Subgroups — C₂×C₃×#64773 #14312901

Order 1920.

472 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3944	C₂	1	2	yes	maximal, complemented
3	C₃×#64773	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64773	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₅×M₁₆	—	2	2	no	—
7	C₂×C₃×C₅×M₁₆	—	2	2	no	—
8	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×#3944	—	2	2	no	—
13	C₃×#3944	C₂²	1	4	yes	complemented
14	C₂×C₃×#672	—	2	2	no	—
15	C₂×C₃×#349	—	5	1	no	maximal, Hall
16	C₂²×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
17	C₂×#3944	C₂×C₃	1	2	yes	complemented
18	#64773	C₂×C₃	1	4	yes	direct product factor
19	C₂⁴×C₃×C₅	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	—	2	1	no	—
21	C₂²×C₃×C₄×C₅	—	2	1	no	—
22	C₂²×C₃×C₄×C₅	—	2	1	no	—
23	C₂²×C₃×C₄×C₅	—	2	1	no	—
24	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₅×M₁₆	—	2	4	no	—
29	C₃×C₅×M₁₆	—	2	4	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₂×C₃×C₅×C₈	—	2	1	no	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₃×#672	—	2	4	no	—
38	C₂²×C₃×M₁₆	D₅	1	1	yes	characteristic
39	C₂×C₃×M₃₂	—	5	2	no	—
40	C₃×#349	—	5	4	no	—
41	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	—
45	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
46	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	#3944	—	2	2	no	—
49	#3944	C₂²×C₃	1	4	yes	complemented
50	C₂×#672	—	2	2	no	—
51	C₂×#349	—	5	1	no	Sylow
52	C₂³×C₃×C₅	—	2	1	no	—
53	C₂³×C₃×C₅	—	2	1	no	—
54	C₂³×C₃×C₅	—	4	2	no	—
55	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₃×C₅×C₈	—	2	2	no	—
71	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
73	C₂³×C₃×C₄	Dic₅	1	1	yes	characteristic
74	C₂×C₃×M₁₆	—	2	2	no	—
75	C₂×C₃×M₁₆	—	2	2	no	—
76	C₂×C₃×M₁₆	Dic₅	1	2	yes	—
77	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
78	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
80	C₃×M₃₂	—	5	4	no	—
81	C₃×M₃₂	—	10	2	no	—
82	C₂×C₃×C₁₆	—	10	2	no	—
83	C₂⁴×C₅	C₃×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	—	2	1	no	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	C₃×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
101	#672	—	2	4	no	—
102	C₂²×M₁₆	C₃×D₅	1	1	yes	characteristic
103	C₂×M₃₂	—	5	2	no	—
104	#349	—	5	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	—	4	2	no	—
108	C₂²×C₃×C₅	—	4	2	no	—
109	C₂²×C₃×C₅	—	4	2	no	—
110	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂⁴×C₃	#168	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	#168	1	1	yes	characteristic
124	C₂²×C₃×C₄	#168	1	2	yes	—
125	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
127	C₃×M₁₆	—	2	4	no	—
128	C₃×M₁₆	—	2	4	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
132	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
134	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
135	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
136	C₃×C₁₆	—	10	4	no	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	—	2	1	no	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
141	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
157	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
158	C₂³×C₄	C₃×Dic₅	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	C₃×Dic₅	1	2	yes	—
162	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
163	C₂²×C₈	C₃×Dic₅	1	1	yes	characteristic
164	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
165	M₃₂	—	5	4	no	—
166	M₃₂	—	10	2	no	—
167	C₂×C₁₆	—	10	2	no	—
168	C₂×C₃×C₅	—	2	1	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	—	4	2	no	—
171	C₂×C₃×C₅	#349	1	2	yes	complemented
172	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	—	4	2	no	—
176	C₂³×C₃	C₂×#168	1	1	yes	characteristic
177	C₂³×C₃	C₂×#168	1	2	yes	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	C₂×#168	1	2	yes	—
185	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
186	C₂×C₃×C₄	#681	1	1	yes	characteristic
187	C₂×C₃×C₄	#681	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
193	C₃×C₈	C₂×C₂²⋊D₅	1	2	yes	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	4	2	no	—
197	C₂²×C₅	—	4	2	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
200	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
201	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
204	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
205	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
207	C₂⁴	C₃×#168	1	1	yes	characteristic
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	C₃×#168	1	1	yes	characteristic
213	C₂²×C₄	C₃×#168	1	2	yes	—
214	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
215	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
221	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
222	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
223	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
224	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
225	C₁₆	—	10	4	no	—
226	C₃×C₅	C₂×#349	1	1	yes	characteristic, complemented, Hall
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	4	2	no	—
230	C₂²×C₃	—	4	2	no	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	C₂²⋊#168	1	1	yes	characteristic
233	C₂²×C₃	C₂²⋊#168	1	1	yes	characteristic
234	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	C₂²⋊#168	1	1	yes	characteristic
237	C₃×C₄	C₂²⋊#168	1	1	yes	characteristic
238	C₃×C₄	C₂×#681	1	1	yes	characteristic
239	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	4	2	no	—
243	C₂×C₅	C₃×#349	1	2	yes	complemented
244	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₃×#168	1	1	yes	characteristic
249	C₂³	C₂×C₃×#168	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	C₂×C₃×#168	1	2	yes	—
257	C₂×C₄	C₂×C₃×#168	1	1	yes	characteristic
258	C₂×C₄	C₃×#681	1	1	yes	characteristic
259	C₂×C₄	C₃×#681	1	1	yes	characteristic
260	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	C₃×C₂²⋊Dic₅	1	2	yes	—
265	C₈	C₂×C₃×C₂²⋊D₅	1	2	yes	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	#64773	1	2	yes	direct product factor
270	C₂×C₃	C₂×C₂²⋊#168	1	1	yes	characteristic
271	C₅	C₂×C₃×#349	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₃×C₂²⋊#168	1	1	yes	characteristic
278	C₂²	C₃×C₂²⋊#168	1	1	yes	characteristic
279	C₂²	C₂²×C₃×#168	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	C₃×C₂²⋊#168	1	1	yes	characteristic
282	C₄	C₃×C₂²⋊#168	1	1	yes	characteristic
283	C₄	C₂×C₃×#681	1	1	yes	characteristic
284	C₄	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
285	C₃	C₂×#64773	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	C₃×#64773	1	2	yes	direct product factor
290	C₂	C₂×C₃×C₂²⋊#168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂²×C₃×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²⋊C₈
prime radical	C₂⁴×C₃×C₅	C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂²×C₃×M₁₆	D₅
hypocenter	C₅	C₂×C₃×#349
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3944	C₂	semidirect
3. C₃×#64773	C₂	direct (N × H)
4. C₂×#64773	C₃	direct (N × H)
13. C₃×#3944	C₂²	semidirect
17. C₂×#3944	C₂×C₃	semidirect
18. #64773	C₂×C₃	direct (N × H)
49. #3944	C₂²×C₃	semidirect
171. C₂×C₃×C₅	#349	semidirect
226. C₃×C₅	C₂×#349	semidirect
243. C₂×C₅	C₃×#349	semidirect
269. C₂×C₃	#64773	direct (N × H)
271. C₅	C₂×C₃×#349	semidirect
285. C₃	C₂×#64773	direct (N × H)
289. C₂	C₃×#64773	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×C₅×M₁₆	2
C₂×C₃×#3944	2
C₃×#64773	2
C₂×#64773	3
C₂×C₃×#349	5

Subgroups — C2×C3×#64773 #14312901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64773 #14312901