Subgroups — C₃×C₅×#1944 #14274978

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#389	C₂	1	2	yes	maximal
5	C₃×C₅×#340	C₂	1	2	yes	maximal
6	C₅×#1944	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	—	2	2	no	—
14	C₃×C₅×#106	C₄	1	4	yes	complemented
15	C₃×C₅×#97	—	2	2	no	—
16	C₃×C₅×#95	—	2	2	no	—
17	C₃×#1944	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
21	C₅×#389	C₂×C₃	1	2	yes	—
22	C₅×#340	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
33	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	4	4	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₃×C₄²×C₅	—	2	1	no	—
40	C₃×C₄²×C₅	—	2	1	no	—
41	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
43	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
44	C₃×#389	C₂×C₅	1	2	yes	—
45	C₃×#340	C₂×C₅	1	2	yes	—
46	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₅×#107	—	2	2	no	—
53	C₅×#106	C₃×C₄	1	4	yes	complemented
54	C₅×#97	—	2	2	no	—
55	C₅×#95	—	2	2	no	—
56	#1944	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
74	C₃×C₅×C₈	—	2	2	no	—
75	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
76	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×C₄⋊C₄	—	2	2	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
81	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
82	C₃×#107	—	2	2	no	—
83	C₃×#106	C₄×C₅	1	4	yes	complemented
84	C₃×#97	—	2	2	no	—
85	C₃×#95	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
95	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
96	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	4	4	no	—
101	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
105	C₂×#106	C₂×C₃×C₅	1	1	yes	characteristic
106	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
107	#389	C₂×C₃×C₅	1	2	yes	—
108	#340	C₂×C₃×C₅	1	2	yes	—
109	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₅	#121	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
113	C₂²×C₃×C₅	#137	1	2	yes	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	—	4	1	no	—
116	C₃×C₄×C₅	—	4	1	no	—
117	C₃×C₄×C₅	—	8	2	no	—
118	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₃×C₄×C₅	#125	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
131	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
132	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	4	4	no	—
137	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
156	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	C₃×C₄⋊C₄	1	4	yes	complemented
159	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
160	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
164	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
165	#107	—	2	2	no	—
166	#106	C₃×C₄×C₅	1	4	yes	complemented
167	#97	—	2	2	no	—
168	#95	—	2	2	no	—
169	C₂×C₃×C₅	#389	1	1	yes	characteristic
170	C₂×C₃×C₅	#440	1	2	yes	—
171	C₂×C₃×C₅	#469	1	1	yes	characteristic
172	C₂×C₃×C₅	#470	1	1	yes	characteristic
173	C₂×C₃×C₅	#501	1	2	yes	—
174	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	2	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
189	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
190	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	complemented
193	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
194	C₂²×C₅	C₃×#121	1	1	yes	characteristic
195	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
197	C₂²×C₅	C₃×#137	1	2	yes	—
198	C₄×C₅	—	4	4	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	—	4	1	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
204	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
205	C₄×C₅	C₃×#125	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
214	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
215	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
216	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₃×C₅	#1944	1	1	yes	characteristic, direct product factor, Hall
225	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
226	C₂²×C₃	C₅×#121	1	1	yes	characteristic
227	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
228	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
229	C₂²×C₃	C₅×#137	1	2	yes	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	—	4	1	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	8	2	no	—
234	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
236	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
237	C₃×C₄	C₅×#125	1	1	yes	characteristic
238	C₂×C₅	C₃×#389	1	1	yes	characteristic
239	C₂×C₅	C₃×#440	1	2	yes	—
240	C₂×C₅	C₃×#469	1	1	yes	characteristic
241	C₂×C₅	C₃×#470	1	1	yes	characteristic
242	C₂×C₅	C₃×#501	1	2	yes	—
243	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
258	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
259	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	C₃×C₅×C₄⋊C₄	1	4	yes	complemented
262	C₂×C₃	C₅×#389	1	1	yes	characteristic
263	C₂×C₃	C₅×#440	1	2	yes	—
264	C₂×C₃	C₅×#469	1	1	yes	characteristic
265	C₂×C₃	C₅×#470	1	1	yes	characteristic
266	C₂×C₃	C₅×#501	1	2	yes	—
267	C₅	C₃×#1944	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
269	C₂²	C₃×C₅×#121	1	1	yes	characteristic
270	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
271	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
272	C₂²	C₃×C₅×#137	1	2	yes	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
278	C₄	C₃×C₄×C₅×Q₈	1	1	yes	characteristic
279	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
280	C₄	C₃×C₅×#125	1	1	yes	characteristic
281	C₃	C₅×#1944	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	C₃×C₅×#389	1	1	yes	characteristic
283	C₂	C₃×C₅×#440	1	2	yes	—
284	C₂	C₃×C₅×#469	1	1	yes	characteristic
285	C₂	C₃×C₅×#470	1	1	yes	characteristic
286	C₂	C₃×C₅×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1944	C₃	direct (N × H)
14. C₃×C₅×#106	C₄	semidirect
17. C₃×#1944	C₅	direct (N × H)
53. C₅×#106	C₃×C₄	semidirect
56. #1944	C₃×C₅	direct (N × H)
75. C₃×C₅×C₈	C₄⋊C₄	semidirect
83. C₃×#106	C₄×C₅	semidirect
158. C₅×C₈	C₃×C₄⋊C₄	semidirect
166. #106	C₃×C₄×C₅	semidirect
192. C₃×C₈	C₅×C₄⋊C₄	semidirect
224. C₃×C₅	#1944	direct (N × H)
261. C₈	C₃×C₅×C₄⋊C₄	semidirect
267. C₅	C₃×#1944	direct (N × H)
281. C₃	C₅×#1944	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×#107	2
C₂×C₃×C₅×#106	2
C₂×C₃×C₅×#97	2
C₃×C₅×#389	2
C₃×C₅×#340	2
C₅×#1944	3
C₃×#1944	5

Subgroups — C3×C5×#1944 #14274978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1944 #14274978