Subgroups — C₃×C₅×#1798 #14274832

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₃×C₅×#336	C₂	1	2	yes	maximal, complemented
5	C₅×#1798	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	4	yes	—
8	C₃×C₅×#117	—	2	4	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	—	2	4	no	—
14	C₃×C₅×C₂²⋊C₈	—	2	4	no	—
15	C₃×#1798	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
18	C₂×C₅×C₂²⋊C₈	C₂×C₃	1	2	yes	—
19	C₅×#336	C₂×C₃	1	2	yes	complemented
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	—	2	4	no	—
30	C₂×C₃×C₅×C₈	—	2	4	no	—
31	C₂×C₃×C₅×C₈	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
34	C₃×C₄²×C₅	—	2	2	no	—
35	C₃×C₄²×C₅	D₄	1	4	yes	—
36	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
38	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	2	yes	—
39	C₃×#336	C₂×C₅	1	2	yes	complemented
40	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	4	yes	—
42	C₅×#117	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
44	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₅×C₄⋊C₈	—	2	4	no	—
48	C₅×C₂²⋊C₈	—	2	4	no	—
49	#1798	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	2	1	no	—
51	C₂³×C₃×C₅	—	2	1	no	—
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₃×C₅×C₈	—	4	4	no	—
69	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂²×C₅	1	4	yes	—
71	C₃×#117	—	2	4	no	—
72	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
73	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	2	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄⋊C₈	—	2	4	no	—
77	C₃×C₂²⋊C₈	—	2	4	no	—
78	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	C₃×D₄	1	4	yes	—
94	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic
95	C₂×C₄⋊C₈	C₂×C₃×C₅	1	2	yes	complemented
96	C₂×C₂²⋊C₈	C₂×C₃×C₅	1	2	yes	—
97	#336	C₂×C₃×C₅	1	2	yes	complemented
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	4	no	—
100	C₂²×C₃×C₅	—	4	2	no	—
101	C₂²×C₃×C₅	—	4	2	no	—
102	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#117	1	1	yes	characteristic
105	C₂²×C₃×C₅	#131	1	4	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	—	4	2	no	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
116	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
118	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
119	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	—	2	4	no	—
122	C₂×C₃×C₈	—	2	4	no	—
123	C₃×C₄⋊C₄	—	2	4	no	—
124	C₃×C₂²⋊C₄	—	2	4	no	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	C₅×D₄	1	4	yes	—
127	C₂³×C₅	—	2	1	no	—
128	C₂³×C₅	—	2	1	no	—
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
131	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
140	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₃×#41	1	2	yes	—
145	C₅×C₈	—	4	4	no	—
146	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₂²×C₃×C₅	1	4	yes	—
148	#117	—	2	4	no	—
149	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
150	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	2	yes	—
151	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₄⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂×C₃×C₅	—	4	2	no	—
156	C₂×C₃×C₅	#380	1	1	yes	characteristic
157	C₂×C₃×C₅	#408	1	2	yes	—
158	C₂×C₃×C₅	#424	1	2	yes	—
159	C₂×C₃×C₅	#433	1	2	yes	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
164	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
173	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
177	C₂×C₃×C₄	C₅×#41	1	2	yes	—
178	C₃×C₈	—	4	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₅	C₃×#117	1	1	yes	characteristic
186	C₂²×C₅	C₃×#131	1	4	yes	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	C₃×#121	1	2	yes	—
191	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
192	C₂⁴	C₂×C₃×C₄×C₅	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
196	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
197	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
198	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
199	C₂²×C₄	C₃×C₅×Q₈	1	2	yes	—
200	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₃×C₅×D₄	1	4	yes	—
208	C₃×C₅	#1798	1	1	yes	characteristic, direct product factor, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	C₅×#117	1	1	yes	characteristic
216	C₂²×C₃	C₅×#131	1	4	yes	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	C₅×#121	1	2	yes	—
221	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	C₃×#380	1	1	yes	characteristic
224	C₂×C₅	C₃×#408	1	2	yes	—
225	C₂×C₅	C₃×#424	1	2	yes	—
226	C₂×C₅	C₃×#433	1	2	yes	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	4	no	—
230	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
231	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	4	yes	—
240	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
243	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
244	C₂×C₄	C₃×C₅×#41	1	2	yes	—
245	C₈	—	4	4	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₅×#380	1	1	yes	characteristic
248	C₂×C₃	C₅×#408	1	2	yes	—
249	C₂×C₃	C₅×#424	1	2	yes	—
250	C₂×C₃	C₅×#433	1	2	yes	—
251	C₅	C₃×#1798	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
258	C₂²	C₃×C₅×#117	1	1	yes	characteristic
259	C₂²	C₃×C₅×#131	1	4	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	C₃×C₅×#121	1	2	yes	—
264	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
265	C₃	C₅×#1798	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	4	2	no	—
267	C₂	C₃×C₅×#380	1	1	yes	characteristic
268	C₂	C₃×C₅×#408	1	2	yes	—
269	C₂	C₃×C₅×#424	1	2	yes	—
270	C₂	C₃×C₅×#433	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
4. C₃×C₅×#336	C₂	semidirect
5. C₅×#1798	C₃	direct (N × H)
15. C₃×#1798	C₅	direct (N × H)
17. C₂×C₅×C₄⋊C₈	C₂×C₃	semidirect
19. C₅×#336	C₂×C₃	semidirect
37. C₂×C₃×C₄⋊C₈	C₂×C₅	semidirect
39. C₃×#336	C₂×C₅	semidirect
49. #1798	C₃×C₅	direct (N × H)
95. C₂×C₄⋊C₈	C₂×C₃×C₅	semidirect
97. #336	C₂×C₃×C₅	semidirect
208. C₃×C₅	#1798	direct (N × H)
251. C₅	C₃×#1798	direct (N × H)
265. C₃	C₅×#1798	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×#117	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₃×C₅×#336	2
C₅×#1798	3
C₃×#1798	5

Subgroups — C3×C5×#1798 #14274832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1798 #14274832