Subgroups — C₃×C₅×#1718 #14274752

Order 1920.

328 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
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic
8	C₅×#1718	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	—	2	2	no	—
14	C₃×C₅×#106	—	2	2	no	—
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×#103	—	2	2	no	—
17	C₃×C₅×#103	—	2	2	no	—
18	C₃×C₅×#103	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#1718	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#500	C₂×C₃	1	1	yes	characteristic
24	C₅×#498	C₂×C₃	1	1	yes	characteristic
25	C₅×#487	C₂×C₃	1	1	yes	characteristic
26	C₅×#487	C₂×C₃	1	1	yes	characteristic
27	C₅×#487	C₂×C₃	1	1	yes	characteristic
28	C₅×#487	C₂×C₃	1	1	yes	characteristic
29	C₅×#322	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×#500	C₂×C₅	1	1	yes	characteristic
47	C₃×#498	C₂×C₅	1	1	yes	characteristic
48	C₃×#487	C₂×C₅	1	1	yes	characteristic
49	C₃×#487	C₂×C₅	1	1	yes	characteristic
50	C₃×#487	C₂×C₅	1	1	yes	characteristic
51	C₃×#487	C₂×C₅	1	1	yes	characteristic
52	C₃×#322	C₂×C₅	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₅×#107	—	2	2	no	—
58	C₅×#106	—	2	2	no	—
59	C₅×#103	—	2	2	no	—
60	C₅×#103	—	2	2	no	—
61	C₅×#103	—	2	2	no	—
62	C₅×#103	—	2	2	no	—
63	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	#1718	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
67	C₃×C₅×Q₈	—	4	2	no	—
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×C₈	—	2	2	no	—
79	C₃×C₅×C₈	—	2	2	no	—
80	C₃×C₅×C₈	—	2	2	no	—
81	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₃×#107	—	2	2	no	—
86	C₃×#106	—	2	2	no	—
87	C₃×#103	—	2	2	no	—
88	C₃×#103	—	2	2	no	—
89	C₃×#103	—	2	2	no	—
90	C₃×#103	—	2	2	no	—
91	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	1	no	—
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	1	no	—
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
110	#500	C₂×C₃×C₅	1	1	yes	characteristic
111	#498	C₂×C₃×C₅	1	1	yes	characteristic
112	#487	C₂×C₃×C₅	1	1	yes	characteristic
113	#487	C₂×C₃×C₅	1	1	yes	characteristic
114	#487	C₂×C₃×C₅	1	1	yes	characteristic
115	#487	C₂×C₃×C₅	1	1	yes	characteristic
116	#322	C₂×C₃×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₅	#127	1	1	yes	characteristic
118	C₃×C₄×C₅	—	8	1	no	—
119	C₃×C₄×C₅	—	8	1	no	—
120	C₃×C₄×C₅	—	8	1	no	—
121	C₃×C₄×C₅	—	8	1	no	—
122	C₃×C₄×C₅	#135	1	2	yes	—
123	C₃×C₄×C₅	#137	1	2	yes	—
124	C₃×C₄×C₅	#137	1	2	yes	—
125	C₂×C₃×Q₈	—	2	1	no	—
126	C₂×C₃×Q₈	—	2	1	no	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
130	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
131	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
132	C₃×C₄⋊C₄	—	2	1	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
141	C₅×Q₈	—	4	2	no	—
142	C₅×Q₈	—	4	2	no	—
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	—	2	2	no	—
155	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
156	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
159	#107	—	2	2	no	—
160	#106	—	2	2	no	—
161	#103	—	2	2	no	—
162	#103	—	2	2	no	—
163	#103	—	2	2	no	—
164	#103	—	2	2	no	—
165	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
168	C₂×C₃×C₅	#495	1	1	yes	characteristic
169	C₂×C₃×C₅	#497	1	1	yes	characteristic
170	C₂×C₃×C₅	#497	1	1	yes	characteristic
171	C₃×Q₈	—	4	2	no	—
172	C₃×Q₈	—	4	2	no	—
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₂×C₃×C₄	—	4	1	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₂²×C₅	C₃×#127	1	1	yes	characteristic
186	C₄×C₅	—	8	1	no	—
187	C₄×C₅	—	8	1	no	—
188	C₄×C₅	—	8	1	no	—
189	C₄×C₅	—	8	1	no	—
190	C₄×C₅	C₃×#135	1	2	yes	—
191	C₄×C₅	C₃×#137	1	2	yes	—
192	C₄×C₅	C₃×#137	1	2	yes	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
198	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
199	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
209	C₃×C₅	#1718	1	1	yes	characteristic, direct product factor, Hall
210	C₂²×C₃	C₅×#127	1	1	yes	characteristic
211	C₃×C₄	—	8	1	no	—
212	C₃×C₄	—	8	1	no	—
213	C₃×C₄	—	8	1	no	—
214	C₃×C₄	—	8	1	no	—
215	C₃×C₄	C₅×#135	1	2	yes	—
216	C₃×C₄	C₅×#137	1	2	yes	—
217	C₃×C₄	C₅×#137	1	2	yes	—
218	C₂×C₅	C₃×#495	1	1	yes	characteristic
219	C₂×C₅	C₃×#497	1	1	yes	characteristic
220	C₂×C₅	C₃×#497	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	Q₈	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₂×C₃	C₅×#495	1	1	yes	characteristic
236	C₂×C₃	C₅×#497	1	1	yes	characteristic
237	C₂×C₃	C₅×#497	1	1	yes	characteristic
238	C₅	C₃×#1718	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	C₃×C₅×#127	1	1	yes	characteristic
240	C₄	—	8	1	no	—
241	C₄	—	8	1	no	—
242	C₄	—	8	1	no	—
243	C₄	—	8	1	no	—
244	C₄	C₃×C₅×#135	1	2	yes	—
245	C₄	C₃×C₅×#137	1	2	yes	—
246	C₄	C₃×C₅×#137	1	2	yes	—
247	C₃	C₅×#1718	1	1	yes	characteristic, direct product factor, Sylow
248	C₂	C₃×C₅×#495	1	1	yes	characteristic
249	C₂	C₃×C₅×#497	1	1	yes	characteristic
250	C₂	C₃×C₅×#497	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#127
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
8. C₅×#1718	C₃	direct (N × H)
22. C₃×#1718	C₅	direct (N × H)
66. #1718	C₃×C₅	direct (N × H)
209. C₃×C₅	#1718	direct (N × H)
238. C₅	C₃×#1718	direct (N × H)
247. C₃	C₅×#1718	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×#498	2
C₃×C₅×#487	2
C₃×C₅×#322	2
C₅×#1718	3
C₃×#1718	5

Subgroups — C3×C5×#1718 #14274752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1718 #14274752