Subgroups — C₂×C₃×#65345 #14437723

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3967	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#3964	C₂	1	2	yes	maximal
5	C₃×#65345	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65345	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	—	2	2	no	—
11	C₃×C₅×#124	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×C₃×#683	—	2	2	no	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₃×#4016	—	2	2	no	—
18	C₃×#4016	C₂²	1	4	yes	complemented
19	C₃×#3967	C₂²	1	8	yes	complemented
20	C₃×#3964	C₂²	1	8	yes	—
21	C₃×C₄×#168	C₂²	1	4	yes	complemented
22	C₂×C₃×#488	—	5	1	no	maximal, Hall
23	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
24	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#3967	C₂×C₃	1	2	yes	complemented
26	C₂×#3964	C₂×C₃	1	2	yes	—
27	#65345	C₂×C₃	1	8	yes	direct product factor
28	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂⁴×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	4	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	4	no	—
35	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	—	2	1	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₅	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	8	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	C₂³	1	4	yes	—
45	C₃×#683	—	2	8	no	—
46	C₂×C₃×#168	—	2	4	no	—
47	C₂×C₃×#168	C₂³	1	4	yes	—
48	C₂×C₃×#125	—	5	1	no	—
49	C₂×C₃×#124	D₅	1	1	yes	characteristic
50	C₃×#488	—	5	8	no	—
51	C₂×C₃×#103	—	5	2	no	—
52	C₂×C₃×#102	—	5	2	no	—
53	C₂×C₃×C₄×C₈	—	5	1	no	—
54	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₅×#124	—	2	2	no	—
57	C₅×#124	C₂²×C₃	1	4	yes	—
58	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
61	C₂×#683	—	2	2	no	—
62	C₂²×#168	C₂²×C₃	1	2	yes	—
63	#4016	—	2	2	no	—
64	#4016	C₂²×C₃	1	4	yes	complemented
65	#3967	C₂²×C₃	1	8	yes	complemented
66	#3964	C₂²×C₃	1	8	yes	—
67	C₄×#168	C₂²×C₃	1	4	yes	complemented
68	C₂×#488	—	5	1	no	Sylow
69	C₂³×C₃×C₅	—	2	2	no	—
70	C₂³×C₃×C₅	—	4	2	no	—
71	C₂³×C₃×C₅	—	4	4	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	2	no	—
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	4	2	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	4	2	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
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×#168	—	2	4	no	—
87	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
88	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
89	C₂²×C₃×C₈	—	5	2	no	—
90	C₃×#125	—	5	4	no	—
91	C₃×#125	—	10	2	no	—
92	C₃×#124	—	2	2	no	—
93	C₃×#124	C₂×D₅	1	4	yes	—
94	C₂×C₃×C₄⋊C₄	—	5	2	no	—
95	C₂×C₃×C₄⋊C₄	—	10	2	no	—
96	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×#103	—	5	8	no	—
99	C₃×#102	—	5	8	no	—
100	C₃×C₄×C₈	—	5	4	no	—
101	C₂⁴×C₅	—	2	1	no	—
102	C₂×C₅×Q₈	—	2	1	no	—
103	C₂×C₅×Q₈	—	2	4	no	—
104	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
105	C₂×C₅×D₄	—	2	1	no	—
106	C₂×C₅×D₄	—	2	4	no	—
107	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₄×C₅	—	2	1	no	—
109	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²×Dic₅	—	2	2	no	—
112	C₅×C₂²⋊C₄	—	2	8	no	—
113	C₄²×C₅	C₃×D₄	1	2	yes	—
114	C₄²×C₅	C₂³×C₃	1	2	yes	—
115	C₄⋊Dic₅	—	2	2	no	—
116	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
117	#683	—	2	8	no	—
118	C₂×#168	—	2	4	no	—
119	C₂×#168	C₂³×C₃	1	4	yes	—
120	C₂×#125	—	5	1	no	—
121	C₂×#124	C₃×D₅	1	1	yes	characteristic
122	#488	—	5	8	no	—
123	C₂×#103	—	5	2	no	—
124	C₂×#102	—	5	2	no	—
125	C₂×C₄×C₈	—	5	1	no	—
126	C₂²×C₃×C₅	—	4	2	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	—	8	4	no	—
129	C₂²×C₃×C₅	#124	1	2	yes	—
130	C₂²×C₃×C₅	#135	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	4	2	no	—
134	C₃×C₄×C₅	#124	1	2	yes	—
135	C₃×C₄×C₅	C₂×#41	1	2	yes	—
136	C₃×Dic₅	—	4	4	no	—
137	C₂⁴×C₃	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×Q₈	—	2	4	no	—
140	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
141	C₂×C₃×D₄	—	2	4	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	—	10	2	no	—
146	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
147	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×C₃×C₈	—	5	4	no	—
149	C₂×C₃×C₈	—	10	4	no	—
150	C₃×C₄⋊C₄	—	5	4	no	—
151	C₃×C₄⋊C₄	—	10	8	no	—
152	C₃×C₄⋊C₄	—	10	2	no	—
153	C₃×C₂²⋊C₄	—	2	8	no	—
154	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
155	C₃×C₄²	C₂²×D₅	1	2	yes	—
156	C₂³×C₅	—	2	2	no	—
157	C₂³×C₅	—	4	2	no	—
158	C₂³×C₅	—	4	4	no	—
159	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₅×Q₈	—	2	4	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×D₄	—	2	4	no	—
163	C₅×D₄	—	4	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
168	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₃×#41	1	4	yes	—
170	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
171	C₂×Dic₅	—	2	4	no	—
172	C₂×Dic₅	—	4	4	no	—
173	#168	—	2	4	no	—
174	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
175	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
176	C₂²×C₈	—	5	2	no	—
177	#125	—	5	4	no	—
178	#125	—	10	2	no	—
179	#124	—	2	2	no	—
180	#124	C₂×C₃×D₅	1	4	yes	—
181	C₂×C₄⋊C₄	—	5	2	no	—
182	C₂×C₄⋊C₄	—	10	2	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
185	#103	—	5	8	no	—
186	#102	—	5	8	no	—
187	C₄×C₈	—	5	4	no	—
188	C₂×C₃×C₅	—	8	2	no	—
189	C₂×C₃×C₅	#488	1	4	yes	complemented
190	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#135	1	2	yes	—
192	C₂³×C₃	—	2	2	no	—
193	C₂³×C₃	—	4	4	no	—
194	C₂³×C₃	—	4	2	no	—
195	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₃×Q₈	—	2	4	no	—
197	C₃×Q₈	—	4	2	no	—
198	C₃×D₄	—	2	4	no	—
199	C₃×D₄	—	4	2	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	—	20	4	no	—
205	C₂×C₃×C₄	#711	1	4	yes	—
206	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₃×C₈	—	10	4	no	—
210	C₂²×C₅	—	4	4	no	—
211	C₂²×C₅	—	4	2	no	—
212	C₂²×C₅	—	8	4	no	—
213	C₂²×C₅	C₃×#124	1	2	yes	—
214	C₂²×C₅	C₃×#135	1	4	yes	—
215	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	C₃×#124	1	2	yes	—
219	C₄×C₅	C₂×C₃×#41	1	2	yes	—
220	Dic₅	—	4	4	no	—
221	C₂⁴	—	2	1	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	C₂²×C₃×D₅	1	2	yes	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₂²×C₃×D₅	1	2	yes	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	10	2	no	—
230	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
231	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
232	C₂×C₈	—	5	4	no	—
233	C₂×C₈	—	10	4	no	—
234	C₄⋊C₄	—	5	4	no	—
235	C₄⋊C₄	—	10	2	no	—
236	C₄⋊C₄	—	10	8	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
239	C₄²	C₂²×C₃×D₅	1	2	yes	—
240	C₃×C₅	C₂×#488	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	4	2	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	—	8	4	no	—
244	C₂²×C₃	#4082	1	2	yes	—
245	C₂²×C₃	#4096	1	4	yes	—
246	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	—	20	4	no	—
250	C₃×C₄	#4082	1	2	yes	—
251	C₃×C₄	C₂×#711	1	2	yes	—
252	C₂×C₅	—	8	2	no	—
253	C₂×C₅	C₃×#488	1	4	yes	complemented
254	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
255	C₂×C₅	C₂×C₃×#135	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	D₄	—	2	4	no	—
263	D₄	—	4	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	10	4	no	—
268	C₂×C₄	—	20	4	no	—
269	C₂×C₄	C₃×#711	1	4	yes	—
270	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
271	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
273	C₈	—	10	4	no	—
274	C₂×C₃	—	8	2	no	—
275	C₂×C₃	#65345	1	4	yes	direct product factor
276	C₂×C₃	C₂×#4082	1	1	yes	characteristic
277	C₂×C₃	C₂×#4096	1	2	yes	—
278	C₅	C₂×C₃×#488	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	4	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₃×#4082	1	2	yes	—
283	C₂²	C₃×#4096	1	4	yes	—
284	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	20	4	no	—
288	C₄	C₃×#4082	1	2	yes	—
289	C₄	C₂×C₃×#711	1	2	yes	—
290	C₃	C₂×#65345	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	8	2	no	—
292	C₂	C₃×#65345	1	4	yes	direct product factor
293	C₂	C₂×C₃×#4082	1	1	yes	characteristic
294	C₂	C₂×C₃×#4096	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₂×C₃×#488
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4016	C₂	semidirect
3. C₂×C₃×#3967	C₂	semidirect
5. C₃×#65345	C₂	direct (N × H)
6. C₂×C₃×C₄×#168	C₂	semidirect
7. C₂×#65345	C₃	direct (N × H)
18. C₃×#4016	C₂²	semidirect
19. C₃×#3967	C₂²	semidirect
21. C₃×C₄×#168	C₂²	semidirect
24. C₂×#4016	C₂×C₃	semidirect
25. C₂×#3967	C₂×C₃	semidirect
27. #65345	C₂×C₃	direct (N × H)
28. C₂×C₄×#168	C₂×C₃	semidirect
64. #4016	C₂²×C₃	semidirect
65. #3967	C₂²×C₃	semidirect
67. C₄×#168	C₂²×C₃	semidirect
189. C₂×C₃×C₅	#488	semidirect
240. C₃×C₅	C₂×#488	semidirect
253. C₂×C₅	C₃×#488	semidirect
275. C₂×C₃	#65345	direct (N × H)
278. C₅	C₂×C₃×#488	semidirect
290. C₃	C₂×#65345	direct (N × H)
292. C₂	C₃×#65345	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×#124	2
C₂×C₃×#4016	2
C₂×C₃×#3967	2
C₂×C₃×#3964	2
C₃×#65345	2
C₂×C₃×C₄×#168	2
C₂×#65345	3
C₂×C₃×#488	5

Subgroups — C2×C3×#65345 #14437723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65345 #14437723