Subgroups — C₃×C₅×#1673 #14274707

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#461	C₂	1	2	yes	maximal
3	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#332	C₂	1	2	yes	maximal, complemented
6	C₅×#1673	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
8	C₂²×C₃×C₅×C₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×#107	—	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×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1673	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#461	C₂×C₃	1	2	yes	—
21	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#332	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₃×C₅×SD₁₆	—	4	4	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	4	2	no	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
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×C₅×C₂²⋊C₄	—	2	1	no	—
37	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#461	C₂×C₅	1	2	yes	—
40	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
41	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#332	C₂×C₅	1	2	yes	complemented
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂²×C₅×C₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
46	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
47	C₅×#107	—	2	1	no	—
48	C₅×#107	—	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×C₂²⋊C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#1673	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	2	1	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×D₄	—	2	1	no	—
59	C₃×C₅×D₄	D₈	1	2	yes	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	—	4	2	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
71	C₂×C₃×SD₁₆	—	2	2	no	—
72	C₂²×C₃×C₈	—	2	1	no	—
73	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
74	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
75	C₃×#107	—	2	1	no	—
76	C₃×#107	—	2	1	no	—
77	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
78	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
79	C₃×#103	—	2	2	no	—
80	C₃×C₂²⋊C₈	—	2	1	no	—
81	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₅×SD₁₆	—	4	4	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
96	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
97	#461	C₂×C₃×C₅	1	2	yes	—
98	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic
99	#335	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	#332	C₂×C₃×C₅	1	2	yes	complemented
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	#121	1	1	yes	characteristic
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	4	1	no	—
107	C₃×C₄×C₅	—	8	2	no	—
108	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
110	C₃×C₄×C₅	#135	1	1	yes	characteristic
111	C₃×C₄×C₅	#137	1	1	yes	characteristic
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₃×SD₁₆	—	4	4	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	4	2	no	—
119	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
124	C₃×C₂²⋊C₄	—	2	1	no	—
125	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
126	C₂³×C₅	—	2	1	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₅×Q₈	—	4	2	no	—
129	C₅×D₄	—	2	1	no	—
130	C₅×D₄	C₃×D₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	4	2	no	—
140	C₅×C₈	—	4	1	no	—
141	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
142	C₂×SD₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
145	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
150	#103	—	2	2	no	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₂×C₃×C₅	—	4	2	no	—
154	C₂×C₃×C₅	#461	1	1	yes	characteristic
155	C₂×C₃×C₅	#466	1	1	yes	characteristic
156	C₂×C₃×C₅	#473	1	1	yes	characteristic
157	C₂³×C₃	—	2	1	no	—
158	C₃×Q₈	—	4	2	no	—
159	C₃×Q₈	—	4	2	no	—
160	C₃×D₄	—	2	1	no	—
161	C₃×D₄	C₅×D₈	1	2	yes	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	4	2	no	—
171	C₃×C₈	—	4	1	no	—
172	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	4	1	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	C₃×#121	1	1	yes	characteristic
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	4	1	no	—
179	C₄×C₅	—	8	2	no	—
180	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
181	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
182	C₄×C₅	C₃×#135	1	1	yes	characteristic
183	C₄×C₅	C₃×#137	1	1	yes	characteristic
184	C₂×Q₈	—	2	2	no	—
185	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	SD₁₆	—	4	4	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
192	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
198	C₃×C₅	#1673	1	1	yes	characteristic, direct product factor, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	C₅×#121	1	1	yes	characteristic
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	—	8	2	no	—
206	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
207	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
208	C₃×C₄	C₅×#135	1	1	yes	characteristic
209	C₃×C₄	C₅×#137	1	1	yes	characteristic
210	C₂×C₅	—	4	2	no	—
211	C₂×C₅	C₃×#461	1	1	yes	characteristic
212	C₂×C₅	C₃×#466	1	1	yes	characteristic
213	C₂×C₅	C₃×#473	1	1	yes	characteristic
214	C₂³	—	2	1	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	2	1	no	—
218	D₄	C₃×C₅×D₈	1	2	yes	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	4	2	no	—
228	C₈	—	4	1	no	—
229	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
230	C₂×C₃	—	4	2	no	—
231	C₂×C₃	C₅×#461	1	1	yes	characteristic
232	C₂×C₃	C₅×#466	1	1	yes	characteristic
233	C₂×C₃	C₅×#473	1	1	yes	characteristic
234	C₅	C₃×#1673	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	C₃×C₅×#121	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
243	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
244	C₄	C₃×C₅×#135	1	1	yes	characteristic
245	C₄	C₃×C₅×#137	1	1	yes	characteristic
246	C₃	C₅×#1673	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	4	2	no	—
248	C₂	C₃×C₅×#461	1	1	yes	characteristic
249	C₂	C₃×C₅×#466	1	1	yes	characteristic
250	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	D₄
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
1. C₃×C₅×#500	C₂	semidirect
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1673	C₃	direct (N × H)
18. C₃×#1673	C₅	direct (N × H)
19. C₅×#500	C₂×C₃	semidirect
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#332	C₂×C₃	semidirect
38. C₃×#500	C₂×C₅	semidirect
41. C₃×#335	C₂×C₅	semidirect
42. C₃×#332	C₂×C₅	semidirect
54. #1673	C₃×C₅	direct (N × H)
70. C₃×C₅×C₈	SD₁₆	semidirect
96. #500	C₂×C₃×C₅	semidirect
99. #335	C₂×C₃×C₅	semidirect
100. #332	C₂×C₃×C₅	semidirect
141. C₅×C₈	C₃×SD₁₆	semidirect
172. C₃×C₈	C₅×SD₁₆	semidirect
198. C₃×C₅	#1673	direct (N × H)
229. C₈	C₃×C₅×SD₁₆	semidirect
234. C₅	C₃×#1673	direct (N × H)
246. C₃	C₅×#1673	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×#500	2
C₃×C₅×#461	2
C₃×C₅×C₈×D₄	2
C₃×C₅×#335	2
C₃×C₅×#332	2
C₅×#1673	3
C₃×#1673	5

Subgroups — C3×C5×#1673 #14274707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1673 #14274707