Subgroups — C₂×C₅×#4524 #14294274

Order 1920.

1056 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	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₅×#4524	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#336	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	—	2	4	no	—
8	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
10	C₂²×C₅×C₃⋊C₈	—	2	4	no	—
11	C₂²×C₅×C₃⋊C₈	—	2	4	no	—
12	C₂²×C₅×C₃⋊C₈	C₄	1	8	yes	complemented
13	C₂²×C₅×C₃⋊C₈	C₂²	1	4	yes	—
14	C₂×#4524	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₅×C₈	—	3	2	no	—
16	C₂²×C₄²×C₅	D₃	1	1	yes	characteristic
17	C₅×#336	—	3	4	no	—
18	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	2	4	no	—
20	C₂²×C₃×C₄×C₅	—	2	4	no	—
21	C₂²×C₃×C₄×C₅	C₈	1	8	yes	complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₄²×C₅	—	2	4	no	—
30	C₃×C₄²×C₅	—	2	4	no	—
31	C₂×C₅×C₃⋊C₈	—	2	16	no	—
32	C₂×C₅×C₃⋊C₈	—	2	4	no	—
33	C₂×C₅×C₃⋊C₈	—	2	4	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂²×C₃×C₄²	C₂×C₅	1	1	yes	characteristic
36	C₂³×C₃⋊C₈	C₂×C₅	1	2	yes	—
37	#4524	C₂×C₅	1	4	yes	direct product factor
38	C₂³×C₄×C₅	Dic₃	1	2	yes	—
39	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₅×C₈	—	3	8	no	—
41	C₂²×C₅×C₈	—	3	4	no	—
42	C₂²×C₅×C₈	—	6	4	no	—
43	C₂²×C₅×C₈	—	6	4	no	—
44	C₂×C₄²×C₅	—	2	4	no	—
45	C₂×C₄²×C₅	Dic₃	1	2	yes	—
46	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
47	C₂×#336	—	3	1	no	Sylow
48	C₂³×C₃×C₅	C₄²	1	2	yes	—
49	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
51	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
52	C₂³×C₃×C₅	M₁₆	1	4	yes	—
53	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	16	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	8	yes	—
60	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₄⋊C₄	1	8	yes	—
62	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	complemented
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₅×C₃⋊C₈	—	2	8	no	—
68	C₂³×C₃×C₄	C₄×C₅	1	2	yes	—
69	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	2	4	no	—
71	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
73	C₂²×C₃⋊C₈	—	2	4	no	—
74	C₂²×C₃⋊C₈	—	2	4	no	—
75	C₂²×C₃⋊C₈	C₄×C₅	1	8	yes	complemented
76	C₂²×C₃⋊C₈	C₂²×C₅	1	4	yes	—
77	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	C₃⋊C₈	1	8	yes	complemented
81	C₂²×C₄×C₅	Dic₆	1	2	yes	—
82	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
83	C₂²×C₄×C₅	D₁₂	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
85	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
86	C₂²×C₄×C₅	#67	1	4	yes	—
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	3	8	no	—
89	C₂×C₅×C₈	—	6	16	no	—
90	C₂×C₅×C₈	—	6	4	no	—
91	C₂×C₅×C₈	—	6	4	no	—
92	C₄²×C₅	—	2	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₂³×C₈	—	3	2	no	—
95	C₂²×C₄²	C₅×D₃	1	1	yes	characteristic
96	#336	—	3	4	no	—
97	C₂²×C₃×C₅	#95	1	4	yes	—
98	C₂²×C₃×C₅	C₄×C₈	1	4	yes	—
99	C₂²×C₃×C₅	#97	1	4	yes	—
100	C₂²×C₃×C₅	C₂²⋊C₈	1	8	yes	—
101	C₂²×C₃×C₅	C₄⋊C₈	1	8	yes	—
102	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
106	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
107	C₃×C₄×C₅	—	2	8	no	—
108	C₃×C₄×C₅	#95	1	4	yes	—
109	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
110	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
111	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	2	4	no	—
114	C₂²×C₃×C₄	C₅×C₈	1	8	yes	complemented
115	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
116	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
117	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
119	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
120	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₃×C₄²	—	2	4	no	—
123	C₃×C₄²	—	2	4	no	—
124	C₂×C₃⋊C₈	—	2	16	no	—
125	C₂×C₃⋊C₈	—	2	4	no	—
126	C₂×C₃⋊C₈	—	2	4	no	—
127	C₂×C₃⋊C₈	C₂×C₄×C₅	1	8	yes	complemented
128	C₂³×C₅	C₂×C₃⋊C₈	1	4	yes	—
129	C₂³×C₅	#208	1	4	yes	—
130	C₂³×C₅	C₄×Dic₃	1	2	yes	—
131	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
132	C₂³×C₅	#217	1	2	yes	—
133	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
134	C₂×C₄×C₅	—	2	16	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	C₂×C₃⋊C₈	1	8	yes	complemented
138	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
139	C₂×C₄×C₅	Dic₃⋊C₄	1	8	yes	—
140	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
141	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	8	yes	—
142	C₂×C₄×C₅	#217	1	2	yes	—
143	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
145	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#67	1	2	yes	—
147	C₅×C₈	—	6	8	no	—
148	C₂³×C₄	C₅×Dic₃	1	2	yes	—
149	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
150	C₂²×C₈	—	3	8	no	—
151	C₂²×C₈	—	3	4	no	—
152	C₂²×C₈	—	6	4	no	—
153	C₂²×C₈	—	6	4	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	C₅×Dic₃	1	2	yes	—
156	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
157	C₂×C₃×C₅	#336	1	8	yes	complemented
158	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
159	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
160	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	2	yes	—
162	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
163	C₂³×C₃	C₄²×C₅	1	2	yes	—
164	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
165	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
166	C₂³×C₃	C₂×C₅×C₈	1	4	yes	—
167	C₂³×C₃	C₅×M₁₆	1	4	yes	—
168	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	16	no	—
172	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
173	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	8	yes	—
174	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
175	C₂×C₃×C₄	C₅×C₄⋊C₄	1	8	yes	—
176	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₅×C₈	1	8	yes	complemented
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
182	C₃⋊C₈	—	2	8	no	—
183	C₂²×C₅	C₄×C₃⋊C₈	1	4	yes	—
184	C₂²×C₅	#803	1	4	yes	—
185	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	8	yes	—
186	C₂²×C₅	#830	1	8	yes	—
187	C₂²×C₅	C₃⋊#95	1	4	yes	—
188	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
189	C₂²×C₅	C₂×#208	1	2	yes	—
190	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
191	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
192	C₂²×C₅	C₂×#217	1	1	yes	characteristic
193	C₄×C₅	—	2	8	no	—
194	C₄×C₅	C₃⋊#95	1	4	yes	—
195	C₄×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
196	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
197	C₂⁴	C₂×C₅×Dic₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₅×C₃⋊C₈	1	8	yes	complemented
201	C₂²×C₄	C₅×Dic₆	1	2	yes	—
202	C₂²×C₄	C₄×C₅×D₃	1	4	yes	—
203	C₂²×C₄	C₅×D₁₂	1	2	yes	—
204	C₂²×C₄	C₂×C₅×Dic₃	1	4	yes	—
205	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
206	C₂²×C₄	C₅×#67	1	4	yes	—
207	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
208	C₂×C₈	—	3	8	no	—
209	C₂×C₈	—	6	16	no	—
210	C₂×C₈	—	6	4	no	—
211	C₂×C₈	—	6	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₃×C₅	C₂×#336	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	C₅×#95	1	4	yes	—
216	C₂²×C₃	C₄×C₅×C₈	1	4	yes	—
217	C₂²×C₃	C₅×#97	1	4	yes	—
218	C₂²×C₃	C₅×C₂²⋊C₈	1	8	yes	—
219	C₂²×C₃	C₅×C₄⋊C₈	1	8	yes	—
220	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
224	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	C₅×#95	1	4	yes	—
227	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
228	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
229	C₂×C₅	#4524	1	8	yes	direct product factor
230	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
231	C₂×C₅	C₂×#803	1	1	yes	characteristic
232	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
233	C₂×C₅	C₂×#830	1	2	yes	—
234	C₂×C₅	C₂×C₃⋊#95	1	1	yes	characteristic
235	C₂³	C₂×C₅×C₃⋊C₈	1	4	yes	—
236	C₂³	C₅×#208	1	4	yes	—
237	C₂³	C₄×C₅×Dic₃	1	2	yes	—
238	C₂³	C₅×C₄⋊Dic₃	1	2	yes	—
239	C₂³	C₅×#217	1	2	yes	—
240	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	16	no	—
244	C₂×C₄	C₂×C₅×C₃⋊C₈	1	8	yes	complemented
245	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
246	C₂×C₄	C₅×Dic₃⋊C₄	1	8	yes	—
247	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
248	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	8	yes	—
249	C₂×C₄	C₅×#217	1	2	yes	—
250	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
252	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
253	C₂×C₄	C₂×C₅×#67	1	2	yes	—
254	C₈	—	6	8	no	—
255	C₂×C₃	C₅×#336	1	8	yes	complemented
256	C₂×C₃	C₂×C₅×#95	1	1	yes	characteristic
257	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
258	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
259	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	2	yes	—
260	C₂×C₃	C₂×C₅×C₄⋊C₈	1	2	yes	—
261	C₅	C₂×#4524	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	C₄×C₅×C₃⋊C₈	1	4	yes	—
263	C₂²	C₅×#803	1	4	yes	—
264	C₂²	C₅×C₄⋊(C₃⋊C₈)	1	8	yes	—
265	C₂²	C₅×#830	1	8	yes	—
266	C₂²	C₅×C₃⋊#95	1	4	yes	—
267	C₂²	C₂²×C₅×C₃⋊C₈	1	2	yes	—
268	C₂²	C₂×C₅×#208	1	2	yes	—
269	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
270	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
271	C₂²	C₂×C₅×#217	1	1	yes	characteristic
272	C₄	—	2	8	no	—
273	C₄	C₅×C₃⋊#95	1	4	yes	—
274	C₄	C₂×C₅×Dic₃⋊C₄	1	2	yes	—
275	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	2	yes	—
276	C₃	C₂×C₅×#336	1	1	yes	characteristic, complemented, Sylow
277	C₂	C₅×#4524	1	8	yes	direct product factor
278	C₂	C₂×C₄×C₅×C₃⋊C₈	1	1	yes	characteristic
279	C₂	C₂×C₅×#803	1	1	yes	characteristic
280	C₂	C₂×C₅×C₄⋊(C₃⋊C₈)	1	2	yes	—
281	C₂	C₂×C₅×#830	1	2	yes	—
282	C₂	C₂×C₅×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle, prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂²×C₄²×C₅	D₃
hypocenter	C₃	C₂×C₅×#336
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅

Complements

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

N	H ≅ G/N	type
3. C₅×#4524	C₂	direct (N × H)
12. C₂²×C₅×C₃⋊C₈	C₄	semidirect
14. C₂×#4524	C₅	direct (N × H)
21. C₂²×C₃×C₄×C₅	C₈	semidirect
34. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
37. #4524	C₂×C₅	direct (N × H)
62. C₂×C₃×C₄×C₅	C₂×C₈	semidirect
75. C₂²×C₃⋊C₈	C₄×C₅	semidirect
80. C₂²×C₄×C₅	C₃⋊C₈	semidirect
114. C₂²×C₃×C₄	C₅×C₈	semidirect
127. C₂×C₃⋊C₈	C₂×C₄×C₅	semidirect
137. C₂×C₄×C₅	C₂×C₃⋊C₈	semidirect
157. C₂×C₃×C₅	#336	semidirect
177. C₂×C₃×C₄	C₂×C₅×C₈	semidirect
200. C₂²×C₄	C₅×C₃⋊C₈	semidirect
214. C₃×C₅	C₂×#336	semidirect
229. C₂×C₅	#4524	direct (N × H)
244. C₂×C₄	C₂×C₅×C₃⋊C₈	semidirect
255. C₂×C₃	C₅×#336	semidirect
261. C₅	C₂×#4524	direct (N × H)
276. C₃	C₂×C₅×#336	semidirect
277. C₂	C₅×#4524	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×C₃×C₄²×C₅	2
C₂³×C₅×C₃⋊C₈	2
C₅×#4524	2
C₂×C₅×#336	3
C₂×#4524	5

Subgroups — C2×C5×#4524 #14294274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4524 #14294274