Subgroups — C₃×C₅×#2166 #14275200

Order 1920.

244 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#364	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#349	C₂	1	2	yes	maximal
4	C₃×C₅×#347	C₂	1	2	yes	maximal, complemented
5	C₅×#2166	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
8	C₃×C₅×#117	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	C₂²	1	4	yes	—
10	C₃×C₅×#97	C₄	1	1	yes	characteristic
11	C₃×C₅×#97	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#2166	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#405	C₂×C₃	1	1	yes	characteristic
16	C₅×#364	C₂×C₃	1	2	yes	complemented
17	C₅×#349	C₂×C₃	1	2	yes	—
18	C₅×#347	C₂×C₃	1	2	yes	complemented
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₃×C₅×M₁₆	—	2	2	no	—
21	C₂×C₃×C₅×C₈	—	2	2	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	—	2	1	no	—
27	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	4	no	—
31	C₃×#405	C₂×C₅	1	1	yes	characteristic
32	C₃×#364	C₂×C₅	1	2	yes	complemented
33	C₃×#349	C₂×C₅	1	2	yes	—
34	C₃×#347	C₂×C₅	1	2	yes	complemented
35	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
37	C₅×#117	C₃×C₄	1	1	yes	characteristic
38	C₅×M₃₂	C₂²×C₃	1	4	yes	—
39	C₅×#97	C₃×C₄	1	1	yes	characteristic
40	C₅×#97	C₂²×C₃	1	1	yes	characteristic
41	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	#2166	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	—	2	1	no	—
46	C₂×C₃×C₄×C₅	—	2	1	no	—
47	C₂×C₃×C₄×C₅	—	2	1	no	—
48	C₂×C₃×C₄×C₅	—	2	1	no	—
49	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₅×C₈	—	2	1	no	—
52	C₃×C₅×C₈	—	2	1	no	—
53	C₃×C₅×C₈	M₁₆	1	2	yes	—
54	C₃×C₅×C₈	#41	1	2	yes	—
55	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
57	C₃×#117	C₄×C₅	1	1	yes	characteristic
58	C₃×M₃₂	C₂²×C₅	1	4	yes	—
59	C₃×#97	C₄×C₅	1	1	yes	characteristic
60	C₃×#97	C₂²×C₅	1	1	yes	characteristic
61	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
62	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
64	C₅×M₁₆	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	1	no	—
71	C₅×C₂²⋊C₄	—	2	1	no	—
72	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₅×C₁₆	—	2	4	no	—
75	#405	C₂×C₃×C₅	1	1	yes	characteristic
76	#364	C₂×C₃×C₅	1	2	yes	complemented
77	#349	C₂×C₃×C₅	1	2	yes	—
78	#347	C₂×C₃×C₅	1	2	yes	complemented
79	C₂²×C₃×C₅	—	2	1	no	—
80	C₂²×C₃×C₅	—	4	1	no	—
81	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	—	4	1	no	—
84	C₃×C₄×C₅	—	4	1	no	—
85	C₃×C₄×C₅	#117	1	1	yes	characteristic
86	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
88	C₃×M₁₆	—	2	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	1	no	—
95	C₃×C₂²⋊C₄	—	2	1	no	—
96	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
98	C₃×C₁₆	—	2	4	no	—
99	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	C₃×M₁₆	1	2	yes	—
109	C₅×C₈	C₃×#41	1	2	yes	—
110	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
111	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
112	#117	C₃×C₄×C₅	1	1	yes	characteristic
113	M₃₂	C₂²×C₃×C₅	1	4	yes	—
114	#97	C₃×C₄×C₅	1	1	yes	characteristic
115	#97	C₂²×C₃×C₅	1	1	yes	characteristic
116	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
117	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
118	C₂×C₃×C₅	—	2	1	no	—
119	C₂×C₃×C₅	—	4	1	no	—
120	C₂×C₃×C₅	#432	1	1	yes	characteristic
121	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₃×C₈	—	2	1	no	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	C₅×M₁₆	1	2	yes	—
131	C₃×C₈	C₅×#41	1	2	yes	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	—	4	1	no	—
134	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	4	1	no	—
137	C₄×C₅	—	4	1	no	—
138	C₄×C₅	C₃×#117	1	1	yes	characteristic
139	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
141	M₁₆	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
144	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
145	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
146	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
150	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
151	C₁₆	—	2	4	no	—
152	C₃×C₅	#2166	1	1	yes	characteristic, direct product factor, Hall
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	4	1	no	—
155	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
156	C₃×C₄	—	2	1	no	—
157	C₃×C₄	—	4	1	no	—
158	C₃×C₄	—	4	1	no	—
159	C₃×C₄	C₅×#117	1	1	yes	characteristic
160	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
161	C₂×C₅	—	2	1	no	—
162	C₂×C₅	—	4	1	no	—
163	C₂×C₅	C₃×#432	1	1	yes	characteristic
164	C₂³	C₃×C₅×M₁₆	1	1	yes	characteristic
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	C₃×C₅×M₁₆	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	C₃×C₅×M₁₆	1	2	yes	—
174	C₈	C₃×C₅×#41	1	2	yes	—
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	—	4	1	no	—
177	C₂×C₃	C₅×#432	1	1	yes	characteristic
178	C₅	C₃×#2166	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	—	2	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	1	no	—
185	C₄	C₃×C₅×#117	1	1	yes	characteristic
186	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
187	C₃	C₅×#2166	1	1	yes	characteristic, direct product factor, Sylow
188	C₂	—	2	1	no	—
189	C₂	—	4	1	no	—
190	C₂	C₃×C₅×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#432
prime radical	C₂³×C₃×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₃×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#364	C₂	semidirect
4. C₃×C₅×#347	C₂	semidirect
5. C₅×#2166	C₃	direct (N × H)
14. C₃×#2166	C₅	direct (N × H)
16. C₅×#364	C₂×C₃	semidirect
18. C₅×#347	C₂×C₃	semidirect
32. C₃×#364	C₂×C₅	semidirect
34. C₃×#347	C₂×C₅	semidirect
43. #2166	C₃×C₅	direct (N × H)
76. #364	C₂×C₃×C₅	semidirect
78. #347	C₂×C₃×C₅	semidirect
152. C₃×C₅	#2166	direct (N × H)
178. C₅	C₃×#2166	direct (N × H)
187. C₃	C₅×#2166	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×#405	2
C₃×C₅×#364	2
C₃×C₅×#349	2
C₃×C₅×#347	2
C₅×#2166	3
C₃×#2166	5

Subgroups — C3×C5×#2166 #14275200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2166 #14275200