Subgroups — C₃×C₅×#1574 #14274608

Order 1920.

316 conjugacy classes of subgroups.

The subgroup lattice has 240 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	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#334	C₂	1	2	yes	maximal, complemented
6	C₅×#1574	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊C₈	—	2	1	no	—
10	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
16	C₃×C₄×C₅×C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₃×#1574	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#431	C₂×C₃	1	1	yes	characteristic
20	C₅×#431	C₂×C₃	1	1	yes	characteristic
21	C₅×#423	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	C₂×C₃	1	2	yes	complemented
23	C₅×#334	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₃×C₅×M₁₆	—	4	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×#431	C₂×C₅	1	1	yes	characteristic
37	C₃×#431	C₂×C₅	1	1	yes	characteristic
38	C₃×#423	C₂×C₅	1	1	yes	characteristic
39	C₃×#335	C₂×C₅	1	2	yes	complemented
40	C₃×#334	C₂×C₅	1	2	yes	complemented
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	C₃×C₄	1	2	yes	—
45	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
47	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
48	C₅×C₂²⋊C₈	—	2	1	no	—
49	C₅×C₂²⋊C₈	C₃×C₄	1	2	yes	—
50	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
52	#1574	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₃×C₅×C₈	—	2	4	no	—
65	C₃×C₅×C₈	—	4	1	no	—
66	C₃×C₅×C₈	—	4	2	no	—
67	C₃×C₅×C₈	—	4	1	no	—
68	C₂×C₃×M₁₆	—	2	1	no	—
69	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄⋊C₈	—	2	1	no	—
71	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
72	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
73	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₂²⋊C₈	—	2	1	no	—
76	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
79	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
82	C₅×M₁₆	—	4	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
87	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
89	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
90	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
91	#431	C₂×C₃×C₅	1	1	yes	characteristic
92	#431	C₂×C₃×C₅	1	1	yes	characteristic
93	#423	C₂×C₃×C₅	1	1	yes	characteristic
94	#335	C₂×C₃×C₅	1	2	yes	complemented
95	#334	C₂×C₃×C₅	1	2	yes	complemented
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	#108	1	2	yes	—
104	C₃×C₄×C₅	#131	1	2	yes	—
105	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₃×C₄×C₅	#137	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
110	C₃×M₁₆	—	4	2	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
115	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
116	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
117	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
118	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
119	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
130	C₅×C₈	—	2	4	no	—
131	C₅×C₈	—	4	1	no	—
132	C₅×C₈	—	4	2	no	—
133	C₅×C₈	—	4	1	no	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	C₃×C₄×C₅	1	2	yes	—
138	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
140	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
141	C₂²⋊C₈	—	2	1	no	—
142	C₂²⋊C₈	C₃×C₄×C₅	1	2	yes	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
145	C₂×C₃×C₅	—	4	1	no	—
146	C₂×C₃×C₅	#424	1	1	yes	characteristic
147	C₂×C₃×C₅	#428	1	1	yes	characteristic
148	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
149	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
160	C₃×C₈	—	2	4	no	—
161	C₃×C₈	—	4	1	no	—
162	C₃×C₈	—	4	2	no	—
163	C₃×C₈	—	4	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	C₃×#108	1	2	yes	—
172	C₄×C₅	C₃×#131	1	2	yes	—
173	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄×C₅	C₃×#137	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	C₂×C₃×C₄×C₅	1	4	yes	—
183	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
184	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
185	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
186	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
187	C₃×C₅	#1574	1	1	yes	characteristic, direct product factor, Hall
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	C₅×#108	1	2	yes	—
196	C₃×C₄	C₅×#131	1	2	yes	—
197	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₃×C₄	C₅×#137	1	1	yes	characteristic
199	C₂×C₅	—	4	1	no	—
200	C₂×C₅	C₃×#424	1	1	yes	characteristic
201	C₂×C₅	C₃×#428	1	1	yes	characteristic
202	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
203	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
212	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	4	2	no	—
216	C₈	—	4	1	no	—
217	C₈	—	4	1	no	—
218	C₂×C₃	—	4	1	no	—
219	C₂×C₃	C₅×#424	1	1	yes	characteristic
220	C₂×C₃	C₅×#428	1	1	yes	characteristic
221	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
222	C₅	C₃×#1574	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	C₃×C₅×#108	1	2	yes	—
231	C₄	C₃×C₅×#131	1	2	yes	—
232	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₄	C₃×C₅×#137	1	1	yes	characteristic
234	C₃	C₅×#1574	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	4	1	no	—
236	C₂	C₃×C₅×#424	1	1	yes	characteristic
237	C₂	C₃×C₅×#428	1	1	yes	characteristic
238	C₂	C₂×C₃×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
6. C₅×#1574	C₃	direct (N × H)
18. C₃×#1574	C₅	direct (N × H)
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#334	C₂×C₃	semidirect
39. C₃×#335	C₂×C₅	semidirect
40. C₃×#334	C₂×C₅	semidirect
52. #1574	C₃×C₅	direct (N × H)
94. #335	C₂×C₃×C₅	semidirect
95. #334	C₂×C₃×C₅	semidirect
187. C₃×C₅	#1574	direct (N × H)
222. C₅	C₃×#1574	direct (N × H)
234. C₃	C₅×#1574	direct (N × H)

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
C₃×C₅×#423	2
C₃×C₅×#335	2
C₃×C₅×#334	2
C₅×#1574	3
C₃×#1574	5

Subgroups — C3×C5×#1574 #14274608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1574 #14274608