Subgroups — C₃×D₅×#498 #14439730

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65368	C₂	1	2	yes	maximal, complemented
4	C₃×D₅×#106	C₂	1	4	yes	maximal
5	C₃×#65144	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
7	C₃×#64969	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×#498	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×Q₈×D₅	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×#4090	—	2	2	no	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	C₃×D₅×C₄⋊C₄	—	2	2	no	—
14	C₃×D₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₃×#4034	—	2	2	no	—
16	C₃×#4034	C₂²	1	4	yes	—
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×#4015	C₂²	1	2	yes	—
19	C₃×C₅×#106	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×#3949	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×#3940	C₂²	1	8	yes	—
24	C₃×C₄×#168	C₂²	1	1	yes	characteristic
25	C₂×C₃×#498	—	5	1	no	maximal, Hall
26	D₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
27	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
28	#65368	C₂×C₃	1	2	yes	complemented
29	D₅×#106	C₂×C₃	1	4	yes	—
30	#65144	C₂×C₃	1	4	yes	complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
32	#64969	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₃×Q₈×D₅	—	4	8	no	—
35	C₂×C₃×C₄×D₅	—	2	4	no	—
36	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
37	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	—	2	2	no	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×C₄⋊Dic₅	—	2	2	no	—
45	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
46	C₃×#683	—	2	4	no	—
47	C₃×#683	—	2	4	no	—
48	C₃×C₄×Dic₅	D₄	1	2	yes	—
49	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×#168	C₂³	1	2	yes	—
51	C₃×C₈×D₅	Q₈	1	8	yes	complemented
52	C₂×C₃×C₄⋊Q₈	—	5	2	no	—
53	C₃×#498	—	5	2	no	—
54	C₃×#498	—	5	4	no	—
55	C₃×#498	—	5	1	no	—
56	C₃×#498	D₅	1	1	yes	characteristic, direct product factor
57	C₂×C₃×#106	—	5	4	no	—
58	C₂×C₃×C₄×C₈	—	5	1	no	—
59	C₂×Q₈×D₅	—	2	2	no	—
60	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	#4090	—	2	2	no	—
62	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
63	D₅×C₄⋊C₄	—	2	2	no	—
64	D₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
65	#4034	—	2	2	no	—
66	#4034	C₂²×C₃	1	4	yes	—
67	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
68	#4015	C₂²×C₃	1	2	yes	—
69	C₅×#106	C₂²×C₃	1	4	yes	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	#3949	C₂²×C₃	1	4	yes	—
72	C₈×Dic₅	C₂²×C₃	1	2	yes	—
73	#3940	C₂²×C₃	1	8	yes	—
74	C₄×#168	C₂²×C₃	1	1	yes	characteristic
75	C₂×#498	—	5	1	no	Sylow
76	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
77	C₃×C₅×Q₈	—	4	4	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₃×Dic₅	—	2	4	no	—
82	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
84	C₃×C₄×D₅	—	4	8	no	—
85	C₃×C₄×D₅	SD₁₆	1	8	yes	—
86	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
87	C₃×Dic₁₀	—	4	4	no	—
88	C₃×Dic₁₀	—	4	8	no	—
89	C₃×C₅×C₈	C₂×Q₈	1	4	yes	complemented
90	C₃×#168	C₂×Q₈	1	4	yes	complemented
91	C₂²×C₃×Q₈	—	10	2	no	—
92	C₂²×C₃×C₈	—	5	2	no	—
93	C₃×C₄⋊Q₈	—	5	4	no	—
94	C₃×C₄⋊Q₈	—	5	2	no	—
95	C₃×C₄⋊Q₈	—	10	2	no	—
96	C₃×C₄⋊Q₈	—	10	2	no	—
97	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
98	C₂×C₃×C₄⋊C₄	—	5	4	no	—
99	C₂×C₃×C₄⋊C₄	—	10	2	no	—
100	C₂×C₃×C₄²	—	5	1	no	—
101	C₃×#106	—	5	4	no	—
102	C₃×#106	—	5	8	no	—
103	C₃×#106	C₂×D₅	1	4	yes	—
104	C₃×C₄×C₈	—	5	2	no	—
105	C₃×C₄×C₈	—	5	1	no	—
106	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₅×Q₈	—	2	2	no	—
108	Q₈×D₅	—	4	8	no	—
109	C₂×C₄×D₅	—	2	4	no	—
110	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
111	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₁₀	—	2	2	no	—
113	C₂×Dic₁₀	—	2	4	no	—
114	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
120	#683	—	2	4	no	—
121	#683	—	2	4	no	—
122	C₄×Dic₅	C₃×D₄	1	2	yes	—
123	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×#168	C₂³×C₃	1	2	yes	—
125	C₈×D₅	C₃×Q₈	1	8	yes	complemented
126	C₂×C₄⋊Q₈	—	5	2	no	—
127	#498	—	5	2	no	—
128	#498	—	5	4	no	—
129	#498	—	5	1	no	—
130	#498	C₃×D₅	1	1	yes	characteristic, direct product factor
131	C₂×#106	—	5	4	no	—
132	C₂×C₄×C₈	—	5	1	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
135	C₂×C₃×D₅	C₂×SD₁₆	1	4	yes	—
136	C₃×C₄×C₅	—	4	4	no	—
137	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
138	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
139	C₃×Dic₅	—	4	4	no	—
140	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
141	C₃×Dic₅	C₂×SD₁₆	1	4	yes	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×Q₈	—	10	4	no	—
144	C₂×C₃×Q₈	—	10	2	no	—
145	C₂×C₃×Q₈	—	20	8	no	—
146	C₂²×C₃×C₄	—	5	2	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₂²×C₃×C₄	—	10	4	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	5	8	no	—
151	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	5	4	no	—
154	C₃×C₄⋊C₄	—	10	4	no	—
155	C₃×C₄⋊C₄	—	10	4	no	—
156	C₃×C₄⋊C₄	—	10	2	no	—
157	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
158	C₃×C₄²	—	5	2	no	—
159	C₃×C₄²	—	5	1	no	—
160	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
161	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₅×Q₈	—	4	4	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
166	C₂×Dic₅	—	2	4	no	—
167	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
169	C₄×D₅	—	4	8	no	—
170	C₄×D₅	C₃×SD₁₆	1	8	yes	—
171	C₄×D₅	C₂×C₃×Q₈	1	4	yes	—
172	Dic₁₀	—	4	4	no	—
173	Dic₁₀	—	4	8	no	—
174	C₅×C₈	C₂×C₃×Q₈	1	4	yes	complemented
175	#168	C₂×C₃×Q₈	1	4	yes	complemented
176	C₂²×Q₈	—	10	2	no	—
177	C₂²×C₈	—	5	2	no	—
178	C₄⋊Q₈	—	5	4	no	—
179	C₄⋊Q₈	—	5	2	no	—
180	C₄⋊Q₈	—	10	2	no	—
181	C₄⋊Q₈	—	10	2	no	—
182	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
183	C₂×C₄⋊C₄	—	5	4	no	—
184	C₂×C₄⋊C₄	—	10	2	no	—
185	C₂×C₄²	—	5	1	no	—
186	#106	—	5	4	no	—
187	#106	—	5	8	no	—
188	#106	C₂×C₃×D₅	1	4	yes	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
194	C₃×D₅	#498	1	4	yes	direct product factor
195	C₂³×C₃	—	5	1	no	—
196	C₃×Q₈	—	4	4	no	—
197	C₃×Q₈	—	20	8	no	—
198	C₃×Q₈	—	20	4	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	2	no	—
202	C₂×C₃×C₄	—	5	4	no	—
203	C₂×C₃×C₄	—	5	8	no	—
204	C₂×C₃×C₄	—	10	4	no	—
205	C₂×C₃×C₄	—	20	8	no	—
206	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
207	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
208	C₃×C₈	—	5	4	no	—
209	C₃×C₈	Q₈×D₅	1	4	yes	complemented
210	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
211	C₂×D₅	C₃×C₄⋊Q₈	1	2	yes	—
212	C₂×D₅	C₂×C₃×SD₁₆	1	4	yes	—
213	C₄×C₅	—	4	4	no	—
214	C₄×C₅	C₂×C₃×SD₁₆	1	4	yes	—
215	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
216	Dic₅	—	4	4	no	—
217	Dic₅	C₃×C₄⋊Q₈	1	2	yes	—
218	Dic₅	C₂×C₃×SD₁₆	1	4	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	10	4	no	—
221	C₂×Q₈	—	10	2	no	—
222	C₂×Q₈	—	20	8	no	—
223	C₂²×C₄	—	5	2	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	—	10	4	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	5	8	no	—
228	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	5	4	no	—
231	C₄⋊C₄	—	10	4	no	—
232	C₄⋊C₄	—	10	4	no	—
233	C₄⋊C₄	—	10	2	no	—
234	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
235	C₄²	—	5	2	no	—
236	C₄²	—	5	1	no	—
237	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
238	C₃×C₅	C₂×#498	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	5	2	no	—
240	C₂²×C₃	—	5	4	no	—
241	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
242	C₃×C₄	—	4	4	no	—
243	C₃×C₄	—	5	2	no	—
244	C₃×C₄	—	5	4	no	—
245	C₃×C₄	—	20	4	no	—
246	C₃×C₄	D₅×SD₁₆	1	4	yes	—
247	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
248	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
249	C₂×C₅	C₂²×C₃×SD₁₆	1	2	yes	—
250	D₅	C₃×#498	1	4	yes	direct product factor
251	C₂³	—	5	1	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	20	8	no	—
254	Q₈	—	20	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	4	no	—
259	C₂×C₄	—	5	8	no	—
260	C₂×C₄	—	10	4	no	—
261	C₂×C₄	—	20	8	no	—
262	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
263	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
264	C₈	—	5	4	no	—
265	C₈	C₃×Q₈×D₅	1	4	yes	complemented
266	C₂×C₃	—	5	4	no	—
267	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
268	C₂×C₃	C₂×D₅×SD₁₆	1	2	yes	—
269	C₅	C₂×C₃×#498	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	5	2	no	—
271	C₂²	—	5	4	no	—
272	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	5	2	no	—
275	C₄	—	5	4	no	—
276	C₄	—	20	4	no	—
277	C₄	C₃×D₅×SD₁₆	1	4	yes	—
278	C₄	C₂×C₃×Q₈×D₅	1	2	yes	—
279	C₃	D₅×#498	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	5	4	no	—
281	C₂	C₃×D₅×C₄⋊Q₈	1	1	yes	characteristic
282	C₂	C₂×C₃×D₅×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#498	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₃×#498	D₅
hypocenter	C₅	C₂×C₃×#498
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#498

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#498	C₂	semidirect
3. C₃×#65368	C₂	semidirect
5. C₃×#65144	C₂	semidirect
7. C₃×#64969	C₂	semidirect
8. D₅×#498	C₃	direct (N × H)
27. C₅×#498	C₂×C₃	semidirect
28. #65368	C₂×C₃	semidirect
30. #65144	C₂×C₃	semidirect
32. #64969	C₂×C₃	semidirect
51. C₃×C₈×D₅	Q₈	semidirect
56. C₃×#498	D₅	direct (N × H)
89. C₃×C₅×C₈	C₂×Q₈	semidirect
90. C₃×#168	C₂×Q₈	semidirect
125. C₈×D₅	C₃×Q₈	semidirect
130. #498	C₃×D₅	direct (N × H)
174. C₅×C₈	C₂×C₃×Q₈	semidirect
175. #168	C₂×C₃×Q₈	semidirect
194. C₃×D₅	#498	direct (N × H)
209. C₃×C₈	Q₈×D₅	semidirect
238. C₃×C₅	C₂×#498	semidirect
250. D₅	C₃×#498	direct (N × H)
265. C₈	C₃×Q₈×D₅	semidirect
269. C₅	C₂×C₃×#498	semidirect
279. C₃	D₅×#498	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₅×C₄⋊Q₈	2
C₃×C₅×#498	2
C₃×#65368	2
C₃×D₅×#106	2
C₃×#65144	2
C₃×C₄×C₈×D₅	2
C₃×#64969	2
D₅×#498	3
C₂×C₃×#498	5

Subgroups — C3×D5×#498 #14439730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×#498 #14439730