Subgroups — C₅×#10615 #11933335

Order 1280.

418 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1954	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1802	C₂	1	2	yes	maximal
4	C₅×#1296	C₂	1	2	yes	maximal, complemented
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#102	—	2	2	no	—
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#340	—	2	2	no	—
17	#10615	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	—	2	1	no	—
20	C₂²×C₅×C₈	—	2	1	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×D₄	—	2	2	no	—
25	C₄×C₅×D₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×#107	—	4	2	no	—
33	C₅×#107	—	4	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×#102	—	4	4	no	—
38	C₅×C₂²⋊C₈	—	4	2	no	—
39	C₅×C₂²⋊C₈	—	4	2	no	—
40	C₄×C₅×C₈	—	2	2	no	—
41	#1954	C₂×C₅	1	1	yes	characteristic
42	#1845	C₂×C₅	1	1	yes	characteristic, complemented
43	#1802	C₂×C₅	1	2	yes	—
44	#1296	C₂×C₅	1	2	yes	complemented
45	#1277	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂⁴×C₅	—	2	1	no	—
47	C₂×C₅×D₄	—	2	2	no	—
48	C₂×C₅×D₄	—	2	1	no	—
49	C₂×C₅×D₄	—	2	2	no	—
50	C₂×C₅×D₄	D₈	1	2	yes	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	4	1	no	—
53	C₂²×C₄×C₅	—	4	1	no	—
54	C₂²×C₄×C₅	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	1	no	—
68	C₂×C₅×C₈	—	4	1	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	—	4	1	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	D₈	1	2	yes	—
73	C₂×C₅×C₈	SD₁₆	1	2	yes	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	8	2	no	—
80	C₅×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₅	#41	1	2	yes	—
82	C₄²×C₅	#41	1	2	yes	—
83	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
84	C₂×#107	—	2	1	no	—
85	C₂×#107	—	2	1	no	—
86	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
87	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×#102	—	2	2	no	—
89	C₂×C₂²⋊C₈	—	2	1	no	—
90	C₂×C₂²⋊C₈	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	#384	C₂²×C₅	1	2	yes	—
93	#340	—	2	2	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	4	1	no	—
96	C₂³×C₅	—	4	1	no	—
97	C₂³×C₅	—	4	1	no	—
98	C₂³×C₅	—	4	1	no	—
99	C₂³×C₅	—	4	1	no	—
100	C₂³×C₅	—	4	1	no	—
101	C₂³×C₅	#121	1	1	yes	characteristic
102	C₅×D₄	—	2	4	no	—
103	C₅×D₄	—	4	2	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	—	4	1	no	—
109	C₂×C₄×C₅	—	8	1	no	—
110	C₂×C₄×C₅	—	8	1	no	—
111	C₂×C₄×C₅	—	8	1	no	—
112	C₂×C₄×C₅	—	8	1	no	—
113	C₂×C₄×C₅	—	8	2	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	—	8	2	no	—
116	C₂×C₄×C₅	#123	1	1	yes	characteristic
117	C₂×C₄×C₅	#123	1	2	yes	—
118	C₂×C₄×C₅	#124	1	1	yes	characteristic
119	C₂×C₄×C₅	#126	1	2	yes	—
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	#135	1	1	yes	characteristic
124	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅	#137	1	2	yes	—
126	C₂×C₄×C₅	#137	1	1	yes	characteristic
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	8	2	no	—
130	C₅×C₈	—	8	1	no	—
131	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
132	C₂³×C₄	—	2	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	2	1	no	—
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	#107	—	4	2	no	—
146	#107	—	4	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	#102	—	4	4	no	—
151	C₂²⋊C₈	—	4	2	no	—
152	C₂²⋊C₈	—	4	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₂²×C₅	—	4	2	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	8	1	no	—
158	C₂²×C₅	—	8	1	no	—
159	C₂²×C₅	—	8	1	no	—
160	C₂²×C₅	—	8	1	no	—
161	C₂²×C₅	#397	1	1	yes	characteristic
162	C₂²×C₅	#459	1	1	yes	characteristic
163	C₂²×C₅	#461	1	1	yes	characteristic
164	C₂²×C₅	#466	1	1	yes	characteristic
165	C₂²×C₅	#470	1	1	yes	characteristic
166	C₂²×C₅	#472	1	1	yes	characteristic
167	C₂²×C₅	#473	1	1	yes	characteristic
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	8	1	no	—
170	C₄×C₅	—	16	2	no	—
171	C₄×C₅	#480	1	1	yes	characteristic
172	C₄×C₅	#480	1	2	yes	—
173	C₄×C₅	#481	1	1	yes	characteristic
174	C₄×C₅	#485	1	2	yes	—
175	C₄×C₅	#486	1	1	yes	characteristic
176	C₄×C₅	#489	1	1	yes	characteristic
177	C₂⁴	—	2	1	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₅×D₈	1	2	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₅×#41	1	1	yes	characteristic
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	C₅×D₈	1	2	yes	—
204	C₂×C₈	C₅×SD₁₆	1	2	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	8	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	C₅×#41	1	2	yes	—
213	C₄²	C₅×#41	1	2	yes	—
214	C₂×C₅	—	8	2	no	—
215	C₂×C₅	#1667	1	1	yes	characteristic
216	C₂×C₅	#1673	1	1	yes	characteristic
217	C₂×C₅	#1689	1	1	yes	characteristic
218	C₂×C₅	#1695	1	1	yes	characteristic
219	C₂×C₅	#2071	1	1	yes	characteristic
220	C₂×C₅	#2075	1	1	yes	characteristic
221	C₂×C₅	#2078	1	1	yes	characteristic
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₅×#121	1	1	yes	characteristic
230	D₄	—	2	4	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	C₅×#123	1	1	yes	characteristic
245	C₂×C₄	C₅×#123	1	2	yes	—
246	C₂×C₄	C₅×#124	1	1	yes	characteristic
247	C₂×C₄	C₅×#126	1	2	yes	—
248	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
251	C₂×C₄	C₅×#135	1	1	yes	characteristic
252	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₅×#137	1	2	yes	—
254	C₂×C₄	C₅×#137	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	8	2	no	—
258	C₈	—	8	1	no	—
259	C₅	#10615	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	C₅×#397	1	1	yes	characteristic
268	C₂²	C₅×#459	1	1	yes	characteristic
269	C₂²	C₅×#461	1	1	yes	characteristic
270	C₂²	C₅×#466	1	1	yes	characteristic
271	C₂²	C₅×#470	1	1	yes	characteristic
272	C₂²	C₅×#472	1	1	yes	characteristic
273	C₂²	C₅×#473	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	8	1	no	—
276	C₄	—	16	2	no	—
277	C₄	C₅×#480	1	1	yes	characteristic
278	C₄	C₅×#480	1	2	yes	—
279	C₄	C₅×#481	1	1	yes	characteristic
280	C₄	C₅×#485	1	2	yes	—
281	C₄	C₅×#486	1	1	yes	characteristic
282	C₄	C₅×#489	1	1	yes	characteristic
283	C₂	—	8	2	no	—
284	C₂	C₅×#1667	1	1	yes	characteristic
285	C₂	C₅×#1673	1	1	yes	characteristic
286	C₂	C₅×#1689	1	1	yes	characteristic
287	C₂	C₅×#1695	1	1	yes	characteristic
288	C₂	C₅×#2071	1	1	yes	characteristic
289	C₂	C₅×#2075	1	1	yes	characteristic
290	C₂	C₅×#2078	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1845	C₂	semidirect
4. C₅×#1296	C₂	semidirect
5. C₅×#1277	C₂	semidirect
17. #10615	C₅	direct (N × H)
42. #1845	C₂×C₅	semidirect
44. #1296	C₂×C₅	semidirect
45. #1277	C₂×C₅	semidirect
259. C₅	#10615	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×#1954	2
C₅×#1845	2
C₅×#1802	2
C₅×#1296	2
C₅×#1277	2
#10615	5

Subgroups — C5×#10615 #11933335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10615 #11933335