Subgroups — C₃×#10601544 #14285900

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₃×#65215	C₂	1	1	yes	maximal, characteristic
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic
4	C₃×#64678	C₂	1	2	yes	maximal, complemented
5	C₃×#64677	C₂	1	2	yes	maximal, complemented
6	#10601544	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
9	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₂²⋊#168	—	2	1	no	—
13	C₃×C₂²⋊#168	—	2	1	no	—
14	C₃×C₄⋊#168	C₂²	1	2	yes	—
15	C₃×C₄⋊#168	C₂²	1	2	yes	—
16	C₃×#3935	—	2	1	no	—
17	C₃×C₄×#168	—	2	1	no	—
18	C₃×#1572	—	5	1	no	maximal, Hall
19	C₅×#431	C₂×C₃	1	1	yes	characteristic
20	#65215	C₂×C₃	1	1	yes	characteristic
21	#65214	C₂×C₃	1	1	yes	characteristic
22	#64678	C₂×C₃	1	2	yes	complemented
23	#64677	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₃×#432	—	5	1	no	—
36	C₃×#431	—	5	1	no	—
37	C₃×#431	D₅	1	1	yes	characteristic
38	C₃×#335	—	5	2	no	—
39	C₃×#334	—	5	2	no	—
40	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊C₈	C₃×C₄	1	2	yes	—
42	C₅×C₂²⋊C₈	C₃×C₄	1	2	yes	—
43	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂²⋊#168	—	2	1	no	—
46	C₂²⋊#168	—	2	1	no	—
47	C₄⋊#168	C₂²×C₃	1	2	yes	—
48	C₄⋊#168	C₂²×C₃	1	2	yes	—
49	#3935	—	2	1	no	—
50	C₄×#168	—	2	1	no	—
51	#1572	—	5	1	no	Sylow
52	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	1	no	—
56	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	2	no	—
63	C₃×C₅×C₈	—	2	2	no	—
64	C₃×#168	—	4	2	no	—
65	C₃×#168	—	4	2	no	—
66	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
67	C₃×C₄⋊C₈	—	5	2	no	—
68	C₃×C₄⋊C₈	—	5	2	no	—
69	C₃×C₄⋊C₈	Dic₅	1	2	yes	—
70	C₃×C₂²⋊C₈	—	10	1	no	—
71	C₃×C₂²⋊C₈	—	10	1	no	—
72	C₃×C₂²⋊C₈	Dic₅	1	2	yes	—
73	C₃×#97	—	10	1	no	—
74	C₃×C₄×C₈	—	10	1	no	—
75	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
82	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
84	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
85	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₂×#168	—	2	2	no	—
87	C₂×#168	—	2	2	no	—
88	#432	—	5	1	no	—
89	#431	—	5	1	no	—
90	#431	C₃×D₅	1	1	yes	characteristic
91	#335	—	5	2	no	—
92	#334	—	5	2	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	#131	1	2	yes	—
102	C₃×C₄×C₅	#135	1	2	yes	—
103	C₂²×C₃×C₄	Dic₁₀	1	1	yes	characteristic
104	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
106	C₂×C₃×C₈	—	10	2	no	—
107	C₂×C₃×C₈	—	10	2	no	—
108	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
109	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
110	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
111	C₃×C₄²	D₂₀	1	1	yes	characteristic
112	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
113	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
114	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	—	2	2	no	—
126	#168	—	4	2	no	—
127	#168	—	4	2	no	—
128	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
129	C₄⋊C₈	—	5	2	no	—
130	C₄⋊C₈	—	5	2	no	—
131	C₄⋊C₈	C₃×Dic₅	1	2	yes	—
132	C₂²⋊C₈	—	10	1	no	—
133	C₂²⋊C₈	—	10	1	no	—
134	C₂²⋊C₈	C₃×Dic₅	1	2	yes	—
135	#97	—	10	1	no	—
136	C₄×C₈	—	10	1	no	—
137	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
138	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
139	C₂×C₃×C₅	—	4	1	no	—
140	C₂×C₃×C₅	#424	1	1	yes	characteristic
141	C₂×C₃×C₅	#427	1	1	yes	characteristic
142	C₂×C₃×C₅	#430	1	1	yes	characteristic
143	C₂³×C₃	C₄⋊Dic₅	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	20	2	no	—
156	C₃×C₈	—	20	2	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	C₃×#131	1	2	yes	—
166	C₄×C₅	C₃×#135	1	2	yes	—
167	C₂²×C₄	C₃×Dic₁₀	1	1	yes	characteristic
168	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
170	C₂×C₈	—	10	2	no	—
171	C₂×C₈	—	10	2	no	—
172	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
173	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
174	C₄²	C₃×Dic₁₀	1	1	yes	characteristic
175	C₄²	C₃×D₂₀	1	1	yes	characteristic
176	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
177	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
178	C₃×C₅	#1572	1	1	yes	characteristic, complemented, Hall
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	#4050	1	2	yes	—
188	C₃×C₄	#4094	1	2	yes	—
189	C₂×C₅	—	4	1	no	—
190	C₂×C₅	C₃×#424	1	1	yes	characteristic
191	C₂×C₅	C₃×#427	1	1	yes	characteristic
192	C₂×C₅	C₃×#430	1	1	yes	characteristic
193	C₂³	C₃×C₄⋊Dic₅	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₃×C₄⋊Dic₅	1	1	yes	characteristic
198	C₂×C₄	C₃×C₄⋊Dic₅	1	1	yes	characteristic
199	C₂×C₄	C₃×C₄⋊Dic₅	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	20	2	no	—
206	C₈	—	20	2	no	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	#65288	1	1	yes	characteristic
209	C₂×C₃	#65395	1	1	yes	characteristic
210	C₂×C₃	#65414	1	1	yes	characteristic
211	C₅	C₃×#1572	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	C₃×#4050	1	2	yes	—
221	C₄	C₃×#4094	1	2	yes	—
222	C₃	#10601544	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	4	1	no	—
224	C₂	C₃×#65288	1	1	yes	characteristic
225	C₂	C₃×#65395	1	1	yes	characteristic
226	C₂	C₃×#65414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#431	D₅
hypocenter	C₅	C₃×#1572
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#431

Complements

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

N	H ≅ G/N	type
4. C₃×#64678	C₂	semidirect
5. C₃×#64677	C₂	semidirect
6. #10601544	C₃	direct (N × H)
22. #64678	C₂×C₃	semidirect
23. #64677	C₂×C₃	semidirect
178. C₃×C₅	#1572	semidirect
211. C₅	C₃×#1572	semidirect
222. C₃	#10601544	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×C₅×#431	2
C₃×#65215	2
C₃×#65214	2
C₃×#64678	2
C₃×#64677	2
#10601544	3
C₃×#1572	5

Subgroups — C3×#10601544 #14285900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601544 #14285900