Subgroups — C₃×C₅×#2309 #14275343

Order 1920.

2208 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#117	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#390	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#379	C₂	1	4	yes	maximal, complemented
5	C₅×#2309	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	2	yes	—
7	C₂²×C₃×C₅×D₄	—	2	8	no	—
8	C₂²×C₃×C₅×D₄	C₄	1	8	yes	complemented
9	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	—	2	8	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	16	yes	complemented
15	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	complemented
16	C₃×#2309	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂³×C₅×D₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#117	C₂×C₃	1	2	yes	complemented
19	C₅×#390	C₂×C₃	1	8	yes	complemented
20	C₅×#379	C₂×C₃	1	4	yes	complemented
21	C₂⁴×C₃×C₅	—	2	8	no	—
22	C₂⁴×C₃×C₅	—	4	8	no	—
23	C₂⁴×C₃×C₅	C₂×C₄	1	8	yes	—
24	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
25	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₅×D₄	—	2	32	no	—
27	C₂×C₃×C₅×D₄	—	2	8	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	C₂×C₄	1	16	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	8	no	—
35	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
36	C₃×C₅×C₂²⋊C₄	—	4	16	no	—
37	C₃×C₄²×C₅	—	2	8	no	—
38	C₂³×C₃×D₄	C₂×C₅	1	1	yes	characteristic
39	C₂×C₃×#117	C₂×C₅	1	2	yes	complemented
40	C₃×#390	C₂×C₅	1	8	yes	complemented
41	C₃×#379	C₂×C₅	1	4	yes	complemented
42	C₂⁵×C₅	C₂²×C₃	1	2	yes	—
43	C₂²×C₅×D₄	—	2	8	no	—
44	C₂²×C₅×D₄	C₃×C₄	1	8	yes	complemented
45	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
46	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₅×#117	—	2	8	no	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
49	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	16	yes	complemented
51	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	complemented
52	#2309	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	—	2	4	no	—
54	C₂³×C₃×C₅	—	2	16	no	—
55	C₂³×C₃×C₅	—	2	8	no	—
56	C₂³×C₃×C₅	—	2	16	no	—
57	C₂³×C₃×C₅	—	4	16	no	—
58	C₂³×C₃×C₅	—	4	32	no	—
59	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
60	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
61	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
62	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
63	C₃×C₅×D₄	—	2	32	no	—
64	C₂×C₃×C₄×C₅	—	2	8	no	—
65	C₂×C₃×C₄×C₅	—	2	16	no	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	16	yes	complemented
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
70	C₂⁵×C₃	C₂²×C₅	1	2	yes	—
71	C₂²×C₃×D₄	—	2	8	no	—
72	C₂²×C₃×D₄	C₄×C₅	1	8	yes	complemented
73	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
74	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
75	C₃×#117	—	2	8	no	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
77	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
78	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	16	yes	complemented
79	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	complemented
80	C₂⁴×C₅	—	2	8	no	—
81	C₂⁴×C₅	—	4	8	no	—
82	C₂⁴×C₅	C₂×C₃×C₄	1	8	yes	—
83	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
84	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₅×D₄	—	2	32	no	—
86	C₂×C₅×D₄	—	2	8	no	—
87	C₂×C₅×D₄	—	2	8	no	—
88	C₂×C₅×D₄	C₂×C₃×C₄	1	16	yes	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
90	C₂²×C₄×C₅	C₃×D₄	1	8	yes	—
91	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
92	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
93	C₅×C₄⋊C₄	—	2	8	no	—
94	C₅×C₂²⋊C₄	—	2	8	no	—
95	C₅×C₂²⋊C₄	—	4	16	no	—
96	C₄²×C₅	—	2	8	no	—
97	C₂³×D₄	C₂×C₃×C₅	1	1	yes	characteristic
98	C₂×#117	C₂×C₃×C₅	1	2	yes	complemented
99	#390	C₂×C₃×C₅	1	8	yes	complemented
100	#379	C₂×C₃×C₅	1	4	yes	complemented
101	C₂²×C₃×C₅	—	2	4	no	—
102	C₂²×C₃×C₅	—	2	8	no	—
103	C₂²×C₃×C₅	—	2	8	no	—
104	C₂²×C₃×C₅	—	2	16	no	—
105	C₂²×C₃×C₅	—	4	32	no	—
106	C₂²×C₃×C₅	—	4	16	no	—
107	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
110	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
111	C₃×C₄×C₅	—	4	8	no	—
112	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	8	yes	—
113	C₂⁴×C₃	—	2	8	no	—
114	C₂⁴×C₃	—	4	8	no	—
115	C₂⁴×C₃	C₂×C₄×C₅	1	8	yes	—
116	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
117	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
118	C₂×C₃×D₄	—	2	32	no	—
119	C₂×C₃×D₄	—	2	8	no	—
120	C₂×C₃×D₄	—	2	8	no	—
121	C₂×C₃×D₄	C₂×C₄×C₅	1	16	yes	—
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
123	C₂²×C₃×C₄	C₅×D₄	1	8	yes	—
124	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
125	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
126	C₃×C₄⋊C₄	—	2	8	no	—
127	C₃×C₂²⋊C₄	—	2	8	no	—
128	C₃×C₂²⋊C₄	—	4	16	no	—
129	C₃×C₄²	—	2	8	no	—
130	C₂³×C₅	—	2	4	no	—
131	C₂³×C₅	—	2	16	no	—
132	C₂³×C₅	—	2	8	no	—
133	C₂³×C₅	—	2	16	no	—
134	C₂³×C₅	—	4	16	no	—
135	C₂³×C₅	—	4	32	no	—
136	C₂³×C₅	C₂²×C₃×C₄	1	8	yes	—
137	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
138	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
139	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
140	C₅×D₄	—	2	32	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	16	no	—
143	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	16	yes	complemented
144	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
145	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
147	C₂⁵	C₂²×C₃×C₅	1	2	yes	—
148	C₂²×D₄	—	2	8	no	—
149	C₂²×D₄	C₃×C₄×C₅	1	8	yes	complemented
150	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
151	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
152	#117	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	complemented
154	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	16	yes	complemented
156	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	complemented
157	C₂×C₃×C₅	—	2	4	no	—
158	C₂×C₃×C₅	—	4	8	no	—
159	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₅	#518	1	2	yes	—
161	C₂×C₃×C₅	#535	1	4	yes	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	—	2	16	no	—
164	C₂³×C₃	—	2	8	no	—
165	C₂³×C₃	—	2	16	no	—
166	C₂³×C₃	—	4	16	no	—
167	C₂³×C₃	—	4	32	no	—
168	C₂³×C₃	C₂²×C₄×C₅	1	8	yes	—
169	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
170	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
171	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
172	C₃×D₄	—	2	32	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	16	no	—
175	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	16	yes	complemented
176	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
177	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
178	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	2	8	no	—
181	C₂²×C₅	—	2	8	no	—
182	C₂²×C₅	—	2	16	no	—
183	C₂²×C₅	—	4	32	no	—
184	C₂²×C₅	—	4	16	no	—
185	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	4	yes	—
186	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
187	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
188	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
189	C₄×C₅	—	4	8	no	—
190	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	8	yes	—
191	C₂⁴	—	2	8	no	—
192	C₂⁴	—	4	8	no	—
193	C₂⁴	C₂×C₃×C₄×C₅	1	8	yes	—
194	C₂⁴	C₂³×C₃×C₅	1	4	yes	—
195	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
196	C₂×D₄	—	2	32	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	C₂×C₃×C₄×C₅	1	16	yes	—
200	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
201	C₂²×C₄	C₃×C₅×D₄	1	8	yes	—
202	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
203	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	4	16	no	—
207	C₄²	—	2	8	no	—
208	C₃×C₅	#2309	1	1	yes	characteristic, direct product factor, Hall
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	—	2	16	no	—
213	C₂²×C₃	—	4	32	no	—
214	C₂²×C₃	—	4	16	no	—
215	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
216	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
217	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
218	C₂²×C₃	C₅×H(5,𝔽₂)	1	4	yes	—
219	C₃×C₄	—	4	8	no	—
220	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	8	yes	—
221	C₂×C₅	—	2	4	no	—
222	C₂×C₅	—	4	8	no	—
223	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₅	C₃×#518	1	2	yes	—
225	C₂×C₅	C₃×#535	1	4	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	2	16	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	32	no	—
232	C₂³	C₂²×C₃×C₄×C₅	1	8	yes	—
233	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
234	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
235	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
236	D₄	—	2	32	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	16	yes	complemented
240	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
241	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
242	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
243	C₂×C₃	—	2	4	no	—
244	C₂×C₃	—	4	8	no	—
245	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₃	C₅×#518	1	2	yes	—
247	C₂×C₃	C₅×#535	1	4	yes	—
248	C₅	C₃×#2309	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	4	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	4	32	no	—
254	C₂²	—	4	16	no	—
255	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	4	yes	—
256	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
257	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
258	C₂²	C₃×C₅×H(5,𝔽₂)	1	4	yes	—
259	C₄	—	4	8	no	—
260	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	8	yes	—
261	C₃	C₅×#2309	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	4	no	—
263	C₂	—	4	8	no	—
264	C₂	C₂²×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂	C₃×C₅×#518	1	2	yes	—
266	C₂	C₃×C₅×#535	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#117	C₂	semidirect
3. C₃×C₅×#390	C₂	semidirect
4. C₃×C₅×#379	C₂	semidirect
5. C₅×#2309	C₃	direct (N × H)
8. C₂²×C₃×C₅×D₄	C₄	semidirect
12. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
14. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
15. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₃×#2309	C₅	direct (N × H)
18. C₂×C₅×#117	C₂×C₃	semidirect
19. C₅×#390	C₂×C₃	semidirect
20. C₅×#379	C₂×C₃	semidirect
39. C₂×C₃×#117	C₂×C₅	semidirect
40. C₃×#390	C₂×C₅	semidirect
41. C₃×#379	C₂×C₅	semidirect
44. C₂²×C₅×D₄	C₃×C₄	semidirect
48. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
50. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
51. C₂×C₄²×C₅	C₂²×C₃	semidirect
52. #2309	C₃×C₅	direct (N × H)
66. C₂×C₃×C₄×C₅	C₂²⋊C₄	semidirect
72. C₂²×C₃×D₄	C₄×C₅	semidirect
76. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
78. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
79. C₂×C₃×C₄²	C₂²×C₅	semidirect
98. C₂×#117	C₂×C₃×C₅	semidirect
99. #390	C₂×C₃×C₅	semidirect
100. #379	C₂×C₃×C₅	semidirect
143. C₂×C₄×C₅	C₃×C₂²⋊C₄	semidirect
149. C₂²×D₄	C₃×C₄×C₅	semidirect
153. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
155. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
156. C₂×C₄²	C₂²×C₃×C₅	semidirect
175. C₂×C₃×C₄	C₅×C₂²⋊C₄	semidirect
208. C₃×C₅	#2309	direct (N × H)
239. C₂×C₄	C₃×C₅×C₂²⋊C₄	semidirect
248. C₅	C₃×#2309	direct (N × H)
261. C₃	C₅×#2309	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂×C₃×C₅×#117	2
C₃×C₅×#390	2
C₃×C₅×#379	2
C₅×#2309	3
C₃×#2309	5

Subgroups — C3×C5×#2309 #14275343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2309 #14275343