Subgroups — C₅×#7980 #11930700

Order 1280.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₅×#2283	C₂	1	2	yes	maximal
3	C₅×#1289	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
5	C₅×#431	—	2	2	no	—
6	C₅×#431	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	complemented
8	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
10	C₅×#383	C₄	1	4	yes	complemented
11	C₅×#382	—	2	2	no	—
12	C₅×#381	—	2	2	no	—
13	C₅×#380	C₂²	1	4	yes	—
14	C₄×C₅×C₄⋊C₄	—	2	2	no	—
15	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
16	C₅×#376	C₄	1	4	yes	complemented
17	#7980	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₂×C₅×C₄⋊C₄	—	2	4	no	—
23	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
24	C₂×C₄²×C₅	—	2	4	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₅×C₂²⋊C₈	—	2	4	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	—	2	2	no	—
37	C₅×#95	—	2	4	no	—
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	4	no	—
40	C₅×#95	C₂×C₄	1	8	yes	—
41	C₂×#431	C₂×C₅	1	1	yes	characteristic
42	#2283	C₂×C₅	1	2	yes	—
43	#1289	C₂×C₅	1	4	yes	—
44	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	4	no	—
46	C₂²×C₄×C₅	—	2	8	no	—
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	—	4	2	no	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	4	4	no	—
63	C₅×C₄⋊C₄	—	4	8	no	—
64	C₅×C₂²⋊C₄	—	4	8	no	—
65	C₄²×C₅	—	4	8	no	—
66	C₄²×C₅	C₄²	1	1	yes	characteristic
67	C₄²×C₅	C₄²	1	2	yes	—
68	C₄²×C₅	C₄²	1	1	yes	characteristic
69	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
77	#431	—	2	2	no	—
78	#431	—	2	2	no	—
79	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	complemented
80	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
81	C₂×C₄×C₈	C₄×C₅	1	2	yes	complemented
82	#383	C₄×C₅	1	4	yes	complemented
83	#382	—	2	2	no	—
84	#381	—	2	2	no	—
85	#380	C₂²×C₅	1	4	yes	—
86	C₄×C₄⋊C₄	—	2	2	no	—
87	C₄×C₂²⋊C₄	—	2	2	no	—
88	#376	C₄×C₅	1	4	yes	complemented
89	C₂³×C₅	—	4	1	no	—
90	C₂³×C₅	—	4	1	no	—
91	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₅	#135	1	4	yes	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	1	no	—
106	C₂×C₄×C₅	—	4	8	no	—
107	C₂×C₄×C₅	—	4	8	no	—
108	C₂×C₄×C₅	—	4	8	no	—
109	C₂×C₄×C₅	#95	1	1	yes	characteristic
110	C₂×C₄×C₅	#95	1	2	yes	—
111	C₂×C₄×C₅	#95	1	2	yes	—
112	C₂×C₄×C₅	#95	1	1	yes	characteristic
113	C₂×C₄×C₅	#95	1	1	yes	characteristic
114	C₂×C₄×C₅	#95	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
118	C₅×C₈	—	4	4	no	—
119	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
120	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
121	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₂²⋊C₄	—	2	4	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₄²	C₅×D₄	1	2	yes	—
131	C₂×C₄²	C₅×D₄	1	2	yes	—
132	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
133	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	#95	—	2	4	no	—
139	#95	—	2	4	no	—
140	#95	—	2	4	no	—
141	#95	C₂×C₄×C₅	1	8	yes	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	4	4	no	—
147	C₂²×C₅	C₂×#95	1	1	yes	characteristic
148	C₂²×C₅	#416	1	4	yes	—
149	C₂²×C₅	#427	1	2	yes	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	2	1	no	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	8	8	no	—
158	C₄×C₅	H(3,ℤ/4)	1	2	yes	—
159	C₄×C₅	#410	1	2	yes	—
160	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
166	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
167	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
169	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
175	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
176	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₄⋊C₄	—	4	8	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	4	8	no	—
182	C₄²	C₄²×C₅	1	1	yes	characteristic
183	C₄²	C₄²×C₅	1	2	yes	—
184	C₄²	C₄²×C₅	1	1	yes	characteristic
185	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
188	C₄²	C₅×C₄⋊C₄	1	2	yes	—
189	C₄²	C₅×C₄⋊C₄	1	2	yes	—
190	C₄²	C₅×C₄⋊C₄	1	2	yes	—
191	C₄²	C₅×C₄⋊C₄	1	2	yes	—
192	C₂×C₅	—	2	4	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	C₂×H(3,ℤ/4)	1	1	yes	characteristic
195	C₂×C₅	#1735	1	1	yes	characteristic
196	C₂×C₅	#1740	1	1	yes	characteristic
197	C₂³	—	4	1	no	—
198	C₂³	—	4	1	no	—
199	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
200	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂³	C₅×#135	1	4	yes	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₅×#95	1	1	yes	characteristic
218	C₂×C₄	C₅×#95	1	2	yes	—
219	C₂×C₄	C₅×#95	1	1	yes	characteristic
220	C₂×C₄	C₅×#95	1	1	yes	characteristic
221	C₂×C₄	C₅×#95	1	2	yes	—
222	C₂×C₄	C₅×#95	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
224	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
226	C₈	—	4	4	no	—
227	C₅	#7980	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	2	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×C₅×#95	1	1	yes	characteristic
234	C₂²	C₅×#416	1	4	yes	—
235	C₂²	C₅×#427	1	2	yes	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	8	8	no	—
244	C₄	C₅×H(3,ℤ/4)	1	2	yes	—
245	C₄	C₅×#410	1	2	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	2	no	—
248	C₂	C₂×C₅×H(3,ℤ/4)	1	1	yes	characteristic
249	C₂	C₅×#1735	1	1	yes	characteristic
250	C₂	C₅×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#95
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂×C₅×C₄⋊C₈	C₄	semidirect
9. C₂×C₄×C₅×C₈	C₄	semidirect
10. C₅×#383	C₄	semidirect
16. C₅×#376	C₄	semidirect
17. #7980	C₅	direct (N × H)
79. C₂×C₄⋊C₈	C₄×C₅	semidirect
81. C₂×C₄×C₈	C₄×C₅	semidirect
82. #383	C₄×C₅	semidirect
88. #376	C₄×C₅	semidirect
227. C₅	#7980	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×#431	2
C₅×#2283	2
C₅×#1289	2
#7980	5

Subgroups — C5×#7980 #11930700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7980 #11930700