Subgroups — C₅×D₃×#498 #14439729

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	C₅×D₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5080	C₂	1	2	yes	maximal, complemented
4	C₅×D₃×#106	C₂	1	4	yes	maximal
5	C₅×#4855	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₅×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#498	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₅×#944	—	2	2	no	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₅×D₃×C₄⋊C₄	—	2	2	no	—
14	C₅×D₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₅×#888	—	2	2	no	—
16	C₅×#888	C₂²	1	4	yes	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	2	yes	—
19	C₃×C₅×#106	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#817	C₂²	1	4	yes	—
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₅×#808	C₂²	1	8	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	D₃×#498	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂×C₅×C₄⋊Q₈	—	3	2	no	—
27	C₅×#498	—	3	4	no	—
28	C₅×#498	—	3	1	no	—
29	C₅×#498	—	3	2	no	—
30	C₅×#498	D₃	1	1	yes	characteristic, direct product factor
31	C₂×C₅×#106	—	3	4	no	—
32	C₂×C₄×C₅×C₈	—	3	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₅×D₃×Q₈	—	4	8	no	—
35	C₂×C₄×C₅×D₃	—	2	4	no	—
36	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
37	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
38	C₂×C₅×Dic₆	—	2	2	no	—
39	C₂×C₅×Dic₆	—	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	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₅×Dic₃⋊C₄	—	2	4	no	—
47	C₅×Dic₃⋊C₄	—	2	4	no	—
48	C₄×C₅×Dic₃	D₄	1	2	yes	—
49	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₅×D₃×C₈	Q₈	1	8	yes	complemented
52	D₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
53	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
54	#5080	C₂×C₅	1	2	yes	complemented
55	D₃×#106	C₂×C₅	1	4	yes	—
56	#4855	C₂×C₅	1	4	yes	complemented
57	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
58	#4680	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂²×C₅×Q₈	—	6	2	no	—
60	C₂²×C₅×C₈	—	3	2	no	—
61	C₅×C₄⋊Q₈	—	3	4	no	—
62	C₅×C₄⋊Q₈	—	3	2	no	—
63	C₅×C₄⋊Q₈	—	6	2	no	—
64	C₅×C₄⋊Q₈	—	6	2	no	—
65	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
66	C₂×C₅×C₄⋊C₄	—	3	4	no	—
67	C₂×C₅×C₄⋊C₄	—	6	2	no	—
68	C₂×C₄²×C₅	—	3	1	no	—
69	C₅×#106	—	3	4	no	—
70	C₅×#106	—	3	8	no	—
71	C₅×#106	C₂×D₃	1	4	yes	—
72	C₄×C₅×C₈	—	3	2	no	—
73	C₄×C₅×C₈	—	3	1	no	—
74	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
75	C₂×#498	—	3	1	no	Sylow
76	C₂²×C₅×D₃	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×C₅×Dic₃	—	2	4	no	—
82	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
84	C₄×C₅×D₃	—	4	8	no	—
85	C₄×C₅×D₃	SD₁₆	1	8	yes	—
86	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
87	C₅×Dic₆	—	4	4	no	—
88	C₅×Dic₆	—	4	8	no	—
89	C₃×C₅×C₈	C₂×Q₈	1	4	yes	complemented
90	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	complemented
91	C₂×D₃×Q₈	—	2	2	no	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
93	#944	—	2	2	no	—
94	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
95	D₃×C₄⋊C₄	—	2	2	no	—
96	D₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
97	#888	—	2	2	no	—
98	#888	C₂²×C₅	1	4	yes	—
99	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
100	#869	C₂²×C₅	1	2	yes	—
101	C₃×#106	C₂²×C₅	1	4	yes	—
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#817	C₂²×C₅	1	4	yes	—
104	C₈×Dic₃	C₂²×C₅	1	2	yes	—
105	#808	C₂²×C₅	1	8	yes	—
106	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	—	6	4	no	—
109	C₂×C₅×Q₈	—	6	2	no	—
110	C₂×C₅×Q₈	—	12	8	no	—
111	C₂²×C₄×C₅	—	3	2	no	—
112	C₂²×C₄×C₅	—	3	1	no	—
113	C₂²×C₄×C₅	—	6	4	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	3	8	no	—
116	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	3	4	no	—
119	C₅×C₄⋊C₄	—	6	4	no	—
120	C₅×C₄⋊C₄	—	6	4	no	—
121	C₅×C₄⋊C₄	—	6	2	no	—
122	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
123	C₄²×C₅	—	3	2	no	—
124	C₄²×C₅	—	3	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	3	2	no	—
127	#498	—	3	2	no	—
128	#498	—	3	4	no	—
129	#498	—	3	1	no	—
130	#498	C₅×D₃	1	1	yes	characteristic, direct product factor
131	C₂×#106	—	3	4	no	—
132	C₂×C₄×C₈	—	3	1	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
135	C₂×C₅×D₃	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	C₅×Dic₃	—	4	4	no	—
140	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
141	C₅×Dic₃	C₂×SD₁₆	1	4	yes	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	D₃×Q₈	—	4	8	no	—
144	C₂×C₄×D₃	—	2	4	no	—
145	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
146	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
147	C₂×Dic₆	—	2	2	no	—
148	C₂×Dic₆	—	2	4	no	—
149	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
152	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₄⋊Dic₃	—	2	2	no	—
154	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
155	Dic₃⋊C₄	—	2	4	no	—
156	Dic₃⋊C₄	—	2	4	no	—
157	C₄×Dic₃	C₅×D₄	1	2	yes	—
158	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
160	D₃×C₈	C₅×Q₈	1	8	yes	complemented
161	C₂³×C₅	—	3	1	no	—
162	C₅×Q₈	—	4	4	no	—
163	C₅×Q₈	—	12	8	no	—
164	C₅×Q₈	—	12	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	3	1	no	—
167	C₂×C₄×C₅	—	3	2	no	—
168	C₂×C₄×C₅	—	3	8	no	—
169	C₂×C₄×C₅	—	3	4	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	—	12	8	no	—
172	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₅×C₈	—	3	4	no	—
175	C₅×C₈	D₃×Q₈	1	4	yes	complemented
176	C₂²×Q₈	—	6	2	no	—
177	C₂²×C₈	—	3	2	no	—
178	C₄⋊Q₈	—	3	4	no	—
179	C₄⋊Q₈	—	3	2	no	—
180	C₄⋊Q₈	—	6	2	no	—
181	C₄⋊Q₈	—	6	2	no	—
182	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
183	C₂×C₄⋊C₄	—	3	4	no	—
184	C₂×C₄⋊C₄	—	6	2	no	—
185	C₂×C₄²	—	3	1	no	—
186	#106	—	3	4	no	—
187	#106	—	3	8	no	—
188	#106	C₂×C₅×D₃	1	4	yes	—
189	C₄×C₈	—	3	2	no	—
190	C₄×C₈	—	3	1	no	—
191	C₄×C₈	C₂×C₅×D₃	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	C₅×D₃	#498	1	4	yes	direct product factor
195	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₃×Q₈	—	4	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
200	C₂×Dic₃	—	2	4	no	—
201	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
203	C₄×D₃	—	4	8	no	—
204	C₄×D₃	C₅×SD₁₆	1	8	yes	—
205	C₄×D₃	C₂×C₅×Q₈	1	4	yes	—
206	Dic₆	—	4	4	no	—
207	Dic₆	—	4	8	no	—
208	C₃×C₈	C₂×C₅×Q₈	1	4	yes	complemented
209	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	complemented
210	C₂²×C₅	—	3	2	no	—
211	C₂²×C₅	—	3	4	no	—
212	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
213	C₄×C₅	—	3	2	no	—
214	C₄×C₅	—	3	4	no	—
215	C₄×C₅	—	4	4	no	—
216	C₄×C₅	—	12	4	no	—
217	C₄×C₅	D₃×SD₁₆	1	4	yes	—
218	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	6	4	no	—
221	C₂×Q₈	—	6	2	no	—
222	C₂×Q₈	—	12	8	no	—
223	C₂²×C₄	—	3	2	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	6	4	no	—
226	C₂×C₈	—	3	2	no	—
227	C₂×C₈	—	3	8	no	—
228	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	3	4	no	—
231	C₄⋊C₄	—	6	4	no	—
232	C₄⋊C₄	—	6	4	no	—
233	C₄⋊C₄	—	6	2	no	—
234	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
235	C₄²	—	3	2	no	—
236	C₄²	—	3	1	no	—
237	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
238	C₃×C₅	C₂×#498	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
240	C₂×D₃	C₅×C₄⋊Q₈	1	2	yes	—
241	C₂×D₃	C₂×C₅×SD₁₆	1	4	yes	—
242	C₃×C₄	—	4	4	no	—
243	C₃×C₄	C₂×C₅×SD₁₆	1	4	yes	—
244	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
245	Dic₃	—	4	4	no	—
246	Dic₃	C₅×C₄⋊Q₈	1	2	yes	—
247	Dic₃	C₂×C₅×SD₁₆	1	4	yes	—
248	C₂×C₅	—	3	4	no	—
249	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
250	C₂×C₅	C₂×D₃×SD₁₆	1	2	yes	—
251	C₂³	—	3	1	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	12	8	no	—
254	Q₈	—	12	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	8	no	—
259	C₂×C₄	—	3	4	no	—
260	C₂×C₄	—	6	4	no	—
261	C₂×C₄	—	12	8	no	—
262	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
263	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
264	C₈	—	3	4	no	—
265	C₈	C₅×D₃×Q₈	1	4	yes	complemented
266	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
267	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
268	D₃	C₅×#498	1	4	yes	direct product factor
269	C₅	D₃×#498	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	3	2	no	—
271	C₂²	—	3	4	no	—
272	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
273	C₄	—	3	2	no	—
274	C₄	—	3	4	no	—
275	C₄	—	4	4	no	—
276	C₄	—	12	4	no	—
277	C₄	C₅×D₃×SD₁₆	1	4	yes	—
278	C₄	C₂×C₅×D₃×Q₈	1	2	yes	—
279	C₃	C₂×C₅×#498	1	1	yes	characteristic, complemented, Sylow
280	C₂	—	3	4	no	—
281	C₂	C₅×D₃×C₄⋊Q₈	1	1	yes	characteristic
282	C₂	C₂×C₅×D₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#498	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#498	D₃
hypocenter	C₃	C₂×C₅×#498
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#498

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#498	C₂	semidirect
3. C₅×#5080	C₂	semidirect
5. C₅×#4855	C₂	semidirect
7. C₅×#4680	C₂	semidirect
25. D₃×#498	C₅	direct (N × H)
30. C₅×#498	D₃	direct (N × H)
51. C₅×D₃×C₈	Q₈	semidirect
53. C₃×#498	C₂×C₅	semidirect
54. #5080	C₂×C₅	semidirect
56. #4855	C₂×C₅	semidirect
58. #4680	C₂×C₅	semidirect
89. C₃×C₅×C₈	C₂×Q₈	semidirect
90. C₅×C₃⋊C₈	C₂×Q₈	semidirect
130. #498	C₅×D₃	direct (N × H)
160. D₃×C₈	C₅×Q₈	semidirect
175. C₅×C₈	D₃×Q₈	semidirect
194. C₅×D₃	#498	direct (N × H)
208. C₃×C₈	C₂×C₅×Q₈	semidirect
209. C₃⋊C₈	C₂×C₅×Q₈	semidirect
238. C₃×C₅	C₂×#498	semidirect
265. C₈	C₅×D₃×Q₈	semidirect
268. D₃	C₅×#498	direct (N × H)
269. C₅	D₃×#498	direct (N × H)
279. C₃	C₂×C₅×#498	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×C₄⋊Q₈	2
C₃×C₅×#498	2
C₅×#5080	2
C₅×D₃×#106	2
C₅×#4855	2
C₄×C₅×D₃×C₈	2
C₅×#4680	2
C₂×C₅×#498	3
D₃×#498	5

Subgroups — C5×D3×#498 #14439729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×#498 #14439729