Subgroups — C₃×C₅×#1568 #14274602

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#432	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#334	C₂	1	2	yes	maximal, complemented
5	C₅×#1568	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
11	C₃×C₅×#97	—	2	2	no	—
12	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#1568	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#432	C₂×C₃	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
19	C₅×#335	C₂×C₃	1	2	yes	complemented
20	C₅×#334	C₂×C₃	1	2	yes	complemented
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	4	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×#432	C₂×C₅	1	2	yes	—
36	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
37	C₃×#335	C₂×C₅	1	2	yes	complemented
38	C₃×#334	C₂×C₅	1	2	yes	complemented
39	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×#97	—	2	2	no	—
45	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#1568	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	—	4	4	no	—
64	C₃×C₅×C₈	M₁₆	1	2	yes	—
65	C₃×C₅×C₈	M₁₆	1	2	yes	—
66	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
67	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
70	C₃×C₂²⋊C₈	—	2	2	no	—
71	C₃×#97	—	2	2	no	—
72	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
73	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
88	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
89	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
90	#432	C₂×C₃×C₅	1	2	yes	—
91	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
92	#335	C₂×C₃×C₅	1	2	yes	complemented
93	#334	C₂×C₃×C₅	1	2	yes	complemented
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	#108	1	2	yes	—
97	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	#135	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
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	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	2	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	1	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	4	4	no	—
133	C₅×C₈	C₃×M₁₆	1	2	yes	—
134	C₅×C₈	C₃×M₁₆	1	2	yes	—
135	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
138	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
139	C₂²⋊C₈	—	2	2	no	—
140	#97	—	2	2	no	—
141	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
142	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
143	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₂×C₃×C₅	—	2	2	no	—
146	C₂×C₃×C₅	#423	1	1	yes	characteristic
147	C₂×C₃×C₅	#427	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	1	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	4	4	no	—
163	C₃×C₈	C₅×M₁₆	1	2	yes	—
164	C₃×C₈	C₅×M₁₆	1	2	yes	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	C₃×#108	1	2	yes	—
168	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
174	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
175	C₄×C₅	C₃×#135	1	2	yes	—
176	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
177	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
183	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
184	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
185	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
186	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
187	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
188	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
189	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
190	C₃×C₅	#1568	1	1	yes	characteristic, direct product factor, Hall
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	C₅×#108	1	2	yes	—
194	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
200	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
201	C₃×C₄	C₅×#135	1	2	yes	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	C₃×#423	1	1	yes	characteristic
204	C₂×C₅	C₃×#427	1	1	yes	characteristic
205	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
206	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
215	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	4	4	no	—
220	C₈	C₃×C₅×M₁₆	1	2	yes	—
221	C₈	C₃×C₅×M₁₆	1	2	yes	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	C₅×#423	1	1	yes	characteristic
224	C₂×C₃	C₅×#427	1	1	yes	characteristic
225	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
226	C₅	C₃×#1568	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	C₃×C₅×#108	1	2	yes	—
230	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
236	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
237	C₄	C₃×C₅×#135	1	2	yes	—
238	C₃	C₅×#1568	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	2	2	no	—
240	C₂	C₃×C₅×#423	1	1	yes	characteristic
241	C₂	C₃×C₅×#427	1	1	yes	characteristic
242	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
3. C₃×C₅×#335	C₂	semidirect
4. C₃×C₅×#334	C₂	semidirect
5. C₅×#1568	C₃	direct (N × H)
16. C₃×#1568	C₅	direct (N × H)
19. C₅×#335	C₂×C₃	semidirect
20. C₅×#334	C₂×C₃	semidirect
37. C₃×#335	C₂×C₅	semidirect
38. C₃×#334	C₂×C₅	semidirect
49. #1568	C₃×C₅	direct (N × H)
92. #335	C₂×C₃×C₅	semidirect
93. #334	C₂×C₃×C₅	semidirect
190. C₃×C₅	#1568	direct (N × H)
226. C₅	C₃×#1568	direct (N × H)
238. C₃	C₅×#1568	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×#432	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#335	2
C₃×C₅×#334	2
C₅×#1568	3
C₃×#1568	5

Subgroups — C3×C5×#1568 #14274602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1568 #14274602