—
169	#102	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×C₈	C₅×D₄	1	1	yes	characteristic
173	C₄×C₈	C₅×D₄	1	1	yes	characteristic
174	#95	—	4	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	8	1	no	—
178	C₂²×C₅	—	8	1	no	—
179	C₂²×C₅	—	8	1	no	—
180	C₂²×C₅	C₂×#102	1	1	yes	characteristic
181	C₂²×C₅	#508	1	2	yes	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	8	4	no	—
187	C₄×C₅	—	8	1	no	—
188	C₄×C₅	—	8	1	no	—
189	C₄×C₅	—	8	1	no	—
190	C₄×C₅	#360	1	2	yes	—
191	C₄×C₅	#410	1	1	yes	characteristic
192	C₄×C₅	C₂×#104	1	1	yes	characteristic
193	C₂×Q₈	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
199	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	Q₁₆	—	4	1	no	—
202	SD₁₆	—	4	2	no	—
203	D₈	—	4	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₄²	—	8	2	no	—
225	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₄²	C₅×SD₁₆	1	1	yes	characteristic
228	C₄²	C₅×SD₁₆	1	1	yes	characteristic
229	C₁₆	—	4	2	no	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	16	1	no	—
232	C₂×C₅	#1502	1	1	yes	characteristic
233	C₂×C₅	#2138	1	1	yes	characteristic
234	C₂×C₅	C₂×#360	1	1	yes	characteristic
235	C₂³	—	4	1	no	—
236	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
237	Q₈	—	4	1	no	—
238	Q₈	—	4	1	no	—
239	D₄	—	4	1	no	—
240	D₄	—	4	1	no	—
241	D₄	—	8	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₅×#102	1	1	yes	characteristic
254	C₂×C₄	C₅×#102	1	1	yes	characteristic
255	C₂×C₄	C₅×#102	1	1	yes	characteristic
256	C₂×C₄	C₅×#102	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	C₅×#104	1	2	yes	—
263	C₅	#11741	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₂×C₅×#102	1	1	yes	characteristic
270	C₂²	C₅×#508	1	2	yes	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	4	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	C₅×#360	1	2	yes	—
280	C₄	C₅×#410	1	1	yes	characteristic
281	C₄	C₂×C₅×#104	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	16	1	no	—
284	C₂	C₅×#1502	1	1	yes	characteristic
285	C₂	C₅×#2138	1	1	yes	characteristic
286	C₂	C₂×C₅×#360	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₅×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#2162	C₂	semidirect
3. C₄×C₅×#106	C₂	semidirect
4. C₅×#1348	C₂	semidirect
7. C₅×#499	C₄	semidirect
8. C₅×#495	C₄	semidirect
13. C₄×C₅×Q₁₆	C₄	semidirect
15. C₄×C₅×D₈	C₄	semidirect
23. #11741	C₅	direct (N × H)
57. #2162	C₂×C₅	semidirect
58. C₄×#106	C₂×C₅	semidirect
59. #1348	C₂×C₅	semidirect
99. #499	C₄×C₅	semidirect
100. #495	C₄×C₅	semidirect
105. C₄×Q₁₆	C₄×C₅	semidirect
107. C₄×D₈	C₄×C₅	semidirect
263. C₅	#11741	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×#2162	2
C₄×C₅×#106	2
C₅×#1348	2
C₅×#1347	2
#11741	5

Subgroups — C5×#11741 #11934461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11741 #11934461