Subgroups — C₃×#10613513 #14438893

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×#4058	C₂	1	4	yes	maximal, complemented
2	C₃×#66050	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65444	C₂	1	4	yes	maximal, complemented
5	C₃×#65365	C₂	1	2	yes	maximal, complemented
6	C₃×#64988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#10613513	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#711	C₂²	1	4	yes	—
10	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
11	C₃×C₅×#127	C₂²	1	2	yes	—
12	C₃×#4084	—	2	4	no	—
13	C₃×#4082	C₂²	1	4	yes	—
14	C₃×D₄×Dic₅	—	2	4	no	—
15	C₂×C₃×#687	C₂²	1	8	yes	—
16	C₃×#4058	—	2	16	no	—
17	C₂×C₃×Dic₂₀	C₂²	1	4	yes	—
18	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
19	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
20	C₃×#4015	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	C₃×#3147	—	5	1	no	maximal, Hall
25	C₂×#4058	C₂×C₃	1	4	yes	complemented
26	#66050	C₂×C₃	1	2	yes	complemented
27	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
28	#65444	C₂×C₃	1	4	yes	complemented
29	#65365	C₂×C₃	1	2	yes	complemented
30	#64988	C₂×C₃	1	1	yes	characteristic, complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×D₄	—	2	4	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
34	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
35	C₂²×C₃×Dic₅	—	2	4	no	—
36	C₃×#711	—	2	16	no	—
37	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
40	C₃×C₅×D₈	—	2	8	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊Dic₅	—	2	8	no	—
44	C₃×#687	—	2	16	no	—
45	C₃×C₄⋊Dic₅	—	2	4	no	—
46	C₃×C₄×Dic₅	D₄	1	2	yes	—
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	2	yes	—
49	C₃×Dic₂₀	—	2	8	no	—
50	C₃×C₈×D₅	D₄	1	8	yes	complemented
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	D₅	1	1	yes	characteristic, complemented
56	C₃×#492	—	5	2	no	—
57	C₂×C₃×C₄×C₈	—	5	1	no	—
58	C₂×#711	C₂²×C₃	1	4	yes	—
59	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
60	C₅×#127	C₂²×C₃	1	2	yes	—
61	#4084	—	2	4	no	—
62	#4082	C₂²×C₃	1	4	yes	—
63	D₄×Dic₅	—	2	4	no	—
64	C₂×#687	C₂²×C₃	1	8	yes	—
65	#4058	—	2	16	no	—
66	C₂×Dic₂₀	C₂²×C₃	1	4	yes	—
67	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
68	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
69	#4015	C₂²×C₃	1	2	yes	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₈×Dic₅	C₂²×C₃	1	2	yes	—
72	C₄×#168	C₂²×C₃	1	1	yes	characteristic
73	#3147	—	5	1	no	Sylow
74	C₂³×C₃×C₅	—	2	4	no	—
75	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	8	no	—
77	C₃×C₅×D₄	—	4	8	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₃×C₂²⋊D₅	—	4	8	no	—
81	C₂×C₃×Dic₅	—	2	4	no	—
82	C₂×C₃×Dic₅	—	4	8	no	—
83	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
85	C₃×C₄×D₅	—	2	4	no	—
86	C₃×C₄×D₅	C₂×D₄	1	4	yes	—
87	C₃×Dic₁₀	—	2	8	no	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₃×#168	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	C₂×D₅	1	4	yes	—
95	C₂²×C₃×C₈	—	5	2	no	—
96	C₃×C₄⋊Q₈	—	5	2	no	—
97	C₃×#127	C₂×D₅	1	2	yes	—
98	C₃×#124	—	5	4	no	—
99	C₃×#121	—	10	4	no	—
100	C₃×C₄×D₄	—	10	4	no	—
101	C₂×C₃×C₄²	—	5	1	no	—
102	C₃×C₄×C₈	—	5	2	no	—
103	C₃×C₄×C₈	—	5	1	no	—
104	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	C₂×C₅×D₄	—	2	4	no	—
106	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
107	C₂×C₂²⋊D₅	—	2	4	no	—
108	C₂²×Dic₅	—	2	4	no	—
109	#711	—	2	16	no	—
110	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
111	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
113	C₅×D₈	—	2	8	no	—
114	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
115	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²⋊Dic₅	—	2	8	no	—
117	#687	—	2	16	no	—
118	C₄⋊Dic₅	—	2	4	no	—
119	C₄×Dic₅	C₃×D₄	1	2	yes	—
120	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
121	C₂×#168	C₂³×C₃	1	2	yes	—
122	Dic₂₀	—	2	8	no	—
123	C₈×D₅	C₃×D₄	1	8	yes	complemented
124	C₂×#135	—	5	4	no	—
125	#532	—	5	2	no	—
126	#495	—	5	4	no	—
127	#494	—	5	1	no	—
128	#493	C₃×D₅	1	1	yes	characteristic, complemented
129	#492	—	5	2	no	—
130	C₂×C₄×C₈	—	5	1	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	—	4	8	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	2	no	—
135	C₂×C₃×D₅	#127	1	2	yes	—
136	C₃×C₄×C₅	#135	1	4	yes	complemented
137	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
138	C₃×Dic₅	—	2	2	no	—
139	C₃×Dic₅	—	4	4	no	—
140	C₃×Dic₅	#127	1	2	yes	—
141	C₃×#41	—	10	16	no	—
142	C₂×C₃×Q₈	—	5	4	no	—
143	C₂×C₃×D₄	—	2	4	no	—
144	C₂×C₃×D₄	—	10	4	no	—
145	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	2	no	—
148	C₂²×C₃×C₄	—	10	4	no	—
149	C₃×Q₁₆	—	10	8	no	—
150	C₃×SD₁₆	—	10	16	no	—
151	C₃×D₈	—	2	8	no	—
152	C₂×C₃×C₈	—	5	2	no	—
153	C₂×C₃×C₈	—	5	8	no	—
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	—	5	2	no	—
158	C₃×C₄²	—	5	1	no	—
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	C₂³×C₅	—	2	4	no	—
161	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₅×D₄	—	2	8	no	—
163	C₅×D₄	—	4	8	no	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
166	C₂²⋊D₅	—	4	8	no	—
167	C₂×Dic₅	—	2	4	no	—
168	C₂×Dic₅	—	4	8	no	—
169	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
170	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₄×D₅	—	2	4	no	—
172	C₄×D₅	C₂×C₃×D₄	1	4	yes	—
173	Dic₁₀	—	2	8	no	—
174	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
175	#168	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	C₂×C₃×D₅	1	4	yes	—
181	C₂²×C₈	—	5	2	no	—
182	C₄⋊Q₈	—	5	2	no	—
183	#127	C₂×C₃×D₅	1	2	yes	—
184	#124	—	5	4	no	—
185	#121	—	10	4	no	—
186	C₄×D₄	—	10	4	no	—
187	C₂×C₄²	—	5	1	no	—
188	C₄×C₈	—	5	2	no	—
189	C₄×C₈	—	5	1	no	—
190	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
191	C₂×C₃×C₅	—	8	4	no	—
192	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×#135	1	2	yes	—
194	C₃×D₅	—	2	2	no	—
195	C₂³×C₃	—	2	4	no	—
196	C₂³×C₃	—	5	1	no	—
197	C₃×Q₈	—	10	8	no	—
198	C₃×D₄	—	2	8	no	—
199	C₃×D₄	—	4	8	no	—
200	C₃×D₄	—	20	8	no	—
201	C₂×C₃×C₄	—	5	1	no	—
202	C₂×C₃×C₄	—	5	4	no	—
203	C₂×C₃×C₄	—	5	2	no	—
204	C₂×C₃×C₄	—	10	4	no	—
205	C₂×C₃×C₄	—	10	4	no	—
206	C₂×C₃×C₄	—	20	8	no	—
207	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₃×C₈	—	5	4	no	—
210	C₃×C₈	D₄×D₅	1	4	yes	—
211	C₂²×C₅	—	4	8	no	—
212	C₂²×C₅	—	4	4	no	—
213	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	C₃×#127	1	2	yes	—
216	C₄×C₅	C₃×#135	1	4	yes	complemented
217	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
218	Dic₅	—	2	2	no	—
219	Dic₅	—	4	4	no	—
220	Dic₅	C₃×#127	1	2	yes	—
221	#41	—	10	16	no	—
222	C₂×Q₈	—	5	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	10	4	no	—
225	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
226	C₂²×C₄	—	5	1	no	—
227	C₂²×C₄	—	5	2	no	—
228	C₂²×C₄	—	10	4	no	—
229	Q₁₆	—	10	8	no	—
230	SD₁₆	—	10	16	no	—
231	D₈	—	2	8	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	8	no	—
234	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
235	C₄⋊C₄	—	10	4	no	—
236	C₂²⋊C₄	—	10	8	no	—
237	C₄²	—	5	2	no	—
238	C₄²	—	5	1	no	—
239	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	4	4	no	—
242	C₂²×C₃	—	4	8	no	—
243	C₂²×C₃	—	5	2	no	—
244	C₂²×C₃	—	10	2	no	—
245	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
246	C₃×C₄	—	5	2	no	—
247	C₃×C₄	—	10	2	no	—
248	C₃×C₄	—	20	4	no	—
249	C₃×C₄	#4058	1	4	yes	complemented
250	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
251	C₂×C₅	—	8	4	no	—
252	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
253	C₂×C₅	C₂×C₃×#135	1	2	yes	—
254	D₅	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	5	1	no	—
257	Q₈	—	10	8	no	—
258	D₄	—	2	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	20	8	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	4	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	10	4	no	—
265	C₂×C₄	—	10	4	no	—
266	C₂×C₄	—	20	8	no	—
267	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
268	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
269	C₈	—	5	4	no	—
270	C₈	C₃×D₄×D₅	1	4	yes	—
271	C₂×C₃	—	8	4	no	—
272	C₂×C₃	—	10	2	no	—
273	C₂×C₃	D₅×#127	1	1	yes	characteristic
274	C₂×C₃	C₂×#4058	1	2	yes	—
275	C₅	C₃×#3147	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	4	4	no	—
277	C₂²	—	4	8	no	—
278	C₂²	—	5	2	no	—
279	C₂²	—	10	2	no	—
280	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
281	C₄	—	5	2	no	—
282	C₄	—	10	2	no	—
283	C₄	—	20	4	no	—
284	C₄	C₃×#4058	1	4	yes	complemented
285	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
286	C₃	#10613513	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	8	4	no	—
288	C₂	—	10	2	no	—
289	C₂	C₃×D₅×#127	1	1	yes	characteristic
290	C₂	C₂×C₃×#4058	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#493	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₃×#493	D₅
hypocenter	C₅	C₃×#3147
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#493

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4058	C₂	semidirect
2. C₃×#66050	C₂	semidirect
3. C₃×C₅×#493	C₂	semidirect
4. C₃×#65444	C₂	semidirect
5. C₃×#65365	C₂	semidirect
6. C₃×#64988	C₂	semidirect
7. C₃×C₄×C₈×D₅	C₂	semidirect
8. #10613513	C₃	direct (N × H)
25. C₂×#4058	C₂×C₃	semidirect
26. #66050	C₂×C₃	semidirect
27. C₅×#493	C₂×C₃	semidirect
28. #65444	C₂×C₃	semidirect
29. #65365	C₂×C₃	semidirect
30. #64988	C₂×C₃	semidirect
31. C₄×C₈×D₅	C₂×C₃	semidirect
50. C₃×C₈×D₅	D₄	semidirect
55. C₃×#493	D₅	semidirect
123. C₈×D₅	C₃×D₄	semidirect
128. #493	C₃×D₅	semidirect
136. C₃×C₄×C₅	#135	semidirect
216. C₄×C₅	C₃×#135	semidirect
240. C₃×C₅	#3147	semidirect
249. C₃×C₄	#4058	semidirect
275. C₅	C₃×#3147	semidirect
284. C₄	C₃×#4058	semidirect
286. C₃	#10613513	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×#4058	2
C₃×#66050	2
C₃×C₅×#493	2
C₃×#65444	2
C₃×#65365	2
C₃×#64988	2
C₃×C₄×C₈×D₅	2
#10613513	3
C₃×#3147	5

Subgroups — C3×#10613513 #14438893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613513 #14438893