Subgroups — C₂²×C₄×#1200 #14478672

Order 1920.

6000 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×#1200	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Dic₃×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#1200	C₂	1	24	yes	maximal, direct product factor
7	C₂²×C₄²×D₅	—	3	1	no	maximal, Hall
8	C₂⁴×D₁₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₅	C₄	1	16	yes	complemented
12	C₂²×C₄×D₁₅	C₂²	1	12	yes	—
13	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
14	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
15	C₂²×#1200	C₄	1	32	yes	direct product factor
16	C₂²×#1200	C₂²	1	24	yes	—
17	C₂×C₄×C₅×Dic₃	C₂²	1	12	yes	complemented
18	C₂×C₃×C₄×Dic₅	C₂²	1	12	yes	complemented
19	C₂×Dic₃×Dic₅	C₂²	1	24	yes	complemented
20	C₄×#1200	C₂²	1	64	yes	direct product factor
21	C₂²×C₄²×D₃	—	5	1	no	maximal, Hall
22	C₂³×C₄×D₅	—	3	1	no	—
23	C₂³×C₄×D₅	—	3	2	no	—
24	C₂²×C₄²×C₅	—	3	1	no	—
25	C₂²×C₄×Dic₅	—	3	2	no	—
26	C₂²×C₄×Dic₅	D₃	1	1	yes	characteristic, complemented
27	C₂×C₄²×D₅	—	3	24	no	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂³×D₁₅	C₂×C₄	1	16	yes	—
30	C₂³×D₁₅	C₂×C₄	1	8	yes	—
31	C₂³×D₁₅	C₂³	1	6	yes	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	complemented
33	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
34	C₂²×Dic₁₅	C₂×C₄	1	8	yes	complemented
35	C₂²×Dic₁₅	C₂³	1	6	yes	—
36	C₂×C₄×D₁₅	C₂×C₄	1	96	yes	complemented
37	C₂×C₄×D₁₅	C₂³	1	16	yes	—
38	C₂²×C₅×Dic₃	C₂×C₄	1	16	yes	complemented
39	C₂²×C₅×Dic₃	C₂³	1	12	yes	—
40	C₂²×C₃×Dic₅	C₂×C₄	1	16	yes	complemented
41	C₂²×C₃×Dic₅	C₂³	1	12	yes	—
42	C₂×#1200	C₂×C₄	1	192	yes	direct product factor
43	C₂×#1200	C₂³	1	32	yes	—
44	C₄×C₅×Dic₃	C₂³	1	16	yes	complemented
45	C₃×C₄×Dic₅	C₂³	1	16	yes	complemented
46	Dic₃×Dic₅	C₂³	1	32	yes	complemented
47	C₂³×C₄×D₃	—	5	1	no	—
48	C₂³×C₄×D₃	—	5	2	no	—
49	C₂²×C₃×C₄²	—	5	1	no	—
50	C₂²×C₄×Dic₃	—	5	2	no	—
51	C₂²×C₄×Dic₃	D₅	1	1	yes	characteristic, complemented
52	C₂×C₄²×D₃	—	5	24	no	—
53	C₂⁴×D₅	—	3	1	no	—
54	C₂³×C₄×C₅	—	3	2	no	—
55	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂³×Dic₅	—	3	1	no	—
57	C₂³×Dic₅	C₂×D₃	1	2	yes	—
58	C₂²×C₄×D₅	—	3	16	no	—
59	C₂²×C₄×D₅	—	3	12	no	—
60	C₂²×C₄×D₅	—	3	32	no	—
61	C₂²×C₄×D₅	—	3	24	no	—
62	C₂×C₄²×C₅	—	3	12	no	—
63	C₂×C₄×Dic₅	—	3	24	no	—
64	C₂×C₄×Dic₅	C₂×D₃	1	12	yes	complemented
65	C₄²×D₅	—	3	64	no	—
66	C₂³×C₄²	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
68	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
69	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
70	C₂²×D₁₅	C₄²	1	64	yes	complemented
71	C₂²×D₁₅	C₂²×C₄	1	48	yes	—
72	C₂²×D₁₅	C₂²×C₄	1	24	yes	—
73	C₂²×D₁₅	C₂⁴	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	24	yes	complemented
75	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
76	C₂×Dic₁₅	C₂²×C₄	1	24	yes	complemented
77	C₂×Dic₁₅	C₂⁴	1	4	yes	—
78	C₄×D₁₅	C₂²×C₄	1	64	yes	complemented
79	C₂×C₅×Dic₃	C₂²×C₄	1	48	yes	complemented
80	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
81	C₂×C₃×Dic₅	C₂²×C₄	1	48	yes	complemented
82	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
83	#1200	C₂²×C₄	1	128	yes	direct product factor
84	C₂⁴×D₃	—	5	1	no	—
85	C₂³×C₃×C₄	—	5	2	no	—
86	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
87	C₂³×Dic₃	—	5	1	no	—
88	C₂³×Dic₃	C₂×D₅	1	2	yes	—
89	C₂²×C₄×D₃	—	5	16	no	—
90	C₂²×C₄×D₃	—	5	12	no	—
91	C₂²×C₄×D₃	—	5	32	no	—
92	C₂²×C₄×D₃	—	5	24	no	—
93	C₂×C₃×C₄²	—	5	12	no	—
94	C₂×C₄×Dic₃	—	5	24	no	—
95	C₂×C₄×Dic₃	C₂×D₅	1	12	yes	complemented
96	C₄²×D₃	—	5	64	no	—
97	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₂³×D₅	—	3	16	no	—
99	C₂³×D₅	—	3	8	no	—
100	C₂³×D₅	—	3	6	no	—
101	C₂²×C₄×C₅	—	3	16	no	—
102	C₂²×C₄×C₅	—	3	12	no	—
103	C₂²×C₄×C₅	C₄×D₃	1	8	yes	complemented
104	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
105	C₂²×Dic₅	—	3	8	no	—
106	C₂²×Dic₅	—	3	6	no	—
107	C₂²×Dic₅	C₄×D₃	1	16	yes	complemented
108	C₂²×Dic₅	C₂²×D₃	1	12	yes	—
109	C₂×C₄×D₅	—	3	96	no	—
110	C₂×C₄×D₅	—	3	16	no	—
111	C₂×C₄×D₅	—	3	192	no	—
112	C₂×C₄×D₅	—	3	32	no	—
113	C₄²×C₅	—	3	16	no	—
114	C₄×Dic₅	—	3	32	no	—
115	C₄×Dic₅	C₂²×D₃	1	16	yes	complemented
116	C₂⁴×C₄	—	15	2	no	—
117	C₂⁴×C₄	—	15	1	no	—
118	C₂²×C₄²	—	3	1	no	—
119	C₂²×C₄²	—	5	1	no	—
120	C₂²×C₄²	—	15	24	no	—
121	C₂²×C₄²	—	15	2	no	—
122	C₂²×C₃×C₅	C₂×C₄²	1	16	yes	complemented
123	C₂²×C₃×C₅	C₂³×C₄	1	12	yes	—
124	C₂²×C₃×C₅	C₂³×C₄	1	6	yes	—
125	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₅	C₂×C₄²	1	96	yes	complemented
127	C₂×D₁₅	C₂³×C₄	1	16	yes	—
128	C₂×D₁₅	C₂³×C₄	1	8	yes	—
129	C₃×C₄×C₅	C₂³×C₄	1	8	yes	complemented
130	Dic₁₅	C₂³×C₄	1	8	yes	complemented
131	C₃×Dic₅	C₂³×C₄	1	16	yes	complemented
132	C₅×Dic₃	C₂³×C₄	1	16	yes	complemented
133	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
134	C₂³×D₃	—	5	16	no	—
135	C₂³×D₃	—	5	8	no	—
136	C₂³×D₃	—	5	6	no	—
137	C₂²×C₃×C₄	—	5	16	no	—
138	C₂²×C₃×C₄	—	5	12	no	—
139	C₂²×C₃×C₄	C₄×D₅	1	8	yes	complemented
140	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
141	C₂²×Dic₃	—	5	8	no	—
142	C₂²×Dic₃	—	5	6	no	—
143	C₂²×Dic₃	C₄×D₅	1	16	yes	complemented
144	C₂²×Dic₃	C₂²×D₅	1	12	yes	—
145	C₂×C₄×D₃	—	5	96	no	—
146	C₂×C₄×D₃	—	5	16	no	—
147	C₂×C₄×D₃	—	5	192	no	—
148	C₂×C₄×D₃	—	5	32	no	—
149	C₃×C₄²	—	5	16	no	—
150	C₄×Dic₃	—	5	32	no	—
151	C₄×Dic₃	C₂²×D₅	1	16	yes	complemented
152	C₂³×C₅	C₂×C₄×D₃	1	8	yes	—
153	C₂³×C₅	C₂×C₄×D₃	1	4	yes	—
154	C₂³×C₅	C₂³×D₃	1	3	yes	—
155	C₂²×D₅	—	3	64	no	—
156	C₂²×D₅	—	3	48	no	—
157	C₂²×D₅	—	3	24	no	—
158	C₂²×D₅	—	3	4	no	—
159	C₂×C₄×C₅	—	3	48	no	—
160	C₂×C₄×C₅	—	3	8	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	24	yes	complemented
162	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
163	C₂×Dic₅	—	3	24	no	—
164	C₂×Dic₅	—	3	4	no	—
165	C₂×Dic₅	C₂×C₄×D₃	1	48	yes	complemented
166	C₂×Dic₅	C₂³×D₃	1	8	yes	—
167	C₄×D₅	—	3	64	no	—
168	C₄×D₅	—	3	128	no	—
169	C₂⁵	—	15	1	no	—
170	C₂³×C₄	—	3	2	no	—
171	C₂³×C₄	—	5	2	no	—
172	C₂³×C₄	—	15	32	no	—
173	C₂³×C₄	—	15	24	no	—
174	C₂³×C₄	—	15	1	no	—
175	C₂³×C₄	—	15	16	no	—
176	C₂³×C₄	—	15	12	no	—
177	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
178	C₂×C₄²	—	3	12	no	—
179	C₂×C₄²	—	5	12	no	—
180	C₂×C₄²	—	15	64	no	—
181	C₂×C₄²	—	15	24	no	—
182	C₂×C₃×C₅	C₂²×C₄²	1	12	yes	complemented
183	C₂×C₃×C₅	C₂⁴×C₄	1	2	yes	—
184	C₂×C₃×C₅	C₂⁴×C₄	1	1	yes	characteristic
185	D₁₅	C₂²×C₄²	1	16	yes	complemented
186	C₂³×C₃	C₂×C₄×D₅	1	8	yes	—
187	C₂³×C₃	C₂×C₄×D₅	1	4	yes	—
188	C₂³×C₃	C₂³×D₅	1	3	yes	—
189	C₂²×D₃	—	5	48	no	—
190	C₂²×D₃	—	5	24	no	—
191	C₂²×D₃	—	5	4	no	—
192	C₂²×D₃	—	5	64	no	—
193	C₂×C₃×C₄	—	5	48	no	—
194	C₂×C₃×C₄	—	5	8	no	—
195	C₂×C₃×C₄	C₂×C₄×D₅	1	24	yes	complemented
196	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
197	C₂×Dic₃	—	5	24	no	—
198	C₂×Dic₃	—	5	4	no	—
199	C₂×Dic₃	C₂×C₄×D₅	1	48	yes	complemented
200	C₂×Dic₃	C₂³×D₅	1	8	yes	—
201	C₄×D₃	—	5	64	no	—
202	C₄×D₃	—	5	128	no	—
203	C₂²×C₅	C₄²×D₃	1	16	yes	complemented
204	C₂²×C₅	C₂²×C₄×D₃	1	12	yes	—
205	C₂²×C₅	C₂²×C₄×D₃	1	6	yes	—
206	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
207	C₂×D₅	—	3	8	no	—
208	C₂×D₅	—	3	16	no	—
209	C₂×D₅	—	3	96	no	—
210	C₄×C₅	—	3	16	no	—
211	C₄×C₅	C₂²×C₄×D₃	1	8	yes	complemented
212	Dic₅	—	3	8	no	—
213	Dic₅	C₂²×C₄×D₃	1	16	yes	complemented
214	C₂⁴	—	15	16	no	—
215	C₂⁴	—	15	8	no	—
216	C₂⁴	—	15	6	no	—
217	C₂⁴	C₂×D₃×D₅	1	1	yes	characteristic
218	C₂²×C₄	—	3	16	no	—
219	C₂²×C₄	—	3	12	no	—
220	C₂²×C₄	—	5	16	no	—
221	C₂²×C₄	—	5	12	no	—
222	C₂²×C₄	—	15	32	no	—
223	C₂²×C₄	—	15	192	no	—
224	C₂²×C₄	—	15	8	no	—
225	C₂²×C₄	—	15	6	no	—
226	C₂²×C₄	—	15	96	no	—
227	C₂²×C₄	—	15	16	no	—
228	C₂²×C₄	#1200	1	8	yes	direct product factor
229	C₂²×C₄	C₂×D₃×D₅	1	6	yes	—
230	C₄²	—	3	16	no	—
231	C₄²	—	5	16	no	—
232	C₄²	—	15	32	no	—
233	C₃×C₅	C₂³×C₄²	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	C₄²×D₅	1	16	yes	complemented
235	C₂²×C₃	C₂²×C₄×D₅	1	12	yes	—
236	C₂²×C₃	C₂²×C₄×D₅	1	6	yes	—
237	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
238	C₂×D₃	—	5	96	no	—
239	C₂×D₃	—	5	8	no	—
240	C₂×D₃	—	5	16	no	—
241	C₃×C₄	—	5	16	no	—
242	C₃×C₄	C₂²×C₄×D₅	1	8	yes	complemented
243	Dic₃	—	5	8	no	—
244	Dic₃	C₂²×C₄×D₅	1	16	yes	complemented
245	C₂×C₅	C₂×C₄²×D₃	1	12	yes	complemented
246	C₂×C₅	C₂³×C₄×D₃	1	1	yes	characteristic
247	C₂×C₅	C₂³×C₄×D₃	1	2	yes	—
248	D₅	—	3	16	no	—
249	C₂³	—	15	64	no	—
250	C₂³	—	15	48	no	—
251	C₂³	—	15	24	no	—
252	C₂³	—	15	4	no	—
253	C₂³	C₄×D₃×D₅	1	8	yes	—
254	C₂³	C₂×#1200	1	4	yes	—
255	C₂³	C₂²×D₃×D₅	1	3	yes	—
256	C₂×C₄	—	3	48	no	—
257	C₂×C₄	—	3	8	no	—
258	C₂×C₄	—	5	48	no	—
259	C₂×C₄	—	5	8	no	—
260	C₂×C₄	—	15	128	no	—
261	C₂×C₄	—	15	24	no	—
262	C₂×C₄	—	15	4	no	—
263	C₂×C₄	—	15	64	no	—
264	C₂×C₄	C₂×#1200	1	24	yes	direct product factor
265	C₂×C₄	C₂²×D₃×D₅	1	4	yes	—
266	C₂×C₃	C₂×C₄²×D₅	1	12	yes	complemented
267	C₂×C₃	C₂³×C₄×D₅	1	1	yes	characteristic
268	C₂×C₃	C₂³×C₄×D₅	1	2	yes	—
269	D₃	—	5	16	no	—
270	C₅	C₂²×C₄²×D₃	1	1	yes	characteristic, complemented, Sylow
271	C₂²	—	15	8	no	—
272	C₂²	—	15	96	no	—
273	C₂²	—	15	16	no	—
274	C₂²	C₄×#1200	1	16	yes	direct product factor
275	C₂²	C₂×C₄×D₃×D₅	1	12	yes	—
276	C₂²	C₂²×#1200	1	6	yes	—
277	C₂²	C₂³×D₃×D₅	1	1	yes	characteristic
278	C₄	—	3	16	no	—
279	C₄	—	5	16	no	—
280	C₄	—	15	8	no	—
281	C₄	C₂²×#1200	1	8	yes	direct product factor
282	C₃	C₂²×C₄²×D₅	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	15	16	no	—
284	C₂	C₂×C₄×#1200	1	12	yes	direct product factor
285	C₂	C₂²×C₄×D₃×D₅	1	2	yes	—
286	C₂	C₂³×#1200	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂³×C₄²
Frattini	C₂²	C₂³×D₃×D₅
Fitting	C₂³×C₃×C₄×C₅	C₂²
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂⁴×D₁₅	C₂²
cosocle	C₂²×C₃×C₅	C₂⁵
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₅×Dic₃	C₂	semidirect
4. C₂²×C₃×C₄×Dic₅	C₂	semidirect
5. C₂²×Dic₃×Dic₅	C₂	semidirect
6. C₂×C₄×#1200	C₂	direct (N × H)
11. C₂²×C₄×D₁₅	C₄	semidirect
15. C₂²×#1200	C₄	direct (N × H)
17. C₂×C₄×C₅×Dic₃	C₂²	semidirect
18. C₂×C₃×C₄×Dic₅	C₂²	semidirect
19. C₂×Dic₃×Dic₅	C₂²	semidirect
20. C₄×#1200	C₂²	direct (N × H)
26. C₂²×C₄×Dic₅	D₃	semidirect
32. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
34. C₂²×Dic₁₅	C₂×C₄	semidirect
36. C₂×C₄×D₁₅	C₂×C₄	semidirect
38. C₂²×C₅×Dic₃	C₂×C₄	semidirect
40. C₂²×C₃×Dic₅	C₂×C₄	semidirect
42. C₂×#1200	C₂×C₄	direct (N × H)
44. C₄×C₅×Dic₃	C₂³	semidirect
45. C₃×C₄×Dic₅	C₂³	semidirect
46. Dic₃×Dic₅	C₂³	semidirect
51. C₂²×C₄×Dic₃	D₅	semidirect
64. C₂×C₄×Dic₅	C₂×D₃	semidirect
70. C₂²×D₁₅	C₄²	semidirect
74. C₂×C₃×C₄×C₅	C₂²×C₄	semidirect
76. C₂×Dic₁₅	C₂²×C₄	semidirect
78. C₄×D₁₅	C₂²×C₄	semidirect
79. C₂×C₅×Dic₃	C₂²×C₄	semidirect
81. C₂×C₃×Dic₅	C₂²×C₄	semidirect
83. #1200	C₂²×C₄	direct (N × H)
95. C₂×C₄×Dic₃	C₂×D₅	semidirect
103. C₂²×C₄×C₅	C₄×D₃	semidirect
107. C₂²×Dic₅	C₄×D₃	semidirect
115. C₄×Dic₅	C₂²×D₃	semidirect
122. C₂²×C₃×C₅	C₂×C₄²	semidirect
126. C₂×D₁₅	C₂×C₄²	semidirect
129. C₃×C₄×C₅	C₂³×C₄	semidirect
130. Dic₁₅	C₂³×C₄	semidirect
131. C₃×Dic₅	C₂³×C₄	semidirect
132. C₅×Dic₃	C₂³×C₄	semidirect
139. C₂²×C₃×C₄	C₄×D₅	semidirect
143. C₂²×Dic₃	C₄×D₅	semidirect
151. C₄×Dic₃	C₂²×D₅	semidirect
161. C₂×C₄×C₅	C₂×C₄×D₃	semidirect
165. C₂×Dic₅	C₂×C₄×D₃	semidirect
182. C₂×C₃×C₅	C₂²×C₄²	semidirect
185. D₁₅	C₂²×C₄²	semidirect
195. C₂×C₃×C₄	C₂×C₄×D₅	semidirect
199. C₂×Dic₃	C₂×C₄×D₅	semidirect
203. C₂²×C₅	C₄²×D₃	semidirect
211. C₄×C₅	C₂²×C₄×D₃	semidirect
213. Dic₅	C₂²×C₄×D₃	semidirect
228. C₂²×C₄	#1200	direct (N × H)
233. C₃×C₅	C₂³×C₄²	semidirect
234. C₂²×C₃	C₄²×D₅	semidirect
242. C₃×C₄	C₂²×C₄×D₅	semidirect
244. Dic₃	C₂²×C₄×D₅	semidirect
245. C₂×C₅	C₂×C₄²×D₃	semidirect
264. C₂×C₄	C₂×#1200	direct (N × H)
266. C₂×C₃	C₂×C₄²×D₅	semidirect
270. C₅	C₂²×C₄²×D₃	semidirect
274. C₂²	C₄×#1200	direct (N × H)
281. C₄	C₂²×#1200	direct (N × H)
282. C₃	C₂²×C₄²×D₅	semidirect
284. C₂	C₂×C₄×#1200	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄×D₁₅	2
C₂³×#1200	2
C₂²×C₄×C₅×Dic₃	2
C₂²×C₃×C₄×Dic₅	2
C₂²×Dic₃×Dic₅	2
C₂×C₄×#1200	2
C₂²×C₄²×D₅	3
C₂²×C₄²×D₃	5

Subgroups — C22×C4×#1200 #14478672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#1200 #14478672