Subgroups — C₅×#70744 #14280667

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic
3	C₅×#4925	C₂	1	1	yes	maximal, characteristic
4	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1454	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
11	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#830	C₄	1	2	yes	—
15	C₅×#830	C₄	1	2	yes	—
16	C₅×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
17	C₅×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#70744	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#431	—	3	1	no	—
24	C₅×#431	—	3	1	no	—
25	C₅×#431	D₃	1	1	yes	characteristic
26	C₅×#322	—	3	1	no	—
27	C₅×#322	—	3	1	no	—
28	C₅×#322	—	3	1	no	—
29	C₅×C₈²	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₃×#431	C₂×C₅	1	1	yes	characteristic
44	#4925	C₂×C₅	1	1	yes	characteristic
45	#4925	C₂×C₅	1	1	yes	characteristic
46	#4455	C₂×C₅	1	1	yes	characteristic, complemented
47	#4454	C₂×C₅	1	1	yes	characteristic, complemented
48	#4454	C₂×C₅	1	1	yes	characteristic, complemented
49	C₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	3	2	no	—
52	C₅×C₄⋊C₈	—	3	2	no	—
53	C₅×C₄⋊C₈	Dic₃	1	2	yes	—
54	C₅×C₂²⋊C₈	—	3	2	no	—
55	C₅×C₂²⋊C₈	—	3	2	no	—
56	C₅×C₂²⋊C₈	Dic₃	1	2	yes	—
57	C₄×C₅×C₈	—	3	1	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	—	3	1	no	—
60	C₄×C₅×C₈	—	3	1	no	—
61	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	#1454	—	3	1	no	Sylow
64	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₅×C₈	—	2	2	no	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₅×C₃⋊C₈	—	2	2	no	—
77	C₅×C₃⋊C₈	—	2	2	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
82	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
83	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
85	#830	C₄×C₅	1	2	yes	—
86	#830	C₄×C₅	1	2	yes	—
87	C₄⋊(C₃⋊C₈)	C₄×C₅	1	2	yes	—
88	C₄⋊(C₃⋊C₈)	C₄×C₅	1	2	yes	—
89	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
90	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
101	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
102	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
105	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
106	#431	—	3	1	no	—
107	#431	—	3	1	no	—
108	#431	C₅×D₃	1	1	yes	characteristic
109	#322	—	3	1	no	—
110	#322	—	3	1	no	—
111	#322	—	3	1	no	—
112	C₈²	—	3	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	2	1	no	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	C₃×C₄×C₅	#131	1	2	yes	—
123	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
128	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
129	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
131	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
134	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
136	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	6	2	no	—
149	C₅×C₈	—	6	2	no	—
150	C₅×C₈	—	6	2	no	—
151	C₅×C₈	—	6	2	no	—
152	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	—	3	2	no	—
155	C₄⋊C₈	C₅×Dic₃	1	2	yes	—
156	C₂²⋊C₈	—	3	2	no	—
157	C₂²⋊C₈	—	3	2	no	—
158	C₂²⋊C₈	C₅×Dic₃	1	2	yes	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
164	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	#405	1	1	yes	characteristic
167	C₂×C₃×C₅	#405	1	1	yes	characteristic
168	C₂×C₃×C₅	#405	1	1	yes	characteristic
169	C₂³×C₃	C₄²×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₃×C₈	—	2	2	no	—
180	C₃×C₈	—	2	2	no	—
181	C₃⋊C₈	—	2	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	2	2	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	#901	1	2	yes	—
193	C₄×C₅	#907	1	2	yes	—
194	C₄×C₅	#948	1	2	yes	—
195	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
196	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	—	3	2	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
203	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
204	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
205	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
206	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
207	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
208	C₃×C₅	#1454	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	C₅×#131	1	2	yes	—
217	C₃×C₄	C₅×#131	1	2	yes	—
218	C₃×C₄	C₅×#131	1	2	yes	—
219	C₂×C₅	—	4	1	no	—
220	C₂×C₅	#4997	1	1	yes	characteristic
221	C₂×C₅	#5101	1	1	yes	characteristic
222	C₂×C₅	#5122	1	1	yes	characteristic
223	C₂³	C₄×C₅×Dic₃	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
228	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
229	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	6	2	no	—
236	C₈	—	6	2	no	—
237	C₈	—	6	2	no	—
238	C₈	—	6	2	no	—
239	C₂×C₃	—	4	1	no	—
240	C₂×C₃	C₅×#405	1	1	yes	characteristic
241	C₂×C₃	C₅×#405	1	1	yes	characteristic
242	C₂×C₃	C₅×#405	1	1	yes	characteristic
243	C₅	#70744	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	C₅×#901	1	2	yes	—
252	C₄	C₅×#907	1	2	yes	—
253	C₄	C₅×#948	1	2	yes	—
254	C₃	C₅×#1454	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	4	1	no	—
256	C₂	C₅×#4997	1	1	yes	characteristic
257	C₂	C₅×#5101	1	1	yes	characteristic
258	C₂	C₅×#5122	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1454
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
4. C₅×#4455	C₂	semidirect
5. C₅×#4454	C₂	semidirect
6. C₅×#4454	C₂	semidirect
7. C₅×C₈×C₃⋊C₈	C₂	semidirect
22. #70744	C₅	direct (N × H)
46. #4455	C₂×C₅	semidirect
47. #4454	C₂×C₅	semidirect
48. #4454	C₂×C₅	semidirect
49. C₈×C₃⋊C₈	C₂×C₅	semidirect
208. C₃×C₅	#1454	semidirect
243. C₅	#70744	direct (N × H)
254. C₃	C₅×#1454	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#431	2
C₅×#4925	2
C₅×#4455	2
C₅×#4454	2
C₅×C₈×C₃⋊C₈	2
C₅×#1454	3
#70744	5

Subgroups — C5×#70744 #14280667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70744 #14280667