Subgroups — C₃×C₅×#2090 #14275124

Order 1920.

352 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#428	C₂	1	2	yes	maximal
2	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#343	C₂	1	2	yes	maximal
5	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic
6	C₅×#2090	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₅×#107	—	2	2	no	—
13	C₃×C₅×#106	—	2	2	no	—
14	C₃×C₅×#106	—	2	2	no	—
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×#2090	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#428	C₂×C₃	1	2	yes	—
18	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic
19	C₅×#424	C₂×C₃	1	1	yes	characteristic
20	C₅×#343	C₂×C₃	1	2	yes	—
21	C₅×#341	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₃×C₅×M₁₆	—	2	2	no	—
25	C₃×C₅×M₁₆	—	2	2	no	—
26	C₃×C₅×M₁₆	Q₈	1	4	yes	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
34	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₃×C₄²×C₅	—	2	1	no	—
36	C₃×#428	C₂×C₅	1	2	yes	—
37	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
38	C₃×#424	C₂×C₅	1	1	yes	characteristic
39	C₃×#343	C₂×C₅	1	2	yes	—
40	C₃×#341	C₂×C₅	1	1	yes	characteristic
41	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
43	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₅×#117	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₅×#107	—	2	2	no	—
47	C₅×#106	—	2	2	no	—
48	C₅×#106	—	2	2	no	—
49	C₅×C₄⋊C₈	—	2	2	no	—
50	#2090	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	#41	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	4	2	no	—
54	C₂×C₃×C₄×C₅	—	8	2	no	—
55	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	#41	1	2	yes	—
59	C₃×C₅×C₈	—	2	4	no	—
60	C₃×C₅×C₈	—	2	2	no	—
61	C₃×C₅×C₈	—	4	1	no	—
62	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
64	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×#117	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
67	C₃×#107	—	2	2	no	—
68	C₃×#106	—	2	2	no	—
69	C₃×#106	—	2	2	no	—
70	C₃×C₄⋊C₈	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
73	C₅×M₁₆	—	2	2	no	—
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	C₃×Q₈	1	4	yes	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
83	C₅×C₂²⋊C₄	—	2	1	no	—
84	C₄²×C₅	—	2	1	no	—
85	#428	C₂×C₃×C₅	1	2	yes	—
86	C₂×#106	C₂×C₃×C₅	1	1	yes	characteristic
87	#424	C₂×C₃×C₅	1	1	yes	characteristic
88	#343	C₂×C₃×C₅	1	2	yes	—
89	#341	C₂×C₃×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	#121	1	1	yes	characteristic
92	C₂²×C₃×C₅	#125	1	2	yes	—
93	C₃×C₄×C₅	—	4	2	no	—
94	C₃×C₄×C₅	—	8	2	no	—
95	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
96	C₃×C₄×C₅	#123	1	1	yes	characteristic
97	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	2	2	no	—
102	C₃×M₁₆	C₅×Q₈	1	4	yes	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
110	C₃×C₂²⋊C₄	—	2	1	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₂³×C₅	C₃×#41	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	8	2	no	—
116	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
119	C₂×C₄×C₅	C₃×#41	1	2	yes	—
120	C₅×C₈	—	2	4	no	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	4	1	no	—
123	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
124	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
125	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
126	#117	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
128	#107	—	2	2	no	—
129	#106	—	2	2	no	—
130	#106	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₂×C₃×C₅	—	2	2	no	—
133	C₂×C₃×C₅	#398	1	1	yes	characteristic
134	C₂×C₃×C₅	#471	1	2	yes	—
135	C₂³×C₃	C₅×#41	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	8	2	no	—
139	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
142	C₂×C₃×C₄	C₅×#41	1	2	yes	—
143	C₃×C₈	—	2	4	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	4	1	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	C₃×#121	1	1	yes	characteristic
148	C₂²×C₅	C₃×#125	1	2	yes	—
149	C₄×C₅	—	4	2	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
152	C₄×C₅	C₃×#123	1	1	yes	characteristic
153	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	C₃×C₅×Q₈	1	4	yes	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₃×C₅	#2090	1	1	yes	characteristic, direct product factor, Hall
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	C₅×#121	1	1	yes	characteristic
171	C₂²×C₃	C₅×#125	1	2	yes	—
172	C₃×C₄	—	4	2	no	—
173	C₃×C₄	—	8	2	no	—
174	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
175	C₃×C₄	C₅×#123	1	1	yes	characteristic
176	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	C₃×#398	1	1	yes	characteristic
179	C₂×C₅	C₃×#471	1	2	yes	—
180	C₂³	C₃×C₅×#41	1	1	yes	characteristic
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
187	C₂×C₄	C₃×C₅×#41	1	2	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	2	2	no	—
190	C₈	—	4	1	no	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	C₅×#398	1	1	yes	characteristic
193	C₂×C₃	C₅×#471	1	2	yes	—
194	C₅	C₃×#2090	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	C₃×C₅×#121	1	1	yes	characteristic
197	C₂²	C₃×C₅×#125	1	2	yes	—
198	C₄	—	4	2	no	—
199	C₄	—	8	2	no	—
200	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
201	C₄	C₃×C₅×#123	1	1	yes	characteristic
202	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
203	C₃	C₅×#2090	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	—	2	2	no	—
205	C₂	C₃×C₅×#398	1	1	yes	characteristic
206	C₂	C₃×C₅×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2090	C₃	direct (N × H)
16. C₃×#2090	C₅	direct (N × H)
50. #2090	C₃×C₅	direct (N × H)
168. C₃×C₅	#2090	direct (N × H)
194. C₅	C₃×#2090	direct (N × H)
203. C₃	C₅×#2090	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×#428	2
C₂×C₃×C₅×#106	2
C₃×C₅×#424	2
C₃×C₅×#343	2
C₃×C₅×#341	2
C₅×#2090	3
C₃×#2090	5

Subgroups — C3×C5×#2090 #14275124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2090 #14275124