Subgroups — C₂×C₃×#64744 #14294275

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×C₃×#168	C₂	1	2	yes	maximal
3	C₃×#64744	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64744	C₃	1	1	yes	maximal, characteristic, direct product factor, 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×C₃×#168	—	2	4	no	—
11	C₂²×C₃×#168	—	2	4	no	—
12	C₂²×C₃×#168	C₄	1	8	yes	complemented
13	C₂²×C₃×#168	C₂²	1	4	yes	—
14	C₂×C₃×#336	—	5	1	no	maximal, Hall
15	C₂²×C₄²×C₅	C₂×C₃	1	1	yes	characteristic
16	C₂³×#168	C₂×C₃	1	2	yes	—
17	#64744	C₂×C₃	1	4	yes	direct product factor
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×C₃×#168	—	2	16	no	—
32	C₂×C₃×#168	—	2	4	no	—
33	C₂×C₃×#168	—	2	4	no	—
34	C₂×C₃×#168	C₂×C₄	1	8	yes	complemented
35	C₂³×C₃×C₈	—	5	2	no	—
36	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic
37	C₃×#336	—	5	4	no	—
38	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
39	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₄²×C₅	—	2	4	no	—
41	C₂×C₄²×C₅	C₃×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
43	C₂²×#168	—	2	4	no	—
44	C₂²×#168	—	2	4	no	—
45	C₂²×#168	C₃×C₄	1	8	yes	complemented
46	C₂²×#168	C₂²×C₃	1	4	yes	—
47	C₂×#336	—	5	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	C₃×#168	—	2	8	no	—
68	C₂³×C₃×C₄	Dic₅	1	2	yes	—
69	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
70	C₂²×C₃×C₈	—	5	8	no	—
71	C₂²×C₃×C₈	—	5	4	no	—
72	C₂²×C₃×C₈	—	10	4	no	—
73	C₂²×C₃×C₈	—	10	4	no	—
74	C₂×C₃×C₄²	—	2	4	no	—
75	C₂×C₃×C₄²	Dic₅	1	2	yes	—
76	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
77	C₂⁴×C₅	C₂×C₃×C₄	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	C₂×C₃×C₄	1	4	yes	—
82	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
83	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
86	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	—	2	4	no	—
90	C₂×#168	—	2	16	no	—
91	C₂×#168	—	2	4	no	—
92	C₂×#168	—	2	4	no	—
93	C₂×#168	C₂×C₃×C₄	1	8	yes	complemented
94	C₂³×C₈	—	5	2	no	—
95	C₂²×C₄²	C₃×D₅	1	1	yes	characteristic
96	#336	—	5	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×Dic₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	2	4	no	—
114	C₂²×C₃×C₄	#168	1	8	yes	complemented
115	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
116	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
117	C₂²×C₃×C₄	D₂₀	1	2	yes	—
118	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
119	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
121	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₂×C₃×C₈	—	5	8	no	—
123	C₂×C₃×C₈	—	10	16	no	—
124	C₂×C₃×C₈	—	10	4	no	—
125	C₂×C₃×C₈	—	10	4	no	—
126	C₃×C₄²	—	2	4	no	—
127	C₃×C₄²	—	2	4	no	—
128	C₂³×C₅	C₃×C₄²	1	2	yes	—
129	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
130	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
131	C₂³×C₅	C₂×C₃×C₈	1	4	yes	—
132	C₂³×C₅	C₃×M₁₆	1	4	yes	—
133	C₂³×C₅	C₂²×C₃×C₄	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	C₃×C₄²	1	2	yes	—
138	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	8	yes	—
139	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₃×C₄⋊C₄	1	8	yes	—
141	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₃×C₈	1	8	yes	complemented
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
147	#168	—	2	8	no	—
148	C₂³×C₄	C₃×Dic₅	1	2	yes	—
149	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
150	C₂²×C₈	—	5	8	no	—
151	C₂²×C₈	—	5	4	no	—
152	C₂²×C₈	—	10	4	no	—
153	C₂²×C₈	—	10	4	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	C₃×Dic₅	1	2	yes	—
156	C₂×C₄²	C₂×C₃×D₅	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	C₂×#168	1	4	yes	—
164	C₂³×C₃	#681	1	4	yes	—
165	C₂³×C₃	C₄×Dic₅	1	2	yes	—
166	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
167	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
168	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	16	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	C₂×#168	1	8	yes	complemented
173	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
174	C₂×C₃×C₄	#683	1	8	yes	—
175	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
176	C₂×C₃×C₄	#685	1	8	yes	—
177	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
178	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
180	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
182	C₃×C₈	—	10	8	no	—
183	C₂²×C₅	C₃×#95	1	4	yes	—
184	C₂²×C₅	C₃×C₄×C₈	1	4	yes	—
185	C₂²×C₅	C₃×#97	1	4	yes	—
186	C₂²×C₅	C₃×C₂²⋊C₈	1	8	yes	—
187	C₂²×C₅	C₃×C₄⋊C₈	1	8	yes	—
188	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
189	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₅	C₂²×C₃×C₈	1	2	yes	—
192	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
193	C₄×C₅	—	2	8	no	—
194	C₄×C₅	C₃×#95	1	4	yes	—
195	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
196	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
197	C₂⁴	C₂×C₃×Dic₅	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₃×#168	1	8	yes	complemented
201	C₂²×C₄	C₃×Dic₁₀	1	2	yes	—
202	C₂²×C₄	C₃×C₄×D₅	1	4	yes	—
203	C₂²×C₄	C₃×D₂₀	1	2	yes	—
204	C₂²×C₄	C₂×C₃×Dic₅	1	4	yes	—
205	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
206	C₂²×C₄	C₃×C₂²⋊D₅	1	4	yes	—
207	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
208	C₂×C₈	—	5	8	no	—
209	C₂×C₈	—	10	16	no	—
210	C₂×C₈	—	10	4	no	—
211	C₂×C₈	—	10	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	C₄×#168	1	4	yes	—
216	C₂²×C₃	#3935	1	4	yes	—
217	C₂²×C₃	C₄⋊#168	1	8	yes	—
218	C₂²×C₃	C₂²⋊#168	1	8	yes	—
219	C₂²×C₃	#3963	1	4	yes	—
220	C₂²×C₃	C₂²×#168	1	2	yes	—
221	C₂²×C₃	C₂×#681	1	2	yes	—
222	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
223	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	#3963	1	4	yes	—
227	C₃×C₄	C₂×#683	1	2	yes	—
228	C₃×C₄	C₂×#685	1	2	yes	—
229	C₂×C₅	C₃×#336	1	8	yes	complemented
230	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
231	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
232	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
233	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	2	yes	—
234	C₂×C₅	C₂×C₃×C₄⋊C₈	1	2	yes	—
235	C₂³	C₂×C₃×#168	1	4	yes	—
236	C₂³	C₃×#681	1	4	yes	—
237	C₂³	C₃×C₄×Dic₅	1	2	yes	—
238	C₂³	C₃×C₄⋊Dic₅	1	2	yes	—
239	C₂³	C₃×C₂²⋊Dic₅	1	2	yes	—
240	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₂×C₃×#168	1	8	yes	complemented
245	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
246	C₂×C₄	C₃×#683	1	8	yes	—
247	C₂×C₄	C₃×C₄⋊Dic₅	1	2	yes	—
248	C₂×C₄	C₃×#685	1	8	yes	—
249	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
250	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
252	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
254	C₈	—	10	8	no	—
255	C₂×C₃	#64744	1	8	yes	direct product factor
256	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
257	C₂×C₃	C₂×#3935	1	1	yes	characteristic
258	C₂×C₃	C₂×C₄⋊#168	1	2	yes	—
259	C₂×C₃	C₂×C₂²⋊#168	1	2	yes	—
260	C₂×C₃	C₂×#3963	1	1	yes	characteristic
261	C₅	C₂×C₃×#336	1	1	yes	characteristic, complemented, Sylow
262	C₂²	C₃×C₄×#168	1	4	yes	—
263	C₂²	C₃×#3935	1	4	yes	—
264	C₂²	C₃×C₄⋊#168	1	8	yes	—
265	C₂²	C₃×C₂²⋊#168	1	8	yes	—
266	C₂²	C₃×#3963	1	4	yes	—
267	C₂²	C₂²×C₃×#168	1	2	yes	—
268	C₂²	C₂×C₃×#681	1	2	yes	—
269	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
270	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
271	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
272	C₄	—	2	8	no	—
273	C₄	C₃×#3963	1	4	yes	—
274	C₄	C₂×C₃×#683	1	2	yes	—
275	C₄	C₂×C₃×#685	1	2	yes	—
276	C₃	C₂×#64744	1	1	yes	characteristic, direct product factor, Sylow
277	C₂	C₃×#64744	1	8	yes	direct product factor
278	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
279	C₂	C₂×C₃×#3935	1	1	yes	characteristic
280	C₂	C₂×C₃×C₄⋊#168	1	2	yes	—
281	C₂	C₂×C₃×C₂²⋊#168	1	2	yes	—
282	C₂	C₂×C₃×#3963	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	C₂×D₅
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle, prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂²×C₃×C₄²	D₅
hypocenter	C₅	C₂×C₃×#336
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ C₂²×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×#64744	C₂	direct (N × H)
4. C₂×#64744	C₃	direct (N × H)
12. C₂²×C₃×#168	C₄	semidirect
17. #64744	C₂×C₃	direct (N × H)
21. C₂²×C₃×C₄×C₅	C₈	semidirect
34. C₂×C₃×#168	C₂×C₄	semidirect
45. C₂²×#168	C₃×C₄	semidirect
62. C₂×C₃×C₄×C₅	C₂×C₈	semidirect
80. C₂²×C₄×C₅	C₃×C₈	semidirect
93. C₂×#168	C₂×C₃×C₄	semidirect
114. C₂²×C₃×C₄	#168	semidirect
142. C₂×C₄×C₅	C₂×C₃×C₈	semidirect
157. C₂×C₃×C₅	#336	semidirect
172. C₂×C₃×C₄	C₂×#168	semidirect
200. C₂²×C₄	C₃×#168	semidirect
214. C₃×C₅	C₂×#336	semidirect
229. C₂×C₅	C₃×#336	semidirect
244. C₂×C₄	C₂×C₃×#168	semidirect
255. C₂×C₃	#64744	direct (N × H)
261. C₅	C₂×C₃×#336	semidirect
276. C₃	C₂×#64744	direct (N × H)
277. C₂	C₃×#64744	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×C₃×#168	2
C₃×#64744	2
C₂×#64744	3
C₂×C₃×#336	5

Subgroups — C2×C3×#64744 #14294275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64744 #14294275