Subgroups — C₃×#10598865 #14280604

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#676	C₂	1	1	yes	maximal, characteristic
2	C₃×#64974	C₂	1	1	yes	maximal, characteristic
3	C₃×#64974	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64674	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598865	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#676	C₄	1	2	yes	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#3952	C₄	1	2	yes	—
15	C₃×#3952	C₄	1	2	yes	—
16	C₃×#3947	C₄	1	2	yes	—
17	C₃×#3946	C₄	1	2	yes	—
18	C₃×#3946	C₄	1	2	yes	—
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	C₃×#1453	—	5	1	no	maximal, Hall
23	C₄×#676	C₂×C₃	1	1	yes	characteristic
24	#64974	C₂×C₃	1	1	yes	characteristic
25	#64974	C₂×C₃	1	1	yes	characteristic
26	C₅×#322	C₂×C₃	1	1	yes	characteristic, complemented
27	#64674	C₂×C₃	1	1	yes	characteristic, complemented
28	#64674	C₂×C₃	1	1	yes	characteristic, complemented
29	C₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×#168	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₃×#676	—	2	4	no	—
44	C₃×#431	—	5	1	no	—
45	C₃×#431	—	5	1	no	—
46	C₃×C₄×M₁₆	—	5	1	no	—
47	C₃×#322	—	5	1	no	—
48	C₃×#322	—	5	1	no	—
49	C₃×#322	D₅	1	1	yes	characteristic, complemented
50	C₃×C₈²	—	5	1	no	—
51	C₂×#676	C₃×C₄	1	2	yes	—
52	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	#3952	C₃×C₄	1	2	yes	—
57	#3952	C₃×C₄	1	2	yes	—
58	#3947	C₃×C₄	1	2	yes	—
59	#3946	C₃×C₄	1	2	yes	—
60	#3946	C₃×C₄	1	2	yes	—
61	C₄×#168	C₂²×C₃	1	1	yes	characteristic
62	C₄×#168	C₂²×C₃	1	1	yes	characteristic
63	C₄×#168	C₂²×C₃	1	1	yes	characteristic
64	#1453	—	5	1	no	Sylow
65	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
72	C₃×C₄×D₅	—	2	2	no	—
73	C₃×C₄×D₅	—	2	2	no	—
74	C₃×C₄×D₅	—	2	2	no	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃×#168	—	2	2	no	—
79	C₃×#168	—	2	2	no	—
80	C₃×#168	M₁₆	1	4	yes	—
81	C₂×C₃×M₁₆	—	5	2	no	—
82	C₂×C₃×C₄²	—	5	1	no	—
83	C₃×C₄⋊C₈	—	5	2	no	—
84	C₃×C₄⋊C₈	—	5	2	no	—
85	C₃×C₂²⋊C₈	—	5	2	no	—
86	C₃×C₂²⋊C₈	—	5	2	no	—
87	C₃×#97	—	5	2	no	—
88	C₃×C₄×C₈	—	5	1	no	—
89	C₃×C₄×C₈	—	5	1	no	—
90	C₃×C₄×C₈	—	5	1	no	—
91	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
100	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×#168	C₂×C₃×C₄	1	2	yes	—
105	C₂×#168	C₂×C₃×C₄	1	2	yes	—
106	C₂×#168	C₂×C₃×C₄	1	2	yes	—
107	#676	—	2	4	no	—
108	#431	—	5	1	no	—
109	#431	—	5	1	no	—
110	C₄×M₁₆	—	5	1	no	—
111	#322	—	5	1	no	—
112	#322	—	5	1	no	—
113	#322	C₃×D₅	1	1	yes	characteristic, complemented
114	C₈²	—	5	1	no	—
115	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	1	no	—
117	C₂×C₃×D₅	—	2	1	no	—
118	C₂×C₃×D₅	—	2	1	no	—
119	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	C₃×Dic₅	—	2	1	no	—
123	C₃×Dic₅	—	2	1	no	—
124	C₃×Dic₅	—	2	1	no	—
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	1	no	—
128	C₃×M₁₆	—	10	4	no	—
129	C₂×C₃×C₈	—	5	2	no	—
130	C₂×C₃×C₈	—	5	2	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
133	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
134	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
135	C₃×C₄²	—	5	1	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	—	5	1	no	—
138	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
139	C₂²×D₅	C₃×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
144	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
145	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	2	no	—
148	C₄×D₅	—	2	2	no	—
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	—	2	2	no	—
152	#168	—	2	2	no	—
153	#168	—	2	2	no	—
154	#168	C₃×M₁₆	1	4	yes	—
155	C₂×M₁₆	—	5	2	no	—
156	C₂×C₄²	—	5	1	no	—
157	C₄⋊C₈	—	5	2	no	—
158	C₄⋊C₈	—	5	2	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	—	5	2	no	—
161	#97	—	5	2	no	—
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	—	5	1	no	—
164	C₄×C₈	—	5	1	no	—
165	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
166	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
167	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
168	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
169	C₂×C₃×C₅	#405	1	1	yes	characteristic
170	C₂×C₃×C₅	#405	1	1	yes	characteristic
171	C₃×D₅	—	4	1	no	—
172	C₂³×C₃	—	5	1	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	10	2	no	—
177	C₂×C₃×C₄	—	10	2	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	5	4	no	—
186	C₃×C₈	—	10	2	no	—
187	C₃×C₈	—	10	2	no	—
188	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
189	C₂×D₅	—	2	1	no	—
190	C₂×D₅	—	2	1	no	—
191	C₂×D₅	—	2	1	no	—
192	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
193	C₄×C₅	C₃×#131	1	2	yes	—
194	C₄×C₅	C₃×#131	1	2	yes	—
195	Dic₅	—	2	1	no	—
196	Dic₅	—	2	1	no	—
197	Dic₅	—	2	1	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	—	5	1	no	—
201	M₁₆	—	10	4	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
206	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
207	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
208	C₄²	—	5	1	no	—
209	C₄²	—	5	1	no	—
210	C₄²	—	5	1	no	—
211	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
212	C₃×C₅	#1453	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	10	1	no	—
214	C₂²×C₃	—	10	1	no	—
215	C₂²×C₃	—	10	1	no	—
216	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
217	C₃×C₄	—	10	1	no	—
218	C₃×C₄	—	10	1	no	—
219	C₃×C₄	—	10	1	no	—
220	C₃×C₄	#4047	1	2	yes	—
221	C₃×C₄	D₅×M₁₆	1	2	yes	—
222	C₃×C₄	#4053	1	2	yes	—
223	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
224	C₂×C₅	C₃×#405	1	1	yes	characteristic
225	C₂×C₅	C₃×#405	1	1	yes	characteristic
226	D₅	—	4	1	no	—
227	C₂³	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	—	10	2	no	—
234	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	5	4	no	—
241	C₈	—	10	2	no	—
242	C₈	—	10	2	no	—
243	C₂×C₃	—	20	1	no	—
244	C₂×C₃	#64978	1	1	yes	characteristic
245	C₂×C₃	#64996	1	1	yes	characteristic
246	C₂×C₃	#64997	1	1	yes	characteristic
247	C₅	C₃×#1453	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	10	1	no	—
249	C₂²	—	10	1	no	—
250	C₂²	—	10	1	no	—
251	C₂²	C₃×C₄²×D₅	1	1	yes	characteristic
252	C₄	—	10	1	no	—
253	C₄	—	10	1	no	—
254	C₄	—	10	1	no	—
255	C₄	C₃×#4047	1	2	yes	—
256	C₄	C₃×D₅×M₁₆	1	2	yes	—
257	C₄	C₃×#4053	1	2	yes	—
258	C₃	#10598865	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	20	1	no	—
260	C₂	C₃×#64978	1	1	yes	characteristic
261	C₂	C₃×#64996	1	1	yes	characteristic
262	C₂	C₃×#64997	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#322	D₅
hypocenter	C₅	C₃×#1453
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄² ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#322	C₂	semidirect
5. C₃×#64674	C₂	semidirect
6. C₃×#64674	C₂	semidirect
7. C₃×C₈×#168	C₂	semidirect
8. #10598865	C₃	direct (N × H)
26. C₅×#322	C₂×C₃	semidirect
27. #64674	C₂×C₃	semidirect
28. #64674	C₂×C₃	semidirect
29. C₈×#168	C₂×C₃	semidirect
49. C₃×#322	D₅	semidirect
113. #322	C₃×D₅	semidirect
212. C₃×C₅	#1453	semidirect
247. C₅	C₃×#1453	semidirect
258. C₃	#10598865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×#676	2
C₃×#64974	2
C₃×C₅×#322	2
C₃×#64674	2
C₃×C₈×#168	2
#10598865	3
C₃×#1453	5

Subgroups — C3×#10598865 #14280604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598865 #14280604