Subgroups — C₃×#10610198 #14438896

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4050	C₂	1	4	yes	maximal, complemented
2	C₃×#65811	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64985	C₂	1	4	yes	maximal, complemented
6	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64982	C₂	1	2	yes	maximal, complemented
8	#10610198	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	4	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×#4076	—	2	4	no	—
13	C₂×C₃×Dic₂₀	C₂²	1	4	yes	—
14	C₃×#4050	—	2	16	no	—
15	C₂×C₃×D₄₀	C₂²	1	4	yes	—
16	C₂×C₃×#677	C₂²	1	8	yes	—
17	C₃×#4021	C₂²	1	4	yes	—
18	C₃×#4020	C₂²	1	2	yes	—
19	C₃×C₄×D₂₀	—	2	4	no	—
20	C₃×#4015	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₃×#3147	—	5	1	no	maximal, Hall
25	C₂×#4050	C₂×C₃	1	4	yes	complemented
26	#65811	C₂×C₃	1	2	yes	complemented
27	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#64988	C₂×C₃	1	1	yes	characteristic, complemented
29	#64985	C₂×C₃	1	4	yes	complemented
30	#64984	C₂×C₃	1	1	yes	characteristic, complemented
31	#64982	C₂×C₃	1	2	yes	complemented
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
35	C₃×#709	—	2	16	no	—
36	C₂×C₃×D₂₀	—	2	4	no	—
37	C₂×C₃×D₂₀	C₂³	1	4	yes	—
38	C₂×C₃×C₄×D₅	—	2	4	no	—
39	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
40	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×#685	—	2	8	no	—
47	C₃×C₄⋊Dic₅	—	2	4	no	—
48	C₃×Dic₂₀	—	2	8	no	—
49	C₃×D₄₀	—	2	8	no	—
50	C₃×#677	—	2	16	no	—
51	C₂×C₃×#135	—	5	4	no	—
52	C₃×#532	—	5	2	no	—
53	C₃×#495	—	5	4	no	—
54	C₃×#494	—	5	1	no	—
55	C₃×#493	—	5	1	no	—
56	C₃×#492	—	5	2	no	—
57	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
58	C₂×#709	C₂²×C₃	1	4	yes	—
59	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	#4076	—	2	4	no	—
62	C₂×Dic₂₀	C₂²×C₃	1	4	yes	—
63	#4050	—	2	16	no	—
64	C₂×D₄₀	C₂²×C₃	1	4	yes	—
65	C₂×#677	C₂²×C₃	1	8	yes	—
66	#4021	C₂²×C₃	1	4	yes	—
67	#4020	C₂²×C₃	1	2	yes	—
68	C₄×D₂₀	—	2	4	no	—
69	#4015	C₂²×C₃	1	2	yes	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
73	#3147	—	5	1	no	Sylow
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×D₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₃×C₂²⋊D₅	—	4	8	no	—
83	C₂×C₃×Dic₅	—	2	4	no	—
84	C₃×D₂₀	—	2	8	no	—
85	C₃×D₂₀	—	4	8	no	—
86	C₃×C₄×D₅	—	4	8	no	—
87	C₃×Dic₁₀	—	2	8	no	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
90	C₂×C₃×#41	—	5	4	no	—
91	C₃×#135	—	10	16	no	—
92	C₂×C₃×Q₁₆	—	5	4	no	—
93	C₂×C₃×SD₁₆	—	5	8	no	—
94	C₂×C₃×D₈	—	5	4	no	—
95	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
96	C₃×C₄⋊Q₈	—	5	2	no	—
97	C₃×#127	—	5	2	no	—
98	C₃×#124	—	5	4	no	—
99	C₃×#121	—	10	4	no	—
100	C₃×C₄×D₄	—	10	4	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
105	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	—	2	4	no	—
108	#709	—	2	16	no	—
109	C₂×D₂₀	—	2	4	no	—
110	C₂×D₂₀	C₂³×C₃	1	4	yes	—
111	C₂×C₄×D₅	—	2	4	no	—
112	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
113	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
114	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
115	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
116	C₄²×C₅	C₃×D₄	1	2	yes	—
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
119	#685	—	2	8	no	—
120	C₄⋊Dic₅	—	2	4	no	—
121	Dic₂₀	—	2	8	no	—
122	D₄₀	—	2	8	no	—
123	#677	—	2	16	no	—
124	C₂×#135	—	5	4	no	—
125	#532	—	5	2	no	—
126	#495	—	5	4	no	—
127	#494	—	5	1	no	—
128	#493	—	5	1	no	—
129	#492	—	5	2	no	—
130	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic, complemented
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	#127	1	2	yes	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	4	4	no	—
135	C₂×C₃×D₅	—	4	8	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	#127	1	2	yes	—
138	C₃×C₄×C₅	#135	1	4	yes	complemented
139	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
140	C₃×Dic₅	—	4	4	no	—
141	C₃×#41	—	10	16	no	—
142	C₂×C₃×Q₈	—	5	4	no	—
143	C₂×C₃×D₄	—	5	4	no	—
144	C₂×C₃×D₄	—	10	4	no	—
145	C₂×C₃×D₄	—	10	4	no	—
146	C₂²×C₃×C₄	—	10	4	no	—
147	C₂²×C₃×C₄	D₂₀	1	2	yes	—
148	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₃×Q₁₆	—	10	8	no	—
150	C₃×SD₁₆	—	10	16	no	—
151	C₃×D₈	—	10	8	no	—
152	C₂×C₃×C₈	D₂₀	1	8	yes	complemented
153	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
154	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
155	C₃×C₄⋊C₄	—	10	4	no	—
156	C₃×C₂²⋊C₄	—	10	8	no	—
157	C₃×C₄²	D₂₀	1	2	yes	—
158	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×D₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
168	C₂²⋊D₅	—	4	8	no	—
169	C₂×Dic₅	—	2	4	no	—
170	D₂₀	—	2	8	no	—
171	D₂₀	—	4	8	no	—
172	C₄×D₅	—	4	8	no	—
173	Dic₁₀	—	2	8	no	—
174	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
175	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
176	C₂×#41	—	5	4	no	—
177	#135	—	10	16	no	—
178	C₂×Q₁₆	—	5	4	no	—
179	C₂×SD₁₆	—	5	8	no	—
180	C₂×D₈	—	5	4	no	—
181	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
182	C₄⋊Q₈	—	5	2	no	—
183	#127	—	5	2	no	—
184	#124	—	5	4	no	—
185	#121	—	10	4	no	—
186	C₄×D₄	—	10	4	no	—
187	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
188	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
189	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
190	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
191	C₂×C₃×C₅	—	2	2	no	—
192	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×#135	1	2	yes	—
194	C₃×D₅	—	8	4	no	—
195	C₂³×C₃	—	10	4	no	—
196	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
197	C₃×Q₈	—	10	8	no	—
198	C₃×D₄	—	10	8	no	—
199	C₃×D₄	—	20	8	no	—
200	C₃×D₄	—	20	8	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	10	4	no	—
203	C₂×C₃×C₄	—	20	8	no	—
204	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
205	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
206	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
207	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₃×C₈	C₂×D₂₀	1	4	yes	—
210	C₃×C₈	C₂×D₂₀	1	4	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	C₃×#127	1	2	yes	—
213	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
214	C₂×D₅	—	4	8	no	—
215	C₂×D₅	—	4	4	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	C₃×#127	1	2	yes	—
218	C₄×C₅	C₃×#135	1	4	yes	complemented
219	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
220	Dic₅	—	4	4	no	—
221	#41	—	10	16	no	—
222	C₂×Q₈	—	5	4	no	—
223	C₂×D₄	—	5	4	no	—
224	C₂×D₄	—	10	4	no	—
225	C₂×D₄	—	10	4	no	—
226	C₂²×C₄	—	10	4	no	—
227	C₂²×C₄	C₃×D₂₀	1	2	yes	—
228	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
229	Q₁₆	—	10	8	no	—
230	SD₁₆	—	10	16	no	—
231	D₈	—	10	8	no	—
232	C₂×C₈	C₃×D₂₀	1	8	yes	complemented
233	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
234	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
235	C₄⋊C₄	—	10	4	no	—
236	C₂²⋊C₄	—	10	8	no	—
237	C₄²	C₃×D₂₀	1	2	yes	—
238	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
239	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	20	4	no	—
243	C₂²×C₃	—	20	8	no	—
244	C₂²×C₃	#4020	1	2	yes	—
245	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	20	4	no	—
248	C₃×C₄	#4020	1	2	yes	—
249	C₃×C₄	#4050	1	4	yes	complemented
250	C₃×C₄	C₂²×D₂₀	1	2	yes	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
253	C₂×C₅	C₂×C₃×#135	1	2	yes	—
254	D₅	—	8	4	no	—
255	C₂³	—	10	4	no	—
256	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
257	Q₈	—	10	8	no	—
258	D₄	—	10	8	no	—
259	D₄	—	20	8	no	—
260	D₄	—	20	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	10	4	no	—
263	C₂×C₄	—	20	8	no	—
264	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
265	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
266	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
267	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
268	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
269	C₈	C₂×C₃×D₂₀	1	4	yes	—
270	C₈	C₂×C₃×D₂₀	1	4	yes	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	40	4	no	—
273	C₂×C₃	C₂×#4020	1	1	yes	characteristic
274	C₂×C₃	C₂×#4050	1	2	yes	—
275	C₅	C₃×#3147	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	20	4	no	—
278	C₂²	—	20	8	no	—
279	C₂²	C₃×#4020	1	2	yes	—
280	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	20	4	no	—
283	C₄	C₃×#4020	1	2	yes	—
284	C₄	C₃×#4050	1	4	yes	complemented
285	C₄	C₂²×C₃×D₂₀	1	2	yes	—
286	C₃	#10610198	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	40	4	no	—
289	C₂	C₂×C₃×#4020	1	1	yes	characteristic
290	C₂	C₂×C₃×#4050	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×#3147
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4050	C₂	semidirect
2. C₃×#65811	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₃×#64988	C₂	semidirect
5. C₃×#64985	C₂	semidirect
6. C₃×#64984	C₂	semidirect
7. C₃×#64982	C₂	semidirect
8. #10610198	C₃	direct (N × H)
25. C₂×#4050	C₂×C₃	semidirect
26. #65811	C₂×C₃	semidirect
27. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
28. #64988	C₂×C₃	semidirect
29. #64985	C₂×C₃	semidirect
30. #64984	C₂×C₃	semidirect
31. #64982	C₂×C₃	semidirect
40. C₂×C₃×C₅×C₈	D₄	semidirect
57. C₂×C₃×C₄×C₈	D₅	semidirect
113. C₂×C₅×C₈	C₃×D₄	semidirect
130. C₂×C₄×C₈	C₃×D₅	semidirect
138. C₃×C₄×C₅	#135	semidirect
152. C₂×C₃×C₈	D₂₀	semidirect
218. C₄×C₅	C₃×#135	semidirect
232. C₂×C₈	C₃×D₂₀	semidirect
240. C₃×C₅	#3147	semidirect
249. C₃×C₄	#4050	semidirect
275. C₅	C₃×#3147	semidirect
284. C₄	C₃×#4050	semidirect
286. C₃	#10610198	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×#4050	2
C₃×#65811	2
C₂×C₃×C₄×C₅×C₈	2
C₃×#64988	2
C₃×#64985	2
C₃×#64984	2
C₃×#64982	2
#10610198	3
C₃×#3147	5

Subgroups — C3×#10610198 #14438896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610198 #14438896