Subgroups — D₅×#4457 #14285494

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	D₅×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745864	C₂	1	2	yes	maximal, complemented
6	D₅×#334	—	3	1	no	maximal, Hall
7	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	C₂×D₅×C₃⋊C₈	—	2	2	no	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
13	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
16	C₃×C₄×#168	C₂²	1	1	yes	characteristic
17	#90982	—	2	2	no	—
18	C₂×#4457	—	5	1	no	maximal, Hall
19	D₅×C₄⋊C₈	—	3	2	no	—
20	C₄×C₈×D₅	D₃	1	1	yes	characteristic
21	C₅×#334	—	3	1	no	—
22	#64700	—	3	2	no	—
23	#64677	—	3	1	no	—
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊#168	—	2	2	no	—
31	C₂×C₅×C₃⋊C₈	—	2	2	no	—
32	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
34	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×D₅	—	2	2	no	—
38	C₃×C₈×D₅	C₈	1	4	yes	complemented
39	D₅×C₃⋊C₈	—	4	4	no	—
40	C₂×C₃×C₄×C₈	—	5	1	no	—
41	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
42	#4457	—	5	1	no	—
43	#4457	—	5	2	no	—
44	#4457	D₅	1	1	yes	characteristic, direct product factor
45	C₂×C₈×D₅	—	6	2	no	—
46	C₂×C₈×D₅	Dic₃	1	1	yes	characteristic
47	C₂×C₈×D₅	Dic₃	1	1	yes	characteristic
48	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	3	2	no	—
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#3946	—	6	2	no	—
52	C₈×Dic₅	Dic₃	1	1	yes	characteristic
53	C₈×Dic₅	Dic₃	1	1	yes	characteristic
54	C₄⋊#168	—	3	2	no	—
55	C₄×#168	C₂×D₃	1	1	yes	characteristic
56	C₂×#334	—	15	1	no	Sylow
57	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₄×D₅	—	2	2	no	—
65	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
66	C₃×C₄×D₅	M₁₆	1	2	yes	—
67	C₃×C₄×D₅	SD₁₆	1	4	yes	—
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
70	C₃⋊#168	—	4	2	no	—
71	C₃×#168	—	2	1	no	—
72	C₃×#168	C₂×C₈	1	2	yes	complemented
73	C₅×C₃⋊C₈	—	4	2	no	—
74	C₂²×C₃×C₈	—	5	1	no	—
75	C₂²×C₃×C₈	—	5	1	no	—
76	C₂×C₃×C₄²	—	5	1	no	—
77	C₂²×C₃⋊C₈	—	10	2	no	—
78	C₃×C₄×C₈	—	5	1	no	—
79	C₃×C₄×C₈	—	5	1	no	—
80	C₃×C₄×C₈	—	5	1	no	—
81	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	C₄⋊(C₃⋊C₈)	—	5	2	no	—
83	C₄⋊(C₃⋊C₈)	—	10	2	no	—
84	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
85	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
86	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	6	2	no	—
89	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
91	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
92	C₄×Dic₅	Dic₆	1	1	yes	characteristic
93	C₄×Dic₅	D₁₂	1	1	yes	characteristic
94	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
95	C₂×#168	—	6	2	no	—
96	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
97	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
98	C₈×D₅	—	2	2	no	—
99	C₈×D₅	—	12	4	no	—
100	C₈×D₅	C₃⋊C₈	1	4	yes	complemented
101	C₂×C₄⋊C₈	—	15	2	no	—
102	C₂×C₄×C₈	—	5	1	no	—
103	#334	—	3	1	no	—
104	#334	—	15	2	no	—
105	#334	—	15	1	no	—
106	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
108	C₂×C₃×D₅	#106	1	2	yes	—
109	C₂×C₃×D₅	#108	1	2	yes	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
114	C₃×Dic₅	—	2	1	no	—
115	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
116	C₃×Dic₅	#106	1	2	yes	—
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₂²×C₃×C₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂×C₃×C₈	—	5	1	no	—
121	C₂×C₃×C₈	—	5	4	no	—
122	C₂×C₃×C₈	—	5	1	no	—
123	C₂×C₃×C₈	—	10	2	no	—
124	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
126	C₃×C₄²	—	5	1	no	—
127	C₃×C₄²	—	5	1	no	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	10	2	no	—
132	C₂×C₃⋊C₈	—	20	4	no	—
133	C₂²×D₅	C₄⋊Dic₃	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
137	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
138	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
139	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
140	C₄×D₅	—	2	2	no	—
141	C₄×D₅	#204	1	4	yes	—
142	C₄×D₅	C₂×C₃⋊C₈	1	2	yes	—
143	C₄×D₅	#208	1	2	yes	—
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	12	2	no	—
146	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
147	#168	—	2	1	no	—
148	#168	—	12	2	no	—
149	#168	C₂×C₃⋊C₈	1	2	yes	complemented
150	C₂²×C₈	—	5	1	no	—
151	C₂²×C₈	—	5	1	no	—
152	C₂²×C₈	—	30	2	no	—
153	C₂×C₄²	—	5	1	no	—
154	C₄⋊C₈	—	3	2	no	—
155	C₄⋊C₈	—	15	2	no	—
156	C₄⋊C₈	—	30	2	no	—
157	C₄×C₈	—	5	1	no	—
158	C₄×C₈	—	5	1	no	—
159	C₄×C₈	—	5	1	no	—
160	C₄×C₈	D₃×D₅	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
164	C₃×D₅	#334	1	4	yes	complemented
165	C₂³×C₃	—	5	1	no	—
166	C₂×C₃×C₄	—	5	1	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	5	2	no	—
170	C₂×C₃×C₄	—	5	4	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	10	2	no	—
173	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	5	2	no	—
178	C₃×C₈	—	10	1	no	—
179	C₃×C₈	C₈×D₅	1	2	yes	complemented
180	C₃⋊C₈	—	4	2	no	—
181	C₃⋊C₈	—	20	2	no	—
182	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
183	C₂×D₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
184	C₂×D₅	#817	1	2	yes	—
185	C₂×D₅	#819	1	2	yes	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	C₂×#204	1	2	yes	—
188	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
189	C₄×C₅	C₂×#208	1	1	yes	characteristic
190	Dic₅	—	2	1	no	—
191	Dic₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
192	Dic₅	#817	1	2	yes	—
193	C₂²×C₄	—	5	1	no	—
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂×C₈	—	5	1	no	—
197	C₂×C₈	—	5	4	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	—	10	2	no	—
201	C₂×C₈	—	30	2	no	—
202	C₂×C₈	—	60	4	no	—
203	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
204	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
205	C₄²	—	5	1	no	—
206	C₄²	—	5	1	no	—
207	C₄²	—	5	1	no	—
208	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
209	C₃×C₅	C₂×#334	1	1	yes	characteristic, complemented, Hall
210	C₂²×C₃	—	5	2	no	—
211	C₂²×C₃	—	5	2	no	—
212	C₂²×C₃	—	5	2	no	—
213	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	5	2	no	—
216	C₃×C₄	—	5	2	no	—
217	C₃×C₄	—	10	1	no	—
218	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
219	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
220	C₃×C₄	D₅×SD₁₆	1	2	yes	—
221	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
222	C₂×C₅	C₂×#817	1	1	yes	characteristic
223	C₂×C₅	C₂×#819	1	1	yes	characteristic
224	D₅	#4457	1	4	yes	direct product factor
225	C₂³	—	5	1	no	—
226	C₂×C₄	—	5	1	no	—
227	C₂×C₄	—	5	1	no	—
228	C₂×C₄	—	5	4	no	—
229	C₂×C₄	—	5	2	no	—
230	C₂×C₄	—	5	2	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
234	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
235	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	5	2	no	—
238	C₈	—	10	1	no	—
239	C₈	—	12	2	no	—
240	C₈	—	60	2	no	—
241	C₈	D₅×C₃⋊C₈	1	2	yes	complemented
242	C₂×C₃	—	5	4	no	—
243	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
244	C₂×C₃	D₅×#106	1	1	yes	characteristic
245	C₂×C₃	D₅×#108	1	1	yes	characteristic
246	C₅	C₂×#4457	1	1	yes	characteristic, complemented, Sylow
247	C₂²	—	5	2	no	—
248	C₂²	—	5	2	no	—
249	C₂²	—	5	2	no	—
250	C₂²	D₅×C₄⋊Dic₃	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	5	2	no	—
253	C₄	—	5	2	no	—
254	C₄	—	10	1	no	—
255	C₄	D₅×#204	1	2	yes	—
256	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
257	C₄	D₅×#208	1	1	yes	characteristic
258	C₃	D₅×#334	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	5	4	no	—
260	C₂	D₅×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
261	C₂	D₅×#817	1	1	yes	characteristic
262	C₂	D₅×#819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#334
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746278	C₂	semidirect
4. C₅×#4457	C₂	semidirect
5. #11745864	C₂	semidirect
38. C₃×C₈×D₅	C₈	semidirect
44. #4457	D₅	direct (N × H)
69. C₃×C₅×C₈	C₂×C₈	semidirect
72. C₃×#168	C₂×C₈	semidirect
100. C₈×D₅	C₃⋊C₈	semidirect
146. C₅×C₈	C₂×C₃⋊C₈	semidirect
149. #168	C₂×C₃⋊C₈	semidirect
164. C₃×D₅	#334	semidirect
179. C₃×C₈	C₈×D₅	semidirect
209. C₃×C₅	C₂×#334	semidirect
224. D₅	#4457	direct (N × H)
241. C₈	D₅×C₃⋊C₈	semidirect
246. C₅	C₂×#4457	semidirect
258. C₃	D₅×#334	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₈×D₅	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746278	2
C₅×#4457	2
#11745864	2
D₅×#334	3
C₂×#4457	5

Subgroups — D5×#4457 #14285494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4457 #14285494