Subgroups — C₂×C₃×#65139 #14430076

Order 1920.

832 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3950	C₂	1	2	yes	maximal
5	C₂×C₃×#3948	C₂	1	2	yes	maximal
6	C₃×#65139	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65139	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×Dic₁₀	—	2	1	no	—
9	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×#683	—	2	1	no	—
16	C₂×C₃×C₄×Dic₅	—	2	1	no	—
17	C₃×#4015	—	2	2	no	—
18	C₃×#4015	C₂²	1	4	yes	—
19	C₃×C₄×Dic₁₀	—	2	2	no	—
20	C₃×C₄×Dic₁₀	C₂²	1	4	yes	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
22	C₃×#3950	C₂²	1	8	yes	—
23	C₃×#3948	C₂²	1	8	yes	—
24	C₂×C₃×#477	—	5	1	no	maximal, Hall
25	C₂×#4015	C₂×C₃	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
27	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₂×#3950	C₂×C₃	1	2	yes	—
29	C₂×#3948	C₂×C₃	1	2	yes	—
30	#65139	C₂×C₃	1	8	yes	direct product factor
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	—	2	1	no	—
35	C₂²×C₃×Dic₅	—	2	2	no	—
36	C₂²×C₃×Dic₅	—	2	1	no	—
37	C₂×C₃×Dic₁₀	—	2	1	no	—
38	C₂×C₃×Dic₁₀	—	2	2	no	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×Dic₁₀	—	4	4	no	—
41	C₂×C₃×Dic₁₀	Q₈	1	4	yes	complemented
42	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	—	2	2	no	—
49	C₃×C₄⋊Dic₅	—	2	4	no	—
50	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
51	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
52	C₃×#683	—	2	4	no	—
53	C₃×C₄×Dic₅	—	2	4	no	—
54	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
55	C₂×C₃×C₄×Q₈	—	5	1	no	—
56	C₃×#477	—	5	8	no	—
57	C₂×C₃×#107	—	5	2	no	—
58	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic
59	C₂×C₃×#103	—	5	2	no	—
60	C₂²×Dic₁₀	—	2	1	no	—
61	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄⋊Dic₅	—	2	1	no	—
65	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
67	C₂×#683	—	2	1	no	—
68	C₂×C₄×Dic₅	—	2	1	no	—
69	#4015	—	2	2	no	—
70	#4015	C₂²×C₃	1	4	yes	—
71	C₄×Dic₁₀	—	2	2	no	—
72	C₄×Dic₁₀	C₂²×C₃	1	4	yes	—
73	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
74	#3950	C₂²×C₃	1	8	yes	—
75	#3948	C₂²×C₃	1	8	yes	—
76	C₂×#477	—	5	1	no	Sylow
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₃×Dic₅	—	2	4	no	—
87	C₂×C₃×Dic₅	—	2	2	no	—
88	C₂×C₃×Dic₅	—	2	2	no	—
89	C₂×C₃×Dic₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	4	4	no	—
91	C₂×C₃×Dic₅	—	4	2	no	—
92	C₃×Dic₁₀	—	2	4	no	—
93	C₃×Dic₁₀	—	2	2	no	—
94	C₃×Dic₁₀	—	4	8	no	—
95	C₃×Dic₁₀	C₂×Q₈	1	4	yes	complemented
96	C₃×C₅×C₈	—	2	4	no	—
97	C₂²×C₃×Q₈	—	5	1	no	—
98	C₂²×C₃×Q₈	—	10	1	no	—
99	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
100	C₃×C₄⋊Q₈	—	5	4	no	—
101	C₃×C₄⋊Q₈	—	10	2	no	—
102	C₃×C₄×Q₈	—	5	4	no	—
103	C₃×C₄×Q₈	—	10	2	no	—
104	C₂×C₃×C₄⋊C₄	—	5	2	no	—
105	C₂×C₃×C₄⋊C₄	—	5	1	no	—
106	C₂×C₃×C₄⋊C₄	—	10	1	no	—
107	C₂×C₃×C₄⋊C₄	—	10	1	no	—
108	C₂×C₃×C₄²	—	10	1	no	—
109	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
110	C₃×#107	—	5	8	no	—
111	C₃×C₄⋊C₈	C₂×D₅	1	4	yes	—
112	C₃×#103	—	5	8	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×Dic₅	—	2	1	no	—
117	C₂²×Dic₅	—	2	2	no	—
118	C₂²×Dic₅	—	2	1	no	—
119	C₂×Dic₁₀	—	2	1	no	—
120	C₂×Dic₁₀	—	2	2	no	—
121	C₂×Dic₁₀	—	2	4	no	—
122	C₂×Dic₁₀	—	4	4	no	—
123	C₂×Dic₁₀	C₃×Q₈	1	4	yes	complemented
124	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
125	C₂×C₅×C₈	—	2	4	no	—
126	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
127	C₄²×C₅	C₃×D₄	1	2	yes	—
128	C₄²×C₅	C₂³×C₃	1	2	yes	—
129	C₄⋊Dic₅	—	2	1	no	—
130	C₄⋊Dic₅	—	2	2	no	—
131	C₄⋊Dic₅	—	2	4	no	—
132	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
133	C₄⋊Dic₅	C₂³×C₃	1	2	yes	—
134	#683	—	2	4	no	—
135	C₄×Dic₅	—	2	4	no	—
136	C₂×C₄⋊Q₈	—	5	1	no	—
137	C₂×C₄×Q₈	—	5	1	no	—
138	#477	—	5	8	no	—
139	C₂×#107	—	5	2	no	—
140	C₂×C₄⋊C₈	C₃×D₅	1	1	yes	characteristic
141	C₂×#103	—	5	2	no	—
142	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
144	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
145	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
148	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
149	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
151	C₃×Dic₅	—	2	4	no	—
152	C₃×Dic₅	—	4	4	no	—
153	C₃×Dic₅	—	4	2	no	—
154	C₂×C₃×Q₈	—	5	4	no	—
155	C₂×C₃×Q₈	—	5	2	no	—
156	C₂×C₃×Q₈	—	10	1	no	—
157	C₂×C₃×Q₈	—	10	4	no	—
158	C₂×C₃×Q₈	—	10	2	no	—
159	C₂×C₃×Q₈	—	20	4	no	—
160	C₂²×C₃×C₄	—	10	1	no	—
161	C₂²×C₃×C₄	—	10	1	no	—
162	C₂²×C₃×C₄	—	10	2	no	—
163	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
164	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
165	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₃×C₈	—	2	4	no	—
167	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
168	C₃×C₄⋊C₄	—	5	4	no	—
169	C₃×C₄⋊C₄	—	5	2	no	—
170	C₃×C₄⋊C₄	—	10	4	no	—
171	C₃×C₄⋊C₄	—	10	1	no	—
172	C₃×C₄⋊C₄	—	10	2	no	—
173	C₃×C₄⋊C₄	—	10	4	no	—
174	C₃×C₄²	—	10	4	no	—
175	C₃×C₄²	D₂₀	1	2	yes	—
176	C₃×C₄²	C₂²×D₅	1	2	yes	—
177	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	—
180	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
184	C₂×C₄×C₅	C₃×#41	1	2	yes	—
185	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×Dic₅	—	2	4	no	—
187	C₂×Dic₅	—	2	2	no	—
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	—	2	4	no	—
190	C₂×Dic₅	—	4	4	no	—
191	C₂×Dic₅	—	4	2	no	—
192	Dic₁₀	—	2	4	no	—
193	Dic₁₀	—	2	2	no	—
194	Dic₁₀	—	4	8	no	—
195	Dic₁₀	C₂×C₃×Q₈	1	4	yes	complemented
196	C₅×C₈	—	2	4	no	—
197	C₂²×Q₈	—	5	1	no	—
198	C₂²×Q₈	—	10	1	no	—
199	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
200	C₄⋊Q₈	—	5	4	no	—
201	C₄⋊Q₈	—	10	2	no	—
202	C₄×Q₈	—	5	4	no	—
203	C₄×Q₈	—	10	2	no	—
204	C₂×C₄⋊C₄	—	5	1	no	—
205	C₂×C₄⋊C₄	—	5	2	no	—
206	C₂×C₄⋊C₄	—	10	1	no	—
207	C₂×C₄⋊C₄	—	10	1	no	—
208	C₂×C₄²	—	10	1	no	—
209	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
210	#107	—	5	8	no	—
211	C₄⋊C₈	C₂×C₃×D₅	1	4	yes	—
212	#103	—	5	8	no	—
213	C₂×C₃×C₅	#477	1	4	yes	complemented
214	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
218	C₃×Q₈	—	5	4	no	—
219	C₃×Q₈	—	10	2	no	—
220	C₃×Q₈	—	10	4	no	—
221	C₃×Q₈	—	20	8	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	10	4	no	—
225	C₂×C₃×C₄	—	10	2	no	—
226	C₂×C₃×C₄	—	10	4	no	—
227	C₂×C₃×C₄	—	20	2	no	—
228	C₂×C₃×C₄	—	20	4	no	—
229	C₂×C₃×C₄	Dic₂₀	1	4	yes	—
230	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
231	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
233	C₂×C₃×C₄	#711	1	2	yes	—
234	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
235	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
236	C₃×C₈	—	2	4	no	—
237	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
238	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
239	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
240	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
243	C₄×C₅	C₂×C₃×Q₁₆	1	4	yes	—
244	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
245	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
246	Dic₅	—	2	4	no	—
247	Dic₅	—	4	4	no	—
248	Dic₅	—	4	2	no	—
249	C₂×Q₈	—	5	4	no	—
250	C₂×Q₈	—	5	2	no	—
251	C₂×Q₈	—	10	1	no	—
252	C₂×Q₈	—	10	4	no	—
253	C₂×Q₈	—	10	2	no	—
254	C₂×Q₈	—	20	4	no	—
255	C₂²×C₄	—	10	1	no	—
256	C₂²×C₄	—	10	1	no	—
257	C₂²×C₄	—	10	2	no	—
258	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
259	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
263	C₄⋊C₄	—	5	4	no	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	10	1	no	—
266	C₄⋊C₄	—	10	2	no	—
267	C₄⋊C₄	—	10	4	no	—
268	C₄⋊C₄	—	10	4	no	—
269	C₄²	—	10	4	no	—
270	C₄²	C₃×D₂₀	1	2	yes	—
271	C₄²	C₂²×C₃×D₅	1	2	yes	—
272	C₃×C₅	C₂×#477	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	#4043	1	2	yes	—
274	C₂²×C₃	C₂×Dic₂₀	1	2	yes	—
275	C₂²×C₃	#4054	1	2	yes	—
276	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	10	4	no	—
279	C₃×C₄	—	20	2	no	—
280	C₃×C₄	—	20	4	no	—
281	C₃×C₄	#4043	1	2	yes	—
282	C₃×C₄	C₂×Dic₂₀	1	4	yes	—
283	C₃×C₄	C₂×#711	1	1	yes	characteristic
284	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
285	C₂×C₅	C₃×#477	1	4	yes	complemented
286	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂×C₅	C₂²×C₃×Q₁₆	1	1	yes	characteristic
288	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
290	Q₈	—	5	4	no	—
291	Q₈	—	10	2	no	—
292	Q₈	—	10	4	no	—
293	Q₈	—	20	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	4	no	—
297	C₂×C₄	—	10	2	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	20	4	no	—
300	C₂×C₄	—	20	2	no	—
301	C₂×C₄	C₃×Dic₂₀	1	4	yes	—
302	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
303	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
305	C₂×C₄	C₃×#711	1	2	yes	—
306	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
307	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₂×C₃	#65139	1	4	yes	direct product factor
310	C₂×C₃	C₂×#4043	1	1	yes	characteristic
311	C₂×C₃	C₂²×Dic₂₀	1	1	yes	characteristic
312	C₂×C₃	C₂×#4054	1	1	yes	characteristic
313	C₅	C₂×C₃×#477	1	1	yes	characteristic, complemented, Sylow
314	C₂²	C₃×#4043	1	2	yes	—
315	C₂²	C₂×C₃×Dic₂₀	1	2	yes	—
316	C₂²	C₃×#4054	1	2	yes	—
317	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	10	4	no	—
320	C₄	—	20	4	no	—
321	C₄	—	20	2	no	—
322	C₄	C₃×#4043	1	2	yes	—
323	C₄	C₂×C₃×Dic₂₀	1	4	yes	—
324	C₄	C₂×C₃×#711	1	1	yes	characteristic
325	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
326	C₃	C₂×#65139	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	C₃×#65139	1	4	yes	direct product factor
328	C₂	C₂×C₃×#4043	1	1	yes	characteristic
329	C₂	C₂²×C₃×Dic₂₀	1	1	yes	characteristic
330	C₂	C₂×C₃×#4054	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×C₄⋊C₈	D₅
hypocenter	C₅	C₂×C₃×#477
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. C₃×#65139	C₂	direct (N × H)
7. C₂×#65139	C₃	direct (N × H)
30. #65139	C₂×C₃	direct (N × H)
41. C₂×C₃×Dic₁₀	Q₈	semidirect
95. C₃×Dic₁₀	C₂×Q₈	semidirect
123. C₂×Dic₁₀	C₃×Q₈	semidirect
195. Dic₁₀	C₂×C₃×Q₈	semidirect
213. C₂×C₃×C₅	#477	semidirect
272. C₃×C₅	C₂×#477	semidirect
285. C₂×C₅	C₃×#477	semidirect
309. C₂×C₃	#65139	direct (N × H)
313. C₅	C₂×C₃×#477	semidirect
326. C₃	C₂×#65139	direct (N × H)
327. C₂	C₃×#65139	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#4015	2
C₂×C₃×C₄×Dic₁₀	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₃×#3950	2
C₂×C₃×#3948	2
C₃×#65139	2
C₂×#65139	3
C₂×C₃×#477	5

Subgroups — C2×C3×#65139 #14430076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65139 #14430076