Subgroups — C₅×#13480 #11936200

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#357	C₂	1	2	yes	maximal
2	C₅×#1970	C₂	1	1	yes	maximal, characteristic
3	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1380	C₂	1	2	yes	maximal, complemented
5	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×#107	C₂²	1	1	yes	characteristic
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₂×C₅×#102	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#390	—	2	1	no	—
14	C₅×#384	—	2	1	no	—
15	C₅×#357	—	2	4	no	—
16	#13480	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×D₄	—	2	1	no	—
18	C₂×C₅×D₈	—	2	2	no	—
19	C₂×C₅×D₈	—	2	2	no	—
20	C₂×C₅×D₈	—	2	1	no	—
21	C₂×C₅×D₈	D₄	1	2	yes	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	4	1	no	—
27	C₂×C₅×C₄⋊C₄	—	4	1	no	—
28	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
29	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₂×C₅×C₁₆	—	2	2	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₅×#107	—	2	2	no	—
35	C₅×#107	—	2	2	no	—
36	C₅×#107	—	2	2	no	—
37	C₅×#107	—	4	2	no	—
38	C₅×#102	—	4	2	no	—
39	C₄×C₅×C₈	—	2	2	no	—
40	C₂×#357	C₂×C₅	1	2	yes	—
41	#1970	C₂×C₅	1	1	yes	characteristic
42	#1845	C₂×C₅	1	1	yes	characteristic, complemented
43	#1380	C₂×C₅	1	2	yes	complemented
44	#1374	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂⁴×C₅	—	4	1	no	—
46	C₂×C₅×D₄	—	2	2	no	—
47	C₂×C₅×D₄	—	4	1	no	—
48	C₂×C₅×D₄	—	4	1	no	—
49	C₂×C₅×D₄	—	4	1	no	—
50	C₂×C₅×D₄	—	4	1	no	—
51	C₂×C₅×D₄	—	4	1	no	—
52	C₂²×C₄×C₅	—	4	1	no	—
53	C₂²×C₄×C₅	—	4	2	no	—
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₅×D₈	—	2	4	no	—
58	C₅×D₈	—	4	2	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	SD₁₆	1	2	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	#41	1	2	yes	—
64	C₂×C₅×C₈	#41	1	2	yes	—
65	C₂×C₅×C₈	#41	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	8	2	no	—
71	C₅×C₄⋊C₄	—	8	2	no	—
72	C₅×C₂²⋊C₄	—	8	2	no	—
73	C₅×C₂²⋊C₄	—	8	2	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₅×C₁₆	—	4	2	no	—
77	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
79	C₂×#107	—	2	1	no	—
80	C₂×#107	C₂²×C₅	1	2	yes	—
81	C₂×#107	C₂²×C₅	1	1	yes	characteristic
82	C₂×#102	—	2	1	no	—
83	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	#390	—	2	1	no	—
85	#384	—	2	1	no	—
86	#357	—	2	4	no	—
87	C₂³×C₅	—	4	2	no	—
88	C₂³×C₅	—	8	1	no	—
89	C₂³×C₅	—	8	1	no	—
90	C₂³×C₅	—	8	1	no	—
91	C₂³×C₅	—	8	1	no	—
92	C₂³×C₅	—	8	1	no	—
93	C₂³×C₅	—	8	1	no	—
94	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
95	C₅×D₄	—	4	2	no	—
96	C₅×D₄	—	4	2	no	—
97	C₅×D₄	—	8	1	no	—
98	C₅×D₄	—	8	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	8	2	no	—
104	C₂×C₄×C₅	—	8	1	no	—
105	C₂×C₄×C₅	—	8	1	no	—
106	C₂×C₄×C₅	—	8	2	no	—
107	C₂×C₄×C₅	—	8	1	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	D₁₆	1	2	yes	—
110	C₂×C₄×C₅	SD₃₂	1	2	yes	—
111	C₂×C₄×C₅	#121	1	1	yes	characteristic
112	C₂×C₄×C₅	#123	1	1	yes	characteristic
113	C₂×C₄×C₅	#124	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₅	#137	1	1	yes	characteristic
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	#123	1	2	yes	—
119	C₅×C₈	#126	1	2	yes	—
120	C₂²×D₄	—	2	1	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	—	2	1	no	—
124	C₂×D₈	C₅×D₄	1	2	yes	—
125	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₂²⋊C₄	—	4	1	no	—
132	C₂×C₂²⋊C₄	—	4	1	no	—
133	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	—	4	2	no	—
141	#102	—	4	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₂²×C₅	—	8	2	no	—
144	C₂²×C₅	—	8	2	no	—
145	C₂²×C₅	—	8	2	no	—
146	C₂²×C₅	—	16	1	no	—
147	C₂²×C₅	—	16	1	no	—
148	C₂²×C₅	—	16	1	no	—
149	C₂²×C₅	—	16	1	no	—
150	C₂²×C₅	#459	1	1	yes	characteristic
151	C₂²×C₅	#480	1	1	yes	characteristic
152	C₂²×C₅	#486	1	1	yes	characteristic
153	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
154	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
155	C₂²×C₅	#509	1	1	yes	characteristic
156	C₂²×C₅	#510	1	1	yes	characteristic
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	16	2	no	—
160	C₄×C₅	—	16	1	no	—
161	C₄×C₅	#397	1	1	yes	characteristic
162	C₄×C₅	#461	1	1	yes	characteristic
163	C₄×C₅	#481	1	1	yes	characteristic
164	C₄×C₅	#489	1	1	yes	characteristic
165	C₂⁴	—	4	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂×D₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	D₈	—	2	4	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	C₅×SD₁₆	1	2	yes	—
182	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×C₈	C₅×#41	1	2	yes	—
184	C₂×C₈	C₅×#41	1	2	yes	—
185	C₂×C₈	C₅×#41	1	1	yes	characteristic
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	8	2	no	—
191	C₄⋊C₄	—	8	2	no	—
192	C₂²⋊C₄	—	8	2	no	—
193	C₂²⋊C₄	—	8	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₁₆	—	4	2	no	—
197	C₂×C₅	—	16	2	no	—
198	C₂×C₅	#2071	1	1	yes	characteristic
199	C₂×C₅	#2206	1	1	yes	characteristic
200	C₂×C₅	#2208	1	1	yes	characteristic
201	C₂×C₅	#2232	1	1	yes	characteristic
202	C₂×C₅	#2233	1	1	yes	characteristic
203	C₂×C₅	#2240	1	1	yes	characteristic
204	C₂×C₅	#2243	1	1	yes	characteristic
205	C₂³	—	4	2	no	—
206	C₂³	—	8	1	no	—
207	C₂³	—	8	1	no	—
208	C₂³	—	8	1	no	—
209	C₂³	—	8	1	no	—
210	C₂³	—	8	1	no	—
211	C₂³	—	8	1	no	—
212	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	8	1	no	—
216	D₄	—	8	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	1	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	C₅×D₁₆	1	2	yes	—
228	C₂×C₄	C₅×SD₃₂	1	2	yes	—
229	C₂×C₄	C₅×#121	1	1	yes	characteristic
230	C₂×C₄	C₅×#123	1	1	yes	characteristic
231	C₂×C₄	C₅×#124	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
233	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂×C₄	C₅×#137	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₅×#123	1	2	yes	—
237	C₈	C₅×#126	1	2	yes	—
238	C₅	#13480	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	16	1	no	—
243	C₂²	—	16	1	no	—
244	C₂²	—	16	1	no	—
245	C₂²	—	16	1	no	—
246	C₂²	C₅×#459	1	1	yes	characteristic
247	C₂²	C₅×#480	1	1	yes	characteristic
248	C₂²	C₅×#486	1	1	yes	characteristic
249	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
250	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
251	C₂²	C₅×#509	1	1	yes	characteristic
252	C₂²	C₅×#510	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	16	2	no	—
256	C₄	—	16	1	no	—
257	C₄	C₅×#397	1	1	yes	characteristic
258	C₄	C₅×#461	1	1	yes	characteristic
259	C₄	C₅×#481	1	1	yes	characteristic
260	C₄	C₅×#489	1	1	yes	characteristic
261	C₂	—	16	2	no	—
262	C₂	C₅×#2071	1	1	yes	characteristic
263	C₂	C₅×#2206	1	1	yes	characteristic
264	C₂	C₅×#2208	1	1	yes	characteristic
265	C₂	C₅×#2232	1	1	yes	characteristic
266	C₂	C₅×#2233	1	1	yes	characteristic
267	C₂	C₅×#2240	1	1	yes	characteristic
268	C₂	C₅×#2243	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
3. C₅×#1845	C₂	semidirect
4. C₅×#1380	C₂	semidirect
5. C₅×#1374	C₂	semidirect
16. #13480	C₅	direct (N × H)
42. #1845	C₂×C₅	semidirect
43. #1380	C₂×C₅	semidirect
44. #1374	C₂×C₅	semidirect
238. C₅	#13480	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#357	2
C₅×#1970	2
C₅×#1845	2
C₅×#1380	2
C₅×#1374	2
#13480	5

Subgroups — C5×#13480 #11936200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13480 #11936200