Subgroups — C₃×C₅×#1670 #14274704

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 232 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#332	C₂	1	2	yes	maximal
6	C₅×#1670	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×C₄×C₅×C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1670	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×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	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×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
41	C₃×#335	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	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#1670	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₅×Q₈	—	2	1	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×Q₈	Q₁₆	1	2	yes	complemented
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₈	—	4	2	no	—
67	C₃×C₅×C₈	—	4	1	no	—
68	C₃×C₅×C₈	SD₁₆	1	2	yes	—
69	C₂×C₃×Q₁₆	—	2	1	no	—
70	C₂×C₃×Q₁₆	—	2	1	no	—
71	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
72	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×#106	—	2	2	no	—
74	C₃×C₄⋊C₈	—	2	1	no	—
75	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
77	C₃×#103	—	2	2	no	—
78	C₃×C₄×C₈	—	2	1	no	—
79	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
82	C₅×Q₁₆	—	4	2	no	—
83	C₅×Q₁₆	—	4	2	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
94	#494	C₂×C₃×C₅	1	1	yes	characteristic
95	#475	C₂×C₃×C₅	1	2	yes	—
96	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
97	#335	C₂×C₃×C₅	1	1	yes	characteristic
98	#332	C₂×C₃×C₅	1	2	yes	—
99	C₂²×C₃×C₅	#121	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	4	1	no	—
103	C₃×C₄×C₅	—	8	2	no	—
104	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
105	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
106	C₃×C₄×C₅	#135	1	1	yes	characteristic
107	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
110	C₃×Q₁₆	—	4	2	no	—
111	C₃×Q₁₆	—	4	2	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
116	C₃×C₄⋊C₄	—	2	1	no	—
117	C₃×C₄⋊C₄	—	2	2	no	—
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
120	C₃×C₄²	—	2	1	no	—
121	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
122	C₅×Q₈	—	2	1	no	—
123	C₅×Q₈	—	4	2	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	C₃×Q₁₆	1	2	yes	complemented
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	4	2	no	—
134	C₅×C₈	—	4	1	no	—
135	C₅×C₈	C₃×SD₁₆	1	2	yes	—
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
139	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
140	#106	—	2	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
144	#103	—	2	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂×C₃×C₅	#460	1	1	yes	characteristic
148	C₂×C₃×C₅	#467	1	1	yes	characteristic
149	C₂×C₃×C₅	#473	1	1	yes	characteristic
150	C₃×Q₈	—	2	1	no	—
151	C₃×Q₈	—	4	2	no	—
152	C₃×Q₈	—	4	2	no	—
153	C₃×Q₈	C₅×Q₁₆	1	2	yes	complemented
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	4	2	no	—
162	C₃×C₈	—	4	1	no	—
163	C₃×C₈	C₅×SD₁₆	1	2	yes	—
164	C₂²×C₅	C₃×#121	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	4	1	no	—
168	C₄×C₅	—	8	2	no	—
169	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
170	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
171	C₄×C₅	C₃×#135	1	1	yes	characteristic
172	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
175	Q₁₆	—	4	2	no	—
176	Q₁₆	—	4	2	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
180	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
185	C₄²	—	2	1	no	—
186	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
187	C₃×C₅	#1670	1	1	yes	characteristic, direct product factor, Hall
188	C₂²×C₃	C₅×#121	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	1	no	—
192	C₃×C₄	—	8	2	no	—
193	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
194	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
195	C₃×C₄	C₅×#135	1	1	yes	characteristic
196	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×C₅	C₃×#460	1	1	yes	characteristic
198	C₂×C₅	C₃×#467	1	1	yes	characteristic
199	C₂×C₅	C₃×#473	1	1	yes	characteristic
200	Q₈	—	2	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	Q₈	C₃×C₅×Q₁₆	1	2	yes	complemented
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
209	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	1	no	—
213	C₈	C₃×C₅×SD₁₆	1	2	yes	—
214	C₂×C₃	C₅×#460	1	1	yes	characteristic
215	C₂×C₃	C₅×#467	1	1	yes	characteristic
216	C₂×C₃	C₅×#473	1	1	yes	characteristic
217	C₅	C₃×#1670	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	C₃×C₅×#121	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	8	2	no	—
223	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
224	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
225	C₄	C₃×C₅×#135	1	1	yes	characteristic
226	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₃	C₅×#1670	1	1	yes	characteristic, direct product factor, Sylow
228	C₂	C₃×C₅×#460	1	1	yes	characteristic
229	C₂	C₃×C₅×#467	1	1	yes	characteristic
230	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×#1670	C₃	direct (N × H)
18. C₃×#1670	C₅	direct (N × H)
54. #1670	C₃×C₅	direct (N × H)
58. C₃×C₅×Q₈	Q₁₆	semidirect
125. C₅×Q₈	C₃×Q₁₆	semidirect
153. C₃×Q₈	C₅×Q₁₆	semidirect
187. C₃×C₅	#1670	direct (N × H)
203. Q₈	C₃×C₅×Q₁₆	semidirect
217. C₅	C₃×#1670	direct (N × H)
227. C₃	C₅×#1670	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×C₈×Q₈	2
C₃×C₅×#335	2
C₃×C₅×#332	2
C₅×#1670	3
C₃×#1670	5

Subgroups — C3×C5×#1670 #14274704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1670 #14274704