Subgroups — C₂×C₃×#65725 #14478122

Order 1920.

736 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4053	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65725	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#3990	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×D₅⋊C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×#65725	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×#709	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₃×#4053	C₄	1	4	yes	—
12	C₃×#4053	C₂²	1	4	yes	—
13	C₂×C₃×#676	C₄	1	2	yes	—
14	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
15	C₃×#3998	C₂²	1	4	yes	complemented
16	C₂×C₃×#674	C₂²	1	1	yes	characteristic
17	C₂×C₃×#674	C₂²	1	2	yes	—
18	C₂×C₃×#674	C₂²	1	1	yes	characteristic
19	C₂×C₃×#674	C₂²	1	4	yes	complemented
20	C₃×#3990	C₂²	1	8	yes	complemented
21	C₃×D₅⋊C₁₆	C₂²	1	8	yes	complemented
22	C₂×C₃×#504	—	5	1	no	maximal, Hall
23	C₂×#4053	C₂×C₃	1	1	yes	characteristic
24	C₂×#3998	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×#674	C₂×C₃	1	1	yes	characteristic, complemented
26	#65725	C₂×C₃	1	8	yes	direct product factor
27	C₂×#3990	C₂×C₃	1	2	yes	complemented
28	C₂×D₅⋊C₁₆	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₂²⋊D₅	C₈	1	2	yes	—
31	C₃×#709	C₈	1	4	yes	—
32	C₃×#709	C₂×C₄	1	4	yes	—
33	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
35	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
36	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	2	yes	—
40	C₂×C₃×#168	C₂³	1	1	yes	characteristic
41	C₂×C₃×#168	C₂³	1	2	yes	—
42	C₃×#676	C₂×C₄	1	8	yes	—
43	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
44	C₃×C₈×D₅	C₂³	1	4	yes	—
45	C₃×#674	C₂³	1	2	yes	—
46	C₃×#674	C₂³	1	4	yes	—
47	C₃×#674	C₂³	1	2	yes	—
48	C₂×C₃×#131	—	5	1	no	—
49	C₃×#504	—	5	8	no	—
50	C₂×C₃×M₃₂	—	5	2	no	—
51	C₂×C₃×M₃₂	—	5	1	no	—
52	C₂²×C₃×C₁₆	—	5	2	no	—
53	C₂²×C₃×C₁₆	—	5	1	no	—
54	C₂×#709	C₃×C₄	1	1	yes	characteristic
55	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
56	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
57	#4053	C₃×C₄	1	4	yes	—
58	#4053	C₂²×C₃	1	4	yes	—
59	C₂×#676	C₃×C₄	1	2	yes	—
60	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
61	#3998	C₂²×C₃	1	4	yes	complemented
62	C₂×#674	C₂²×C₃	1	1	yes	characteristic
63	C₂×#674	C₂²×C₃	1	2	yes	—
64	C₂×#674	C₂²×C₃	1	1	yes	characteristic
65	C₂×#674	C₂²×C₃	1	4	yes	complemented
66	#3990	C₂²×C₃	1	8	yes	complemented
67	D₅⋊C₁₆	C₂²×C₃	1	8	yes	complemented
68	C₂×#504	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₃×D₅	C₂×C₈	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₂²⋊D₅	C₂×C₈	1	8	yes	—
76	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
77	C₃×D₂₀	C₂×C₈	1	4	yes	—
78	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
79	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
80	C₃×Dic₁₀	C₂×C₈	1	4	yes	—
81	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₃×#168	C₂²×C₄	1	1	yes	characteristic
83	C₃×#168	C₂²×C₄	1	2	yes	—
84	C₃×#168	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×#41	—	5	1	no	—
86	C₃×#131	—	5	4	no	—
87	C₃×#131	—	5	4	no	—
88	C₂×C₃×M₁₆	—	5	2	no	—
89	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₂²×C₃×C₈	—	5	1	no	—
91	C₂²×C₃×C₈	—	5	2	no	—
92	C₃×M₃₂	—	5	8	no	—
93	C₃×M₃₂	—	5	4	no	—
94	C₂×C₃×C₁₆	—	5	1	no	—
95	C₂×C₃×C₁₆	—	5	2	no	—
96	C₂×C₃×C₁₆	—	5	1	no	—
97	C₂×C₃×C₁₆	—	5	8	no	—
98	C₂×C₃×C₁₆	—	5	4	no	—
99	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	C₃×C₈	1	2	yes	—
101	#709	C₃×C₈	1	4	yes	—
102	#709	C₂×C₃×C₄	1	4	yes	—
103	C₂×D₂₀	C₃×C₈	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
105	C₂×Dic₁₀	C₃×C₈	1	1	yes	characteristic
106	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
107	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
108	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×#168	C₂×C₃×C₄	1	2	yes	—
110	C₂×#168	C₂³×C₃	1	1	yes	characteristic
111	C₂×#168	C₂³×C₃	1	2	yes	—
112	#676	C₂×C₃×C₄	1	8	yes	—
113	C₈×D₅	C₂×C₃×C₄	1	4	yes	—
114	C₈×D₅	C₂³×C₃	1	4	yes	—
115	#674	C₂³×C₃	1	2	yes	—
116	#674	C₂³×C₃	1	4	yes	—
117	#674	C₂³×C₃	1	2	yes	—
118	C₂×#131	—	5	1	no	—
119	#504	—	5	8	no	—
120	C₂×M₃₂	—	5	2	no	—
121	C₂×M₃₂	—	5	1	no	—
122	C₂²×C₁₆	—	5	2	no	—
123	C₂²×C₁₆	—	5	1	no	—
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
126	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₃×D₅	—	2	4	no	—
128	C₂×C₃×D₅	C₂²×C₈	1	4	yes	—
129	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
130	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
131	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₃×Dic₅	C₂²×C₈	1	4	yes	—
133	C₃×#41	—	5	4	no	—
134	C₃×#41	—	5	4	no	—
135	C₂×C₃×Q₈	—	5	1	no	—
136	C₂×C₃×D₄	—	5	2	no	—
137	C₂×C₃×D₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	2	no	—
139	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₃×M₁₆	—	5	8	no	—
141	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	4	yes	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	—	5	4	no	—
147	C₂×C₃×C₈	—	5	4	no	—
148	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
149	C₃×C₁₆	—	5	2	no	—
150	C₃×C₁₆	—	5	4	no	—
151	C₃×C₁₆	—	5	2	no	—
152	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
153	C₂²×D₅	C₂×C₃×C₈	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
156	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
158	C₂²⋊D₅	C₂×C₃×C₈	1	8	yes	—
159	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
160	D₂₀	C₂×C₃×C₈	1	4	yes	—
161	C₄×D₅	C₂×C₃×C₈	1	4	yes	—
162	C₄×D₅	C₂²×C₃×C₄	1	4	yes	—
163	Dic₁₀	C₂×C₃×C₈	1	4	yes	—
164	C₅×C₈	C₂²×C₃×C₄	1	4	yes	—
165	#168	C₂²×C₃×C₄	1	1	yes	characteristic
166	#168	C₂²×C₃×C₄	1	2	yes	—
167	#168	C₂⁴×C₃	1	1	yes	characteristic
168	C₂×#41	—	5	1	no	—
169	#131	—	5	4	no	—
170	#131	—	5	4	no	—
171	C₂×M₁₆	—	5	2	no	—
172	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂²×C₈	—	5	1	no	—
174	C₂²×C₈	—	5	2	no	—
175	M₃₂	—	5	8	no	—
176	M₃₂	—	5	4	no	—
177	C₂×C₁₆	—	5	8	no	—
178	C₂×C₁₆	—	5	1	no	—
179	C₂×C₁₆	—	5	2	no	—
180	C₂×C₁₆	—	5	1	no	—
181	C₂×C₁₆	—	5	4	no	—
182	C₂×C₃×C₅	—	2	2	no	—
183	C₂×C₃×C₅	#504	1	2	yes	complemented
184	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
185	C₃×D₅	—	2	4	no	—
186	C₂³×C₃	—	5	2	no	—
187	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
188	C₃×Q₈	—	5	4	no	—
189	C₃×D₄	—	5	8	no	—
190	C₃×D₄	—	5	4	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	4	no	—
193	C₂×C₃×C₄	—	5	4	no	—
194	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
196	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
197	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₃×C₈	—	5	1	no	—
199	C₃×C₈	—	5	2	no	—
200	C₃×C₈	—	5	1	no	—
201	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	C₂²×C₃×C₈	1	2	yes	—
204	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
205	C₂×D₅	—	2	4	no	—
206	C₂×D₅	C₂²×C₃×C₈	1	4	yes	—
207	C₄×C₅	C₂²×C₃×C₈	1	2	yes	—
208	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
209	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
210	Dic₅	C₂²×C₃×C₈	1	4	yes	—
211	#41	—	5	4	no	—
212	#41	—	5	4	no	—
213	C₂×Q₈	—	5	1	no	—
214	C₂×D₄	—	5	2	no	—
215	C₂×D₄	—	5	1	no	—
216	C₂²×C₄	—	5	2	no	—
217	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
218	M₁₆	—	5	8	no	—
219	M₁₆	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
220	C₂×C₈	—	5	1	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	2	no	—
224	C₂×C₈	—	5	4	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
227	C₁₆	—	5	2	no	—
228	C₁₆	—	5	4	no	—
229	C₁₆	—	5	2	no	—
230	C₃×C₅	C₂×#504	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	5	4	no	—
233	C₂²×C₃	—	10	4	no	—
234	C₂²×C₃	C₂×D₅⋊C₈	1	2	yes	—
235	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
236	C₃×C₄	—	5	4	no	—
237	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
238	C₃×C₄	C₂×D₅⋊C₈	1	2	yes	—
239	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	C₃×#504	1	2	yes	complemented
242	C₂×C₅	C₂³×C₃×C₈	1	1	yes	characteristic
243	D₅	—	2	4	no	—
244	C₂³	—	5	2	no	—
245	C₂³	C₃×D₅⋊C₈	1	1	yes	characteristic
246	Q₈	—	5	4	no	—
247	D₄	—	5	8	no	—
248	D₄	—	5	4	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	5	4	no	—
251	C₂×C₄	—	5	4	no	—
252	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₃×D₅⋊C₈	1	2	yes	—
254	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
256	C₈	—	5	1	no	—
257	C₈	—	5	1	no	—
258	C₈	—	5	2	no	—
259	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	10	4	no	—
262	C₂×C₃	#65725	1	2	yes	direct product factor
263	C₂×C₃	C₂²×D₅⋊C₈	1	1	yes	characteristic
264	C₅	C₂×C₃×#504	1	1	yes	characteristic, complemented, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	5	4	no	—
267	C₂²	—	10	4	no	—
268	C₂²	C₂×C₃×D₅⋊C₈	1	2	yes	—
269	C₂²	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
270	C₄	—	5	4	no	—
271	C₄	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
272	C₄	C₂×C₃×D₅⋊C₈	1	2	yes	—
273	C₄	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
274	C₃	C₂×#65725	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	10	4	no	—
277	C₂	C₃×#65725	1	2	yes	direct product factor
278	C₂	C₂²×C₃×D₅⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₃×C₈
Frattini	C₄	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×M₁₆	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×#709	C₄
cosocle	#168	C₂⁴×C₃
hypercenter	C₂×C₃×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#504
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3998	C₂	semidirect
3. C₂²×C₃×#674	C₂	semidirect
4. C₃×#65725	C₂	direct (N × H)
5. C₂×C₃×#3990	C₂	semidirect
6. C₂×C₃×D₅⋊C₁₆	C₂	semidirect
7. C₂×#65725	C₃	direct (N × H)
15. C₃×#3998	C₂²	semidirect
19. C₂×C₃×#674	C₂²	semidirect
20. C₃×#3990	C₂²	semidirect
21. C₃×D₅⋊C₁₆	C₂²	semidirect
24. C₂×#3998	C₂×C₃	semidirect
25. C₂²×#674	C₂×C₃	semidirect
26. #65725	C₂×C₃	direct (N × H)
27. C₂×#3990	C₂×C₃	semidirect
28. C₂×D₅⋊C₁₆	C₂×C₃	semidirect
61. #3998	C₂²×C₃	semidirect
65. C₂×#674	C₂²×C₃	semidirect
66. #3990	C₂²×C₃	semidirect
67. D₅⋊C₁₆	C₂²×C₃	semidirect
183. C₂×C₃×C₅	#504	semidirect
230. C₃×C₅	C₂×#504	semidirect
241. C₂×C₅	C₃×#504	semidirect
262. C₂×C₃	#65725	direct (N × H)
264. C₅	C₂×C₃×#504	semidirect
274. C₃	C₂×#65725	direct (N × H)
277. C₂	C₃×#65725	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×#4053	2
C₂×C₃×#3998	2
C₂²×C₃×#674	2
C₃×#65725	2
C₂×C₃×#3990	2
C₂×C₃×D₅⋊C₁₆	2
C₂×#65725	3
C₂×C₃×#504	5

Subgroups — C2×C3×#65725 #14478122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65725 #14478122