Subgroups — C₂×C₃×#65201 #14478264

Order 1920.

736 conjugacy classes of subgroups.

The subgroup lattice has 272 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	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3959	C₂	1	2	yes	maximal, complemented
4	C₃×#65201	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×Dic₄₀	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65201	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×#711	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₃×#687	—	2	2	no	—
11	C₃×#4058	—	2	4	no	—
12	C₃×#4058	C₂²	1	8	yes	complemented
13	C₂×C₃×Dic₂₀	C₂²	1	2	yes	—
14	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₃×#3959	C₂²	1	8	yes	complemented
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic
19	C₃×Dic₄₀	C₂²	1	4	yes	complemented
20	C₃×D₅×C₁₆	C₂²	1	4	yes	complemented
21	C₂×C₃×#508	—	5	1	no	maximal, Hall
22	C₂×#4058	C₂×C₃	1	2	yes	complemented
23	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3959	C₂×C₃	1	2	yes	complemented
25	#65201	C₂×C₃	1	8	yes	direct product factor
26	C₂×Dic₄₀	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₂²⋊D₅	—	4	2	no	—
30	C₂²×C₃×Dic₅	—	4	2	no	—
31	C₃×#711	—	2	8	no	—
32	C₃×#711	—	4	4	no	—
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×Dic₁₀	—	2	2	no	—
35	C₃×C₅×D₈	—	2	2	no	—
36	C₃×C₅×D₈	C₂³	1	4	yes	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×#687	—	2	8	no	—
39	C₂×C₃×#168	D₄	1	1	yes	characteristic
40	C₃×Dic₂₀	—	2	2	no	—
41	C₃×Dic₂₀	C₂³	1	4	yes	—
42	C₃×C₈×D₅	D₄	1	2	yes	—
43	C₃×C₈×D₅	C₂³	1	2	yes	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	—
45	C₃×#672	C₂³	1	2	yes	—
46	C₂×C₃×#135	—	5	2	no	—
47	C₃×#508	—	5	8	no	—
48	C₂×C₃×Q₃₂	—	5	1	no	—
49	C₂×C₃×SD₃₂	—	5	2	no	—
50	C₂×C₃×D₁₆	D₅	1	1	yes	characteristic, complemented
51	C₂²×C₃×C₁₆	—	5	1	no	—
52	C₂×#711	—	2	2	no	—
53	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
54	C₂×#687	—	2	2	no	—
55	#4058	—	2	4	no	—
56	#4058	C₂²×C₃	1	8	yes	complemented
57	C₂×Dic₂₀	C₂²×C₃	1	2	yes	—
58	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×D₁₆	C₂²×C₃	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
61	#3959	C₂²×C₃	1	8	yes	complemented
62	C₂×#672	C₂²×C₃	1	1	yes	characteristic
63	Dic₄₀	C₂²×C₃	1	4	yes	complemented
64	D₅×C₁₆	C₂²×C₃	1	4	yes	complemented
65	C₂×#508	—	5	1	no	Sylow
66	C₂³×C₃×C₅	—	4	2	no	—
67	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
68	C₃×C₅×D₄	—	2	4	no	—
69	C₃×C₅×D₄	—	4	2	no	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₂²⋊D₅	—	4	8	no	—
72	C₂×C₃×Dic₅	—	4	2	no	—
73	C₂×C₃×Dic₅	—	4	8	no	—
74	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
75	C₃×C₄×D₅	D₈	1	2	yes	—
76	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
77	C₃×Dic₁₀	—	2	4	no	—
78	C₃×Dic₁₀	—	4	2	no	—
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
81	C₃×#168	C₂×D₄	1	2	yes	—
82	C₂×C₃×#41	—	10	2	no	—
83	C₃×#135	—	5	8	no	—
84	C₃×#135	—	10	4	no	—
85	C₂×C₃×Q₁₆	—	5	2	no	—
86	C₂×C₃×SD₁₆	—	10	2	no	—
87	C₂×C₃×D₈	C₂×D₅	1	2	yes	—
88	C₂²×C₃×C₈	—	5	1	no	—
89	C₃×Q₃₂	—	5	4	no	—
90	C₃×SD₃₂	—	5	8	no	—
91	C₃×D₁₆	C₂×D₅	1	4	yes	complemented
92	C₂×C₃×C₁₆	—	5	1	no	—
93	C₂×C₃×C₁₆	—	5	4	no	—
94	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₂²⋊D₅	—	4	2	no	—
97	C₂²×Dic₅	—	4	2	no	—
98	#711	—	2	8	no	—
99	#711	—	4	4	no	—
100	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
101	C₂×Dic₁₀	—	2	2	no	—
102	C₅×D₈	—	2	2	no	—
103	C₅×D₈	C₂³×C₃	1	4	yes	—
104	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
105	#687	—	2	8	no	—
106	C₂×#168	C₃×D₄	1	1	yes	characteristic
107	Dic₂₀	—	2	2	no	—
108	Dic₂₀	C₂³×C₃	1	4	yes	—
109	C₈×D₅	C₃×D₄	1	2	yes	—
110	C₈×D₅	C₂³×C₃	1	2	yes	—
111	C₅×C₁₆	C₂³×C₃	1	2	yes	—
112	#672	C₂³×C₃	1	2	yes	—
113	C₂×#135	—	5	2	no	—
114	#508	—	5	8	no	—
115	C₂×Q₃₂	—	5	1	no	—
116	C₂×SD₃₂	—	5	2	no	—
117	C₂×D₁₆	C₃×D₅	1	1	yes	characteristic, complemented
118	C₂²×C₁₆	—	5	1	no	—
119	C₂²×C₃×C₅	—	4	4	no	—
120	C₂²×C₃×C₅	—	8	4	no	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₃×D₅	—	2	2	no	—
123	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
124	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₃×Dic₅	—	4	4	no	—
127	C₃×Dic₅	C₂×D₈	1	2	yes	—
128	C₃×#41	—	10	8	no	—
129	C₃×#41	—	20	4	no	—
130	C₂×C₃×Q₈	—	10	2	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	20	2	no	—
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×C₃×C₄	—	20	2	no	—
135	C₃×Q₁₆	—	5	4	no	—
136	C₃×Q₁₆	—	10	2	no	—
137	C₃×SD₁₆	—	10	8	no	—
138	C₃×D₈	—	2	2	no	—
139	C₃×D₈	C₂²×D₅	1	4	yes	—
140	C₂×C₃×C₈	—	5	1	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₃×C₁₆	—	5	2	no	—
145	C₃×C₁₆	C₂²×D₅	1	2	yes	—
146	C₂³×C₅	—	4	2	no	—
147	C₂²×D₅	C₃×D₈	1	1	yes	characteristic
148	C₅×D₄	—	2	4	no	—
149	C₅×D₄	—	4	2	no	—
150	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂²⋊D₅	—	4	8	no	—
152	C₂×Dic₅	—	4	2	no	—
153	C₂×Dic₅	—	4	8	no	—
154	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
155	C₄×D₅	C₃×D₈	1	2	yes	—
156	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
157	Dic₁₀	—	2	4	no	—
158	Dic₁₀	—	4	2	no	—
159	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
160	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
161	#168	C₂×C₃×D₄	1	2	yes	—
162	C₂×#41	—	10	2	no	—
163	#135	—	5	8	no	—
164	#135	—	10	4	no	—
165	C₂×Q₁₆	—	5	2	no	—
166	C₂×SD₁₆	—	10	2	no	—
167	C₂×D₈	C₂×C₃×D₅	1	2	yes	—
168	C₂²×C₈	—	5	1	no	—
169	Q₃₂	—	5	4	no	—
170	SD₃₂	—	5	8	no	—
171	D₁₆	C₂×C₃×D₅	1	4	yes	complemented
172	C₂×C₁₆	—	5	1	no	—
173	C₂×C₁₆	—	5	4	no	—
174	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
175	C₂×C₃×C₅	—	8	4	no	—
176	C₂×C₃×C₅	#508	1	2	yes	complemented
177	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
178	C₃×D₅	—	2	2	no	—
179	C₂³×C₃	—	4	2	no	—
180	C₂³×C₃	—	5	1	no	—
181	C₃×Q₈	—	10	4	no	—
182	C₃×Q₈	—	20	2	no	—
183	C₃×D₄	—	2	4	no	—
184	C₃×D₄	—	4	2	no	—
185	C₃×D₄	—	20	8	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	5	2	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	20	8	no	—
190	C₂×C₃×C₄	—	20	2	no	—
191	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
192	C₃×C₈	—	5	2	no	—
193	C₃×C₈	D₄×D₅	1	1	yes	characteristic
194	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
195	C₂²×C₅	—	4	4	no	—
196	C₂²×C₅	—	8	4	no	—
197	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
198	C₂×D₅	—	2	2	no	—
199	C₂×D₅	C₂×C₃×D₈	1	2	yes	—
200	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
201	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
202	Dic₅	—	4	4	no	—
203	Dic₅	C₂×C₃×D₈	1	2	yes	—
204	#41	—	10	8	no	—
205	#41	—	20	4	no	—
206	C₂×Q₈	—	10	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	20	2	no	—
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	20	2	no	—
211	Q₁₆	—	5	4	no	—
212	Q₁₆	—	10	2	no	—
213	SD₁₆	—	10	8	no	—
214	D₈	—	2	2	no	—
215	D₈	C₂²×C₃×D₅	1	4	yes	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	5	2	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
220	C₁₆	—	5	2	no	—
221	C₁₆	C₂²×C₃×D₅	1	2	yes	—
222	C₃×C₅	C₂×#508	1	1	yes	characteristic, complemented, Hall
223	C₂²×C₃	—	4	4	no	—
224	C₂²×C₃	—	5	2	no	—
225	C₂²×C₃	—	8	4	no	—
226	C₂²×C₃	—	10	2	no	—
227	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
228	C₃×C₄	—	5	2	no	—
229	C₃×C₄	—	20	4	no	—
230	C₃×C₄	D₅×D₈	1	1	yes	characteristic
231	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
232	C₂×C₅	—	8	4	no	—
233	C₂×C₅	C₃×#508	1	2	yes	complemented
234	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
235	D₅	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	5	1	no	—
238	Q₈	—	10	4	no	—
239	Q₈	—	20	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	4	2	no	—
242	D₄	—	20	8	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	2	no	—
245	C₂×C₄	—	5	2	no	—
246	C₂×C₄	—	20	8	no	—
247	C₂×C₄	—	20	2	no	—
248	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
249	C₈	—	5	2	no	—
250	C₈	C₃×D₄×D₅	1	1	yes	characteristic
251	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
252	C₂×C₃	—	8	4	no	—
253	C₂×C₃	—	10	2	no	—
254	C₂×C₃	#65201	1	2	yes	direct product factor
255	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
256	C₅	C₂×C₃×#508	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	4	4	no	—
258	C₂²	—	5	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	—	10	2	no	—
261	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
262	C₄	—	5	2	no	—
263	C₄	—	20	4	no	—
264	C₄	C₃×D₅×D₈	1	1	yes	characteristic
265	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
266	C₃	C₂×#65201	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	8	4	no	—
268	C₂	—	10	2	no	—
269	C₂	C₃×#65201	1	2	yes	direct product factor
270	C₂	C₂×C₃×D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×D₈
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₃×D₁₆	D₅
hypocenter	C₅	C₂×C₃×#508
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4058	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₂×C₃×#3959	C₂	semidirect
4. C₃×#65201	C₂	direct (N × H)
5. C₂×C₃×Dic₄₀	C₂	semidirect
6. C₂×C₃×D₅×C₁₆	C₂	semidirect
7. C₂×#65201	C₃	direct (N × H)
12. C₃×#4058	C₂²	semidirect
15. C₃×C₅×D₁₆	C₂²	semidirect
17. C₃×#3959	C₂²	semidirect
19. C₃×Dic₄₀	C₂²	semidirect
20. C₃×D₅×C₁₆	C₂²	semidirect
22. C₂×#4058	C₂×C₃	semidirect
23. C₂×C₅×D₁₆	C₂×C₃	semidirect
24. C₂×#3959	C₂×C₃	semidirect
25. #65201	C₂×C₃	direct (N × H)
26. C₂×Dic₄₀	C₂×C₃	semidirect
27. C₂×D₅×C₁₆	C₂×C₃	semidirect
50. C₂×C₃×D₁₆	D₅	semidirect
56. #4058	C₂²×C₃	semidirect
59. C₅×D₁₆	C₂²×C₃	semidirect
61. #3959	C₂²×C₃	semidirect
63. Dic₄₀	C₂²×C₃	semidirect
64. D₅×C₁₆	C₂²×C₃	semidirect
91. C₃×D₁₆	C₂×D₅	semidirect
117. C₂×D₁₆	C₃×D₅	semidirect
171. D₁₆	C₂×C₃×D₅	semidirect
176. C₂×C₃×C₅	#508	semidirect
222. C₃×C₅	C₂×#508	semidirect
233. C₂×C₅	C₃×#508	semidirect
254. C₂×C₃	#65201	direct (N × H)
256. C₅	C₂×C₃×#508	semidirect
266. C₃	C₂×#65201	direct (N × H)
269. C₂	C₃×#65201	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
C₂×C₃×C₅×D₁₆	2
C₂×C₃×#3959	2
C₃×#65201	2
C₂×C₃×Dic₄₀	2
C₂×C₃×D₅×C₁₆	2
C₂×#65201	3
C₂×C₃×#508	5

Subgroups — C2×C3×#65201 #14478264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65201 #14478264