Subgroups — C₅×#70560 #14305049

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#4984	C₂	1	2	yes	maximal
5	C₅×#4529	C₂	1	2	yes	maximal
6	C₅×#1941	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#106	C₄	1	4	yes	complemented
14	C₅×C₃⋊#95	—	2	2	no	—
15	C₅×#817	—	2	2	no	—
16	C₅×C₈×Dic₃	—	2	2	no	—
17	#70560	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#106	—	3	1	no	—
19	C₂×C₅×#106	D₃	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	—	3	1	no	—
21	C₅×#389	—	3	2	no	—
22	C₅×#340	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	—	2	2	no	—
26	C₂²×C₅×Dic₃	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₅×C₄⋊Dic₃	—	2	2	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₅×C₄⋊Dic₃	—	4	4	no	—
35	C₄×C₅×Dic₃	—	2	1	no	—
36	C₄×C₅×Dic₃	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	—	2	1	no	—
39	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
40	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
41	C₂×#817	C₂×C₅	1	1	yes	characteristic
42	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
43	#4984	C₂×C₅	1	2	yes	—
44	#4529	C₂×C₅	1	2	yes	—
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂×C₅×C₄⋊C₄	—	6	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₄²×C₅	—	3	1	no	—
51	C₅×#106	—	6	2	no	—
52	C₅×#106	Dic₃	1	4	yes	complemented
53	C₄×C₅×C₈	—	6	2	no	—
54	C₅×#95	—	6	2	no	—
55	#1941	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	2	no	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×C₅×Dic₃	—	2	2	no	—
67	C₂×C₅×Dic₃	—	2	2	no	—
68	C₂×C₅×Dic₃	—	4	2	no	—
69	C₂×C₅×Dic₃	—	4	2	no	—
70	C₂×C₅×Dic₃	—	4	2	no	—
71	C₂×C₅×Dic₃	SD₁₆	1	4	yes	—
72	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
73	C₅×C₃⋊C₈	—	2	2	no	—
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
76	C₂×C₄⋊Dic₃	—	2	2	no	—
77	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
78	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×#106	C₄×C₅	1	4	yes	complemented
81	C₃⋊#95	—	2	2	no	—
82	#817	—	2	2	no	—
83	C₈×Dic₃	—	2	2	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	3	2	no	—
86	C₂²×C₄×C₅	—	6	2	no	—
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	3	2	no	—
89	C₂×C₅×C₈	—	6	1	no	—
90	C₂×C₅×C₈	—	6	1	no	—
91	C₂×C₅×C₈	Dic₆	1	2	yes	—
92	C₂×C₅×C₈	D₁₂	1	2	yes	—
93	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	6	2	no	—
96	C₅×C₄⋊C₄	—	6	2	no	—
97	C₅×C₄⋊C₄	—	12	4	no	—
98	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
99	C₄²×C₅	—	6	1	no	—
100	C₄²×C₅	—	6	1	no	—
101	C₂×#106	—	3	1	no	—
102	C₂×#106	C₅×D₃	1	1	yes	characteristic
103	C₂×C₄×C₈	—	3	1	no	—
104	#389	—	3	2	no	—
105	#340	—	3	2	no	—
106	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	#121	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
110	C₂²×C₃×C₅	#135	1	2	yes	—
111	C₃×C₄×C₅	—	4	4	no	—
112	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
115	C₃×C₄×C₅	#125	1	1	yes	characteristic
116	C₅×Dic₃	—	2	4	no	—
117	C₅×Dic₃	—	8	2	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	2	2	no	—
121	C₂²×Dic₃	C₅×D₄	1	2	yes	—
122	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
123	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
127	C₄⋊Dic₃	—	2	2	no	—
128	C₄⋊Dic₃	—	2	2	no	—
129	C₄⋊Dic₃	—	4	4	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₄×Dic₃	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	1	no	—
133	C₂×C₃⋊C₈	—	2	1	no	—
134	C₂×C₃⋊C₈	C₅×Q₈	1	2	yes	—
135	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	3	4	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	6	2	no	—
141	C₂×C₄×C₅	—	6	2	no	—
142	C₂×C₄×C₅	—	12	2	no	—
143	C₂×C₄×C₅	—	12	2	no	—
144	C₂×C₄×C₅	—	12	2	no	—
145	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
147	C₂×C₄×C₅	#237	1	1	yes	characteristic
148	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	#239	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
151	C₅×C₈	—	6	2	no	—
152	C₅×C₈	C₄⋊Dic₃	1	4	yes	complemented
153	C₂²×C₈	—	3	1	no	—
154	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	3	2	no	—
156	C₂×C₄⋊C₄	—	6	2	no	—
157	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
158	C₂×C₄²	—	3	1	no	—
159	#106	—	6	2	no	—
160	#106	C₅×Dic₃	1	4	yes	complemented
161	C₄×C₈	—	6	2	no	—
162	#95	—	6	2	no	—
163	C₂×C₃×C₅	#389	1	1	yes	characteristic
164	C₂×C₃×C₅	C₄×SD₁₆	1	2	yes	—
165	C₂×C₃×C₅	#465	1	2	yes	—
166	C₂×C₃×C₅	#498	1	1	yes	characteristic
167	C₂×C₃×C₅	#499	1	1	yes	characteristic
168	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
177	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
178	C₂×Dic₃	—	2	2	no	—
179	C₂×Dic₃	—	2	2	no	—
180	C₂×Dic₃	—	4	2	no	—
181	C₂×Dic₃	—	4	2	no	—
182	C₂×Dic₃	—	4	2	no	—
183	C₂×Dic₃	C₅×SD₁₆	1	4	yes	—
184	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	complemented
185	C₃⋊C₈	—	2	2	no	—
186	C₂²×C₅	#888	1	1	yes	characteristic
187	C₂²×C₅	#895	1	1	yes	characteristic
188	C₂²×C₅	D₃×SD₁₆	1	2	yes	—
189	C₂²×C₅	#916	1	2	yes	—
190	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
191	C₄×C₅	—	4	4	no	—
192	C₄×C₅	—	6	4	no	—
193	C₄×C₅	—	24	2	no	—
194	C₄×C₅	#890	1	1	yes	characteristic
195	C₄×C₅	#897	1	1	yes	characteristic
196	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
197	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	—	6	2	no	—
201	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	6	1	no	—
204	C₂×C₈	—	6	1	no	—
205	C₂×C₈	C₅×Dic₆	1	2	yes	—
206	C₂×C₈	C₅×D₁₂	1	2	yes	—
207	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	6	2	no	—
210	C₄⋊C₄	—	6	2	no	—
211	C₄⋊C₄	—	12	4	no	—
212	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
213	C₄²	—	6	1	no	—
214	C₄²	—	6	1	no	—
215	C₃×C₅	#1941	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₃	C₅×#121	1	1	yes	characteristic
218	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
220	C₂²×C₃	C₅×#135	1	2	yes	—
221	C₃×C₄	—	4	4	no	—
222	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
224	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
225	C₃×C₄	C₅×#125	1	1	yes	characteristic
226	Dic₃	—	2	4	no	—
227	Dic₃	—	8	2	no	—
228	C₂×C₅	#4855	1	1	yes	characteristic
229	C₂×C₅	#4858	1	1	yes	characteristic
230	C₂×C₅	#4864	1	2	yes	—
231	C₂×C₅	#4984	1	1	yes	characteristic
232	C₂×C₅	Dic₃×SD₁₆	1	2	yes	—
233	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	3	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	—	12	2	no	—
241	C₂×C₄	—	12	2	no	—
242	C₂×C₄	—	12	2	no	—
243	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
245	C₂×C₄	C₅×#237	1	1	yes	characteristic
246	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
247	C₂×C₄	C₅×#239	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
249	C₈	—	6	2	no	—
250	C₈	C₅×C₄⋊Dic₃	1	4	yes	complemented
251	C₂×C₃	C₅×#389	1	1	yes	characteristic
252	C₂×C₃	C₄×C₅×SD₁₆	1	2	yes	—
253	C₂×C₃	C₅×#465	1	2	yes	—
254	C₂×C₃	C₅×#498	1	1	yes	characteristic
255	C₂×C₃	C₅×#499	1	1	yes	characteristic
256	C₅	#70560	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	C₅×#888	1	1	yes	characteristic
258	C₂²	C₅×#895	1	1	yes	characteristic
259	C₂²	C₅×D₃×SD₁₆	1	2	yes	—
260	C₂²	C₅×#916	1	2	yes	—
261	C₂²	C₅×D₄×Dic₃	1	1	yes	characteristic
262	C₄	—	4	4	no	—
263	C₄	—	6	4	no	—
264	C₄	—	24	2	no	—
265	C₄	C₅×#890	1	1	yes	characteristic
266	C₄	C₅×#897	1	1	yes	characteristic
267	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
268	C₄	C₅×Q₈×Dic₃	1	1	yes	characteristic
269	C₃	C₅×#1941	1	1	yes	characteristic, complemented, Sylow
270	C₂	C₅×#4855	1	1	yes	characteristic
271	C₂	C₅×#4858	1	1	yes	characteristic
272	C₂	C₅×#4864	1	2	yes	—
273	C₂	C₅×#4984	1	1	yes	characteristic
274	C₂	C₅×Dic₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#106	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#106	D₃
hypocenter	C₃	C₅×#1941
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#106

Complements

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

N	H ≅ G/N	type
13. C₃×C₅×#106	C₄	semidirect
17. #70560	C₅	direct (N × H)
52. C₅×#106	Dic₃	semidirect
72. C₃×C₅×C₈	C₄⋊C₄	semidirect
80. C₃×#106	C₄×C₅	semidirect
152. C₅×C₈	C₄⋊Dic₃	semidirect
160. #106	C₅×Dic₃	semidirect
184. C₃×C₈	C₅×C₄⋊C₄	semidirect
215. C₃×C₅	#1941	semidirect
250. C₈	C₅×C₄⋊Dic₃	semidirect
256. C₅	#70560	direct (N × H)
269. C₃	C₅×#1941	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#106	2
C₂×C₅×#817	2
C₂×C₅×C₈×Dic₃	2
C₅×#4984	2
C₅×#4529	2
C₅×#1941	3
#70560	5

Subgroups — C5×#70560 #14305049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70560 #14305049