Subgroups — C₂×C₅×#4869 #14416477

Order 1920.

1216 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#893	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#815	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#809	C₂	1	2	yes	maximal, complemented
6	C₅×#4869	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#442	—	3	1	no	maximal, Hall
8	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
9	C₂²×C₄×C₅×D₃	—	2	2	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₂₄	C₄	1	8	yes	complemented
16	C₂×C₅×D₂₄	C₂²	1	4	yes	—
17	C₅×#893	—	2	4	no	—
18	C₅×#893	C₂²	1	8	yes	complemented
19	C₃×C₅×#106	C₂²	1	4	yes	complemented
20	C₅×#815	C₂²	1	4	yes	complemented
21	C₅×#809	C₂²	1	8	yes	complemented
22	C₂×#4869	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₅×D₈	—	3	1	no	—
24	C₂×C₄×C₅×D₄	—	3	2	no	—
25	C₅×#442	—	3	8	no	—
26	C₂×C₅×#106	D₃	1	1	yes	characteristic, complemented
27	C₂×C₅×#102	—	3	2	no	—
28	C₂×C₅×#97	—	3	1	no	—
29	C₂³×C₅×D₃	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	D₄	1	2	yes	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	C₂×C₄	1	8	yes	complemented
36	C₂×C₅×D₁₂	C₂³	1	4	yes	—
37	C₂×C₄×C₅×D₃	—	2	8	no	—
38	C₂×C₄×C₅×D₃	—	4	8	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
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	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
44	C₄×C₅×Dic₃	D₄	1	2	yes	—
45	C₄×C₅×Dic₃	C₂³	1	2	yes	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
48	C₅×D₂₄	C₂×C₄	1	16	yes	complemented
49	C₂²×D₂₄	C₂×C₅	1	1	yes	characteristic
50	C₂×#893	C₂×C₅	1	2	yes	complemented
51	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#815	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×#809	C₂×C₅	1	2	yes	complemented
54	#4869	C₂×C₅	1	8	yes	direct product factor
55	C₂²×C₅×D₄	—	3	2	no	—
56	C₂³×C₄×C₅	—	6	2	no	—
57	C₂×C₅×D₈	—	3	8	no	—
58	C₂×C₅×D₈	—	3	4	no	—
59	C₂²×C₅×C₈	—	3	1	no	—
60	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₅×D₄	—	3	8	no	—
62	C₄×C₅×D₄	—	6	4	no	—
63	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
64	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
65	C₂×C₄²×C₅	—	3	1	no	—
66	C₅×#106	C₂×D₃	1	4	yes	complemented
67	C₅×#102	—	3	8	no	—
68	C₅×#97	—	3	4	no	—
69	C₂×#442	—	3	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₅×D₃	—	2	8	no	—
72	C₂²×C₅×D₃	—	2	4	no	—
73	C₂²×C₅×D₃	—	4	4	no	—
74	C₂²×C₅×D₃	—	4	4	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	4	4	no	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	#41	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	4	yes	—
83	C₅×D₁₂	—	2	8	no	—
84	C₅×D₁₂	—	2	4	no	—
85	C₅×D₁₂	C₂²×C₄	1	8	yes	complemented
86	C₄×C₅×D₃	—	4	16	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
88	C₅×C₃⋊C₈	—	2	2	no	—
89	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
90	C₂²×C₄×D₃	—	2	2	no	—
91	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
94	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×D₂₄	C₄×C₅	1	8	yes	complemented
97	C₂×D₂₄	C₂²×C₅	1	4	yes	—
98	#893	—	2	4	no	—
99	#893	C₂²×C₅	1	8	yes	complemented
100	C₃×#106	C₂²×C₅	1	4	yes	complemented
101	#815	C₂²×C₅	1	4	yes	complemented
102	#809	C₂²×C₅	1	8	yes	complemented
103	C₂⁴×C₅	—	6	2	no	—
104	C₂×C₅×D₄	—	3	8	no	—
105	C₂×C₅×D₄	—	3	4	no	—
106	C₂×C₅×D₄	—	6	2	no	—
107	C₂×C₅×D₄	—	6	4	no	—
108	C₂²×C₄×C₅	—	2	2	no	—
109	C₂²×C₄×C₅	—	3	2	no	—
110	C₂²×C₄×C₅	—	6	8	no	—
111	C₂²×C₄×C₅	—	12	8	no	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₅×D₈	—	3	16	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	6	2	no	—
116	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
117	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
120	C₅×C₂²⋊C₄	—	6	8	no	—
121	C₄²×C₅	—	3	2	no	—
122	C₄²×C₅	—	3	2	no	—
123	C₂²×D₈	—	3	1	no	—
124	C₂×C₄×D₄	—	3	2	no	—
125	#442	—	3	8	no	—
126	C₂×#106	C₅×D₃	1	1	yes	characteristic, complemented
127	C₂×#102	—	3	2	no	—
128	C₂×#97	—	3	1	no	—
129	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
130	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	2	8	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	4	4	no	—
135	C₂×C₅×D₃	—	4	16	no	—
136	C₃×C₄×C₅	—	4	4	no	—
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
140	C₅×Dic₃	—	2	4	no	—
141	C₂³×D₃	—	2	2	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂²×Dic₃	C₅×D₄	1	2	yes	—
145	C₂×D₁₂	—	2	2	no	—
146	C₂×D₁₂	—	2	4	no	—
147	C₂×D₁₂	C₂×C₄×C₅	1	8	yes	complemented
148	C₂×D₁₂	C₂³×C₅	1	4	yes	—
149	C₂×C₄×D₃	—	2	8	no	—
150	C₂×C₄×D₃	—	4	8	no	—
151	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	complemented
152	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
155	(C₂×D₃)⋊C₄	—	2	8	no	—
156	C₄×Dic₃	C₅×D₄	1	2	yes	—
157	C₄×Dic₃	C₂³×C₅	1	2	yes	—
158	C₂×C₃⋊C₈	—	2	2	no	—
159	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
160	D₂₄	C₂×C₄×C₅	1	16	yes	complemented
161	C₂³×C₅	—	6	8	no	—
162	C₂³×C₅	—	6	4	no	—
163	C₂³×C₅	—	12	4	no	—
164	C₂³×C₅	—	12	4	no	—
165	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
166	C₅×D₄	—	3	8	no	—
167	C₅×D₄	—	6	8	no	—
168	C₅×D₄	—	6	4	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	3	4	no	—
171	C₂×C₄×C₅	—	4	4	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	—	12	16	no	—
174	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
175	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	#239	1	2	yes	—
177	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
178	C₅×C₈	—	6	2	no	—
179	C₅×C₈	C₂×C₄×D₃	1	4	yes	complemented
180	C₂²×D₄	—	3	2	no	—
181	C₂³×C₄	—	6	2	no	—
182	C₂×D₈	—	3	8	no	—
183	C₂×D₈	—	3	4	no	—
184	C₂²×C₈	—	3	1	no	—
185	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
186	C₄×D₄	—	3	8	no	—
187	C₄×D₄	—	6	4	no	—
188	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
189	C₂×C₂²⋊C₄	—	6	2	no	—
190	C₂×C₄²	—	3	1	no	—
191	#106	C₂×C₅×D₃	1	4	yes	complemented
192	#102	—	3	8	no	—
193	#97	—	3	4	no	—
194	C₂×C₃×C₅	#442	1	4	yes	complemented
195	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
197	C₅×D₃	—	4	8	no	—
198	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×D₃	—	2	8	no	—
200	C₂²×D₃	—	2	4	no	—
201	C₂²×D₃	—	4	4	no	—
202	C₂²×D₃	—	4	4	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	4	4	no	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₅×#41	1	2	yes	—
208	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	C₂×Dic₃	—	2	4	no	—
210	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
211	D₁₂	—	2	8	no	—
212	D₁₂	—	2	4	no	—
213	D₁₂	C₂²×C₄×C₅	1	8	yes	complemented
214	C₄×D₃	—	4	16	no	—
215	C₃×C₈	C₂²×C₄×C₅	1	4	yes	complemented
216	C₃⋊C₈	—	2	2	no	—
217	C₂²×C₅	—	6	8	no	—
218	C₂²×C₅	—	12	16	no	—
219	C₂²×C₅	—	12	4	no	—
220	C₂²×C₅	—	12	4	no	—
221	C₂²×C₅	#893	1	2	yes	—
222	C₂²×C₅	#914	1	4	yes	—
223	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	—	6	4	no	—
226	C₄×C₅	#893	1	2	yes	—
227	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
228	C₄×C₅	C₂×#239	1	1	yes	characteristic
229	C₂⁴	—	6	2	no	—
230	C₂×D₄	—	3	8	no	—
231	C₂×D₄	—	3	4	no	—
232	C₂×D₄	—	6	2	no	—
233	C₂×D₄	—	6	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	—	6	8	no	—
237	C₂²×C₄	—	12	8	no	—
238	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
239	D₈	—	3	16	no	—
240	C₂×C₈	—	3	2	no	—
241	C₂×C₈	—	6	2	no	—
242	C₂×C₈	C₄×C₅×D₃	1	4	yes	complemented
243	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
246	C₂²⋊C₄	—	6	8	no	—
247	C₄²	—	3	2	no	—
248	C₄²	—	3	2	no	—
249	C₃×C₅	C₂×#442	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
251	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	4	yes	—
252	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
253	C₂×D₃	—	2	8	no	—
254	C₂×D₃	—	4	16	no	—
255	C₂×D₃	—	4	4	no	—
256	C₂×D₃	—	4	4	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
259	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
260	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
261	Dic₃	—	2	4	no	—
262	C₂×C₅	—	12	8	no	—
263	C₂×C₅	#4869	1	4	yes	direct product factor
264	C₂×C₅	C₂×#893	1	1	yes	characteristic
265	C₂×C₅	C₂×#914	1	2	yes	—
266	C₂³	—	6	8	no	—
267	C₂³	—	6	4	no	—
268	C₂³	—	12	4	no	—
269	C₂³	—	12	4	no	—
270	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
271	D₄	—	3	8	no	—
272	D₄	—	6	8	no	—
273	D₄	—	6	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	3	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	6	4	no	—
278	C₂×C₄	—	12	16	no	—
279	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
280	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
281	C₂×C₄	C₅×#239	1	2	yes	—
282	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
283	C₈	—	6	2	no	—
284	C₈	C₂×C₄×C₅×D₃	1	4	yes	complemented
285	C₂×C₃	C₅×#442	1	4	yes	complemented
286	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
287	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
288	D₃	—	4	8	no	—
289	C₅	C₂×#4869	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	6	8	no	—
291	C₂²	—	12	4	no	—
292	C₂²	—	12	4	no	—
293	C₂²	—	12	16	no	—
294	C₂²	C₅×#893	1	2	yes	—
295	C₂²	C₅×#914	1	4	yes	—
296	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	6	4	no	—
299	C₄	C₅×#893	1	2	yes	—
300	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
301	C₄	C₂×C₅×#239	1	1	yes	characteristic
302	C₃	C₂×C₅×#442	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	12	8	no	—
304	C₂	C₅×#4869	1	4	yes	direct product factor
305	C₂	C₂×C₅×#893	1	1	yes	characteristic
306	C₂	C₂×C₅×#914	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₂×C₅×#106	D₃
hypocenter	C₃	C₂×C₅×#442
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#893	C₂	semidirect
3. C₂×C₃×C₅×#106	C₂	semidirect
4. C₂×C₅×#815	C₂	semidirect
5. C₂×C₅×#809	C₂	semidirect
6. C₅×#4869	C₂	direct (N × H)
15. C₂×C₅×D₂₄	C₄	semidirect
18. C₅×#893	C₂²	semidirect
19. C₃×C₅×#106	C₂²	semidirect
20. C₅×#815	C₂²	semidirect
21. C₅×#809	C₂²	semidirect
22. C₂×#4869	C₅	direct (N × H)
26. C₂×C₅×#106	D₃	semidirect
35. C₂×C₅×D₁₂	C₂×C₄	semidirect
39. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
48. C₅×D₂₄	C₂×C₄	semidirect
50. C₂×#893	C₂×C₅	semidirect
51. C₂×C₃×#106	C₂×C₅	semidirect
52. C₂×#815	C₂×C₅	semidirect
53. C₂×#809	C₂×C₅	semidirect
54. #4869	C₂×C₅	direct (N × H)
66. C₅×#106	C₂×D₃	semidirect
85. C₅×D₁₂	C₂²×C₄	semidirect
87. C₃×C₅×C₈	C₂²×C₄	semidirect
96. C₂×D₂₄	C₄×C₅	semidirect
99. #893	C₂²×C₅	semidirect
100. C₃×#106	C₂²×C₅	semidirect
101. #815	C₂²×C₅	semidirect
102. #809	C₂²×C₅	semidirect
116. C₂×C₅×C₈	C₄×D₃	semidirect
126. C₂×#106	C₅×D₃	semidirect
147. C₂×D₁₂	C₂×C₄×C₅	semidirect
151. C₂×C₃×C₈	C₂×C₄×C₅	semidirect
160. D₂₄	C₂×C₄×C₅	semidirect
179. C₅×C₈	C₂×C₄×D₃	semidirect
191. #106	C₂×C₅×D₃	semidirect
194. C₂×C₃×C₅	#442	semidirect
213. D₁₂	C₂²×C₄×C₅	semidirect
215. C₃×C₈	C₂²×C₄×C₅	semidirect
242. C₂×C₈	C₄×C₅×D₃	semidirect
249. C₃×C₅	C₂×#442	semidirect
263. C₂×C₅	#4869	direct (N × H)
284. C₈	C₂×C₄×C₅×D₃	semidirect
285. C₂×C₃	C₅×#442	semidirect
289. C₅	C₂×#4869	direct (N × H)
302. C₃	C₂×C₅×#442	semidirect
304. C₂	C₅×#4869	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₂₄	2
C₂×C₅×#893	2
C₂×C₃×C₅×#106	2
C₂×C₅×#815	2
C₂×C₅×#809	2
C₅×#4869	2
C₂×C₅×#442	3
C₂×#4869	5

Subgroups — C2×C5×#4869 #14416477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4869 #14416477