Subgroups — C₃×C₅×#1690 #14274724

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#475	C₂	1	2	yes	maximal
3	C₃×C₅×#446	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#332	C₂	1	2	yes	maximal
6	C₅×#1690	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#106	—	2	2	no	—
12	C₃×C₅×C₄⋊C₈	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1690	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#494	C₂×C₃	1	1	yes	characteristic
20	C₅×#475	C₂×C₃	1	2	yes	—
21	C₅×#446	C₂×C₃	1	1	yes	characteristic
22	C₅×#334	C₂×C₃	1	1	yes	characteristic
23	C₅×#332	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
26	C₃×C₅×Q₁₆	—	4	2	no	—
27	C₃×C₅×Q₁₆	—	4	2	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	1	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	—	2	1	no	—
37	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₃×#494	C₂×C₅	1	1	yes	characteristic
39	C₃×#475	C₂×C₅	1	2	yes	—
40	C₃×#446	C₂×C₅	1	1	yes	characteristic
41	C₃×#334	C₂×C₅	1	1	yes	characteristic
42	C₃×#332	C₂×C₅	1	2	yes	—
43	C₂×C₅×Q₁₆	—	2	1	no	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
46	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
47	C₅×#106	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
50	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
51	C₅×#103	—	2	2	no	—
52	C₅×#97	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#1690	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₅×Q₈	—	2	2	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₃×C₅×C₈	—	2	1	no	—
65	C₃×C₅×C₈	—	4	1	no	—
66	C₃×C₅×C₈	—	4	2	no	—
67	C₃×C₅×C₈	SD₁₆	1	2	yes	—
68	C₂×C₃×Q₁₆	—	2	1	no	—
69	C₂×C₃×Q₁₆	—	2	1	no	—
70	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
71	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
72	C₃×#106	—	2	2	no	—
73	C₃×C₄⋊C₈	—	2	1	no	—
74	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
76	C₃×#103	—	2	2	no	—
77	C₃×#97	—	2	1	no	—
78	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	4	2	no	—
82	C₅×Q₁₆	—	4	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
91	C₄²×C₅	—	2	1	no	—
92	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
93	#494	C₂×C₃×C₅	1	1	yes	characteristic
94	#475	C₂×C₃×C₅	1	2	yes	—
95	#446	C₂×C₃×C₅	1	1	yes	characteristic
96	#334	C₂×C₃×C₅	1	1	yes	characteristic
97	#332	C₂×C₃×C₅	1	2	yes	—
98	C₂²×C₃×C₅	#121	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	4	1	no	—
101	C₃×C₄×C₅	—	4	1	no	—
102	C₃×C₄×C₅	—	8	2	no	—
103	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₃×C₄×C₅	#137	1	2	yes	—
106	C₂×C₃×Q₈	—	2	2	no	—
107	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
108	C₃×Q₁₆	—	4	2	no	—
109	C₃×Q₁₆	—	4	2	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	2	1	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
120	C₅×Q₈	—	2	2	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×Q₈	—	4	2	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	1	no	—
131	C₅×C₈	—	4	2	no	—
132	C₅×C₈	C₃×SD₁₆	1	2	yes	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	—	2	1	no	—
135	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
136	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	#106	—	2	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
140	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
141	#103	—	2	2	no	—
142	#97	—	2	1	no	—
143	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂×C₃×C₅	#460	1	1	yes	characteristic
145	C₂×C₃×C₅	#470	1	1	yes	characteristic
146	C₂×C₃×C₅	#473	1	1	yes	characteristic
147	C₃×Q₈	—	2	2	no	—
148	C₃×Q₈	—	4	2	no	—
149	C₃×Q₈	—	4	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	4	1	no	—
158	C₃×C₈	—	4	2	no	—
159	C₃×C₈	C₅×SD₁₆	1	2	yes	—
160	C₂²×C₅	C₃×#121	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	4	1	no	—
163	C₄×C₅	—	4	1	no	—
164	C₄×C₅	—	8	2	no	—
165	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
166	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₅	C₃×#137	1	2	yes	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	2	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
175	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
180	C₄²	—	2	1	no	—
181	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
182	C₃×C₅	#1690	1	1	yes	characteristic, direct product factor, Hall
183	C₂²×C₃	C₅×#121	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	8	2	no	—
188	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
189	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₃×C₄	C₅×#137	1	2	yes	—
191	C₂×C₅	C₃×#460	1	1	yes	characteristic
192	C₂×C₅	C₃×#470	1	1	yes	characteristic
193	C₂×C₅	C₃×#473	1	1	yes	characteristic
194	Q₈	—	2	2	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
202	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	2	no	—
206	C₈	C₃×C₅×SD₁₆	1	2	yes	—
207	C₂×C₃	C₅×#460	1	1	yes	characteristic
208	C₂×C₃	C₅×#470	1	1	yes	characteristic
209	C₂×C₃	C₅×#473	1	1	yes	characteristic
210	C₅	C₃×#1690	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	C₃×C₅×#121	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	8	2	no	—
216	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
217	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₄	C₃×C₅×#137	1	2	yes	—
219	C₃	C₅×#1690	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	C₃×C₅×#460	1	1	yes	characteristic
221	C₂	C₃×C₅×#470	1	1	yes	characteristic
222	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1690	C₃	direct (N × H)
18. C₃×#1690	C₅	direct (N × H)
54. #1690	C₃×C₅	direct (N × H)
182. C₃×C₅	#1690	direct (N × H)
210. C₅	C₃×#1690	direct (N × H)
219. C₃	C₅×#1690	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×#494	2
C₃×C₅×#475	2
C₃×C₅×#446	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₅×#1690	3
C₃×#1690	5

Subgroups — C3×C5×#1690 #14274724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1690 #14274724