Subgroups — C₂×C₃×#64715 #14314170

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64715	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64715	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	—	2	1	no	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄₀	—	2	1	no	—
10	C₂×C₃×D₄₀	—	2	1	no	—
11	C₂×C₃×D₄₀	—	2	1	no	—
12	C₂×C₃×D₄₀	C₄	1	2	yes	—
13	C₂×C₃×D₄₀	C₄	1	2	yes	—
14	C₂×C₃×D₄₀	C₂²	1	2	yes	—
15	C₃×C₅×#108	—	2	1	no	—
16	C₃×C₅×#108	C₂²	1	2	yes	complemented
17	C₃×#3944	—	2	1	no	—
18	C₃×#3944	C₂²	1	2	yes	complemented
19	C₂×C₃×#672	—	2	1	no	—
20	C₂×C₃×#361	—	5	1	no	maximal, Hall
21	C₂²×D₄₀	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#3944	C₂×C₃	1	1	yes	characteristic, complemented
24	#64715	C₂×C₃	1	4	yes	direct product factor
25	C₂³×C₃×D₅	—	4	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₃×D₂₀	—	2	2	no	—
28	C₂×C₃×D₂₀	—	2	2	no	—
29	C₂×C₃×D₂₀	—	2	2	no	—
30	C₂×C₃×D₂₀	—	4	1	no	—
31	C₂×C₃×D₂₀	—	4	1	no	—
32	C₂×C₃×D₂₀	—	4	1	no	—
33	C₃×C₅×M₁₆	—	2	2	no	—
34	C₃×C₅×M₁₆	—	4	1	no	—
35	C₂×C₃×C₅×C₈	—	4	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×D₄₀	—	2	2	no	—
43	C₃×D₄₀	—	2	2	no	—
44	C₃×D₄₀	—	2	2	no	—
45	C₃×D₄₀	C₂×C₄	1	4	yes	—
46	C₃×#672	—	2	2	no	—
47	C₂²×C₃×D₈	—	5	1	no	—
48	C₂×C₃×M₃₂	—	5	1	no	—
49	C₂×C₃×#108	D₅	1	1	yes	characteristic, complemented
50	C₃×#361	—	5	4	no	—
51	C₂²×D₂₀	—	2	1	no	—
52	C₂×C₅×M₁₆	—	2	1	no	—
53	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×D₄₀	—	2	1	no	—
55	C₂×D₄₀	—	2	1	no	—
56	C₂×D₄₀	—	2	1	no	—
57	C₂×D₄₀	C₃×C₄	1	2	yes	—
58	C₂×D₄₀	C₃×C₄	1	2	yes	—
59	C₂×D₄₀	C₂²×C₃	1	2	yes	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	C₂²×C₃	1	2	yes	complemented
62	#3944	—	2	1	no	—
63	#3944	C₂²×C₃	1	2	yes	complemented
64	C₂×#672	—	2	1	no	—
65	C₂×#361	—	5	1	no	Sylow
66	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₃×D₅	—	4	2	no	—
68	C₂²×C₃×D₅	—	4	2	no	—
69	C₂²×C₃×D₅	—	4	2	no	—
70	C₂²×C₃×D₅	—	8	4	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃×D₂₀	—	2	4	no	—
78	C₃×D₂₀	—	4	2	no	—
79	C₃×D₂₀	—	4	2	no	—
80	C₃×D₂₀	—	4	2	no	—
81	C₃×C₅×C₈	—	4	2	no	—
82	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₃×D₄	—	10	1	no	—
87	C₂×C₃×D₈	—	5	2	no	—
88	C₂×C₃×D₈	—	5	2	no	—
89	C₂×C₃×D₈	—	5	2	no	—
90	C₂×C₃×D₈	—	10	1	no	—
91	C₂×C₃×D₈	—	10	1	no	—
92	C₂×C₃×D₈	—	10	1	no	—
93	C₂×C₃×M₁₆	—	2	1	no	—
94	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
95	C₃×M₃₂	—	5	2	no	—
96	C₃×M₃₂	—	10	1	no	—
97	C₂×C₃×C₁₆	—	10	1	no	—
98	C₃×#108	—	2	1	no	—
99	C₃×#108	C₂×D₅	1	2	yes	complemented
100	C₂³×D₅	—	4	1	no	—
101	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂×D₂₀	—	2	2	no	—
103	C₂×D₂₀	—	2	2	no	—
104	C₂×D₂₀	—	2	2	no	—
105	C₂×D₂₀	—	4	1	no	—
106	C₂×D₂₀	—	4	1	no	—
107	C₂×D₂₀	—	4	1	no	—
108	C₅×M₁₆	—	2	2	no	—
109	C₅×M₁₆	—	4	1	no	—
110	C₂×C₅×C₈	—	4	1	no	—
111	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
117	D₄₀	—	2	2	no	—
118	D₄₀	—	2	2	no	—
119	D₄₀	—	2	2	no	—
120	D₄₀	C₂×C₃×C₄	1	4	yes	—
121	#672	—	2	2	no	—
122	C₂²×D₈	—	5	1	no	—
123	C₂×M₃₂	—	5	1	no	—
124	C₂×#108	C₃×D₅	1	1	yes	characteristic, complemented
125	#361	—	5	4	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	#102	1	1	yes	characteristic
128	C₂²×C₃×C₅	#102	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₃×D₅	—	4	4	no	—
131	C₂×C₃×D₅	—	8	4	no	—
132	C₂×C₃×D₅	—	8	4	no	—
133	C₂×C₃×D₅	—	8	4	no	—
134	C₃×C₄×C₅	#102	1	1	yes	characteristic
135	C₃×C₄×C₅	#102	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂⁴×C₃	—	20	1	no	—
139	C₂×C₃×D₄	—	10	2	no	—
140	C₂×C₃×D₄	—	10	2	no	—
141	C₂×C₃×D₄	—	10	2	no	—
142	C₂×C₃×D₄	—	20	1	no	—
143	C₂×C₃×D₄	—	20	1	no	—
144	C₂×C₃×D₄	—	20	1	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₃×D₈	—	5	4	no	—
147	C₃×D₈	—	10	2	no	—
148	C₃×D₈	—	10	2	no	—
149	C₃×D₈	—	10	2	no	—
150	C₃×M₁₆	—	2	2	no	—
151	C₃×M₁₆	—	4	1	no	—
152	C₂×C₃×C₈	—	4	1	no	—
153	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
154	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
156	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
159	C₃×C₁₆	—	10	2	no	—
160	C₂³×C₅	C₃×SD₁₆	1	1	yes	characteristic
161	C₂²×D₅	—	4	2	no	—
162	C₂²×D₅	—	4	2	no	—
163	C₂²×D₅	—	4	2	no	—
164	C₂²×D₅	—	8	4	no	—
165	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	D₂₀	—	2	4	no	—
172	D₂₀	—	4	2	no	—
173	D₂₀	—	4	2	no	—
174	D₂₀	—	4	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂²×D₄	—	10	1	no	—
181	C₂×D₈	—	5	2	no	—
182	C₂×D₈	—	5	2	no	—
183	C₂×D₈	—	5	2	no	—
184	C₂×D₈	—	10	1	no	—
185	C₂×D₈	—	10	1	no	—
186	C₂×D₈	—	10	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
189	M₃₂	—	5	2	no	—
190	M₃₂	—	10	1	no	—
191	C₂×C₁₆	—	10	1	no	—
192	#108	—	2	1	no	—
193	#108	C₂×C₃×D₅	1	2	yes	complemented
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	#361	1	2	yes	complemented
197	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
198	C₃×D₅	—	8	4	no	—
199	C₂³×C₃	—	20	2	no	—
200	C₂³×C₃	—	20	2	no	—
201	C₂³×C₃	—	20	2	no	—
202	C₂³×C₃	—	40	4	no	—
203	C₂³×C₃	#688	1	1	yes	characteristic
204	C₃×D₄	—	10	4	no	—
205	C₃×D₄	—	20	2	no	—
206	C₃×D₄	—	20	2	no	—
207	C₃×D₄	—	20	2	no	—
208	C₂×C₃×C₄	#685	1	1	yes	characteristic
209	C₂×C₃×C₄	#685	1	1	yes	characteristic
210	C₂×C₃×C₄	#686	1	1	yes	characteristic
211	C₂×C₃×C₄	#686	1	1	yes	characteristic
212	C₂×C₃×C₄	#688	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₃×C₈	—	4	2	no	—
215	C₃×C₈	#685	1	1	yes	characteristic
216	C₃×C₈	#685	1	1	yes	characteristic
217	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
218	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	C₃×#102	1	1	yes	characteristic
221	C₂²×C₅	C₃×#102	1	1	yes	characteristic
222	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
223	C₂×D₅	—	4	4	no	—
224	C₂×D₅	—	8	4	no	—
225	C₂×D₅	—	8	4	no	—
226	C₂×D₅	—	8	4	no	—
227	C₄×C₅	C₃×#102	1	1	yes	characteristic
228	C₄×C₅	C₃×#102	1	1	yes	characteristic
229	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
231	C₂⁴	—	20	1	no	—
232	C₂×D₄	—	10	2	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂×D₄	—	10	2	no	—
235	C₂×D₄	—	20	1	no	—
236	C₂×D₄	—	20	1	no	—
237	C₂×D₄	—	20	1	no	—
238	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
239	D₈	—	5	4	no	—
240	D₈	—	10	2	no	—
241	D₈	—	10	2	no	—
242	D₈	—	10	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
247	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
248	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
249	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
250	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
251	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
252	C₁₆	—	10	2	no	—
253	C₃×C₅	C₂×#361	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	20	4	no	—
256	C₂²×C₃	—	40	4	no	—
257	C₂²×C₃	—	40	4	no	—
258	C₂²×C₃	—	40	4	no	—
259	C₂²×C₃	#3941	1	1	yes	characteristic
260	C₂²×C₃	#3941	1	1	yes	characteristic
261	C₂²×C₃	C₂×#688	1	1	yes	characteristic
262	C₃×C₄	#3941	1	1	yes	characteristic
263	C₃×C₄	#3941	1	1	yes	characteristic
264	C₃×C₄	C₂×#685	1	1	yes	characteristic
265	C₃×C₄	C₂×#686	1	1	yes	characteristic
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	—	2	1	no	—
268	C₂×C₅	C₃×#361	1	2	yes	complemented
269	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
270	D₅	—	8	4	no	—
271	C₂³	—	20	2	no	—
272	C₂³	—	20	2	no	—
273	C₂³	—	20	2	no	—
274	C₂³	—	40	4	no	—
275	C₂³	C₃×#688	1	1	yes	characteristic
276	D₄	—	10	4	no	—
277	D₄	—	20	2	no	—
278	D₄	—	20	2	no	—
279	D₄	—	20	2	no	—
280	C₂×C₄	C₃×#685	1	1	yes	characteristic
281	C₂×C₄	C₃×#685	1	1	yes	characteristic
282	C₂×C₄	C₃×#686	1	1	yes	characteristic
283	C₂×C₄	C₃×#686	1	1	yes	characteristic
284	C₂×C₄	C₃×#688	1	1	yes	characteristic
285	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
286	C₈	—	4	2	no	—
287	C₈	C₃×#685	1	1	yes	characteristic
288	C₈	C₃×#685	1	1	yes	characteristic
289	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
290	C₈	C₂×C₃×D₂₀	1	1	yes	characteristic
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	40	4	no	—
294	C₂×C₃	#64715	1	2	yes	direct product factor
295	C₂×C₃	C₂×#3941	1	1	yes	characteristic
296	C₅	C₂×C₃×#361	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	20	4	no	—
299	C₂²	—	40	4	no	—
300	C₂²	—	40	4	no	—
301	C₂²	—	40	4	no	—
302	C₂²	C₃×#3941	1	1	yes	characteristic
303	C₂²	C₃×#3941	1	1	yes	characteristic
304	C₂²	C₂×C₃×#688	1	1	yes	characteristic
305	C₄	C₃×#3941	1	1	yes	characteristic
306	C₄	C₃×#3941	1	1	yes	characteristic
307	C₄	C₂×C₃×#685	1	1	yes	characteristic
308	C₄	C₂×C₃×#686	1	1	yes	characteristic
309	C₃	C₂×#64715	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	40	4	no	—
313	C₂	C₃×#64715	1	2	yes	direct product factor
314	C₂	C₂×C₃×#3941	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#3941
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	#102
prime radical	C₂²×C₃×D₄₀	C₂
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×#108	D₅
hypocenter	C₅	C₂×C₃×#361
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
3. C₂×C₃×#3944	C₂	semidirect
4. C₃×#64715	C₂	direct (N × H)
5. C₂×#64715	C₃	direct (N × H)
16. C₃×C₅×#108	C₂²	semidirect
18. C₃×#3944	C₂²	semidirect
22. C₂×C₅×#108	C₂×C₃	semidirect
23. C₂×#3944	C₂×C₃	semidirect
24. #64715	C₂×C₃	direct (N × H)
49. C₂×C₃×#108	D₅	semidirect
61. C₅×#108	C₂²×C₃	semidirect
63. #3944	C₂²×C₃	semidirect
99. C₃×#108	C₂×D₅	semidirect
124. C₂×#108	C₃×D₅	semidirect
193. #108	C₂×C₃×D₅	semidirect
196. C₂×C₃×C₅	#361	semidirect
253. C₃×C₅	C₂×#361	semidirect
268. C₂×C₅	C₃×#361	semidirect
294. C₂×C₃	#64715	direct (N × H)
296. C₅	C₂×C₃×#361	semidirect
309. C₃	C₂×#64715	direct (N × H)
313. C₂	C₃×#64715	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×D₄₀	2
C₂×C₃×C₅×#108	2
C₂×C₃×#3944	2
C₃×#64715	2
C₂×#64715	3
C₂×C₃×#361	5

Subgroups — C2×C3×#64715 #14314170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64715 #14314170