Subgroups — C₂×C₃×#65359 #14437639

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3964	C₂	1	4	yes	maximal, complemented
4	C₃×#65359	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65359	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	1	no	—
8	C₂²×C₃×C₅×D₄	—	2	1	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₁₀	—	2	1	no	—
11	C₃×C₅×#127	—	2	2	no	—
12	C₃×C₅×#127	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
15	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₃×#4015	—	2	2	no	—
18	C₃×#4015	C₂²	1	4	yes	complemented
19	C₃×#3964	C₂²	1	16	yes	complemented
20	C₃×C₄×#168	C₂²	1	4	yes	complemented
21	C₂×C₃×#486	—	5	1	no	maximal, Hall
22	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
23	C₂×#4015	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3964	C₂×C₃	1	4	yes	complemented
25	#65359	C₂×C₃	1	8	yes	direct product factor
26	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁴×C₃×C₅	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	4	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	8	no	—
32	C₂×C₃×C₅×D₄	—	4	4	no	—
33	C₂×C₃×C₅×D₄	—	4	4	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₃×Dic₅	—	2	2	no	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×Dic₁₀	—	4	4	no	—
41	C₃×C₄²×C₅	D₄	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₃×C₄⋊Dic₅	—	2	2	no	—
44	C₃×C₄⋊Dic₅	—	2	4	no	—
45	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
46	C₂×C₃×#168	—	2	4	no	—
47	C₂×C₃×#168	C₂³	1	4	yes	—
48	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
49	C₂×C₃×#127	D₅	1	1	yes	characteristic
50	C₃×#486	—	5	8	no	—
51	C₂×C₃×#102	—	5	4	no	—
52	C₂×C₃×C₄×C₈	—	5	1	no	—
53	C₂²×C₅×D₄	—	2	1	no	—
54	C₂²×C₅×D₄	—	2	1	no	—
55	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
56	C₂²×Dic₁₀	—	2	1	no	—
57	C₅×#127	—	2	2	no	—
58	C₅×#127	C₂²×C₃	1	4	yes	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₅	—	2	1	no	—
61	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
62	C₂²×#168	C₂²×C₃	1	2	yes	—
63	#4015	—	2	2	no	—
64	#4015	C₂²×C₃	1	4	yes	complemented
65	#3964	C₂²×C₃	1	16	yes	complemented
66	C₄×#168	C₂²×C₃	1	4	yes	complemented
67	C₂×#486	—	5	1	no	Sylow
68	C₂³×C₃×C₅	—	2	4	no	—
69	C₂³×C₃×C₅	—	4	4	no	—
70	C₂³×C₃×C₅	—	4	4	no	—
71	C₂³×C₃×C₅	—	4	4	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	8	no	—
74	C₃×C₅×D₄	—	4	4	no	—
75	C₃×C₅×D₄	—	4	8	no	—
76	C₃×C₅×D₄	—	4	8	no	—
77	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
78	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	#41	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	4	4	no	—
86	C₃×Dic₁₀	—	4	8	no	—
87	C₃×#168	—	2	4	no	—
88	C₂²×C₃×Q₈	—	10	1	no	—
89	C₂²×C₃×D₄	—	2	1	no	—
90	C₂²×C₃×D₄	—	2	1	no	—
91	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
92	C₂²×C₃×C₈	—	5	2	no	—
93	C₃×C₄⋊Q₈	—	5	4	no	—
94	C₃×C₄⋊Q₈	—	10	2	no	—
95	C₃×#127	—	2	2	no	—
96	C₃×#127	C₂×D₅	1	4	yes	—
97	C₂×C₃×C₄⋊C₄	—	5	2	no	—
98	C₂×C₃×C₄⋊C₄	—	10	1	no	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₃×#102	—	5	16	no	—
101	C₃×C₄×C₈	—	5	4	no	—
102	C₂⁴×C₅	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	2	4	no	—
105	C₂×C₅×D₄	—	2	4	no	—
106	C₂×C₅×D₄	—	2	8	no	—
107	C₂×C₅×D₄	—	4	4	no	—
108	C₂×C₅×D₄	—	4	4	no	—
109	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
110	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₂²×Dic₅	—	2	2	no	—
114	C₂×Dic₁₀	—	2	4	no	—
115	C₂×Dic₁₀	—	4	4	no	—
116	C₄²×C₅	C₃×D₄	1	2	yes	—
117	C₄²×C₅	C₂³×C₃	1	2	yes	—
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄⋊Dic₅	—	2	4	no	—
120	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
121	C₂×#168	—	2	4	no	—
122	C₂×#168	C₂³×C₃	1	4	yes	—
123	C₂×C₄⋊Q₈	—	5	1	no	—
124	C₂×#127	C₃×D₅	1	1	yes	characteristic
125	#486	—	5	8	no	—
126	C₂×#102	—	5	4	no	—
127	C₂×C₄×C₈	—	5	1	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	8	8	no	—
132	C₂²×C₃×C₅	#124	1	2	yes	—
133	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
134	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
135	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄×C₅	#124	1	2	yes	—
137	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
138	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
139	C₃×C₄×C₅	C₂×#41	1	2	yes	—
140	C₃×Dic₅	—	4	4	no	—
141	C₂⁴×C₃	—	2	2	no	—
142	C₂×C₃×Q₈	—	10	4	no	—
143	C₂×C₃×Q₈	—	20	4	no	—
144	C₂×C₃×D₄	—	2	8	no	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	4	no	—
147	C₂×C₃×D₄	—	2	4	no	—
148	C₂×C₃×D₄	—	4	4	no	—
149	C₂×C₃×D₄	—	4	4	no	—
150	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
151	C₂²×C₃×C₄	—	10	2	no	—
152	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	—	5	4	no	—
156	C₂×C₃×C₈	—	10	4	no	—
157	C₃×C₄⋊C₄	—	5	4	no	—
158	C₃×C₄⋊C₄	—	10	2	no	—
159	C₃×C₄⋊C₄	—	10	4	no	—
160	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
161	C₃×C₄²	C₂²×D₅	1	2	yes	—
162	C₂³×C₅	—	2	4	no	—
163	C₂³×C₅	—	4	4	no	—
164	C₂³×C₅	—	4	4	no	—
165	C₂³×C₅	—	4	4	no	—
166	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₅×D₄	—	2	8	no	—
168	C₅×D₄	—	4	4	no	—
169	C₅×D₄	—	4	8	no	—
170	C₅×D₄	—	4	8	no	—
171	C₂×C₄×C₅	C₃×D₈	1	4	yes	—
172	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	C₃×#41	1	4	yes	—
177	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
178	C₂×Dic₅	—	2	4	no	—
179	C₂×Dic₅	—	4	4	no	—
180	Dic₁₀	—	4	8	no	—
181	#168	—	2	4	no	—
182	C₂²×Q₈	—	10	1	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
186	C₂²×C₈	—	5	2	no	—
187	C₄⋊Q₈	—	5	4	no	—
188	C₄⋊Q₈	—	10	2	no	—
189	#127	—	2	2	no	—
190	#127	C₂×C₃×D₅	1	4	yes	—
191	C₂×C₄⋊C₄	—	5	2	no	—
192	C₂×C₄⋊C₄	—	10	1	no	—
193	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
194	#102	—	5	16	no	—
195	C₄×C₈	—	5	4	no	—
196	C₂×C₃×C₅	—	8	4	no	—
197	C₂×C₃×C₅	#486	1	4	yes	complemented
198	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
201	C₂³×C₃	—	2	4	no	—
202	C₂³×C₃	—	4	4	no	—
203	C₂³×C₃	—	4	4	no	—
204	C₂³×C₃	—	4	4	no	—
205	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₃×Q₈	—	20	8	no	—
207	C₃×D₄	—	2	8	no	—
208	C₃×D₄	—	4	8	no	—
209	C₃×D₄	—	4	8	no	—
210	C₃×D₄	—	4	4	no	—
211	C₂×C₃×C₄	—	10	4	no	—
212	C₂×C₃×C₄	—	20	4	no	—
213	C₂×C₃×C₄	#686	1	4	yes	—
214	C₂×C₃×C₄	#687	1	4	yes	—
215	C₂×C₃×C₄	#711	1	4	yes	—
216	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
217	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
220	C₃×C₈	—	10	4	no	—
221	C₂²×C₅	—	4	4	no	—
222	C₂²×C₅	—	4	4	no	—
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	8	8	no	—
225	C₂²×C₅	C₃×#124	1	2	yes	—
226	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
227	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
228	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₄×C₅	C₃×#124	1	2	yes	—
230	C₄×C₅	C₂×C₃×D₈	1	4	yes	—
231	C₄×C₅	C₂×C₃×SD₁₆	1	4	yes	—
232	C₄×C₅	C₂×C₃×#41	1	2	yes	—
233	Dic₅	—	4	4	no	—
234	C₂⁴	—	2	2	no	—
235	C₂×Q₈	—	10	4	no	—
236	C₂×Q₈	—	20	4	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
246	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	10	4	no	—
250	C₄⋊C₄	—	5	4	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₄⋊C₄	—	10	4	no	—
253	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
254	C₄²	C₂²×C₃×D₅	1	2	yes	—
255	C₃×C₅	C₂×#486	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	4	4	no	—
257	C₂²×C₃	—	4	4	no	—
258	C₂²×C₃	—	4	4	no	—
259	C₂²×C₃	—	8	8	no	—
260	C₂²×C₃	C₂×#686	1	2	yes	—
261	C₂²×C₃	C₂×#687	1	2	yes	—
262	C₂²×C₃	#4082	1	2	yes	—
263	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
264	C₃×C₄	—	20	4	no	—
265	C₃×C₄	C₂×#686	1	4	yes	—
266	C₃×C₄	C₂×#687	1	4	yes	—
267	C₃×C₄	#4082	1	2	yes	—
268	C₃×C₄	C₂×#711	1	2	yes	—
269	C₂×C₅	—	8	4	no	—
270	C₂×C₅	C₃×#486	1	4	yes	complemented
271	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
272	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
273	C₂×C₅	C₂²×C₃×SD₁₆	1	1	yes	characteristic
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	4	no	—
278	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
279	Q₈	—	20	8	no	—
280	D₄	—	2	8	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	10	4	no	—
285	C₂×C₄	—	20	4	no	—
286	C₂×C₄	C₃×#686	1	4	yes	—
287	C₂×C₄	C₃×#687	1	4	yes	—
288	C₂×C₄	C₃×#711	1	4	yes	—
289	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
290	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
291	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
292	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
293	C₈	—	10	4	no	—
294	C₂×C₃	—	8	4	no	—
295	C₂×C₃	#65359	1	4	yes	direct product factor
296	C₂×C₃	C₂²×#686	1	1	yes	characteristic
297	C₂×C₃	C₂²×#687	1	1	yes	characteristic
298	C₂×C₃	C₂×#4082	1	1	yes	characteristic
299	C₅	C₂×C₃×#486	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	8	no	—
304	C₂²	C₂×C₃×#686	1	2	yes	—
305	C₂²	C₂×C₃×#687	1	2	yes	—
306	C₂²	C₃×#4082	1	2	yes	—
307	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
308	C₄	—	20	4	no	—
309	C₄	C₂×C₃×#686	1	4	yes	—
310	C₄	C₂×C₃×#687	1	4	yes	—
311	C₄	C₃×#4082	1	2	yes	—
312	C₄	C₂×C₃×#711	1	2	yes	—
313	C₃	C₂×#65359	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	8	4	no	—
315	C₂	C₃×#65359	1	4	yes	direct product factor
316	C₂	C₂²×C₃×#686	1	1	yes	characteristic
317	C₂	C₂²×C₃×#687	1	1	yes	characteristic
318	C₂	C₂×C₃×#4082	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₂×C₃×#486
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4015	C₂	semidirect
3. C₂×C₃×#3964	C₂	semidirect
4. C₃×#65359	C₂	direct (N × H)
5. C₂×C₃×C₄×#168	C₂	semidirect
6. C₂×#65359	C₃	direct (N × H)
18. C₃×#4015	C₂²	semidirect
19. C₃×#3964	C₂²	semidirect
20. C₃×C₄×#168	C₂²	semidirect
23. C₂×#4015	C₂×C₃	semidirect
24. C₂×#3964	C₂×C₃	semidirect
25. #65359	C₂×C₃	direct (N × H)
26. C₂×C₄×#168	C₂×C₃	semidirect
64. #4015	C₂²×C₃	semidirect
65. #3964	C₂²×C₃	semidirect
66. C₄×#168	C₂²×C₃	semidirect
197. C₂×C₃×C₅	#486	semidirect
255. C₃×C₅	C₂×#486	semidirect
270. C₂×C₅	C₃×#486	semidirect
295. C₂×C₃	#65359	direct (N × H)
299. C₅	C₂×C₃×#486	semidirect
313. C₃	C₂×#65359	direct (N × H)
315. C₂	C₃×#65359	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×C₅×#127	2
C₂×C₃×#4015	2
C₂×C₃×#3964	2
C₃×#65359	2
C₂×C₃×C₄×#168	2
C₂×#65359	3
C₂×C₃×#486	5

Subgroups — C2×C3×#65359 #14437639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65359 #14437639