Subgroups — C₂×C₃×#64844 #14312548

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×#64844	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64844	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
7	C₂×C₃×D₅⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#674	C₄	1	2	yes	complemented
10	C₃×C₈×AGL(1,𝔽₅)	C₄	1	2	yes	—
11	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
12	C₃×D₅⋊C₁₆	C₄	1	2	yes	complemented
13	C₃×D₅⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×#672	C₄	1	1	yes	characteristic, complemented
16	C₃×D₅×C₁₆	C₄	1	2	yes	complemented
17	C₃×D₅×C₁₆	C₂²	1	2	yes	—
18	C₂×C₃×#346	—	5	1	no	maximal, Hall
19	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
20	C₂×D₅⋊C₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic
22	#64844	C₂×C₃	1	4	yes	direct product factor
23	C₂²×C₃×AGL(1,𝔽₅)	C₈	1	2	yes	—
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×#170	C₈	1	2	yes	—
26	C₃×C₄×AGL(1,𝔽₅)	C₈	1	2	yes	—
27	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
28	C₃×D₅⋊C₈	C₈	1	2	yes	—
29	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
36	C₃×#674	C₂×C₄	1	4	yes	complemented
37	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
38	C₃×#672	C₂×C₄	1	2	yes	complemented
39	C₂²×C₃×C₁₆	—	5	1	no	—
40	C₂²×C₃×C₁₆	—	5	1	no	—
41	C₂×C₃×C₄×C₈	—	5	1	no	—
42	C₃×#346	—	5	4	no	—
43	C₂×C₄×AGL(1,𝔽₅)	C₃×C₄	1	1	yes	characteristic
44	C₂×D₅⋊C₈	C₃×C₄	1	1	yes	characteristic
45	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×#674	C₃×C₄	1	2	yes	complemented
47	C₈×AGL(1,𝔽₅)	C₃×C₄	1	2	yes	—
48	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
49	D₅⋊C₁₆	C₃×C₄	1	2	yes	complemented
50	D₅⋊C₁₆	C₂²×C₃	1	2	yes	—
51	C₂×C₅×C₁₆	C₃×C₄	1	1	yes	characteristic, complemented
52	C₂×#672	C₃×C₄	1	1	yes	characteristic, complemented
53	D₅×C₁₆	C₃×C₄	1	2	yes	complemented
54	D₅×C₁₆	C₂²×C₃	1	2	yes	—
55	C₂×#346	—	5	1	no	Sylow
56	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
58	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
61	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
62	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
63	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
64	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
65	C₃×#170	C₂×C₈	1	4	yes	—
66	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
67	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₃×#168	C₄²	1	1	yes	characteristic
69	C₃×#168	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₈	—	5	1	no	—
71	C₂²×C₃×C₈	—	5	1	no	—
72	C₂×C₃×C₄²	—	5	1	no	—
73	C₂×C₃×C₁₆	—	5	2	no	—
74	C₂×C₃×C₁₆	—	5	2	no	—
75	C₂×C₃×C₁₆	—	5	2	no	—
76	C₂×C₃×C₁₆	—	5	1	no	—
77	C₂×C₃×C₁₆	—	5	2	no	—
78	C₂×C₃×C₁₆	—	5	2	no	—
79	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₃×C₄×C₈	—	5	2	no	—
81	C₃×C₄×C₈	—	5	2	no	—
82	C₂²×AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
83	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×#170	C₃×C₈	1	2	yes	—
85	C₄×AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
86	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
87	D₅⋊C₈	C₃×C₈	1	2	yes	—
88	D₅⋊C₈	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
91	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
95	#674	C₂×C₃×C₄	1	4	yes	complemented
96	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
97	#672	C₂×C₃×C₄	1	2	yes	complemented
98	C₂²×C₁₆	—	5	1	no	—
99	C₂²×C₁₆	—	5	1	no	—
100	C₂×C₄×C₈	—	5	1	no	—
101	#346	—	5	4	no	—
102	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
103	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
104	C₂×C₃×D₅	M₃₂	1	4	yes	—
105	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
106	C₃×AGL(1,𝔽₅)	—	2	4	no	—
107	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
109	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
110	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	5	1	no	—
112	C₂²×C₃×C₄	—	5	2	no	—
113	C₂×C₃×C₈	—	5	2	no	—
114	C₂×C₃×C₈	—	5	2	no	—
115	C₂×C₃×C₈	—	5	2	no	—
116	C₂×C₃×C₈	—	5	1	no	—
117	C₂×C₃×C₈	—	5	1	no	—
118	C₂×C₃×C₈	—	5	1	no	—
119	C₂×C₃×C₈	—	5	1	no	—
120	C₂×C₃×C₈	—	5	1	no	—
121	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₃×C₄²	—	5	2	no	—
123	C₃×C₄²	—	5	2	no	—
124	C₃×C₁₆	—	5	4	no	—
125	C₃×C₁₆	—	5	2	no	—
126	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
127	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
128	C₂×AGL(1,𝔽₅)	—	2	4	no	—
129	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	4	yes	—
130	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
131	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
132	C₄×D₅	C₃×C₄²	1	1	yes	characteristic
133	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
134	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
135	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	#170	C₂×C₃×C₈	1	4	yes	—
137	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
138	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
139	#168	C₃×C₄²	1	1	yes	characteristic
140	#168	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	5	1	no	—
143	C₂×C₄²	—	5	1	no	—
144	C₂×C₁₆	—	5	2	no	—
145	C₂×C₁₆	—	5	2	no	—
146	C₂×C₁₆	—	5	2	no	—
147	C₂×C₁₆	—	5	1	no	—
148	C₂×C₁₆	—	5	2	no	—
149	C₂×C₁₆	—	5	2	no	—
150	C₂×C₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
151	C₄×C₈	—	5	2	no	—
152	C₄×C₈	—	5	2	no	—
153	C₂×C₃×C₅	#346	1	2	yes	complemented
154	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
155	C₃×D₅	#346	1	2	yes	complemented
156	C₃×D₅	C₂×M₃₂	1	2	yes	—
157	C₂³×C₃	—	5	1	no	—
158	C₂×C₃×C₄	—	5	1	no	—
159	C₂×C₃×C₄	—	5	1	no	—
160	C₂×C₃×C₄	—	5	1	no	—
161	C₂×C₃×C₄	—	5	1	no	—
162	C₂×C₃×C₄	—	5	1	no	—
163	C₂×C₃×C₄	—	5	4	no	—
164	C₂×C₃×C₄	—	10	4	no	—
165	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₃×C₈	—	5	4	no	—
167	C₃×C₈	—	5	1	no	—
168	C₃×C₈	—	5	1	no	—
169	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
172	C₂×D₅	C₃×C₄×C₈	1	1	yes	characteristic
173	C₂×D₅	C₃×M₃₂	1	4	yes	—
174	C₂×D₅	C₂²×C₃×C₈	1	1	yes	characteristic
175	AGL(1,𝔽₅)	—	2	4	no	—
176	C₄×C₅	C₃×C₄×C₈	1	1	yes	characteristic
177	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
178	Dic₅	C₃×C₄×C₈	1	1	yes	characteristic
179	Dic₅	C₂²×C₃×C₈	1	1	yes	characteristic
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂×C₈	—	5	2	no	—
183	C₂×C₈	—	5	2	no	—
184	C₂×C₈	—	5	2	no	—
185	C₂×C₈	—	5	1	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	5	1	no	—
188	C₂×C₈	—	5	1	no	—
189	C₂×C₈	—	5	1	no	—
190	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
191	C₄²	—	5	2	no	—
192	C₄²	—	5	2	no	—
193	C₁₆	—	5	4	no	—
194	C₁₆	—	5	2	no	—
195	C₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
196	C₃×C₅	C₂×#346	1	1	yes	characteristic, complemented, Hall
197	C₂²×C₃	—	5	4	no	—
198	C₂²×C₃	—	5	1	no	—
199	C₂²×C₃	—	5	1	no	—
200	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₃×C₄	—	5	1	no	—
202	C₃×C₄	—	5	1	no	—
203	C₃×C₄	—	10	4	no	—
204	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₂×C₅	C₃×#346	1	2	yes	complemented
207	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
208	D₅	C₃×#346	1	2	yes	complemented
209	D₅	C₂×C₃×M₃₂	1	2	yes	—
210	C₂³	—	5	1	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	5	1	no	—
213	C₂×C₄	—	5	1	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	5	4	no	—
217	C₂×C₄	—	10	4	no	—
218	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₈	—	5	4	no	—
220	C₈	—	5	1	no	—
221	C₈	—	5	1	no	—
222	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₂×C₃	—	5	2	no	—
225	C₂×C₃	—	5	2	no	—
226	C₂×C₃	#64844	1	2	yes	direct product factor
227	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₅	C₂×C₃×#346	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	5	1	no	—
230	C₂²	—	5	1	no	—
231	C₂²	—	5	4	no	—
232	C₂²	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₄	—	5	1	no	—
234	C₄	—	5	1	no	—
235	C₄	—	10	4	no	—
236	C₄	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
237	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
238	C₃	C₂×#64844	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	5	2	no	—
240	C₂	—	5	2	no	—
241	C₂	C₃×#64844	1	2	yes	direct product factor
242	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₈	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂²×C₃×C₅	C₄×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₈×D₅	C₂³×C₃
hypercenter	C₂×C₃×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#346
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×#64844	C₂	direct (N × H)
5. C₂×#64844	C₃	direct (N × H)
9. C₂×C₃×#674	C₄	semidirect
12. C₃×D₅⋊C₁₆	C₄	semidirect
14. C₂×C₃×C₅×C₁₆	C₄	semidirect
15. C₂×C₃×#672	C₄	semidirect
16. C₃×D₅×C₁₆	C₄	semidirect
22. #64844	C₂×C₃	direct (N × H)
36. C₃×#674	C₂×C₄	semidirect
37. C₃×C₅×C₁₆	C₂×C₄	semidirect
38. C₃×#672	C₂×C₄	semidirect
46. C₂×#674	C₃×C₄	semidirect
49. D₅⋊C₁₆	C₃×C₄	semidirect
51. C₂×C₅×C₁₆	C₃×C₄	semidirect
52. C₂×#672	C₃×C₄	semidirect
53. D₅×C₁₆	C₃×C₄	semidirect
79. C₂×C₃×C₁₆	AGL(1,𝔽₅)	semidirect
95. #674	C₂×C₃×C₄	semidirect
96. C₅×C₁₆	C₂×C₃×C₄	semidirect
97. #672	C₂×C₃×C₄	semidirect
126. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
150. C₂×C₁₆	C₃×AGL(1,𝔽₅)	semidirect
153. C₂×C₃×C₅	#346	semidirect
155. C₃×D₅	#346	semidirect
195. C₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
196. C₃×C₅	C₂×#346	semidirect
206. C₂×C₅	C₃×#346	semidirect
208. D₅	C₃×#346	semidirect
226. C₂×C₃	#64844	direct (N × H)
228. C₅	C₂×C₃×#346	semidirect
238. C₃	C₂×#64844	direct (N × H)
241. C₂	C₃×#64844	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×C₈×AGL(1,𝔽₅)	2
C₂×C₃×D₅⋊C₁₆	2
C₂×C₃×D₅×C₁₆	2
C₃×#64844	2
C₂×#64844	3
C₂×C₃×#346	5

Subgroups — C2×C3×#64844 #14312548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64844 #14312548