Subgroups — C₅×#20815 #11943535

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1842	C₂	1	2	yes	maximal, complemented
5	C₅×#1831	C₂	1	2	yes	maximal, complemented
6	C₅×#1798	C₂	1	4	yes	maximal
7	C₅×#1768	C₂	1	4	yes	maximal, complemented
8	C₂×C₅×#117	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#432	—	2	4	no	—
11	C₅×#431	—	2	4	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
22	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
23	C₅×#376	C₂²	1	2	yes	—
24	C₄³×C₅	C₂²	1	1	yes	characteristic
25	C₄³×C₅	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	8	yes	—
27	#20815	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₅×#117	—	2	8	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	4	yes	—
43	C₂×C₄²×C₅	Q₈	1	4	yes	—
44	C₂×C₄²×C₅	C₂³	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×C₄⋊C₈	—	2	4	no	—
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×C₄⋊C₈	—	2	16	no	—
51	C₅×C₂²⋊C₈	—	2	4	no	—
52	C₅×C₂²⋊C₈	—	2	4	no	—
53	C₅×#97	—	2	4	no	—
54	C₄×C₅×C₈	—	2	4	no	—
55	C₅×#95	C₂×C₄	1	4	yes	—
56	C₅×#95	C₂×C₄	1	4	yes	—
57	C₂×#432	C₂×C₅	1	1	yes	characteristic
58	C₂×#431	C₂×C₅	1	1	yes	characteristic
59	C₄×#117	C₂×C₅	1	1	yes	characteristic
60	#1842	C₂×C₅	1	2	yes	complemented
61	#1831	C₂×C₅	1	2	yes	complemented
62	#1798	C₂×C₅	1	4	yes	—
63	#1768	C₂×C₅	1	4	yes	complemented
64	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	8	no	—
83	C₂×C₅×C₈	—	2	8	no	—
84	C₅×C₄⋊C₄	—	2	8	no	—
85	C₅×C₂²⋊C₄	—	2	8	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	—	4	2	no	—
90	C₄²×C₅	—	4	2	no	—
91	C₄²×C₅	C₄⋊C₄	1	4	yes	—
92	C₄²×C₅	C₄⋊C₄	1	4	yes	—
93	C₄²×C₅	C₂×D₄	1	2	yes	—
94	C₄²×C₅	C₂×D₄	1	2	yes	—
95	C₄²×C₅	C₂×Q₈	1	2	yes	—
96	C₄²×C₅	C₂×Q₈	1	2	yes	—
97	C₄²×C₅	#41	1	8	yes	—
98	C₂×#117	C₂²×C₅	1	2	yes	—
99	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
100	#432	—	2	4	no	—
101	#431	—	2	4	no	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
105	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
106	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
107	C₂×#97	C₂²×C₅	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
109	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
110	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
111	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
112	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
113	#376	C₂²×C₅	1	2	yes	—
114	C₄³	C₂²×C₅	1	1	yes	characteristic
115	C₄³	C₂²×C₅	1	1	yes	characteristic
116	#336	C₂²×C₅	1	8	yes	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	8	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	#117	1	8	yes	—
141	C₂×C₄×C₅	#131	1	4	yes	—
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	4	yes	—
145	C₅×C₈	—	4	4	no	—
146	C₅×C₈	—	4	4	no	—
147	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₅	1	4	yes	—
151	C₂²×C₈	C₂³×C₅	1	4	yes	—
152	#117	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
155	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×C₄²	C₅×D₄	1	4	yes	—
162	C₂×C₄²	C₅×Q₈	1	4	yes	—
163	C₂×C₄²	C₂³×C₅	1	2	yes	—
164	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₅	1	2	yes	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	2	16	no	—
170	C₂²⋊C₈	—	2	4	no	—
171	C₂²⋊C₈	—	2	4	no	—
172	#97	—	2	4	no	—
173	C₄×C₈	—	2	4	no	—
174	#95	C₂×C₄×C₅	1	4	yes	—
175	#95	C₂×C₄×C₅	1	4	yes	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	4	no	—
180	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₅	C₂×#117	1	2	yes	—
182	C₂²×C₅	C₂×#131	1	2	yes	—
183	C₂²×C₅	#568	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	8	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	#424	1	4	yes	—
191	C₄×C₅	#532	1	2	yes	—
192	C₄×C₅	#533	1	2	yes	—
193	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
198	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
207	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
208	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	2	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₂²⋊C₄	—	2	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	C₅×C₄⋊C₄	1	4	yes	—
221	C₄²	C₅×C₄⋊C₄	1	4	yes	—
222	C₄²	C₂×C₅×D₄	1	2	yes	—
223	C₄²	C₂×C₅×D₄	1	2	yes	—
224	C₄²	C₂×C₅×Q₈	1	2	yes	—
225	C₄²	C₂×C₅×Q₈	1	2	yes	—
226	C₄²	C₅×#41	1	8	yes	—
227	C₂×C₅	—	4	2	no	—
228	C₂×C₅	#2285	1	1	yes	characteristic
229	C₂×C₅	C₂×#424	1	1	yes	characteristic
230	C₂×C₅	#2905	1	1	yes	characteristic
231	C₂×C₅	#2925	1	4	yes	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
236	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
252	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
253	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
254	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
255	C₂×C₄	C₅×#117	1	8	yes	—
256	C₂×C₄	C₅×#131	1	4	yes	—
257	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×#41	1	4	yes	—
260	C₈	—	4	4	no	—
261	C₈	—	4	4	no	—
262	C₅	#20815	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
268	C₂²	C₂×C₅×#117	1	2	yes	—
269	C₂²	C₂×C₅×#131	1	2	yes	—
270	C₂²	C₅×#568	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	8	no	—
276	C₄	—	4	4	no	—
277	C₄	C₅×#424	1	4	yes	—
278	C₄	C₅×#532	1	2	yes	—
279	C₄	C₅×#533	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	C₅×#2285	1	1	yes	characteristic
282	C₂	C₂×C₅×#424	1	1	yes	characteristic
283	C₂	C₅×#2905	1	1	yes	characteristic
284	C₂	C₅×#2925	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1842	C₂	semidirect
5. C₅×#1831	C₂	semidirect
7. C₅×#1768	C₂	semidirect
27. #20815	C₅	direct (N × H)
60. #1842	C₂×C₅	semidirect
61. #1831	C₂×C₅	semidirect
63. #1768	C₂×C₅	semidirect
262. C₅	#20815	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×#432	2
C₂×C₅×#431	2
C₄×C₅×#117	2
C₅×#1842	2
C₅×#1831	2
C₅×#1798	2
C₅×#1768	2
#20815	5

Subgroups — C5×#20815 #11943535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20815 #11943535