Subgroups — C₂×C₃×#6714 #13088024

Order 1440.

582 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6714	C₂	1	4	yes	maximal, direct product factor
5	C₂×#6714	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×#685	—	3	1	no	maximal
7	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₁₅	—	2	2	no	—
9	C₂²×C₃×D₁₅	—	2	2	no	—
10	C₂²×C₃×D₁₅	C₄	1	4	yes	complemented
11	C₂²×C₃×D₁₅	C₂²	1	2	yes	—
12	C₂×C₃²×C₄×C₅	—	2	2	no	—
13	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	complemented
14	C₂×C₃×Dic₁₅	—	2	2	no	—
15	C₂×C₃×Dic₁₅	C₂²	1	2	yes	complemented
16	C₂×C₃×(C₂×D₃)⋊C₄	—	5	1	no	maximal, Hall
17	C₂³×D₁₅	C₂×C₃	1	1	yes	characteristic
18	C₂³×C₃×D₅	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	—	2	1	no	—
20	C₂²×C₃×C₄×C₅	D₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×Dic₅	—	3	1	no	—
24	#6714	C₂×C₃	1	4	yes	direct product factor
25	C₃×#685	—	3	4	no	—
26	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₅	D₄	1	2	yes	—
28	C₂²×C₃²×C₅	D₄	1	2	yes	—
29	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×D₁₅	—	2	8	no	—
31	C₂×C₃×D₁₅	—	2	2	no	—
32	C₂×C₃×D₁₅	—	2	2	no	—
33	C₂×C₃×D₁₅	C₂×C₄	1	4	yes	complemented
34	C₃²×C₄×C₅	—	2	2	no	—
35	C₃×Dic₁₅	—	2	2	no	—
36	C₂×#685	—	3	1	no	Hall
37	C₂³×C₃×D₃	—	5	1	no	—
38	C₂²×C₃²×C₄	D₅	1	1	yes	characteristic, complemented
39	C₂²×C₃×Dic₃	—	5	1	no	—
40	C₃×(C₂×D₃)⋊C₄	—	5	4	no	—
41	C₂³×C₃×C₅	—	2	1	no	—
42	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂²×D₁₅	—	2	2	no	—
45	C₂²×D₁₅	—	2	2	no	—
46	C₂²×D₁₅	C₃×C₄	1	4	yes	complemented
47	C₂²×D₁₅	C₂²×C₃	1	2	yes	—
48	C₂²×C₃×D₅	—	3	4	no	—
49	C₂²×C₃×D₅	—	3	2	no	—
50	C₂²×C₃×D₅	—	6	2	no	—
51	C₂²×C₃×D₅	—	6	2	no	—
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	complemented
57	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	complemented
58	C₂×Dic₁₅	—	2	2	no	—
59	C₂×Dic₁₅	C₂²×C₃	1	2	yes	complemented
60	C₂×C₃×Dic₅	—	3	2	no	—
61	C₂×C₃×Dic₅	—	6	2	no	—
62	C₂×C₃×C₂²⋊C₄	—	15	1	no	—
63	C₂×(C₂×D₃)⋊C₄	—	5	1	no	—
64	C₂×C₃²×C₅	C₂²⋊C₄	1	4	yes	complemented
65	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×D₁₅	—	2	4	no	—
69	C₂³×D₅	—	3	1	no	—
70	C₂²×C₄×C₅	C₃×D₃	1	1	yes	characteristic, complemented
71	C₂²×Dic₅	—	3	1	no	—
72	#685	—	3	4	no	—
73	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₃×D₃	—	5	4	no	—
75	C₂²×C₃×D₃	—	5	2	no	—
76	C₂²×C₃×D₃	—	10	2	no	—
77	C₂²×C₃×D₃	—	10	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	C₂×D₅	1	2	yes	complemented
80	C₂×C₃×Dic₃	—	5	2	no	—
81	C₂×C₃×Dic₃	—	10	2	no	—
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
87	C₂²×C₃×C₅	D₁₂	1	2	yes	—
88	C₂²×C₃×C₅	#67	1	2	yes	—
89	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₃×C₅	C₃×D₄	1	2	yes	—
91	C₂²×C₃×C₅	C₃×D₄	1	2	yes	—
92	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×D₁₅	—	2	8	no	—
95	C₂×D₁₅	—	2	2	no	—
96	C₂×D₁₅	—	2	2	no	—
97	C₂×D₁₅	C₂×C₃×C₄	1	4	yes	complemented
98	C₂×C₃×D₅	—	3	4	no	—
99	C₂×C₃×D₅	—	6	8	no	—
100	C₂×C₃×D₅	—	6	2	no	—
101	C₂×C₃×D₅	—	6	2	no	—
102	C₃×C₄×C₅	—	2	4	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	Dic₁₅	—	2	2	no	—
106	C₃×Dic₅	—	6	2	no	—
107	C₂⁴×C₃	—	15	1	no	—
108	C₂³×D₃	—	5	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	15	1	no	—
111	C₂²×C₃×C₄	C₃×D₅	1	1	yes	characteristic, complemented
112	C₂²×C₃×C₄	D₁₅	1	1	yes	characteristic, complemented
113	C₂²×Dic₃	—	5	1	no	—
114	C₃×C₂²⋊C₄	—	15	4	no	—
115	(C₂×D₃)⋊C₄	—	5	4	no	—
116	C₃²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
117	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂²×D₅	—	3	4	no	—
119	C₂²×D₅	—	3	2	no	—
120	C₂²×D₅	—	6	2	no	—
121	C₂²×D₅	—	6	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	complemented
124	C₂×Dic₅	—	3	2	no	—
125	C₂×Dic₅	—	6	2	no	—
126	C₂²×C₃²	C₄×D₅	1	2	yes	—
127	C₂²×C₃²	D₂₀	1	2	yes	—
128	C₂²×C₃²	C₂²⋊D₅	1	2	yes	—
129	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₃×D₃	—	5	4	no	—
131	C₂×C₃×D₃	—	10	8	no	—
132	C₂×C₃×D₃	—	10	2	no	—
133	C₂×C₃×D₃	—	10	2	no	—
134	C₃²×C₄	—	2	2	no	—
135	C₃×Dic₃	—	10	2	no	—
136	C₂×C₂²⋊C₄	—	15	1	no	Sylow
137	C₂×C₃×C₅	—	2	4	no	—
138	C₂×C₃×C₅	—	2	1	no	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	4	yes	complemented
142	C₂×C₃×C₅	C₃×C₂²⋊C₄	1	4	yes	complemented
143	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
149	D₁₅	—	2	4	no	—
150	C₃×D₅	—	6	4	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	—	15	4	no	—
153	C₂³×C₃	—	15	2	no	—
154	C₂³×C₃	—	30	2	no	—
155	C₂³×C₃	—	30	2	no	—
156	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
157	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
158	C₂²×D₃	—	5	4	no	—
159	C₂²×D₃	—	5	2	no	—
160	C₂²×D₃	—	10	2	no	—
161	C₂²×D₃	—	10	2	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	15	2	no	—
167	C₂×C₃×C₄	—	30	2	no	—
168	C₂×C₃×C₄	C₂×C₃×D₅	1	2	yes	complemented
169	C₂×C₃×C₄	C₂×D₁₅	1	2	yes	complemented
170	C₂×Dic₃	—	5	2	no	—
171	C₂×Dic₃	—	10	2	no	—
172	C₂²×C₅	C₃×C₄×D₃	1	2	yes	—
173	C₂²×C₅	C₃×D₁₂	1	2	yes	—
174	C₂²×C₅	C₃×#67	1	2	yes	—
175	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
176	C₂×D₅	—	3	4	no	—
177	C₂×D₅	—	6	8	no	—
178	C₂×D₅	—	6	2	no	—
179	C₂×D₅	—	6	2	no	—
180	C₄×C₅	—	2	2	no	—
181	Dic₅	—	6	2	no	—
182	C₂×C₃²	#685	1	4	yes	complemented
183	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
185	C₂×C₃²	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₃×D₃	—	10	4	no	—
187	C₂⁴	—	15	1	no	—
188	C₂²×C₄	—	15	1	no	—
189	C₂²×C₄	C₃×D₁₅	1	1	yes	characteristic, complemented
190	C₂²⋊C₄	—	15	4	no	—
191	C₃×C₅	—	2	1	no	—
192	C₃×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
193	C₃×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	15	4	no	—
199	C₂²×C₃	—	30	8	no	—
200	C₂²×C₃	—	30	2	no	—
201	C₂²×C₃	—	30	2	no	—
202	C₂²×C₃	C₃×C₄×D₅	1	2	yes	—
203	C₂²×C₃	C₃×D₂₀	1	2	yes	—
204	C₂²×C₃	C₃×C₂²⋊D₅	1	2	yes	—
205	C₂²×C₃	C₄×D₁₅	1	2	yes	—
206	C₂²×C₃	D₆₀	1	2	yes	—
207	C₂²×C₃	#1220	1	2	yes	—
208	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
209	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
210	C₂×D₃	—	5	4	no	—
211	C₂×D₃	—	10	8	no	—
212	C₂×D₃	—	10	2	no	—
213	C₂×D₃	—	10	2	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	30	2	no	—
218	Dic₃	—	10	2	no	—
219	C₂×C₅	C₃×(C₂×D₃)⋊C₄	1	4	yes	complemented
220	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
221	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
222	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
223	D₅	—	6	4	no	—
224	C₃²	C₂×#685	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	15	4	no	—
226	C₂³	—	15	2	no	—
227	C₂³	—	30	2	no	—
228	C₂³	—	30	2	no	—
229	C₂³	C₂×C₃×D₁₅	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	15	2	no	—
232	C₂×C₄	—	30	2	no	—
233	C₂×C₄	C₂×C₃×D₁₅	1	2	yes	complemented
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	4	no	—
238	C₂×C₃	—	30	4	no	—
239	C₂×C₃	C₃×#685	1	4	yes	complemented
240	C₂×C₃	#6714	1	4	yes	direct product factor
241	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
242	C₂×C₃	C₂×C₃×D₂₀	1	1	yes	characteristic
243	C₂×C₃	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
244	C₂×C₃	C₂×C₄×D₁₅	1	1	yes	characteristic
245	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
246	C₂×C₃	C₂×#1220	1	1	yes	characteristic
247	D₃	—	10	4	no	—
248	C₅	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	15	4	no	—
250	C₂²	—	30	2	no	—
251	C₂²	—	30	2	no	—
252	C₂²	—	30	8	no	—
253	C₂²	C₃×C₄×D₁₅	1	2	yes	—
254	C₂²	C₃×D₆₀	1	2	yes	—
255	C₂²	C₃×#1220	1	2	yes	—
256	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	30	2	no	—
259	C₃	—	2	1	no	—
260	C₃	C₂×C₃×#685	1	1	yes	characteristic, complemented
261	C₃	C₂×#6714	1	1	yes	characteristic, direct product factor
262	C₂	—	30	4	no	—
263	C₂	C₃×#6714	1	4	yes	direct product factor
264	C₂	C₂×C₃×C₄×D₁₅	1	1	yes	characteristic
265	C₂	C₂×C₃×D₆₀	1	1	yes	characteristic
266	C₂	C₂×C₃×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting	C₂²×C₃²×C₄×C₅	C₂
socle	C₂³×C₃²×C₅	C₂²
prime radical	C₂³×C₃×D₁₅	C₂
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₂²×C₃×C₄	D₁₅
hypocenter	C₃×C₅	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄×C₅	C₂	semidirect
3. C₂²×C₃×Dic₁₅	C₂	semidirect
4. C₃×#6714	C₂	direct (N × H)
5. C₂×#6714	C₃	direct (N × H)
10. C₂²×C₃×D₁₅	C₄	semidirect
13. C₂×C₃²×C₄×C₅	C₂²	semidirect
15. C₂×C₃×Dic₁₅	C₂²	semidirect
20. C₂²×C₃×C₄×C₅	D₃	semidirect
21. C₂²×C₃×C₄×C₅	C₂×C₃	semidirect
22. C₂²×Dic₁₅	C₂×C₃	semidirect
24. #6714	C₂×C₃	direct (N × H)
33. C₂×C₃×D₁₅	C₂×C₄	semidirect
38. C₂²×C₃²×C₄	D₅	semidirect
46. C₂²×D₁₅	C₃×C₄	semidirect
56. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
57. C₂×C₃×C₄×C₅	C₂²×C₃	semidirect
59. C₂×Dic₁₅	C₂²×C₃	semidirect
64. C₂×C₃²×C₅	C₂²⋊C₄	semidirect
70. C₂²×C₄×C₅	C₃×D₃	semidirect
79. C₂×C₃²×C₄	C₂×D₅	semidirect
97. C₂×D₁₅	C₂×C₃×C₄	semidirect
111. C₂²×C₃×C₄	C₃×D₅	semidirect
112. C₂²×C₃×C₄	D₁₅	semidirect
116. C₃²×C₅	C₂×C₂²⋊C₄	semidirect
123. C₂×C₄×C₅	C₂×C₃×D₃	semidirect
141. C₂×C₃×C₅	(C₂×D₃)⋊C₄	semidirect
142. C₂×C₃×C₅	C₃×C₂²⋊C₄	semidirect
168. C₂×C₃×C₄	C₂×C₃×D₅	semidirect
169. C₂×C₃×C₄	C₂×D₁₅	semidirect
182. C₂×C₃²	#685	semidirect
189. C₂²×C₄	C₃×D₁₅	semidirect
192. C₃×C₅	C₂×(C₂×D₃)⋊C₄	semidirect
193. C₃×C₅	C₂×C₃×C₂²⋊C₄	semidirect
219. C₂×C₅	C₃×(C₂×D₃)⋊C₄	semidirect
224. C₃²	C₂×#685	semidirect
233. C₂×C₄	C₂×C₃×D₁₅	semidirect
239. C₂×C₃	C₃×#685	semidirect
240. C₂×C₃	#6714	direct (N × H)
248. C₅	C₂×C₃×(C₂×D₃)⋊C₄	semidirect
260. C₃	C₂×C₃×#685	semidirect
261. C₃	C₂×#6714	direct (N × H)
263. C₂	C₃×#6714	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×C₃×D₁₅	2
C₂²×C₃²×C₄×C₅	2
C₂²×C₃×Dic₁₅	2
C₃×#6714	2
C₂×#6714	3
C₂×C₃×#685	3
C₂×C₃×(C₂×D₃)⋊C₄	5

Subgroups — C2×C3×#6714 #13088024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6714 #13088024