Subgroups — C₃×C₅×#1684 #14274718

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#477	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#475	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic
8	C₅×#1684	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×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	—	2	1	no	—
13	C₃×C₅×#106	—	2	1	no	—
14	C₃×C₅×C₄⋊C₈	—	2	1	no	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×#103	—	2	1	no	—
19	C₃×C₅×#103	—	2	1	no	—
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₅×#103	—	2	1	no	—
22	C₃×C₄×C₅×C₈	—	2	1	no	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×#1684	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#487	C₂×C₃	1	1	yes	characteristic
26	C₅×#477	C₂×C₃	1	1	yes	characteristic
27	C₅×#475	C₂×C₃	1	1	yes	characteristic
28	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
29	C₅×#335	C₂×C₃	1	1	yes	characteristic
30	C₅×#333	C₂×C₃	1	1	yes	characteristic
31	C₅×#332	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
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×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	—	2	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#487	C₂×C₅	1	1	yes	characteristic
49	C₃×#477	C₂×C₅	1	1	yes	characteristic
50	C₃×#475	C₂×C₅	1	1	yes	characteristic
51	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
52	C₃×#335	C₂×C₅	1	1	yes	characteristic
53	C₃×#333	C₂×C₅	1	1	yes	characteristic
54	C₃×#332	C₂×C₅	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	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×#107	—	2	1	no	—
59	C₅×#106	—	2	1	no	—
60	C₅×C₄⋊C₈	—	2	1	no	—
61	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₅×#103	—	2	1	no	—
65	C₅×#103	—	2	1	no	—
66	C₅×#103	—	2	1	no	—
67	C₅×#103	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
70	#1684	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
71	C₃×C₅×Q₈	—	2	1	no	—
72	C₃×C₅×Q₈	—	4	1	no	—
73	C₃×C₅×Q₈	—	4	1	no	—
74	C₃×C₅×Q₈	—	8	1	no	—
75	C₃×C₅×Q₈	Q₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
89	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×#107	—	2	1	no	—
92	C₃×#106	—	2	1	no	—
93	C₃×C₄⋊C₈	—	2	1	no	—
94	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
97	C₃×#103	—	2	1	no	—
98	C₃×#103	—	2	1	no	—
99	C₃×#103	—	2	1	no	—
100	C₃×#103	—	2	1	no	—
101	C₃×C₄×C₈	—	2	1	no	—
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₅×Q₈	—	2	1	no	—
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	—	2	1	no	—
107	C₂×C₅×C₈	—	2	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
117	C₄²×C₅	—	2	1	no	—
118	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
119	#487	C₂×C₃×C₅	1	1	yes	characteristic
120	#477	C₂×C₃×C₅	1	1	yes	characteristic
121	#475	C₂×C₃×C₅	1	1	yes	characteristic
122	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
123	#335	C₂×C₃×C₅	1	1	yes	characteristic
124	#333	C₂×C₃×C₅	1	1	yes	characteristic
125	#332	C₂×C₃×C₅	1	1	yes	characteristic
126	C₂²×C₃×C₅	#121	1	1	yes	characteristic
127	C₃×C₄×C₅	—	2	2	no	—
128	C₃×C₄×C₅	—	2	1	no	—
129	C₃×C₄×C₅	—	4	1	no	—
130	C₃×C₄×C₅	—	8	1	no	—
131	C₃×C₄×C₅	—	8	1	no	—
132	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
133	C₃×C₄×C₅	#135	1	1	yes	characteristic
134	C₃×C₄×C₅	#135	1	1	yes	characteristic
135	C₃×C₄×C₅	#137	1	1	yes	characteristic
136	C₂×C₃×Q₈	—	2	1	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
150	C₃×C₄²	—	2	1	no	—
151	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₅×Q₈	—	2	1	no	—
153	C₅×Q₈	—	4	1	no	—
154	C₅×Q₈	—	4	1	no	—
155	C₅×Q₈	—	8	1	no	—
156	C₅×Q₈	C₃×Q₁₆	1	2	yes	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	4	1	no	—
167	C₅×C₈	—	4	1	no	—
168	C₅×C₈	—	4	1	no	—
169	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
172	#107	—	2	1	no	—
173	#106	—	2	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
177	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
178	#103	—	2	1	no	—
179	#103	—	2	1	no	—
180	#103	—	2	1	no	—
181	#103	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
184	C₂×C₃×C₅	#464	1	1	yes	characteristic
185	C₂×C₃×C₅	#467	1	1	yes	characteristic
186	C₂×C₃×C₅	#471	1	1	yes	characteristic
187	C₃×Q₈	—	2	1	no	—
188	C₃×Q₈	—	4	1	no	—
189	C₃×Q₈	—	4	1	no	—
190	C₃×Q₈	—	8	1	no	—
191	C₃×Q₈	C₅×Q₁₆	1	2	yes	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	—	4	1	no	—
196	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃×C₈	—	4	1	no	—
202	C₃×C₈	—	4	1	no	—
203	C₃×C₈	—	4	1	no	—
204	C₂²×C₅	C₃×#121	1	1	yes	characteristic
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	4	1	no	—
208	C₄×C₅	—	8	1	no	—
209	C₄×C₅	—	8	1	no	—
210	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
211	C₄×C₅	C₃×#135	1	1	yes	characteristic
212	C₄×C₅	C₃×#135	1	1	yes	characteristic
213	C₄×C₅	C₃×#137	1	1	yes	characteristic
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
221	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
228	C₄²	—	2	1	no	—
229	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
230	C₃×C₅	#1684	1	1	yes	characteristic, direct product factor, Hall
231	C₂²×C₃	C₅×#121	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	8	1	no	—
236	C₃×C₄	—	8	1	no	—
237	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
238	C₃×C₄	C₅×#135	1	1	yes	characteristic
239	C₃×C₄	C₅×#135	1	1	yes	characteristic
240	C₃×C₄	C₅×#137	1	1	yes	characteristic
241	C₂×C₅	C₃×#464	1	1	yes	characteristic
242	C₂×C₅	C₃×#467	1	1	yes	characteristic
243	C₂×C₅	C₃×#471	1	1	yes	characteristic
244	Q₈	—	2	1	no	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	Q₈	—	8	1	no	—
248	Q₈	C₃×C₅×Q₁₆	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂×C₃	C₅×#464	1	1	yes	characteristic
262	C₂×C₃	C₅×#467	1	1	yes	characteristic
263	C₂×C₃	C₅×#471	1	1	yes	characteristic
264	C₅	C₃×#1684	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	C₃×C₅×#121	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	8	1	no	—
270	C₄	—	8	1	no	—
271	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
272	C₄	C₃×C₅×#135	1	1	yes	characteristic
273	C₄	C₃×C₅×#135	1	1	yes	characteristic
274	C₄	C₃×C₅×#137	1	1	yes	characteristic
275	C₃	C₅×#1684	1	1	yes	characteristic, direct product factor, Sylow
276	C₂	C₃×C₅×#464	1	1	yes	characteristic
277	C₂	C₃×C₅×#467	1	1	yes	characteristic
278	C₂	C₃×C₅×#471	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
8. C₅×#1684	C₃	direct (N × H)
24. C₃×#1684	C₅	direct (N × H)
70. #1684	C₃×C₅	direct (N × H)
230. C₃×C₅	#1684	direct (N × H)
264. C₅	C₃×#1684	direct (N × H)
275. C₃	C₅×#1684	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×#487	2
C₃×C₅×#477	2
C₃×C₅×#475	2
C₃×C₅×C₈×Q₈	2
C₃×C₅×#335	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1684	3
C₃×#1684	5

Subgroups — C3×C5×#1684 #14274718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1684 #14274718