Subgroups — C₃×#91819 #13088027

Order 1440.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1220	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6714	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#91819	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4076	—	3	1	no	maximal
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×D₁₅	C₂²	1	2	yes	—
10	C₂×C₃²×C₄×C₅	—	2	2	no	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₃×#1220	—	2	4	no	—
14	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
15	C₃×D₆₀	—	2	2	no	—
16	C₃×#930	—	5	1	no	maximal, Hall
17	C₂²×C₃×C₄×C₅	—	2	1	no	—
18	C₂²×C₃×C₄×C₅	D₃	1	1	yes	characteristic, complemented
19	C₂²×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×#1220	C₂×C₃	1	2	yes	complemented
21	C₂×D₆₀	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₂²⋊D₅	—	3	2	no	—
23	C₂×C₃×D₂₀	—	3	1	no	—
24	#6714	C₂×C₃	1	2	yes	complemented
25	C₄⋊Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#685	—	3	2	no	—
27	C₃×C₄⋊Dic₅	—	3	1	no	—
28	C₂²×C₃²×C₅	—	2	1	no	—
29	C₂²×C₃²×C₅	—	2	1	no	—
30	C₂²×C₃²×C₅	D₄	1	2	yes	complemented
31	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×D₁₅	—	2	2	no	—
33	C₂×C₃×D₁₅	—	2	2	no	—
34	C₂×C₃×D₁₅	—	2	2	no	—
35	C₃²×C₄×C₅	—	2	1	no	—
36	C₃²×C₄×C₅	D₄	1	2	yes	complemented
37	C₃×Dic₁₅	—	2	2	no	—
38	#4076	—	3	1	no	Hall
39	C₂²×C₃²×C₄	D₅	1	1	yes	characteristic, complemented
40	C₂×C₃×#67	—	5	2	no	—
41	C₂×C₃×D₁₂	—	5	1	no	—
42	C₃×(C₂×D₃)⋊C₄	—	5	2	no	—
43	C₃×C₄⋊Dic₃	—	5	1	no	—
44	C₂³×C₃×C₅	—	2	1	no	—
45	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₂²×D₁₅	C₂²×C₃	1	2	yes	—
48	C₂²×C₃×D₅	—	3	2	no	—
49	C₂×C₃×C₄×C₅	—	2	4	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	2	1	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
58	#1220	—	2	4	no	—
59	C₂×Dic₁₅	C₂²×C₃	1	2	yes	—
60	D₆₀	—	2	2	no	—
61	C₃×C₂²⋊D₅	—	6	4	no	—
62	C₂×C₃×Dic₅	—	3	2	no	—
63	C₃×D₂₀	—	6	2	no	—
64	C₃×#121	—	15	1	no	—
65	#930	—	5	1	no	—
66	C₂×C₃²×C₅	—	2	2	no	—
67	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃²×C₅	#41	1	1	yes	characteristic
70	C₃×D₁₅	—	4	2	no	—
71	C₂²×C₄×C₅	C₃×D₃	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊D₅	—	3	2	no	—
73	C₂×D₂₀	—	3	1	no	—
74	#685	—	3	2	no	—
75	C₄⋊Dic₅	—	3	1	no	—
76	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	5	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
81	C₃×#67	—	10	4	no	—
82	C₂×C₃×Dic₃	—	5	2	no	—
83	C₃×D₁₂	—	10	2	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	—	2	1	no	—
87	C₂²×C₃×C₅	—	2	1	no	—
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	D₁₂	1	2	yes	complemented
93	C₂²×C₃×C₅	C₃×D₄	1	2	yes	complemented
94	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₂×D₁₅	—	2	2	no	—
97	C₂×D₁₅	—	2	2	no	—
98	C₂×D₁₅	—	2	2	no	—
99	C₂×C₃×D₅	—	6	2	no	—
100	C₂×C₃×D₅	—	6	2	no	—
101	C₂×C₃×D₅	—	6	2	no	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	#67	1	2	yes	complemented
107	C₃×C₄×C₅	C₃×D₄	1	2	yes	complemented
108	Dic₁₅	—	2	2	no	—
109	C₃×Dic₅	—	6	2	no	—
110	C₂×C₃×D₄	—	15	2	no	—
111	C₂×C₃×D₄	—	15	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₃×D₅	1	1	yes	characteristic, complemented
114	C₂²×C₃×C₄	D₁₅	1	1	yes	characteristic, complemented
115	C₂×#67	—	5	2	no	—
116	C₂×D₁₂	—	5	1	no	—
117	C₃×C₄⋊C₄	—	15	1	no	—
118	C₃×C₂²⋊C₄	—	15	2	no	—
119	(C₂×D₃)⋊C₄	—	5	2	no	—
120	C₄⋊Dic₃	—	5	1	no	—
121	C₃²×C₅	#121	1	1	yes	characteristic, complemented, Hall
122	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂²×D₅	—	3	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂²⋊D₅	—	6	4	no	—
128	C₂×Dic₅	—	3	2	no	—
129	D₂₀	—	6	2	no	—
130	C₂²×C₃²	—	2	1	no	—
131	C₂²×C₃²	—	2	1	no	—
132	C₂²×C₃²	D₂₀	1	2	yes	complemented
133	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₃×D₃	—	10	2	no	—
135	C₂×C₃×D₃	—	10	2	no	—
136	C₂×C₃×D₃	—	10	2	no	—
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	C₂²⋊D₅	1	2	yes	complemented
139	C₃×Dic₃	—	10	2	no	—
140	#121	—	15	1	no	Sylow
141	C₂×C₃×C₅	—	2	4	no	—
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	—	2	2	no	—
144	C₂×C₃×C₅	—	2	1	no	—
145	C₂×C₃×C₅	—	2	1	no	—
146	C₂×C₃×C₅	—	2	1	no	—
147	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₃×C₅	#235	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
153	D₁₅	—	4	2	no	—
154	C₃×D₅	—	12	2	no	—
155	C₂³×C₃	—	2	1	no	—
156	C₂³×C₃	—	15	2	no	—
157	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
158	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
159	C₂²×D₃	—	5	2	no	—
160	C₃×D₄	—	30	4	no	—
161	C₃×D₄	—	30	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	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	15	2	no	—
168	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
172	#67	—	10	4	no	—
173	C₂×Dic₃	—	5	2	no	—
174	D₁₂	—	10	2	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	C₃×D₁₂	1	2	yes	complemented
178	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
179	C₂×D₅	—	6	2	no	—
180	C₂×D₅	—	6	2	no	—
181	C₂×D₅	—	6	2	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	C₃×#67	1	2	yes	complemented
184	Dic₅	—	6	2	no	—
185	C₂×C₃²	—	2	2	no	—
186	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
187	C₂×C₃²	#709	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₂²⋊D₅	1	1	yes	characteristic
189	C₃×D₃	—	20	2	no	—
190	C₂×D₄	—	15	2	no	—
191	C₂×D₄	—	15	1	no	—
192	C₂²×C₄	C₃×D₁₅	1	1	yes	characteristic, complemented
193	C₄⋊C₄	—	15	1	no	—
194	C₂²⋊C₄	—	15	2	no	—
195	C₃×C₅	—	2	1	no	—
196	C₃×C₅	#930	1	1	yes	characteristic, complemented
197	C₃×C₅	C₃×#121	1	1	yes	characteristic, complemented
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	30	2	no	—
207	C₂²×C₃	—	30	2	no	—
208	C₂²×C₃	—	30	2	no	—
209	C₂²×C₃	C₃×D₂₀	1	2	yes	complemented
210	C₂²×C₃	D₆₀	1	2	yes	complemented
211	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
212	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
213	C₂×D₃	—	10	2	no	—
214	C₂×D₃	—	10	2	no	—
215	C₂×D₃	—	10	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	30	2	no	—
221	C₃×C₄	C₃×C₂²⋊D₅	1	2	yes	complemented
222	C₃×C₄	#1220	1	2	yes	complemented
223	Dic₃	—	10	2	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
226	C₂×C₅	C₃×#235	1	1	yes	characteristic
227	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
228	D₅	—	12	2	no	—
229	C₃²	#4076	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	15	2	no	—
231	C₂³	C₂×C₃×D₁₅	1	1	yes	characteristic
232	D₄	—	30	4	no	—
233	D₄	—	30	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	15	2	no	—
236	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
237	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	60	2	no	—
245	C₂×C₃	C₂×C₃×D₂₀	1	1	yes	characteristic
246	C₂×C₃	C₃×#709	1	1	yes	characteristic
247	C₂×C₃	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
248	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
249	C₂×C₃	#6817	1	1	yes	characteristic
250	C₂×C₃	C₂×#1220	1	1	yes	characteristic
251	D₃	—	20	2	no	—
252	C₅	C₃×#930	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	30	2	no	—
256	C₂²	—	30	2	no	—
257	C₂²	—	30	2	no	—
258	C₂²	C₃×D₆₀	1	2	yes	complemented
259	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	30	2	no	—
262	C₄	C₃×#1220	1	2	yes	complemented
263	C₃	—	2	1	no	—
264	C₃	C₃×#4076	1	1	yes	characteristic, complemented
265	C₃	#91819	1	1	yes	characteristic, direct product factor
266	C₂	—	2	2	no	—
267	C₂	—	60	2	no	—
268	C₂	C₂×C₃×D₆₀	1	1	yes	characteristic
269	C₂	C₃×#6817	1	1	yes	characteristic
270	C₂	C₂×C₃×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting	C₂²×C₃²×C₄×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂²×C₃×C₄	D₁₅
hypocenter	C₃×C₅	C₃×#121
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₄×C₅	C₂	semidirect
2. C₂×C₃×#1220	C₂	semidirect
3. C₂×C₃×D₆₀	C₂	semidirect
4. C₃×#6714	C₂	semidirect
5. C₃×C₄⋊Dic₁₅	C₂	semidirect
6. #91819	C₃	direct (N × H)
18. C₂²×C₃×C₄×C₅	D₃	semidirect
19. C₂²×C₃×C₄×C₅	C₂×C₃	semidirect
20. C₂×#1220	C₂×C₃	semidirect
21. C₂×D₆₀	C₂×C₃	semidirect
24. #6714	C₂×C₃	semidirect
25. C₄⋊Dic₁₅	C₂×C₃	semidirect
30. C₂²×C₃²×C₅	D₄	semidirect
36. C₃²×C₄×C₅	D₄	semidirect
39. C₂²×C₃²×C₄	D₅	semidirect
71. C₂²×C₄×C₅	C₃×D₃	semidirect
92. C₂²×C₃×C₅	D₁₂	semidirect
93. C₂²×C₃×C₅	C₃×D₄	semidirect
106. C₃×C₄×C₅	#67	semidirect
107. C₃×C₄×C₅	C₃×D₄	semidirect
113. C₂²×C₃×C₄	C₃×D₅	semidirect
114. C₂²×C₃×C₄	D₁₅	semidirect
121. C₃²×C₅	#121	semidirect
132. C₂²×C₃²	D₂₀	semidirect
138. C₃²×C₄	C₂²⋊D₅	semidirect
177. C₂²×C₅	C₃×D₁₂	semidirect
183. C₄×C₅	C₃×#67	semidirect
192. C₂²×C₄	C₃×D₁₅	semidirect
196. C₃×C₅	#930	semidirect
197. C₃×C₅	C₃×#121	semidirect
209. C₂²×C₃	C₃×D₂₀	semidirect
210. C₂²×C₃	D₆₀	semidirect
221. C₃×C₄	C₃×C₂²⋊D₅	semidirect
222. C₃×C₄	#1220	semidirect
229. C₃²	#4076	semidirect
252. C₅	C₃×#930	semidirect
258. C₂²	C₃×D₆₀	semidirect
262. C₄	C₃×#1220	semidirect
264. C₃	C₃×#4076	semidirect
265. C₃	#91819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₅	2
C₂×C₃×#1220	2
C₂×C₃×D₆₀	2
C₃×#6714	2
C₃×C₄⋊Dic₁₅	2
#91819	3
C₃×#4076	3
C₃×#930	5

Subgroups — C3×#91819 #13088027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#91819 #13088027