Subgroups — C₃×C₅×#1817 #14274851

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₅×#336	C₂	1	4	yes	maximal, complemented
4	C₅×#1817	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	4	yes	—
9	C₂×C₃×C₄²×C₅	—	2	4	no	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
12	C₃×C₅×C₂²⋊C₈	—	2	4	no	—
13	C₃×#1817	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₄²×C₅	C₂×C₃	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊C₈	C₂×C₃	1	2	yes	—
16	C₅×#336	C₂×C₃	1	4	yes	complemented
17	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	—	2	4	no	—
22	C₂²×C₃×C₄×C₅	—	2	4	no	—
23	C₂²×C₃×C₄×C₅	C₈	1	4	yes	complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	D₄	1	4	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	8	no	—
31	C₃×C₄²×C₅	—	2	4	no	—
32	C₃×C₄²×C₅	—	2	4	no	—
33	C₂²×C₃×C₄²	C₂×C₅	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	2	yes	—
35	C₃×#336	C₂×C₅	1	4	yes	complemented
36	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂²×C₃	1	4	yes	—
40	C₂×C₄²×C₅	—	2	4	no	—
41	C₂×C₄²×C₅	C₃×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
43	C₅×C₂²⋊C₈	—	2	4	no	—
44	#1817	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₅	—	2	2	no	—
46	C₂³×C₃×C₅	—	2	2	no	—
47	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
49	C₂³×C₃×C₅	M₁₆	1	2	yes	—
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	8	no	—
52	C₂×C₃×C₄×C₅	—	2	8	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	8	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
61	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	#41	1	4	yes	—
64	C₃×C₅×C₈	—	4	4	no	—
65	C₂³×C₃×C₄	C₄×C₅	1	1	yes	characteristic
66	C₂³×C₃×C₄	C₄×C₅	1	1	yes	characteristic
67	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₃×C₈	C₂²×C₅	1	4	yes	—
69	C₂×C₃×C₄²	—	2	4	no	—
70	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
71	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
72	C₃×C₂²⋊C₈	—	2	4	no	—
73	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	C₃×C₈	1	4	yes	complemented
80	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	8	no	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₂²×C₄²	C₂×C₃×C₅	1	1	yes	characteristic
90	C₂×C₂²⋊C₈	C₂×C₃×C₅	1	2	yes	—
91	#336	C₂×C₃×C₅	1	4	yes	complemented
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	—	2	2	no	—
94	C₂²×C₃×C₅	—	2	4	no	—
95	C₂²×C₃×C₅	—	2	4	no	—
96	C₂²×C₃×C₅	C₂²⋊C₈	1	4	yes	—
97	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₃×C₅	#117	1	2	yes	—
99	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	4	no	—
104	C₃×C₄×C₅	—	2	4	no	—
105	C₃×C₄×C₅	#123	1	4	yes	—
106	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	2	4	no	—
112	C₂²×C₃×C₄	C₅×C₈	1	4	yes	complemented
113	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
116	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₃×C₄²	—	2	4	no	—
121	C₃×C₄²	—	2	4	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
125	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
126	C₂³×C₅	C₃×M₁₆	1	2	yes	—
127	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	—	2	8	no	—
129	C₂×C₄×C₅	—	2	8	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	8	no	—
136	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
138	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	—
139	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₃×#41	1	4	yes	—
141	C₅×C₈	—	4	4	no	—
142	C₂³×C₄	C₃×C₄×C₅	1	1	yes	characteristic
143	C₂³×C₄	C₃×C₄×C₅	1	1	yes	characteristic
144	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₂²×C₈	C₂²×C₃×C₅	1	4	yes	—
146	C₂×C₄²	—	2	4	no	—
147	C₂×C₄²	C₃×C₄×C₅	1	2	yes	—
148	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
149	C₂²⋊C₈	—	2	4	no	—
150	C₂×C₃×C₅	—	2	4	no	—
151	C₂×C₃×C₅	#381	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₃×C₅	#407	1	1	yes	characteristic
154	C₂×C₃×C₅	#431	1	2	yes	—
155	C₂×C₃×C₅	#432	1	2	yes	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	2	2	no	—
158	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
159	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
160	C₂³×C₃	C₅×M₁₆	1	2	yes	—
161	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
172	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
174	C₂×C₃×C₄	C₅×#41	1	4	yes	—
175	C₃×C₈	—	4	4	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	4	no	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	C₃×C₂²⋊C₈	1	4	yes	—
181	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₅	C₃×#117	1	2	yes	—
183	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
184	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	C₃×#123	1	4	yes	—
190	C₂⁴	C₂×C₃×C₄×C₅	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₃×C₅×C₈	1	4	yes	complemented
197	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
199	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
200	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
201	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	8	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	4	no	—
206	C₃×C₅	#1817	1	1	yes	characteristic, direct product factor, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	C₅×C₂²⋊C₈	1	4	yes	—
212	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₃	C₅×#117	1	2	yes	—
214	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
215	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
216	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	C₅×#123	1	4	yes	—
221	C₂×C₅	—	2	4	no	—
222	C₂×C₅	C₃×#381	1	1	yes	characteristic
223	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
224	C₂×C₅	C₃×#407	1	1	yes	characteristic
225	C₂×C₅	C₃×#431	1	2	yes	—
226	C₂×C₅	C₃×#432	1	2	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₃×C₅×C₂²⋊C₄	1	2	yes	—
230	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
231	C₂³	C₃×C₅×M₁₆	1	2	yes	—
232	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×C₃×C₅×C₈	1	4	yes	—
243	C₂×C₄	C₃×C₅×M₁₆	1	4	yes	—
244	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
245	C₂×C₄	C₃×C₅×#41	1	4	yes	—
246	C₈	—	4	4	no	—
247	C₂×C₃	—	2	4	no	—
248	C₂×C₃	C₅×#381	1	1	yes	characteristic
249	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₃	C₅×#407	1	1	yes	characteristic
251	C₂×C₃	C₅×#431	1	2	yes	—
252	C₂×C₃	C₅×#432	1	2	yes	—
253	C₅	C₃×#1817	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	C₃×C₅×C₂²⋊C₈	1	4	yes	—
259	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂²	C₃×C₅×#117	1	2	yes	—
261	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
262	C₂²	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
263	C₂²	C₂×C₃×C₅×M₁₆	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	C₃×C₅×#123	1	4	yes	—
268	C₃	C₅×#1817	1	1	yes	characteristic, direct product factor, Sylow
269	C₂	—	2	4	no	—
270	C₂	C₃×C₅×#381	1	1	yes	characteristic
271	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic
272	C₂	C₃×C₅×#407	1	1	yes	characteristic
273	C₂	C₃×C₅×#431	1	2	yes	—
274	C₂	C₃×C₅×#432	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#336	C₂	semidirect
4. C₅×#1817	C₃	direct (N × H)
13. C₃×#1817	C₅	direct (N × H)
16. C₅×#336	C₂×C₃	semidirect
23. C₂²×C₃×C₄×C₅	C₈	semidirect
35. C₃×#336	C₂×C₅	semidirect
44. #1817	C₃×C₅	direct (N × H)
79. C₂²×C₄×C₅	C₃×C₈	semidirect
91. #336	C₂×C₃×C₅	semidirect
112. C₂²×C₃×C₄	C₅×C₈	semidirect
196. C₂²×C₄	C₃×C₅×C₈	semidirect
206. C₃×C₅	#1817	direct (N × H)
253. C₅	C₃×#1817	direct (N × H)
268. C₃	C₅×#1817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²×C₅	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₃×C₅×#336	2
C₅×#1817	3
C₃×#1817	5

Subgroups — C3×C5×#1817 #14274851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1817 #14274851