Subgroups — C₃×C₅×#2140 #14275174

Order 1920.

520 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	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	4	yes	maximal, complemented
5	C₅×#2140	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	—	2	1	no	—
7	C₂×C₃×C₅×D₈	—	2	1	no	—
8	C₂×C₃×C₅×D₈	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	—	2	1	no	—
13	C₃×C₅×M₃₂	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
15	C₃×C₅×#107	C₂²	1	2	yes	complemented
16	C₃×C₅×#106	—	2	1	no	—
17	C₃×#2140	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×D₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#427	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#357	C₂×C₃	1	4	yes	complemented
22	C₂⁴×C₃×C₅	—	4	1	no	—
23	C₂×C₃×C₅×D₄	—	2	4	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	4	1	no	—
26	C₂×C₃×C₅×D₄	—	4	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₃×C₅×D₈	—	2	4	no	—
29	C₃×C₅×D₈	—	2	2	no	—
30	C₃×C₅×D₈	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
37	C₃×C₄²×C₅	—	4	1	no	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
41	C₃×#427	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#357	C₂×C₅	1	4	yes	complemented
43	C₂²×C₅×D₄	—	2	1	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	C₃×C₄	1	4	yes	complemented
47	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×#117	—	2	1	no	—
50	C₅×M₃₂	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	complemented
52	C₅×#107	C₂²×C₃	1	2	yes	complemented
53	C₅×#106	—	2	1	no	—
54	#2140	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	4	no	—
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₂³×C₃×C₅	—	8	4	no	—
58	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
59	C₃×C₅×D₄	—	2	4	no	—
60	C₃×C₅×D₄	—	4	4	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×D₄	—	2	1	no	—
71	C₂×C₃×D₈	—	2	1	no	—
72	C₂×C₃×D₈	—	2	2	no	—
73	C₂×C₃×D₈	C₄×C₅	1	4	yes	complemented
74	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
75	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×#117	—	2	1	no	—
77	C₃×M₃₂	—	2	2	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
79	C₃×#107	C₂²×C₅	1	2	yes	complemented
80	C₃×#106	—	2	1	no	—
81	C₂⁴×C₅	—	4	1	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
87	C₅×D₈	—	2	4	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
93	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₂²⋊C₄	—	4	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₅×C₁₆	—	2	2	no	—
98	C₂²×D₈	C₂×C₃×C₅	1	1	yes	characteristic
99	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	#427	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	#357	C₂×C₃×C₅	1	4	yes	complemented
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	4	4	no	—
104	C₂²×C₃×C₅	—	8	8	no	—
105	C₂²×C₃×C₅	—	8	4	no	—
106	C₂²×C₃×C₅	#102	1	2	yes	—
107	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
108	C₃×C₄×C₅	—	8	2	no	—
109	C₃×C₄×C₅	#102	1	2	yes	—
110	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₂⁴×C₃	—	4	1	no	—
113	C₂×C₃×D₄	—	2	4	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	4	1	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
118	C₃×D₈	—	2	4	no	—
119	C₃×D₈	—	2	2	no	—
120	C₃×D₈	C₂×C₄×C₅	1	4	yes	—
121	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
122	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₂²⋊C₄	—	4	1	no	—
127	C₃×C₄²	—	4	1	no	—
128	C₃×C₁₆	—	2	2	no	—
129	C₂³×C₅	—	4	4	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	—	8	4	no	—
132	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
133	C₅×D₄	—	2	4	no	—
134	C₅×D₄	—	4	4	no	—
135	C₅×D₄	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
142	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂²×D₄	—	2	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	C₃×C₄×C₅	1	4	yes	complemented
148	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
149	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	#117	—	2	1	no	—
151	M₃₂	—	2	2	no	—
152	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	complemented
153	#107	C₂²×C₃×C₅	1	2	yes	complemented
154	#106	—	2	1	no	—
155	C₂×C₃×C₅	—	2	2	no	—
156	C₂×C₃×C₅	—	8	4	no	—
157	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
158	C₂×C₃×C₅	#509	1	2	yes	—
159	C₂³×C₃	—	4	2	no	—
160	C₂³×C₃	—	4	4	no	—
161	C₂³×C₃	—	8	4	no	—
162	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
163	C₃×D₄	—	2	4	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	4	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
172	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	4	4	no	—
176	C₂²×C₅	—	8	8	no	—
177	C₂²×C₅	—	8	4	no	—
178	C₂²×C₅	C₃×#102	1	2	yes	—
179	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
180	C₄×C₅	—	8	2	no	—
181	C₄×C₅	C₃×#102	1	2	yes	—
182	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
184	C₂⁴	—	4	1	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
190	D₈	—	2	4	no	—
191	D₈	—	2	2	no	—
192	D₈	C₂×C₃×C₄×C₅	1	4	yes	—
193	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
194	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
195	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
196	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₁₆	—	2	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, direct product factor, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	4	4	no	—
204	C₂²×C₃	—	8	8	no	—
205	C₂²×C₃	—	8	4	no	—
206	C₂²×C₃	C₅×#102	1	2	yes	—
207	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
208	C₃×C₄	—	8	2	no	—
209	C₃×C₄	C₅×#102	1	2	yes	—
210	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	8	4	no	—
214	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
215	C₂×C₅	C₃×#509	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	4	no	—
219	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
226	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
227	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
228	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
229	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
230	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	8	4	no	—
233	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
234	C₂×C₃	C₅×#509	1	2	yes	—
235	C₅	C₃×#2140	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	8	no	—
239	C₂²	—	8	4	no	—
240	C₂²	C₃×C₅×#102	1	2	yes	—
241	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
242	C₄	—	8	2	no	—
243	C₄	C₃×C₅×#102	1	2	yes	—
244	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
246	C₃	C₅×#2140	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
250	C₂	C₃×C₅×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₃×C₅×#427	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₅×#2140	C₃	direct (N × H)
9. C₂×C₃×C₅×D₈	C₄	semidirect
14. C₂×C₃×C₅×C₁₆	C₂²	semidirect
15. C₃×C₅×#107	C₂²	semidirect
17. C₃×#2140	C₅	direct (N × H)
19. C₂×C₅×M₃₂	C₂×C₃	semidirect
20. C₅×#427	C₂×C₃	semidirect
21. C₅×#357	C₂×C₃	semidirect
40. C₂×C₃×M₃₂	C₂×C₅	semidirect
41. C₃×#427	C₂×C₅	semidirect
42. C₃×#357	C₂×C₅	semidirect
46. C₂×C₅×D₈	C₃×C₄	semidirect
51. C₂×C₅×C₁₆	C₂²×C₃	semidirect
52. C₅×#107	C₂²×C₃	semidirect
54. #2140	C₃×C₅	direct (N × H)
73. C₂×C₃×D₈	C₄×C₅	semidirect
78. C₂×C₃×C₁₆	C₂²×C₅	semidirect
79. C₃×#107	C₂²×C₅	semidirect
99. C₂×M₃₂	C₂×C₃×C₅	semidirect
100. #427	C₂×C₃×C₅	semidirect
101. #357	C₂×C₃×C₅	semidirect
147. C₂×D₈	C₃×C₄×C₅	semidirect
152. C₂×C₁₆	C₂²×C₃×C₅	semidirect
153. #107	C₂²×C₃×C₅	semidirect
201. C₃×C₅	#2140	direct (N × H)
235. C₅	C₃×#2140	direct (N × H)
246. C₃	C₅×#2140	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
C₂×C₃×C₅×M₃₂	2
C₃×C₅×#427	2
C₃×C₅×#357	2
C₅×#2140	3
C₃×#2140	5

Subgroups — C3×C5×#2140 #14275174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2140 #14275174