Subgroups — C₂×D₅×D₄₈ #14478207

Order 1920.

880 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×D₂₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90939	C₂	1	2	yes	maximal, complemented
6	D₅×D₄₈	C₂	1	8	yes	maximal, direct product factor
7	C₂×D₅×D₁₆	—	3	1	no	maximal, Hall
8	C₂×D₅×D₁₂	—	2	2	no	—
9	C₂×D₁₂₀	C₂²	1	2	yes	complemented
10	C₂×C₅×D₂₄	C₂²	1	2	yes	complemented
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₂×#6654	—	2	2	no	—
13	D₅×D₂₄	—	2	4	no	—
14	D₅×D₂₄	C₂²	1	8	yes	complemented
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
16	D₂₄₀	C₂²	1	4	yes	complemented
17	C₅×D₄₈	C₂²	1	4	yes	complemented
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic, complemented
19	C₃×D₅×C₁₆	C₂²	1	4	yes	complemented
20	#90939	C₂²	1	8	yes	complemented
21	C₂²×D₄₈	—	5	1	no	maximal, Hall
22	C₂×D₅×D₈	—	3	2	no	—
23	C₂×C₅×D₁₆	—	3	1	no	—
24	C₂×#3958	—	3	2	no	—
25	D₅×D₁₆	—	3	8	no	—
26	C₂×D₈₀	—	3	1	no	—
27	C₂×D₅×C₁₆	D₃	1	1	yes	characteristic, complemented
28	C₂²×D₃×D₅	—	4	2	no	—
29	C₂×D₆₀	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
32	C₂×#1203	—	4	2	no	—
33	D₅×D₁₂	—	2	8	no	—
34	D₅×D₁₂	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	D₁₂₀	—	2	2	no	—
37	D₁₂₀	C₂³	1	4	yes	complemented
38	C₅×D₂₄	—	2	2	no	—
39	C₅×D₂₄	C₂³	1	4	yes	complemented
40	C₂×C₃×#168	D₄	1	1	yes	characteristic
41	C₃×C₈×D₅	D₄	1	2	yes	—
42	C₃×C₈×D₅	C₂³	1	2	yes	—
43	#6654	—	2	8	no	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	complemented
45	C₃×#672	C₂³	1	2	yes	complemented
46	C₂²×D₂₄	—	5	2	no	—
47	C₂²×C₃×C₁₆	—	5	1	no	—
48	C₂×D₄₈	—	5	2	no	—
49	C₂×D₄₈	—	5	1	no	—
50	C₂×D₄₈	—	5	8	no	—
51	C₂×D₄₈	D₅	1	1	yes	characteristic, direct product factor
52	C₂×D₄×D₅	—	6	2	no	—
53	C₂×C₅×D₈	—	3	2	no	—
54	C₂×#686	—	6	2	no	—
55	D₅×D₈	—	3	8	no	—
56	D₅×D₈	—	6	4	no	—
57	C₂×D₄₀	—	3	2	no	—
58	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
59	C₅×D₁₆	—	3	4	no	—
60	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
61	#3958	—	3	8	no	—
62	C₂×#672	C₂×D₃	1	1	yes	characteristic, complemented
63	D₈₀	—	3	4	no	—
64	D₅×C₁₆	C₂×D₃	1	4	yes	complemented
65	C₂²×D₁₆	—	15	1	no	Sylow
66	C₂²×D₁₅	—	4	2	no	—
67	C₂²×C₅×D₃	—	4	2	no	—
68	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
69	C₂×D₃×D₅	—	4	8	no	—
70	C₂×D₃×D₅	—	8	8	no	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	D₆₀	—	2	4	no	—
73	D₆₀	—	4	2	no	—
74	C₅×D₁₂	—	2	4	no	—
75	C₅×D₁₂	—	4	2	no	—
76	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
77	C₃×C₄×D₅	D₈	1	2	yes	—
78	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
79	#1203	—	4	8	no	—
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
82	C₃×#168	C₂×D₄	1	2	yes	—
83	C₂²×D₁₂	—	10	2	no	—
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₂×D₂₄	—	5	2	no	—
86	C₂×D₂₄	—	5	8	no	—
87	C₂×D₂₄	—	10	4	no	—
88	C₂×D₂₄	—	10	2	no	—
89	C₂×D₂₄	C₂×D₅	1	2	yes	complemented
90	C₂×C₃×C₁₆	—	5	1	no	—
91	C₂×C₃×C₁₆	—	5	4	no	—
92	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
93	D₄₈	—	5	8	no	—
94	D₄₈	—	5	4	no	—
95	D₄₈	C₂×D₅	1	4	yes	direct product factor
96	C₂³×D₅	—	12	2	no	—
97	C₂×C₅×D₄	—	6	2	no	—
98	C₂×C₂²⋊D₅	—	12	2	no	—
99	D₄×D₅	—	6	8	no	—
100	D₄×D₅	—	12	4	no	—
101	C₂×D₂₀	—	6	2	no	—
102	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
103	C₅×D₈	—	3	4	no	—
104	C₅×D₈	—	6	2	no	—
105	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
106	#686	—	6	8	no	—
107	C₂×#168	D₁₂	1	1	yes	characteristic
108	D₄₀	—	3	4	no	—
109	D₄₀	—	6	2	no	—
110	C₈×D₅	D₁₂	1	2	yes	—
111	C₈×D₅	C₂²×D₃	1	2	yes	—
112	C₅×C₁₆	C₂²×D₃	1	2	yes	complemented
113	#672	C₂²×D₃	1	2	yes	complemented
114	C₂²×D₈	—	15	2	no	—
115	C₂×D₁₆	—	3	1	no	—
116	C₂×D₁₆	—	15	8	no	—
117	C₂×D₁₆	—	15	2	no	—
118	C₂×D₁₆	—	15	1	no	—
119	C₂²×C₁₆	—	5	1	no	—
120	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×D₁₅	—	4	4	no	—
122	C₂×D₁₅	—	8	4	no	—
123	C₂×C₅×D₃	—	4	4	no	—
124	C₂×C₅×D₃	—	8	4	no	—
125	C₂×C₃×D₅	D₁₆	1	4	yes	complemented
126	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
127	D₃×D₅	—	8	16	no	—
128	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₃×Dic₅	C₂×D₈	1	2	yes	—
131	C₂³×D₃	—	20	2	no	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₂×D₁₂	—	2	2	no	—
134	C₂×D₁₂	—	10	8	no	—
135	C₂×D₁₂	—	10	2	no	—
136	C₂×D₁₂	—	20	4	no	—
137	C₂×D₁₂	—	20	2	no	—
138	C₂×C₃×C₈	—	5	1	no	—
139	C₂×C₃×C₈	—	5	2	no	—
140	C₂×C₃×C₈	—	5	2	no	—
141	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
142	D₂₄	—	2	2	no	—
143	D₂₄	—	5	4	no	—
144	D₂₄	—	10	8	no	—
145	D₂₄	—	10	2	no	—
146	D₂₄	C₂²×D₅	1	4	yes	complemented
147	C₃×C₁₆	—	5	2	no	—
148	C₃×C₁₆	C₂²×D₅	1	2	yes	complemented
149	C₂³×C₅	—	12	2	no	—
150	C₂²×D₅	—	12	2	no	—
151	C₂²×D₅	—	12	8	no	—
152	C₂²×D₅	—	24	8	no	—
153	C₂²×D₅	D₂₄	1	1	yes	characteristic
154	C₅×D₄	—	6	4	no	—
155	C₅×D₄	—	12	2	no	—
156	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
157	C₂²⋊D₅	—	12	8	no	—
158	C₂×Dic₅	D₂₄	1	1	yes	characteristic
159	D₂₀	—	6	4	no	—
160	D₂₀	—	12	2	no	—
161	C₄×D₅	D₂₄	1	2	yes	—
162	C₄×D₅	C₂×D₁₂	1	2	yes	—
163	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
164	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
165	#168	C₂×D₁₂	1	2	yes	—
166	C₂²×D₄	—	30	2	no	—
167	C₂×D₈	—	3	2	no	—
168	C₂×D₈	—	15	8	no	—
169	C₂×D₈	—	15	2	no	—
170	C₂×D₈	—	30	4	no	—
171	C₂×D₈	—	30	2	no	—
172	C₂²×C₈	—	5	1	no	—
173	D₁₆	—	3	4	no	—
174	D₁₆	—	15	8	no	—
175	D₁₆	—	15	4	no	—
176	C₂×C₁₆	—	5	1	no	—
177	C₂×C₁₆	—	5	4	no	—
178	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic, complemented
179	C₂×C₃×C₅	C₂×D₁₆	1	2	yes	complemented
180	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
181	D₁₅	—	8	4	no	—
182	C₃×D₅	C₂×D₁₆	1	4	yes	complemented
183	C₅×D₃	—	8	4	no	—
184	C₂³×C₃	—	5	1	no	—
185	C₂²×D₃	—	4	2	no	—
186	C₂²×D₃	—	20	8	no	—
187	C₂²×D₃	—	20	2	no	—
188	C₂²×D₃	—	40	8	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	2	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
193	D₁₂	—	2	4	no	—
194	D₁₂	—	4	2	no	—
195	D₁₂	—	10	4	no	—
196	D₁₂	—	20	8	no	—
197	D₁₂	—	20	2	no	—
198	C₃×C₈	—	5	2	no	—
199	C₃×C₈	D₄×D₅	1	1	yes	characteristic
200	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
201	C₂²×C₅	—	12	4	no	—
202	C₂²×C₅	—	24	4	no	—
203	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
204	C₂×D₅	—	12	4	no	—
205	C₂×D₅	—	24	4	no	—
206	C₂×D₅	—	24	16	no	—
207	C₂×D₅	D₄₈	1	4	yes	direct product factor
208	C₂×D₅	C₂×D₂₄	1	2	yes	—
209	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
210	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
211	Dic₅	C₂×D₂₄	1	2	yes	—
212	C₂⁴	—	60	2	no	—
213	C₂×D₄	—	6	2	no	—
214	C₂×D₄	—	30	8	no	—
215	C₂×D₄	—	30	2	no	—
216	C₂×D₄	—	60	4	no	—
217	C₂×D₄	—	60	2	no	—
218	C₂²×C₄	—	5	1	no	—
219	D₈	—	3	4	no	—
220	D₈	—	6	2	no	—
221	D₈	—	15	4	no	—
222	D₈	—	30	8	no	—
223	D₈	—	30	2	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
228	C₁₆	—	5	2	no	—
229	C₁₆	C₂×D₃×D₅	1	2	yes	complemented
230	C₃×C₅	C₂²×D₁₆	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	5	2	no	—
232	C₂²×C₃	—	5	4	no	—
233	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
234	C₂×D₃	—	4	4	no	—
235	C₂×D₃	—	8	4	no	—
236	C₂×D₃	—	20	4	no	—
237	C₂×D₃	—	40	16	no	—
238	C₂×D₃	—	40	4	no	—
239	C₃×C₄	—	5	2	no	—
240	C₃×C₄	D₅×D₈	1	1	yes	characteristic
241	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
242	C₂×C₅	—	24	4	no	—
243	C₂×C₅	C₂×D₄₈	1	2	yes	complemented
244	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
245	D₅	—	24	4	no	—
246	D₅	C₂×D₄₈	1	4	yes	direct product factor
247	C₂³	—	5	1	no	—
248	C₂³	—	12	2	no	—
249	C₂³	—	60	8	no	—
250	C₂³	—	60	2	no	—
251	C₂³	—	120	8	no	—
252	D₄	—	6	4	no	—
253	D₄	—	12	2	no	—
254	D₄	—	30	4	no	—
255	D₄	—	60	8	no	—
256	D₄	—	60	2	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
261	C₈	—	5	2	no	—
262	C₈	D₅×D₁₂	1	1	yes	characteristic
263	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₂×C₃	—	5	4	no	—
265	C₂×C₃	D₅×D₁₆	1	2	yes	complemented
266	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
267	D₃	—	8	4	no	—
268	D₃	—	40	4	no	—
269	C₅	C₂²×D₄₈	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	5	2	no	—
271	C₂²	—	5	4	no	—
272	C₂²	—	12	4	no	—
273	C₂²	—	24	4	no	—
274	C₂²	—	60	4	no	—
275	C₂²	—	120	16	no	—
276	C₂²	—	120	4	no	—
277	C₂²	D₅×D₂₄	1	1	yes	characteristic
278	C₄	—	5	2	no	—
279	C₄	D₅×D₂₄	1	1	yes	characteristic
280	C₄	C₂×D₅×D₁₂	1	1	yes	characteristic
281	C₃	C₂×D₅×D₁₆	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	5	4	no	—
283	C₂	—	24	4	no	—
284	C₂	—	120	4	no	—
285	C₂	D₅×D₄₈	1	2	yes	direct product factor
286	C₂	C₂×D₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×D₂₄
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×C₁₆	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	C₂²×D₁₆
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₅×D₂₄	C₂	semidirect
2. C₂×D₂₄₀	C₂	semidirect
3. C₂×C₅×D₄₈	C₂	semidirect
4. C₂×C₃×D₅×C₁₆	C₂	semidirect
5. C₂×#90939	C₂	semidirect
6. D₅×D₄₈	C₂	direct (N × H)
9. C₂×D₁₂₀	C₂²	semidirect
10. C₂×C₅×D₂₄	C₂²	semidirect
14. D₅×D₂₄	C₂²	semidirect
15. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. D₂₄₀	C₂²	semidirect
17. C₅×D₄₈	C₂²	semidirect
18. C₂×C₃×#672	C₂²	semidirect
19. C₃×D₅×C₁₆	C₂²	semidirect
20. #90939	C₂²	semidirect
27. C₂×D₅×C₁₆	D₃	semidirect
37. D₁₂₀	C₂³	semidirect
39. C₅×D₂₄	C₂³	semidirect
44. C₃×C₅×C₁₆	C₂³	semidirect
45. C₃×#672	C₂³	semidirect
51. C₂×D₄₈	D₅	direct (N × H)
60. C₂×C₅×C₁₆	C₂×D₃	semidirect
62. C₂×#672	C₂×D₃	semidirect
64. D₅×C₁₆	C₂×D₃	semidirect
89. C₂×D₂₄	C₂×D₅	semidirect
92. C₂×C₃×C₁₆	C₂×D₅	semidirect
95. D₄₈	C₂×D₅	direct (N × H)
112. C₅×C₁₆	C₂²×D₃	semidirect
113. #672	C₂²×D₃	semidirect
125. C₂×C₃×D₅	D₁₆	semidirect
146. D₂₄	C₂²×D₅	semidirect
148. C₃×C₁₆	C₂²×D₅	semidirect
178. C₂×C₁₆	D₃×D₅	semidirect
179. C₂×C₃×C₅	C₂×D₁₆	semidirect
182. C₃×D₅	C₂×D₁₆	semidirect
207. C₂×D₅	D₄₈	direct (N × H)
229. C₁₆	C₂×D₃×D₅	semidirect
230. C₃×C₅	C₂²×D₁₆	semidirect
243. C₂×C₅	C₂×D₄₈	semidirect
246. D₅	C₂×D₄₈	direct (N × H)
265. C₂×C₃	D₅×D₁₆	semidirect
269. C₅	C₂²×D₄₈	semidirect
281. C₃	C₂×D₅×D₁₆	semidirect
285. C₂	D₅×D₄₈	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×D₅×D₂₄	2
C₂×D₂₄₀	2
C₂×C₅×D₄₈	2
C₂×C₃×D₅×C₁₆	2
C₂×#90939	2
D₅×D₄₈	2
C₂×D₅×D₁₆	3
C₂²×D₄₈	5

Subgroups — C2×D5×D48 #14478207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×D₄₈ #14478207