Subgroups — C₂×C₃×#65357 #14437732

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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
3	C₃×#65357	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#3942	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#3941	C₂	1	2	yes	maximal
6	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65357	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	2	no	—
11	C₃×C₅×#125	C₂²	1	4	yes	complemented
12	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×#685	—	2	2	no	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₃×#4021	—	2	2	no	—
18	C₃×#4021	C₂²	1	4	yes	—
19	C₃×#3942	C₂²	1	8	yes	complemented
20	C₃×#3941	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×#125	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#4021	C₂×C₃	1	1	yes	characteristic
25	#65357	C₂×C₃	1	8	yes	direct product factor
26	C₂×#3942	C₂×C₃	1	2	yes	complemented
27	C₂×#3941	C₂×C₃	1	2	yes	—
28	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×D₅	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	4	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	C₂³	1	2	yes	—
37	C₂×C₃×Dic₁₀	—	2	4	no	—
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
40	C₃×C₅×C₄⋊C₄	—	2	8	no	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₃×#685	—	2	8	no	—
46	C₂×C₃×#168	—	2	4	no	—
47	C₂×C₃×#168	C₂³	1	4	yes	—
48	C₂×C₃×#125	D₅	1	1	yes	characteristic, complemented
49	C₂×C₃×#124	—	5	1	no	—
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×D₂₀	C₂²×C₃	1	1	yes	characteristic
55	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
56	C₅×#125	—	2	2	no	—
57	C₅×#125	C₂²×C₃	1	4	yes	complemented
58	C₂×C₅×C₄⋊C₄	—	2	2	no	—
59	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×#685	—	2	2	no	—
62	C₂²×#168	C₂²×C₃	1	2	yes	—
63	#4021	—	2	2	no	—
64	#4021	C₂²×C₃	1	4	yes	—
65	#3942	C₂²×C₃	1	8	yes	complemented
66	#3941	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	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂²×C₃×D₅	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	—	4	2	no	—
82	C₃×D₂₀	—	2	4	no	—
83	C₃×D₂₀	—	4	2	no	—
84	C₃×Dic₁₀	—	2	4	no	—
85	C₃×Dic₁₀	—	4	2	no	—
86	C₃×#168	—	2	4	no	—
87	C₂²×C₃×Q₈	—	5	1	no	—
88	C₂²×C₃×D₄	—	5	1	no	—
89	C₂²×C₃×C₈	—	5	2	no	—
90	C₃×#125	—	2	2	no	—
91	C₃×#125	C₂×D₅	1	4	yes	complemented
92	C₃×#124	—	5	4	no	—
93	C₃×#124	—	10	2	no	—
94	C₂×C₃×C₄⋊C₄	—	2	2	no	—
95	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
96	C₂×C₃×C₂²⋊C₄	—	10	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	C₂³×D₅	—	2	1	no	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂²×Dic₅	—	2	1	no	—
106	C₂×D₂₀	—	2	4	no	—
107	C₂×D₂₀	—	2	1	no	—
108	C₂×D₂₀	C₂³×C₃	1	2	yes	—
109	C₂×Dic₁₀	—	2	4	no	—
110	C₂×Dic₁₀	—	2	1	no	—
111	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
112	C₅×C₄⋊C₄	—	2	8	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
115	C₄²×C₅	C₃×D₄	1	2	yes	—
116	C₄²×C₅	C₂³×C₃	1	2	yes	—
117	#685	—	2	8	no	—
118	C₂×#168	—	2	4	no	—
119	C₂×#168	C₂³×C₃	1	4	yes	—
120	C₂×#125	C₃×D₅	1	1	yes	characteristic, complemented
121	C₂×#124	—	5	1	no	—
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	#124	1	2	yes	—
127	C₂²×C₃×C₅	#135	1	4	yes	—
128	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₃×D₅	—	4	2	no	—
130	C₂×C₃×D₅	—	4	4	no	—
131	C₂×C₃×D₅	—	8	4	no	—
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	4	4	no	—
134	C₃×C₄×C₅	#124	1	2	yes	—
135	C₃×C₄×C₅	C₂×#41	1	2	yes	—
136	C₃×Dic₅	—	4	2	no	—
137	C₂⁴×C₃	—	10	1	no	—
138	C₂×C₃×Q₈	—	5	2	no	—
139	C₂×C₃×Q₈	—	10	1	no	—
140	C₂×C₃×Q₈	—	10	4	no	—
141	C₂×C₃×D₄	—	5	2	no	—
142	C₂×C₃×D₄	—	10	4	no	—
143	C₂×C₃×D₄	—	10	1	no	—
144	C₂²×C₃×C₄	—	2	2	no	—
145	C₂²×C₃×C₄	—	10	1	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	—	2	8	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
153	C₃×C₂²⋊C₄	—	10	8	no	—
154	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
155	C₃×C₄²	C₂²×D₅	1	2	yes	—
156	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂²×D₅	—	2	2	no	—
158	C₂²×D₅	—	4	2	no	—
159	C₂²×D₅	—	4	4	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₃×#41	1	4	yes	—
166	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
167	C₂×Dic₅	—	2	2	no	—
168	C₂×Dic₅	—	4	2	no	—
169	D₂₀	—	2	4	no	—
170	D₂₀	—	4	2	no	—
171	Dic₁₀	—	2	4	no	—
172	Dic₁₀	—	4	2	no	—
173	#168	—	2	4	no	—
174	C₂²×Q₈	—	5	1	no	—
175	C₂²×D₄	—	5	1	no	—
176	C₂²×C₈	—	5	2	no	—
177	#125	—	2	2	no	—
178	#125	C₂×C₃×D₅	1	4	yes	complemented
179	#124	—	5	4	no	—
180	#124	—	10	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
183	C₂×C₂²⋊C₄	—	10	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	#488	1	4	yes	complemented
189	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×#135	1	2	yes	—
191	C₃×D₅	—	8	2	no	—
192	C₂³×C₃	—	10	2	no	—
193	C₂³×C₃	—	20	4	no	—
194	C₂³×C₃	—	20	2	no	—
195	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₃×Q₈	—	10	4	no	—
197	C₃×Q₈	—	20	2	no	—
198	C₃×D₄	—	10	4	no	—
199	C₃×D₄	—	20	2	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	4	4	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	—	20	2	no	—
205	C₂×C₃×C₄	#713	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	C₃×#124	1	2	yes	—
211	C₂²×C₅	C₃×#135	1	4	yes	—
212	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂×D₅	—	4	4	no	—
214	C₂×D₅	—	4	2	no	—
215	C₂×D₅	—	8	4	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	C₃×#124	1	2	yes	—
219	C₄×C₅	C₂×C₃×#41	1	2	yes	—
220	Dic₅	—	4	2	no	—
221	C₂⁴	—	10	1	no	—
222	C₂×Q₈	—	5	2	no	—
223	C₂×Q₈	—	10	1	no	—
224	C₂×Q₈	—	10	4	no	—
225	C₂×D₄	—	5	2	no	—
226	C₂×D₄	—	10	4	no	—
227	C₂×D₄	—	10	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	10	1	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	—	2	8	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
237	C₂²⋊C₄	—	10	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	—	20	2	no	—
242	C₂²×C₃	—	20	4	no	—
243	C₂²×C₃	—	40	4	no	—
244	C₂²×C₃	#4093	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	4	no	—
249	C₃×C₄	—	20	2	no	—
250	C₃×C₄	#4093	1	2	yes	—
251	C₃×C₄	C₂×#713	1	2	yes	—
252	C₂×C₅	C₃×#488	1	4	yes	complemented
253	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
254	C₂×C₅	C₂×C₃×#135	1	2	yes	—
255	D₅	—	8	2	no	—
256	C₂³	—	10	2	no	—
257	C₂³	—	20	2	no	—
258	C₂³	—	20	4	no	—
259	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
260	Q₈	—	10	4	no	—
261	Q₈	—	20	2	no	—
262	D₄	—	10	4	no	—
263	D₄	—	20	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	20	2	no	—
269	C₂×C₄	C₃×#713	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	—	40	2	no	—
275	C₂×C₃	#65357	1	4	yes	direct product factor
276	C₂×C₃	C₂×#4093	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	—	20	2	no	—
280	C₂²	—	20	4	no	—
281	C₂²	—	40	4	no	—
282	C₂²	C₃×#4093	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	4	no	—
287	C₄	—	20	2	no	—
288	C₄	C₃×#4093	1	2	yes	—
289	C₄	C₂×C₃×#713	1	2	yes	—
290	C₃	C₂×#65357	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	40	2	no	—
292	C₂	C₃×#65357	1	4	yes	direct product factor
293	C₂	C₂×C₃×#4093	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×#125	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₂₀	C₂²
hypercenter	C₂×C₃×#125	D₅
hypocenter	C₅	C₂×C₃×#488
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
3. C₃×#65357	C₂	direct (N × H)
4. C₂×C₃×#3942	C₂	semidirect
6. C₂×C₃×C₄×#168	C₂	semidirect
7. C₂×#65357	C₃	direct (N × H)
11. C₃×C₅×#125	C₂²	semidirect
19. C₃×#3942	C₂²	semidirect
21. C₃×C₄×#168	C₂²	semidirect
23. C₂×C₅×#125	C₂×C₃	semidirect
25. #65357	C₂×C₃	direct (N × H)
26. C₂×#3942	C₂×C₃	semidirect
28. C₂×C₄×#168	C₂×C₃	semidirect
48. C₂×C₃×#125	D₅	semidirect
57. C₅×#125	C₂²×C₃	semidirect
65. #3942	C₂²×C₃	semidirect
67. C₄×#168	C₂²×C₃	semidirect
91. C₃×#125	C₂×D₅	semidirect
120. C₂×#125	C₃×D₅	semidirect
178. #125	C₂×C₃×D₅	semidirect
188. C₂×C₃×C₅	#488	semidirect
240. C₃×C₅	C₂×#488	semidirect
252. C₂×C₅	C₃×#488	semidirect
275. C₂×C₃	#65357	direct (N × H)
278. C₅	C₂×C₃×#488	semidirect
290. C₃	C₂×#65357	direct (N × H)
292. C₂	C₃×#65357	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×#125	2
C₂×C₃×#4021	2
C₃×#65357	2
C₂×C₃×#3942	2
C₂×C₃×#3941	2
C₂×C₃×C₄×#168	2
C₂×#65357	3
C₂×C₃×#488	5

Subgroups — C2×C3×#65357 #14437732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65357 #14437732