Subgroups — C₃×C₅×#1567 #14274601

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#423	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	4	yes	maximal, complemented
4	C₅×#1567	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	—	2	2	no	—
6	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊C₈	—	2	2	no	—
9	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
13	C₃×#1567	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₅×#423	C₂×C₃	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₅×#335	C₂×C₃	1	4	yes	complemented
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
20	C₃×C₅×M₁₆	—	4	4	no	—
21	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₂×C₃×C₅×C₈	—	2	4	no	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₃×#423	C₂×C₅	1	2	yes	—
30	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
31	C₃×#335	C₂×C₅	1	4	yes	complemented
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
37	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
38	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
39	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
40	#1567	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
48	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₃×C₅×C₈	—	2	2	no	—
53	C₃×C₅×C₈	—	4	4	no	—
54	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
55	C₂×C₃×M₁₆	—	2	2	no	—
56	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
57	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
58	C₃×C₄⋊C₈	—	2	2	no	—
59	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
60	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
61	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
62	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
63	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
66	C₅×M₁₆	—	4	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
71	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
72	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
74	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
75	#423	C₂×C₃×C₅	1	2	yes	—
76	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
77	#335	C₂×C₃×C₅	1	4	yes	complemented
78	C₂²×C₃×C₅	—	2	1	no	—
79	C₂²×C₃×C₅	—	2	1	no	—
80	C₂²×C₃×C₅	#107	1	2	yes	—
81	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	—	2	1	no	—
84	C₃×C₄×C₅	#107	1	2	yes	—
85	C₃×C₄×C₅	#108	1	2	yes	—
86	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
87	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
88	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
92	C₃×M₁₆	—	4	4	no	—
93	C₂×C₃×C₈	—	2	4	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
96	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
97	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
98	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
100	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
101	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
108	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
109	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	—	4	4	no	—
114	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
115	C₂×M₁₆	—	2	2	no	—
116	C₂²×C₈	C₃×C₄×C₅	1	2	yes	—
117	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
120	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
121	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
122	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
123	C₂×C₃×C₅	—	2	2	no	—
124	C₂×C₃×C₅	#423	1	1	yes	characteristic
125	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
126	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
127	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
134	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
135	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	4	4	no	—
140	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	C₃×#107	1	2	yes	—
144	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	C₃×#107	1	2	yes	—
148	C₄×C₅	C₃×#108	1	2	yes	—
149	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
150	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
151	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
154	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
155	M₁₆	—	4	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
159	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
160	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
161	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
162	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
163	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
164	C₃×C₅	#1567	1	1	yes	characteristic, direct product factor, Hall
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	C₅×#107	1	2	yes	—
168	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	C₅×#107	1	2	yes	—
172	C₃×C₄	C₅×#108	1	2	yes	—
173	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
174	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
175	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	C₃×#423	1	1	yes	characteristic
178	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
179	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
180	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
187	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
188	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	C₃×C₅×M₁₆	1	4	yes	complemented
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	C₅×#423	1	1	yes	characteristic
196	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
197	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
198	C₅	C₃×#1567	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₃×C₅×#107	1	2	yes	—
202	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₃×C₅×#107	1	2	yes	—
206	C₄	C₃×C₅×#108	1	2	yes	—
207	C₄	C₂×C₃×C₅×M₁₆	1	2	yes	—
208	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
209	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
210	C₃	C₅×#1567	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	2	no	—
212	C₂	C₃×C₅×#423	1	1	yes	characteristic
213	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
214	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. C₅×#1567	C₃	direct (N × H)
13. C₃×#1567	C₅	direct (N × H)
16. C₅×#335	C₂×C₃	semidirect
31. C₃×#335	C₂×C₅	semidirect
40. #1567	C₃×C₅	direct (N × H)
54. C₃×C₅×C₈	M₁₆	semidirect
77. #335	C₂×C₃×C₅	semidirect
114. C₅×C₈	C₃×M₁₆	semidirect
140. C₃×C₈	C₅×M₁₆	semidirect
164. C₃×C₅	#1567	direct (N × H)
193. C₈	C₃×C₅×M₁₆	semidirect
198. C₅	C₃×#1567	direct (N × H)
210. C₃	C₅×#1567	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×#423	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#335	2
C₅×#1567	3
C₃×#1567	5

Subgroups — C3×C5×#1567 #14274601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1567 #14274601