Subgroups — C₃×D₃×#681 #13086486

Order 1440.

292 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6650	C₂	1	2	yes	maximal, complemented
5	C₃×D₃×#168	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#681	—	3	1	no	maximal
7	D₃×#681	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Dic₃	C₄	1	1	yes	characteristic
11	C₃×C₄×C₅×D₃	C₄	1	1	yes	characteristic
12	C₃×C₄×C₅×D₃	C₄	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₃×C₃⋊#168	C₂²	1	2	yes	complemented
16	C₃²×#168	C₂²	1	2	yes	complemented
17	C₃×D₃×M₁₆	—	5	1	no	maximal, Hall
18	C₂²×C₃×C₄×C₅	—	3	1	no	—
19	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
20	#6710	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×#681	—	2	1	no	—
22	C₃×#681	—	3	1	no	—
23	C₃×#681	—	3	2	no	—
24	C₃×#681	D₃	1	1	yes	characteristic, direct product factor
25	C₃×#681	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₃×#168	—	3	2	no	—
27	#6650	C₂×C₃	1	2	yes	complemented
28	D₃×#168	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₃	—	2	2	no	—
31	C₂×C₃×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#681	—	3	1	no	Hall
38	C₂×C₃×C₄×D₃	D₅	1	1	yes	characteristic
39	C₃²×M₁₆	—	5	1	no	—
40	C₃×#208	—	5	1	no	—
41	C₃×ΓL(1,𝔽₂₅)	—	5	2	no	—
42	C₃×D₃×C₈	—	5	2	no	—
43	C₂³×C₃×C₅	—	3	1	no	—
44	C₂²×C₅×D₃	C₃×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	—	2	1	no	—
46	C₂×C₃×C₄×C₅	—	3	1	no	—
47	C₂×C₃×C₄×C₅	—	3	1	no	—
48	C₂×C₃×C₄×C₅	—	3	1	no	—
49	C₂×C₃×C₄×C₅	—	3	1	no	—
50	C₂×C₃×C₄×C₅	—	3	1	no	—
51	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₅×Dic₃	C₃×C₄	1	1	yes	characteristic
54	C₄×C₅×D₃	C₃×C₄	1	1	yes	characteristic
55	C₄×C₅×D₃	C₃×C₄	1	1	yes	characteristic
56	C₄×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₄×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
58	C₃⋊#168	C₂²×C₃	1	2	yes	complemented
59	C₃×#168	—	2	2	no	—
60	C₃×#168	—	3	2	no	—
61	C₃×#168	C₂×D₃	1	2	yes	complemented
62	C₃×#168	C₂²×C₃	1	2	yes	complemented
63	C₂×C₃×M₁₆	—	15	1	no	—
64	D₃×M₁₆	—	5	1	no	—
65	C₂×C₃²×C₅	—	2	1	no	—
66	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₃×C₅×D₃	—	2	1	no	—
68	C₃×C₅×D₃	M₁₆	1	2	yes	complemented
69	C₂²×C₄×C₅	—	3	1	no	—
70	#681	—	3	1	no	—
71	#681	—	3	2	no	—
72	#681	C₃×D₃	1	1	yes	characteristic, direct product factor
73	C₂×#168	—	3	2	no	—
74	C₂²×C₃×D₃	Dic₅	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
76	C₂×C₃×Dic₃	Dic₅	1	1	yes	characteristic
77	C₃×C₄×D₃	Dic₅	1	1	yes	characteristic
78	C₃×C₄×D₃	Dic₅	1	1	yes	characteristic
79	C₃×C₄×D₃	C₂×D₅	1	1	yes	characteristic
80	C₃×C₄×D₃	C₂×D₅	1	1	yes	characteristic
81	C₃²×C₈	—	5	2	no	—
82	C₃×C₃⋊C₈	—	5	2	no	—
83	C₂²×C₃×C₅	—	2	1	no	—
84	C₂²×C₃×C₅	—	3	1	no	—
85	C₂²×C₃×C₅	—	3	1	no	—
86	C₂²×C₃×C₅	—	6	2	no	—
87	C₂²×C₃×C₅	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₅×D₃	—	2	2	no	—
90	C₂×C₅×D₃	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₅×D₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	3	1	no	—
95	C₃×C₄×C₅	—	3	1	no	—
96	C₃×C₄×C₅	C₄×D₃	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
99	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₅×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
101	C₅×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	3	1	no	—
103	C₂×C₄×D₃	C₃×D₅	1	1	yes	characteristic
104	C₃×M₁₆	—	5	1	no	—
105	C₃×M₁₆	—	5	1	no	—
106	C₃×M₁₆	—	10	1	no	—
107	C₃×M₁₆	—	15	2	no	—
108	C₃×M₁₆	—	15	1	no	—
109	C₂×C₃×C₈	—	15	2	no	—
110	#208	—	5	1	no	—
111	ΓL(1,𝔽₂₅)	—	5	2	no	—
112	D₃×C₈	—	5	2	no	—
113	C₃²×C₅	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
114	C₂³×C₅	—	3	1	no	—
115	C₂×C₄×C₅	—	3	1	no	—
116	C₂×C₄×C₅	—	3	1	no	—
117	C₂×C₄×C₅	—	3	1	no	—
118	C₂×C₄×C₅	—	3	1	no	—
119	C₂×C₄×C₅	—	3	1	no	—
120	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
121	#168	—	3	2	no	—
122	#168	C₂×C₃×D₃	1	2	yes	complemented
123	C₂²×C₃²	C₂×Dic₅	1	1	yes	characteristic
124	C₂×C₃×D₃	—	2	2	no	—
125	C₂×C₃×D₃	C₂×Dic₅	1	1	yes	characteristic
126	C₂×C₃×D₃	C₂×Dic₅	1	1	yes	characteristic
127	C₃²×C₄	C₂×Dic₅	1	1	yes	characteristic
128	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₃×Dic₃	C₂×Dic₅	1	1	yes	characteristic
130	C₃×Dic₃	C₂×Dic₅	1	1	yes	characteristic
131	C₂×M₁₆	—	15	1	no	Sylow
132	C₂×C₃×C₅	—	2	1	no	—
133	C₂×C₃×C₅	—	2	1	no	—
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	—	3	2	no	—
136	C₂×C₃×C₅	—	4	1	no	—
137	C₂×C₃×C₅	—	6	1	no	—
138	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₅×D₃	—	2	1	no	—
141	C₅×D₃	C₃×M₁₆	1	2	yes	complemented
142	C₂³×C₃	—	3	1	no	—
143	C₂²×D₃	C₃×Dic₅	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	3	1	no	—
146	C₂×C₃×C₄	—	3	1	no	—
147	C₂×C₃×C₄	—	3	1	no	—
148	C₂×C₃×C₄	—	3	1	no	—
149	C₂×C₃×C₄	—	3	1	no	—
150	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
152	C₂×Dic₃	C₃×Dic₅	1	1	yes	characteristic
153	C₄×D₃	C₃×Dic₅	1	1	yes	characteristic
154	C₄×D₃	C₃×Dic₅	1	1	yes	characteristic
155	C₄×D₃	C₂×C₃×D₅	1	1	yes	characteristic
156	C₄×D₃	C₂×C₃×D₅	1	1	yes	characteristic
157	C₃×C₈	—	5	2	no	—
158	C₃×C₈	—	5	2	no	—
159	C₃×C₈	—	10	2	no	—
160	C₃×C₈	—	15	2	no	—
161	C₃⋊C₈	—	5	2	no	—
162	C₂²×C₅	—	3	1	no	—
163	C₂²×C₅	—	3	1	no	—
164	C₂²×C₅	—	6	2	no	—
165	C₂²×C₅	C₃×C₄×D₃	1	1	yes	characteristic
166	C₄×C₅	—	3	1	no	—
167	C₄×C₅	—	3	1	no	—
168	C₄×C₅	C₃×C₄×D₃	1	1	yes	characteristic
169	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
172	C₃×D₃	—	2	1	no	—
173	C₃×D₃	#681	1	2	yes	direct product factor
174	C₂²×C₄	—	3	1	no	—
175	M₁₆	—	5	1	no	—
176	M₁₆	—	15	2	no	—
177	M₁₆	—	15	1	no	—
178	C₂×C₈	—	15	2	no	—
179	C₃×C₅	—	2	1	no	—
180	C₃×C₅	D₃×M₁₆	1	1	yes	characteristic, complemented
181	C₃×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	3	1	no	—
184	C₂²×C₃	—	3	1	no	—
185	C₂²×C₃	—	6	2	no	—
186	C₂²×C₃	D₃×Dic₅	1	1	yes	characteristic
187	C₂²×C₃	C₂×C₃×Dic₅	1	1	yes	characteristic
188	C₂×D₃	—	2	2	no	—
189	C₂×D₃	C₂×C₃×Dic₅	1	1	yes	characteristic
190	C₂×D₃	C₂×C₃×Dic₅	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	3	1	no	—
194	C₃×C₄	—	3	1	no	—
195	C₃×C₄	D₃×Dic₅	1	1	yes	characteristic
196	C₃×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
197	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
198	C₃×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
199	Dic₃	C₂×C₃×Dic₅	1	1	yes	characteristic
200	Dic₃	C₂×C₃×Dic₅	1	1	yes	characteristic
201	C₂×C₅	—	2	1	no	—
202	C₂×C₅	—	3	2	no	—
203	C₂×C₅	—	6	1	no	—
204	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
205	C₃²	C₂×#681	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	3	1	no	—
207	C₂×C₄	—	3	1	no	—
208	C₂×C₄	—	3	1	no	—
209	C₂×C₄	—	3	1	no	—
210	C₂×C₄	—	3	1	no	—
211	C₂×C₄	—	3	1	no	—
212	C₂×C₄	C₃×D₃×D₅	1	1	yes	characteristic
213	C₈	—	5	2	no	—
214	C₈	—	15	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	3	2	no	—
219	C₂×C₃	—	4	1	no	—
220	C₂×C₃	—	6	1	no	—
221	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
222	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
223	D₃	—	2	1	no	—
224	D₃	C₃×#681	1	2	yes	direct product factor
225	C₅	C₃×D₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
226	C₂²	—	3	1	no	—
227	C₂²	—	3	1	no	—
228	C₂²	—	6	2	no	—
229	C₂²	C₃×D₃×Dic₅	1	1	yes	characteristic
230	C₄	—	3	1	no	—
231	C₄	—	3	1	no	—
232	C₄	C₃×D₃×Dic₅	1	1	yes	characteristic
233	C₄	C₂×C₃×D₃×D₅	1	1	yes	characteristic
234	C₃	—	2	1	no	—
235	C₃	D₃×#681	1	1	yes	characteristic, direct product factor
236	C₃	C₂×C₃×#681	1	1	yes	characteristic, complemented
237	C₂	—	2	1	no	—
238	C₂	—	3	2	no	—
239	C₂	—	6	1	no	—
240	C₂	C₂×C₃×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₄	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₂²×C₃×C₅×D₃	C₄
cosocle	C₃×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×#6710	C₂	semidirect
3. C₃²×#681	C₂	semidirect
4. C₃×#6650	C₂	semidirect
5. C₃×D₃×#168	C₂	semidirect
7. D₃×#681	C₃	direct (N × H)
15. C₃×C₃⋊#168	C₂²	semidirect
16. C₃²×#168	C₂²	semidirect
20. #6710	C₂×C₃	semidirect
24. C₃×#681	D₃	direct (N × H)
25. C₃×#681	C₂×C₃	semidirect
27. #6650	C₂×C₃	semidirect
28. D₃×#168	C₂×C₃	semidirect
58. C₃⋊#168	C₂²×C₃	semidirect
61. C₃×#168	C₂×D₃	semidirect
62. C₃×#168	C₂²×C₃	semidirect
68. C₃×C₅×D₃	M₁₆	semidirect
72. #681	C₃×D₃	direct (N × H)
113. C₃²×C₅	C₂×M₁₆	semidirect
122. #168	C₂×C₃×D₃	semidirect
141. C₅×D₃	C₃×M₁₆	semidirect
173. C₃×D₃	#681	direct (N × H)
180. C₃×C₅	D₃×M₁₆	semidirect
181. C₃×C₅	C₂×C₃×M₁₆	semidirect
205. C₃²	C₂×#681	semidirect
224. D₃	C₃×#681	direct (N × H)
225. C₅	C₃×D₃×M₁₆	semidirect
235. C₃	D₃×#681	direct (N × H)
236. C₃	C₂×C₃×#681	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₃×#6710	2
C₃²×#681	2
C₃×#6650	2
C₃×D₃×#168	2
C₂×C₃×#681	3
D₃×#681	3
C₃×D₃×M₁₆	5

Subgroups — C3×D3×#681 #13086486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#681 #13086486