Subgroups — C₃×#91265 #13086413

Order 1440.

270 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	C₃×#6771	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6651	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6650	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6649	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4047	—	3	1	no	maximal
9	#91265	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×C₄×D₁₅	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₄×D₅	C₂²	1	1	yes	characteristic
13	C₃×#1204	C₄	1	1	yes	characteristic
14	C₃×#1203	C₄	1	1	yes	characteristic
15	C₃×#1202	C₄	1	1	yes	characteristic
16	C₃×#1201	C₄	1	1	yes	characteristic
17	C₃²×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₃⋊#168	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃²×#168	C₂²	1	1	yes	characteristic
21	C₃×#901	—	5	1	no	maximal, Hall
22	C₃×#709	—	3	1	no	—
23	#6771	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	—	3	1	no	—
25	C₈×D₁₅	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×D₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#681	—	3	1	no	—
28	C₃×#676	—	3	1	no	—
29	C₃×#676	—	3	1	no	—
30	C₃×C₈×D₅	—	2	1	no	—
31	C₃×C₈×D₅	—	3	1	no	—
32	C₃×C₈×D₅	D₃	1	1	yes	characteristic, complemented
33	C₃×C₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
34	#6651	C₂×C₃	1	1	yes	characteristic, complemented
35	#6650	C₂×C₃	1	1	yes	characteristic, complemented
36	#6649	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₃×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×D₅	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₃×Dic₁₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₃²×Dic₅	C₂×C₄	1	1	yes	characteristic
44	#4047	—	3	1	no	Hall
45	C₃×#235	—	5	1	no	—
46	C₂×C₃²×C₈	—	5	1	no	—
47	C₃×#208	—	5	1	no	—
48	C₃×ΓL(1,𝔽₂₅)	—	5	1	no	—
49	C₃×ΓL(1,𝔽₂₅)	—	5	1	no	—
50	C₃×D₃×C₈	—	5	1	no	—
51	C₃×D₃×C₈	D₅	1	1	yes	characteristic, complemented
52	C₂×C₃×C₄×C₅	—	3	1	no	—
53	C₄×D₁₅	C₂²×C₃	1	1	yes	characteristic
54	C₄×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₂²⋊D₅	—	3	1	no	—
56	C₃×C₂²⋊D₅	—	3	1	no	—
57	C₃×D₂₀	—	3	1	no	—
58	C₃×C₄×D₅	—	2	1	no	—
59	C₃×C₄×D₅	—	3	1	no	—
60	C₃×C₄×D₅	C₂×D₃	1	1	yes	characteristic
61	C₃×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
62	C₃×Dic₁₀	—	3	1	no	—
63	#1204	C₃×C₄	1	1	yes	characteristic
64	#1203	C₃×C₄	1	1	yes	characteristic
65	#1202	C₃×C₄	1	1	yes	characteristic
66	#1201	C₃×C₄	1	1	yes	characteristic
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	—	3	1	no	—
69	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	C₃×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₃⋊#168	C₂²×C₃	1	1	yes	characteristic
72	C₃×#168	—	2	1	no	—
73	C₃×#168	—	3	1	no	—
74	C₃×#168	C₂×D₃	1	1	yes	characteristic
75	C₃×#168	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×#131	—	15	1	no	—
78	#901	—	5	1	no	—
79	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₃×D₁₅	—	2	1	no	—
81	C₃×C₅×D₃	—	2	1	no	—
82	C₃²×D₅	—	2	1	no	—
83	#709	—	3	1	no	—
84	C₂×C₅×C₈	—	3	1	no	—
85	#681	—	3	1	no	—
86	#676	—	3	1	no	—
87	#676	—	3	1	no	—
88	C₈×D₅	—	3	1	no	—
89	C₈×D₅	C₃×D₃	1	1	yes	characteristic, complemented
90	C₂×C₃²×C₄	—	5	1	no	—
91	C₃×#67	—	5	1	no	—
92	C₃×#67	—	5	1	no	—
93	C₃×D₁₂	—	5	1	no	—
94	C₃×C₄×D₃	—	5	1	no	—
95	C₃×C₄×D₃	C₂×D₅	1	1	yes	characteristic
96	C₃×Dic₆	—	5	1	no	—
97	C₃²×C₈	—	5	1	no	—
98	C₃²×C₈	C₂×D₅	1	1	yes	characteristic
99	C₃×C₃⋊C₈	—	5	1	no	—
100	C₃×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₃×C₅	—	3	1	no	—
102	C₂×D₁₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×C₅×D₃	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₃×D₅	—	2	1	no	—
105	C₂×C₃×D₅	—	3	1	no	—
106	C₂×C₃×D₅	C₄×D₃	1	1	yes	characteristic
107	C₂×C₃×D₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	3	1	no	—
110	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	Dic₁₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	3	1	no	—
115	C₃×Dic₅	C₄×D₃	1	1	yes	characteristic
116	C₃×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₅×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃×#41	—	15	1	no	—
119	#235	—	5	1	no	—
120	C₃×M₁₆	—	15	1	no	—
121	C₃×M₁₆	—	15	1	no	—
122	C₃×M₁₆	—	15	1	no	—
123	C₂×C₃×C₈	—	3	1	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	10	1	no	—
127	C₂×C₃×C₈	—	15	1	no	—
128	#208	—	5	1	no	—
129	ΓL(1,𝔽₂₅)	—	5	1	no	—
130	ΓL(1,𝔽₂₅)	—	5	1	no	—
131	D₃×C₈	—	5	1	no	—
132	D₃×C₈	C₃×D₅	1	1	yes	characteristic, complemented
133	C₃²×C₅	#131	1	1	yes	characteristic, complemented, Hall
134	C₂×C₄×C₅	—	3	1	no	—
135	C₂²⋊D₅	—	3	1	no	—
136	C₂²⋊D₅	—	3	1	no	—
137	D₂₀	—	3	1	no	—
138	C₄×D₅	—	3	1	no	—
139	C₄×D₅	C₂×C₃×D₃	1	1	yes	characteristic
140	Dic₁₀	—	3	1	no	—
141	C₅×C₈	—	3	1	no	—
142	C₅×C₈	C₂×C₃×D₃	1	1	yes	characteristic
143	#168	—	3	1	no	—
144	#168	C₂×C₃×D₃	1	1	yes	characteristic
145	C₂²×C₃²	—	5	1	no	—
146	C₂×C₃×D₃	—	5	1	no	—
147	C₂×C₃×D₃	C₄×D₅	1	1	yes	characteristic
148	C₃²×C₄	—	5	1	no	—
149	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₃×Dic₃	—	5	1	no	—
151	C₃×Dic₃	C₄×D₅	1	1	yes	characteristic
152	#131	—	15	1	no	Sylow
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	6	1	no	—
155	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	D₁₅	—	2	1	no	—
158	C₃×D₅	—	2	2	no	—
159	C₃×D₅	—	2	1	no	—
160	C₃×D₅	—	2	1	no	—
161	C₃×D₅	—	6	1	no	—
162	C₅×D₃	—	2	1	no	—
163	C₃×Q₈	—	15	1	no	—
164	C₃×D₄	—	15	1	no	—
165	C₃×D₄	—	15	1	no	—
166	C₃×D₄	—	15	1	no	—
167	C₂×C₃×C₄	—	3	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	10	1	no	—
171	C₂×C₃×C₄	—	15	1	no	—
172	#67	—	5	1	no	—
173	#67	—	5	1	no	—
174	D₁₂	—	5	1	no	—
175	C₄×D₃	—	5	1	no	—
176	C₄×D₃	C₂×C₃×D₅	1	1	yes	characteristic
177	Dic₆	—	5	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	3	1	no	—
180	C₃×C₈	—	5	1	no	—
181	C₃×C₈	—	5	1	no	—
182	C₃×C₈	—	10	1	no	—
183	C₃×C₈	—	15	1	no	—
184	C₃×C₈	D₃×D₅	1	1	yes	characteristic
185	C₃×C₈	C₂×C₃×D₅	1	1	yes	characteristic
186	C₃⋊C₈	—	5	1	no	—
187	C₃⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂²×C₅	—	3	1	no	—
189	C₂×D₅	—	3	1	no	—
190	C₂×D₅	C₃×C₄×D₃	1	1	yes	characteristic
191	C₄×C₅	—	3	1	no	—
192	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
193	Dic₅	—	3	1	no	—
194	Dic₅	C₃×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃²	—	10	1	no	—
196	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
197	C₃×D₃	—	2	1	no	—
198	C₃×D₃	—	10	1	no	—
199	#41	—	15	1	no	—
200	M₁₆	—	15	1	no	—
201	M₁₆	—	15	1	no	—
202	M₁₆	—	15	1	no	—
203	C₂×C₈	—	3	1	no	—
204	C₂×C₈	—	5	1	no	—
205	C₂×C₈	—	15	1	no	—
206	C₃×C₅	—	2	1	no	—
207	C₃×C₅	#901	1	1	yes	characteristic, complemented
208	C₃×C₅	C₃×#131	1	1	yes	characteristic, complemented
209	C₂²×C₃	—	3	1	no	—
210	C₂²×C₃	—	5	1	no	—
211	C₂²×C₃	—	5	1	no	—
212	C₂²×C₃	—	10	1	no	—
213	C₂²×C₃	—	15	1	no	—
214	C₂×D₃	—	5	1	no	—
215	C₂×D₃	C₃×C₄×D₅	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	3	1	no	—
218	C₃×C₄	—	5	1	no	—
219	C₃×C₄	—	5	1	no	—
220	C₃×C₄	—	10	1	no	—
221	C₃×C₄	—	15	1	no	—
222	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
223	C₃×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
224	Dic₃	—	5	1	no	—
225	Dic₃	C₃×C₄×D₅	1	1	yes	characteristic
226	C₂×C₅	—	6	1	no	—
227	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
228	D₅	—	2	1	no	—
229	D₅	—	6	1	no	—
230	C₃²	#4047	1	1	yes	characteristic, complemented, Sylow
231	Q₈	—	15	1	no	—
232	D₄	—	15	1	no	—
233	D₄	—	15	1	no	—
234	D₄	—	15	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	5	1	no	—
237	C₂×C₄	—	15	1	no	—
238	C₈	—	3	1	no	—
239	C₈	—	5	1	no	—
240	C₈	—	15	1	no	—
241	C₈	C₃×D₃×D₅	1	1	yes	characteristic
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	6	1	no	—
244	C₂×C₃	—	10	2	no	—
245	C₂×C₃	—	10	1	no	—
246	C₂×C₃	—	10	1	no	—
247	C₂×C₃	—	30	1	no	—
248	C₂×C₃	C₄×D₃×D₅	1	1	yes	characteristic
249	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
250	D₃	—	2	1	no	—
251	D₃	—	10	1	no	—
252	C₅	C₃×#901	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	3	1	no	—
254	C₂²	—	5	1	no	—
255	C₂²	—	15	1	no	—
256	C₄	—	3	1	no	—
257	C₄	—	5	1	no	—
258	C₄	—	15	1	no	—
259	C₄	C₂×C₃×D₃×D₅	1	1	yes	characteristic
260	C₃	—	2	1	no	—
261	C₃	#91265	1	1	yes	characteristic, direct product factor
262	C₃	C₃×#4047	1	1	yes	characteristic, complemented
263	C₂	—	6	1	no	—
264	C₂	—	10	1	no	—
265	C₂	—	30	1	no	—
266	C₂	C₃×C₄×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₈	D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₄	C₂×C₃×D₃×D₅
Fitting	C₃²×C₅×C₈	C₂²
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₃×#6771	C₂
cosocle	C₃×C₄×C₅	C₂³×C₃
hypocenter	C₃×C₅	C₃×#131
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×D₁₅	C₂	semidirect
3. C₃×C₅×D₃×C₈	C₂	semidirect
4. C₃²×C₈×D₅	C₂	semidirect
5. C₃×#6651	C₂	semidirect
6. C₃×#6650	C₂	semidirect
7. C₃×#6649	C₂	semidirect
9. #91265	C₃	direct (N × H)
25. C₈×D₁₅	C₂×C₃	semidirect
26. C₅×D₃×C₈	C₂×C₃	semidirect
32. C₃×C₈×D₅	D₃	semidirect
33. C₃×C₈×D₅	C₂×C₃	semidirect
34. #6651	C₂×C₃	semidirect
35. #6650	C₂×C₃	semidirect
36. #6649	C₂×C₃	semidirect
51. C₃×D₃×C₈	D₅	semidirect
89. C₈×D₅	C₃×D₃	semidirect
132. D₃×C₈	C₃×D₅	semidirect
133. C₃²×C₅	#131	semidirect
207. C₃×C₅	#901	semidirect
208. C₃×C₅	C₃×#131	semidirect
230. C₃²	#4047	semidirect
252. C₅	C₃×#901	semidirect
261. C₃	#91265	direct (N × H)
262. C₃	C₃×#4047	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6771	2
C₃×C₈×D₁₅	2
C₃×C₅×D₃×C₈	2
C₃²×C₈×D₅	2
C₃×#6651	2
C₃×#6650	2
C₃×#6649	2
C₃×#4047	3
#91265	3
C₃×#901	5

Subgroups — C3×#91265 #13086413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#91265 #13086413