Subgroups — C₅×#20782 #11943502

Order 1280.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₅×#1840	C₂	1	2	yes	maximal, complemented
3	C₅×#1832	C₂	1	2	yes	maximal, complemented
4	C₅×#1818	C₂	1	2	yes	maximal
5	C₅×#1798	C₂	1	2	yes	maximal
6	C₄×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
8	C₅×#1744	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	—
16	C₄×C₅×C₂²⋊C₄	C₄	1	4	yes	—
17	C₅×#376	C₂²	1	1	yes	characteristic
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#336	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	C₅×#336	C₂²	1	4	yes	—
24	C₅×#336	C₂²	1	4	yes	—
25	#20782	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	8	yes	—
29	C₅×#117	—	2	4	no	—
30	C₅×#117	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	—	2	4	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	8	no	—
46	C₅×C₂²⋊C₈	—	2	8	no	—
47	C₄×C₅×C₈	—	2	8	no	—
48	C₅×#95	C₂×C₄	1	4	yes	—
49	C₅×#95	C₂×C₄	1	4	yes	—
50	C₄×#117	C₂×C₅	1	1	yes	characteristic
51	#1840	C₂×C₅	1	2	yes	complemented
52	#1832	C₂×C₅	1	2	yes	complemented
53	#1818	C₂×C₅	1	2	yes	—
54	#1798	C₂×C₅	1	2	yes	—
55	C₄×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
56	C₄×C₂²⋊C₈	C₂×C₅	1	2	yes	—
57	#1744	C₂×C₅	1	2	yes	complemented
58	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	8	no	—
63	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	4	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₂×C₅×C₈	—	2	8	no	—
73	C₂×C₅×C₈	—	2	8	no	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₄⋊C₄	—	2	4	no	—
77	C₅×C₂²⋊C₄	—	2	4	no	—
78	C₅×C₂²⋊C₄	—	2	4	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	2	8	no	—
81	C₄²×C₅	—	2	8	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	#41	1	4	yes	—
87	C₄²×C₅	#41	1	4	yes	—
88	C₂×#117	C₂²×C₅	1	1	yes	characteristic
89	C₂×#117	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
92	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
93	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
94	C₄×C₄⋊C₄	C₄×C₅	1	4	yes	—
95	C₄×C₂²⋊C₄	C₄×C₅	1	4	yes	—
96	#376	C₂²×C₅	1	1	yes	characteristic
97	#376	C₂²×C₅	1	1	yes	characteristic
98	C₄³	C₂²×C₅	1	1	yes	characteristic
99	C₄³	C₂²×C₅	1	1	yes	characteristic
100	#336	C₂²×C₅	1	4	yes	—
101	#336	C₂²×C₅	1	4	yes	—
102	#336	C₂²×C₅	1	4	yes	—
103	#336	C₂²×C₅	1	4	yes	—
104	C₂³×C₅	—	2	1	no	—
105	C₂³×C₅	—	2	1	no	—
106	C₂³×C₅	—	2	4	no	—
107	C₂³×C₅	#117	1	2	yes	—
108	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	4	8	no	—
122	C₂×C₄×C₅	#117	1	4	yes	—
123	C₂×C₄×C₅	#117	1	2	yes	—
124	C₂×C₄×C₅	#117	1	4	yes	—
125	C₂×C₄×C₅	#131	1	8	yes	—
126	C₂×C₄×C₅	C₂×#41	1	2	yes	—
127	C₂×C₄×C₅	C₂×#41	1	2	yes	—
128	C₂×C₄×C₅	C₂×#41	1	2	yes	—
129	C₅×C₈	—	4	8	no	—
130	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
131	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₅	1	8	yes	—
133	#117	—	2	4	no	—
134	#117	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
141	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
142	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
143	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
149	C₄⋊C₈	—	2	8	no	—
150	C₂²⋊C₈	—	2	8	no	—
151	C₄×C₈	—	2	8	no	—
152	#95	C₂×C₄×C₅	1	4	yes	—
153	#95	C₂×C₄×C₅	1	4	yes	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	4	2	no	—
158	C₂²×C₅	C₂×#117	1	1	yes	characteristic
159	C₂²×C₅	C₂×#117	1	1	yes	characteristic
160	C₂²×C₅	C₂×#117	1	1	yes	characteristic
161	C₂²×C₅	C₂×#131	1	4	yes	—
162	C₄×C₅	—	2	4	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	—	4	2	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	#433	1	8	yes	—
169	C₄×C₅	#529	1	4	yes	—
170	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
182	C₂²×C₄	C₅×#41	1	4	yes	—
183	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	8	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	8	no	—
193	C₄²	—	2	8	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₅×#41	1	4	yes	—
199	C₄²	C₅×#41	1	4	yes	—
200	C₂×C₅	—	4	2	no	—
201	C₂×C₅	#2283	1	1	yes	characteristic
202	C₂×C₅	C₂×#433	1	2	yes	—
203	C₂×C₅	#2922	1	2	yes	—
204	C₂×C₅	#2922	1	2	yes	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	4	no	—
208	C₂³	C₅×#117	1	2	yes	—
209	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	C₅×#117	1	4	yes	—
224	C₂×C₄	C₅×#117	1	2	yes	—
225	C₂×C₄	C₅×#117	1	4	yes	—
226	C₂×C₄	C₅×#131	1	8	yes	—
227	C₂×C₄	C₂×C₅×#41	1	2	yes	—
228	C₂×C₄	C₂×C₅×#41	1	2	yes	—
229	C₂×C₄	C₂×C₅×#41	1	2	yes	—
230	C₈	—	4	8	no	—
231	C₅	#20782	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×C₅×#117	1	1	yes	characteristic
237	C₂²	C₂×C₅×#117	1	1	yes	characteristic
238	C₂²	C₂×C₅×#117	1	1	yes	characteristic
239	C₂²	C₂×C₅×#131	1	4	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₅×#433	1	8	yes	—
247	C₄	C₅×#529	1	4	yes	—
248	C₂	—	4	2	no	—
249	C₂	C₅×#2283	1	1	yes	characteristic
250	C₂	C₂×C₅×#433	1	2	yes	—
251	C₂	C₅×#2922	1	2	yes	—
252	C₂	C₅×#2922	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂³
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 ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1840	C₂	semidirect
3. C₅×#1832	C₂	semidirect
6. C₄×C₅×C₄⋊C₈	C₂	semidirect
8. C₅×#1744	C₂	semidirect
25. #20782	C₅	direct (N × H)
51. #1840	C₂×C₅	semidirect
52. #1832	C₂×C₅	semidirect
55. C₄×C₄⋊C₈	C₂×C₅	semidirect
57. #1744	C₂×C₅	semidirect
231. C₅	#20782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#117	2
C₅×#1840	2
C₅×#1832	2
C₅×#1818	2
C₅×#1798	2
C₄×C₅×C₄⋊C₈	2
C₄×C₅×C₂²⋊C₈	2
C₅×#1744	2
#20782	5

Subgroups — C5×#20782 #11943502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20782 #11943502