Subgroups — C₃×C₅×#3147 #14276181

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#135	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#532	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#495	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#492	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3147	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	C₂²	1	4	yes	—
10	C₃×C₅×#135	—	2	16	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
12	C₂×C₃×C₅×SD₁₆	C₂²	1	8	yes	—
13	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
14	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₃×C₅×#127	C₂²	1	2	yes	—
17	C₃×C₅×#124	C₂²	1	4	yes	—
18	C₃×C₅×#121	—	2	4	no	—
19	C₃×C₄×C₅×D₄	—	2	4	no	—
20	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₃×#3147	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#135	C₂×C₃	1	4	yes	complemented
26	C₅×#532	C₂×C₃	1	2	yes	complemented
27	C₅×#495	C₂×C₃	1	4	yes	complemented
28	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
30	C₅×#492	C₂×C₃	1	2	yes	complemented
31	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×C₅×#41	—	2	16	no	—
33	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
34	C₂×C₃×C₅×D₄	—	2	4	no	—
35	C₂×C₃×C₅×D₄	—	2	4	no	—
36	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
37	C₂²×C₃×C₄×C₅	—	2	4	no	—
38	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×Q₁₆	—	2	8	no	—
41	C₃×C₅×SD₁₆	—	2	16	no	—
42	C₃×C₅×D₈	—	2	8	no	—
43	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	2	4	no	—
47	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
48	C₃×C₄²×C₅	D₄	1	2	yes	—
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×#135	C₂×C₅	1	4	yes	complemented
52	C₃×#532	C₂×C₅	1	2	yes	complemented
53	C₃×#495	C₂×C₅	1	4	yes	complemented
54	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#492	C₂×C₅	1	2	yes	complemented
57	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×#41	C₂²×C₃	1	4	yes	—
59	C₅×#135	—	2	16	no	—
60	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
61	C₂×C₅×SD₁₆	C₂²×C₃	1	8	yes	—
62	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
63	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
64	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
65	C₅×#127	C₂²×C₃	1	2	yes	—
66	C₅×#124	C₂²×C₃	1	4	yes	—
67	C₅×#121	—	2	4	no	—
68	C₄×C₅×D₄	—	2	4	no	—
69	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
73	#3147	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₃×C₅	—	2	4	no	—
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	2	8	no	—
77	C₃×C₅×D₄	—	2	8	no	—
78	C₃×C₅×D₄	—	4	8	no	—
79	C₃×C₅×D₄	—	4	8	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	4	8	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
90	C₂×C₃×#41	C₂²×C₅	1	4	yes	—
91	C₃×#135	—	2	16	no	—
92	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
93	C₂×C₃×SD₁₆	C₂²×C₅	1	8	yes	—
94	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
95	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
96	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
97	C₃×#127	C₂²×C₅	1	2	yes	—
98	C₃×#124	C₂²×C₅	1	4	yes	—
99	C₃×#121	—	2	4	no	—
100	C₃×C₄×D₄	—	2	4	no	—
101	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
105	C₅×#41	—	2	16	no	—
106	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
107	C₂×C₅×D₄	—	2	4	no	—
108	C₂×C₅×D₄	—	2	4	no	—
109	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
110	C₂²×C₄×C₅	—	2	4	no	—
111	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₅×Q₁₆	—	2	8	no	—
114	C₅×SD₁₆	—	2	16	no	—
115	C₅×D₈	—	2	8	no	—
116	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
117	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
118	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₂²⋊C₄	—	2	8	no	—
121	C₄²×C₅	C₃×D₄	1	2	yes	—
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×#135	C₂×C₃×C₅	1	4	yes	complemented
125	#532	C₂×C₃×C₅	1	2	yes	complemented
126	#495	C₂×C₃×C₅	1	4	yes	complemented
127	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
128	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
129	#492	C₂×C₃×C₅	1	2	yes	complemented
130	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	—	4	4	no	—
133	C₂²×C₃×C₅	—	4	8	no	—
134	C₂²×C₃×C₅	#127	1	2	yes	—
135	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	—	4	4	no	—
138	C₃×C₄×C₅	#127	1	2	yes	—
139	C₃×C₄×C₅	#135	1	4	yes	complemented
140	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
141	C₃×#41	—	2	16	no	—
142	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
143	C₂×C₃×D₄	—	2	4	no	—
144	C₂×C₃×D₄	—	2	4	no	—
145	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
146	C₂²×C₃×C₄	—	2	4	no	—
147	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
148	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₃×Q₁₆	—	2	8	no	—
150	C₃×SD₁₆	—	2	16	no	—
151	C₃×D₈	—	2	8	no	—
152	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
153	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
154	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
155	C₃×C₄⋊C₄	—	2	4	no	—
156	C₃×C₂²⋊C₄	—	2	8	no	—
157	C₃×C₄²	C₅×D₄	1	2	yes	—
158	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
160	C₂³×C₅	—	2	4	no	—
161	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₅×Q₈	—	2	8	no	—
163	C₅×D₄	—	2	8	no	—
164	C₅×D₄	—	4	8	no	—
165	C₅×D₄	—	4	8	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	4	8	no	—
169	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
174	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
175	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
176	C₂×#41	C₂²×C₃×C₅	1	4	yes	—
177	#135	—	2	16	no	—
178	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
179	C₂×SD₁₆	C₂²×C₃×C₅	1	8	yes	—
180	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
181	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
182	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
183	#127	C₂²×C₃×C₅	1	2	yes	—
184	#124	C₂²×C₃×C₅	1	4	yes	—
185	#121	—	2	4	no	—
186	C₄×D₄	—	2	4	no	—
187	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
188	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
189	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
190	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
191	C₂×C₃×C₅	—	2	2	no	—
192	C₂×C₃×C₅	—	8	4	no	—
193	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#135	1	2	yes	—
195	C₂³×C₃	—	2	4	no	—
196	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
197	C₃×Q₈	—	2	8	no	—
198	C₃×D₄	—	2	8	no	—
199	C₃×D₄	—	4	8	no	—
200	C₃×D₄	—	4	8	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	4	8	no	—
204	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
210	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	4	8	no	—
213	C₂²×C₅	—	4	4	no	—
214	C₂²×C₅	C₃×#127	1	2	yes	—
215	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	C₃×#127	1	2	yes	—
219	C₄×C₅	C₃×#135	1	4	yes	complemented
220	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
221	#41	—	2	16	no	—
222	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
228	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
229	Q₁₆	—	2	8	no	—
230	SD₁₆	—	2	16	no	—
231	D₈	—	2	8	no	—
232	C₂×C₈	C₃×C₅×D₄	1	8	yes	complemented
233	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
234	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	C₃×C₅×D₄	1	2	yes	—
238	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
239	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, direct product factor, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	—	4	8	no	—
244	C₂²×C₃	C₅×#127	1	2	yes	—
245	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	C₅×#127	1	2	yes	—
249	C₃×C₄	C₅×#135	1	4	yes	complemented
250	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	8	4	no	—
253	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
254	C₂×C₅	C₂×C₃×#135	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	D₄	—	2	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	4	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
265	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
266	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
268	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
269	C₈	C₂×C₃×C₅×D₄	1	4	yes	—
270	C₈	C₂×C₃×C₅×D₄	1	4	yes	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	8	4	no	—
273	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
274	C₂×C₃	C₂×C₅×#135	1	2	yes	—
275	C₅	C₃×#3147	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	8	no	—
279	C₂²	C₃×C₅×#127	1	2	yes	—
280	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	4	no	—
283	C₄	C₃×C₅×#127	1	2	yes	—
284	C₄	C₃×C₅×#135	1	4	yes	complemented
285	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
286	C₃	C₅×#3147	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	8	4	no	—
289	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
290	C₂	C₂×C₃×C₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#135	C₂	semidirect
2. C₃×C₅×#532	C₂	semidirect
3. C₃×C₅×#495	C₂	semidirect
4. C₃×C₅×#494	C₂	semidirect
5. C₃×C₅×#493	C₂	semidirect
6. C₃×C₅×#492	C₂	semidirect
7. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
8. C₅×#3147	C₃	direct (N × H)
24. C₃×#3147	C₅	direct (N × H)
25. C₂×C₅×#135	C₂×C₃	semidirect
26. C₅×#532	C₂×C₃	semidirect
27. C₅×#495	C₂×C₃	semidirect
28. C₅×#494	C₂×C₃	semidirect
29. C₅×#493	C₂×C₃	semidirect
30. C₅×#492	C₂×C₃	semidirect
31. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
43. C₂×C₃×C₅×C₈	D₄	semidirect
51. C₂×C₃×#135	C₂×C₅	semidirect
52. C₃×#532	C₂×C₅	semidirect
53. C₃×#495	C₂×C₅	semidirect
54. C₃×#494	C₂×C₅	semidirect
55. C₃×#493	C₂×C₅	semidirect
56. C₃×#492	C₂×C₅	semidirect
57. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
73. #3147	C₃×C₅	direct (N × H)
116. C₂×C₅×C₈	C₃×D₄	semidirect
124. C₂×#135	C₂×C₃×C₅	semidirect
125. #532	C₂×C₃×C₅	semidirect
126. #495	C₂×C₃×C₅	semidirect
127. #494	C₂×C₃×C₅	semidirect
128. #493	C₂×C₃×C₅	semidirect
129. #492	C₂×C₃×C₅	semidirect
130. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
139. C₃×C₄×C₅	#135	semidirect
152. C₂×C₃×C₈	C₅×D₄	semidirect
219. C₄×C₅	C₃×#135	semidirect
232. C₂×C₈	C₃×C₅×D₄	semidirect
240. C₃×C₅	#3147	direct (N × H)
249. C₃×C₄	C₅×#135	semidirect
275. C₅	C₃×#3147	direct (N × H)
284. C₄	C₃×C₅×#135	semidirect
286. C₃	C₅×#3147	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#135	2
C₃×C₅×#532	2
C₃×C₅×#495	2
C₃×C₅×#494	2
C₃×C₅×#493	2
C₃×C₅×#492	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#3147	3
C₃×#3147	5

Subgroups — C3×C5×#3147 #14276181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3147 #14276181