Subgroups — C₂×C₃×#65365 #14438728

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₂²×C₃×#687	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65365	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65365	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₃×C₅×#127	—	2	2	no	—
11	C₃×C₅×#127	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#687	—	2	16	no	—
14	C₂×C₃×#687	C₂²	1	16	yes	complemented
15	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₃×#4015	—	2	2	no	—
18	C₃×#4015	C₂²	1	4	yes	complemented
19	C₃×C₄×#168	C₂²	1	4	yes	complemented
20	C₂×C₃×#492	—	5	1	no	maximal, Hall
21	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
22	C₂²×#687	C₂×C₃	1	4	yes	complemented
23	C₂×#4015	C₂×C₃	1	1	yes	characteristic, complemented
24	#65365	C₂×C₃	1	8	yes	direct product factor
25	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	8	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	4	8	no	—
31	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	—	2	2	no	—
35	C₂×C₃×Dic₁₀	—	2	8	no	—
36	C₂×C₃×Dic₁₀	—	2	2	no	—
37	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₃×#687	—	2	32	no	—
41	C₃×C₄⋊Dic₅	—	2	8	no	—
42	C₂×C₃×#168	D₄	1	8	yes	complemented
43	C₂×C₃×#168	C₂³	1	4	yes	—
44	C₂²×C₃×SD₁₆	—	5	4	no	—
45	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
46	C₂×C₃×#127	D₅	1	1	yes	characteristic
47	C₃×#492	—	5	8	no	—
48	C₂×C₃×C₄×C₈	—	5	1	no	—
49	C₂²×C₅×D₄	—	2	2	no	—
50	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
51	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
52	C₅×#127	—	2	2	no	—
53	C₅×#127	C₂²×C₃	1	4	yes	—
54	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×#687	—	2	16	no	—
56	C₂×#687	C₂²×C₃	1	16	yes	complemented
57	C₂×C₄⋊Dic₅	—	2	2	no	—
58	C₂²×#168	C₂²×C₃	1	2	yes	—
59	#4015	—	2	2	no	—
60	#4015	C₂²×C₃	1	4	yes	complemented
61	C₄×#168	C₂²×C₃	1	4	yes	complemented
62	C₂×#492	—	5	1	no	Sylow
63	C₂³×C₃×C₅	—	2	4	no	—
64	C₂³×C₃×C₅	—	4	4	no	—
65	C₂³×C₃×C₅	—	4	8	no	—
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×D₄	—	2	8	no	—
68	C₃×C₅×D₄	—	4	4	no	—
69	C₃×C₅×D₄	—	4	16	no	—
70	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×Dic₅	—	2	4	no	—
76	C₂×C₃×Dic₅	—	4	4	no	—
77	C₃×Dic₁₀	—	2	8	no	—
78	C₃×Dic₁₀	—	4	4	no	—
79	C₃×#168	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×Q₈	—	5	2	no	—
81	C₂²×C₃×D₄	—	2	2	no	—
82	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
83	C₂×C₃×SD₁₆	—	5	16	no	—
84	C₂×C₃×SD₁₆	—	10	16	no	—
85	C₂²×C₃×C₈	—	5	2	no	—
86	C₃×C₄⋊Q₈	—	5	4	no	—
87	C₃×C₄⋊Q₈	—	10	2	no	—
88	C₃×#127	—	2	2	no	—
89	C₃×#127	C₂×D₅	1	4	yes	—
90	C₂×C₃×C₄⋊C₄	—	10	2	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄×C₈	—	5	4	no	—
93	C₂⁴×C₅	—	2	2	no	—
94	C₂×C₅×D₄	—	2	8	no	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	2	8	no	—
97	C₂×C₅×D₄	—	4	8	no	—
98	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
99	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
100	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₂²×Dic₅	—	2	2	no	—
102	C₂×Dic₁₀	—	2	8	no	—
103	C₂×Dic₁₀	—	2	2	no	—
104	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
105	C₄²×C₅	C₃×D₄	1	2	yes	—
106	C₄²×C₅	C₂³×C₃	1	2	yes	—
107	#687	—	2	32	no	—
108	C₄⋊Dic₅	—	2	8	no	—
109	C₂×#168	C₃×D₄	1	8	yes	complemented
110	C₂×#168	C₂³×C₃	1	4	yes	—
111	C₂²×SD₁₆	—	5	4	no	—
112	C₂×C₄⋊Q₈	—	5	1	no	—
113	C₂×#127	C₃×D₅	1	1	yes	characteristic
114	#492	—	5	8	no	—
115	C₂×C₄×C₈	—	5	1	no	—
116	C₂²×C₃×C₅	—	4	4	no	—
117	C₂²×C₃×C₅	—	4	8	no	—
118	C₂²×C₃×C₅	—	8	8	no	—
119	C₂²×C₃×C₅	#127	1	2	yes	—
120	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
121	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₃×C₄×C₅	#127	1	2	yes	—
123	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
124	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
125	C₃×Dic₅	—	4	4	no	—
126	C₂⁴×C₃	—	2	2	no	—
127	C₂×C₃×Q₈	—	5	4	no	—
128	C₂×C₃×Q₈	—	10	2	no	—
129	C₂×C₃×Q₈	—	10	8	no	—
130	C₂×C₃×D₄	—	2	8	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	8	no	—
133	C₂×C₃×D₄	—	4	8	no	—
134	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
135	C₂²×C₃×C₄	—	10	2	no	—
136	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
138	C₃×SD₁₆	—	10	32	no	—
139	C₂×C₃×C₈	—	5	8	no	—
140	C₂×C₃×C₈	—	5	4	no	—
141	C₃×C₄⋊C₄	—	10	8	no	—
142	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
143	C₃×C₄²	C₂²×D₅	1	2	yes	—
144	C₂³×C₅	—	2	4	no	—
145	C₂³×C₅	—	4	4	no	—
146	C₂³×C₅	—	4	8	no	—
147	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₅×D₄	—	2	8	no	—
149	C₅×D₄	—	4	4	no	—
150	C₅×D₄	—	4	16	no	—
151	C₂×C₄×C₅	C₃×SD₁₆	1	8	yes	complemented
152	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
155	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
156	C₂×Dic₅	—	2	4	no	—
157	C₂×Dic₅	—	4	4	no	—
158	Dic₁₀	—	2	8	no	—
159	Dic₁₀	—	4	4	no	—
160	#168	C₂×C₃×D₄	1	8	yes	complemented
161	C₂²×Q₈	—	5	2	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
164	C₂×SD₁₆	—	5	16	no	—
165	C₂×SD₁₆	—	10	16	no	—
166	C₂²×C₈	—	5	2	no	—
167	C₄⋊Q₈	—	5	4	no	—
168	C₄⋊Q₈	—	10	2	no	—
169	#127	—	2	2	no	—
170	#127	C₂×C₃×D₅	1	4	yes	—
171	C₂×C₄⋊C₄	—	10	2	no	—
172	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
173	C₄×C₈	—	5	4	no	—
174	C₂×C₃×C₅	—	8	4	no	—
175	C₂×C₃×C₅	#492	1	4	yes	complemented
176	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	—	4	8	no	—
180	C₂³×C₃	—	4	4	no	—
181	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₃×Q₈	—	10	8	no	—
183	C₃×Q₈	—	20	4	no	—
184	C₃×D₄	—	2	8	no	—
185	C₃×D₄	—	4	16	no	—
186	C₃×D₄	—	4	4	no	—
187	C₂×C₃×C₄	—	10	4	no	—
188	C₂×C₃×C₄	—	20	4	no	—
189	C₂×C₃×C₄	#687	1	8	yes	complemented
190	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
191	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
192	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
194	C₃×C₈	—	5	8	no	—
195	C₂²×C₅	—	4	8	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	8	no	—
198	C₂²×C₅	C₃×#127	1	2	yes	—
199	C₂²×C₅	C₂×C₃×SD₁₆	1	4	yes	—
200	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₄×C₅	C₃×#127	1	2	yes	—
202	C₄×C₅	C₂×C₃×SD₁₆	1	8	yes	complemented
203	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
204	Dic₅	—	4	4	no	—
205	C₂⁴	—	2	2	no	—
206	C₂×Q₈	—	5	4	no	—
207	C₂×Q₈	—	10	2	no	—
208	C₂×Q₈	—	10	8	no	—
209	C₂×D₄	—	2	8	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
214	C₂²×C₄	—	10	2	no	—
215	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
216	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
217	SD₁₆	—	10	32	no	—
218	C₂×C₈	—	5	8	no	—
219	C₂×C₈	—	5	4	no	—
220	C₄⋊C₄	—	10	8	no	—
221	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
222	C₄²	C₂²×C₃×D₅	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	4	8	no	—
225	C₂²×C₃	—	4	4	no	—
226	C₂²×C₃	—	8	8	no	—
227	C₂²×C₃	C₂×#687	1	4	yes	—
228	C₂²×C₃	#4086	1	2	yes	—
229	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
230	C₃×C₄	—	20	4	no	—
231	C₃×C₄	C₂×#687	1	8	yes	complemented
232	C₃×C₄	#4086	1	2	yes	—
233	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
234	C₂×C₅	—	8	4	no	—
235	C₂×C₅	C₃×#492	1	4	yes	complemented
236	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
237	C₂×C₅	C₂²×C₃×SD₁₆	1	2	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	4	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
242	Q₈	—	10	8	no	—
243	Q₈	—	20	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	10	4	no	—
248	C₂×C₄	—	20	4	no	—
249	C₂×C₄	C₃×#687	1	8	yes	complemented
250	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
251	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
252	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
254	C₈	—	5	8	no	—
255	C₂×C₃	—	8	4	no	—
256	C₂×C₃	#65365	1	4	yes	direct product factor
257	C₂×C₃	C₂²×#687	1	2	yes	—
258	C₂×C₃	C₂×#4086	1	1	yes	characteristic
259	C₅	C₂×C₃×#492	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	4	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	8	8	no	—
263	C₂²	C₂×C₃×#687	1	4	yes	—
264	C₂²	C₃×#4086	1	2	yes	—
265	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
266	C₄	—	20	4	no	—
267	C₄	C₂×C₃×#687	1	8	yes	complemented
268	C₄	C₃×#4086	1	2	yes	—
269	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
270	C₃	C₂×#65365	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	8	4	no	—
272	C₂	C₃×#65365	1	4	yes	direct product factor
273	C₂	C₂²×C₃×#687	1	2	yes	—
274	C₂	C₂×C₃×#4086	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×#492
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. C₂²×C₃×#687	C₂	semidirect
3. C₂×C₃×#4015	C₂	semidirect
4. C₃×#65365	C₂	direct (N × H)
5. C₂×C₃×C₄×#168	C₂	semidirect
6. C₂×#65365	C₃	direct (N × H)
14. C₂×C₃×#687	C₂²	semidirect
18. C₃×#4015	C₂²	semidirect
19. C₃×C₄×#168	C₂²	semidirect
22. C₂²×#687	C₂×C₃	semidirect
23. C₂×#4015	C₂×C₃	semidirect
24. #65365	C₂×C₃	direct (N × H)
25. C₂×C₄×#168	C₂×C₃	semidirect
42. C₂×C₃×#168	D₄	semidirect
56. C₂×#687	C₂²×C₃	semidirect
60. #4015	C₂²×C₃	semidirect
61. C₄×#168	C₂²×C₃	semidirect
70. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₃×#168	C₂×D₄	semidirect
109. C₂×#168	C₃×D₄	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
151. C₂×C₄×C₅	C₃×SD₁₆	semidirect
160. #168	C₂×C₃×D₄	semidirect
175. C₂×C₃×C₅	#492	semidirect
189. C₂×C₃×C₄	#687	semidirect
202. C₄×C₅	C₂×C₃×SD₁₆	semidirect
223. C₃×C₅	C₂×#492	semidirect
231. C₃×C₄	C₂×#687	semidirect
235. C₂×C₅	C₃×#492	semidirect
249. C₂×C₄	C₃×#687	semidirect
256. C₂×C₃	#65365	direct (N × H)
259. C₅	C₂×C₃×#492	semidirect
267. C₄	C₂×C₃×#687	semidirect
270. C₃	C₂×#65365	direct (N × H)
272. C₂	C₃×#65365	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
C₂²×C₃×#687	2
C₂×C₃×#4015	2
C₃×#65365	2
C₂×C₃×C₄×#168	2
C₂×#65365	3
C₂×C₃×#492	5

Subgroups — C2×C3×#65365 #14438728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65365 #14438728