Subgroups — C₃×#10601455 #14305230

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#65288	C₂	1	2	yes	maximal
5	C₃×#64752	C₂	1	2	yes	maximal
6	#10601455	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×#681	C₂²	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₄	1	2	yes	complemented
13	C₃×C₅×#106	C₄	1	2	yes	complemented
14	C₃×#3951	—	2	2	no	—
15	C₃×C₈×Dic₅	—	2	2	no	—
16	C₃×C₄⋊#168	—	2	4	no	—
17	C₃×#1945	—	5	1	no	maximal, Hall
18	C₅×#427	C₂×C₃	1	1	yes	characteristic
19	C₂×#3951	C₂×C₃	1	1	yes	characteristic
20	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
21	#65288	C₂×C₃	1	2	yes	—
22	#64752	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₃×Dic₅	—	2	1	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
29	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄×Dic₅	—	2	1	no	—
33	C₃×C₄×Dic₅	Q₈	1	2	yes	—
34	C₃×#681	—	2	2	no	—
35	C₃×#681	—	2	2	no	—
36	C₂×C₃×#168	—	2	1	no	—
37	C₂×C₃×#168	—	2	1	no	—
38	C₂×C₃×#168	—	2	2	no	—
39	C₂×C₃×#168	D₄	1	2	yes	—
40	C₂×C₃×#108	—	5	1	no	—
41	C₃×#427	D₅	1	1	yes	characteristic
42	C₃×#424	—	5	2	no	—
43	C₂×C₃×C₄×C₈	—	5	1	no	—
44	C₃×#339	—	5	2	no	—
45	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₅×#117	C₂²×C₃	1	2	yes	—
47	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×#681	C₂²×C₃	1	2	yes	—
49	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
50	C₅×#107	C₃×C₄	1	2	yes	complemented
51	C₅×#106	C₃×C₄	1	2	yes	complemented
52	#3951	—	2	2	no	—
53	C₈×Dic₅	—	2	2	no	—
54	C₄⋊#168	—	2	4	no	—
55	#1945	—	5	1	no	Sylow
56	C₂³×C₃×C₅	#41	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	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	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₂×C₃×Dic₅	—	2	4	no	—
65	C₂×C₃×Dic₅	—	2	2	no	—
66	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
67	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
68	C₃×#168	—	2	2	no	—
69	C₃×#168	—	4	2	no	—
70	C₂×C₃×M₁₆	—	5	2	no	—
71	C₂²×C₃×C₈	—	5	1	no	—
72	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
73	C₃×#117	C₂×D₅	1	2	yes	—
74	C₂×C₃×C₄²	—	5	1	no	—
75	C₃×#108	—	10	2	no	—
76	C₃×#107	Dic₅	1	2	yes	complemented
77	C₃×#106	Dic₅	1	2	yes	complemented
78	C₃×C₄⋊C₈	—	10	4	no	—
79	C₃×C₄×C₈	—	10	2	no	—
80	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₂²×Dic₅	—	2	1	no	—
82	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
86	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
87	C₅×C₂²⋊C₄	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄×Dic₅	—	2	1	no	—
90	C₄×Dic₅	C₃×Q₈	1	2	yes	—
91	#681	—	2	2	no	—
92	#681	—	2	2	no	—
93	C₂×#168	—	2	1	no	—
94	C₂×#168	—	2	1	no	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	C₃×D₄	1	2	yes	—
97	C₂×#108	—	5	1	no	—
98	#427	C₃×D₅	1	1	yes	characteristic
99	#424	—	5	2	no	—
100	C₂×C₄×C₈	—	5	1	no	—
101	#339	—	5	2	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#125	1	1	yes	characteristic
106	C₃×C₄×C₅	—	4	4	no	—
107	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₃×C₄×C₅	#121	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
111	C₃×Dic₅	—	2	4	no	—
112	C₂²×C₃×C₄	—	10	1	no	—
113	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₃×M₁₆	—	10	2	no	—
115	C₃×M₁₆	—	10	2	no	—
116	C₂×C₃×C₈	—	5	2	no	—
117	C₂×C₃×C₈	—	10	1	no	—
118	C₂×C₃×C₈	—	10	1	no	—
119	C₂×C₃×C₈	—	10	2	no	—
120	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
121	C₂×C₃×C₈	D₂₀	1	2	yes	—
122	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
125	C₃×C₂²⋊C₄	—	2	2	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₃×C₄²	—	5	2	no	—
128	C₃×C₄²	—	10	1	no	—
129	C₂³×C₅	C₃×#41	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
137	C₂×Dic₅	—	2	4	no	—
138	C₂×Dic₅	—	2	2	no	—
139	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
140	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
141	#168	—	2	2	no	—
142	#168	—	4	2	no	—
143	C₂×M₁₆	—	5	2	no	—
144	C₂²×C₈	—	5	1	no	—
145	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
146	#117	C₂×C₃×D₅	1	2	yes	—
147	C₂×C₄²	—	5	1	no	—
148	#108	—	10	2	no	—
149	#107	C₃×Dic₅	1	2	yes	complemented
150	#106	C₃×Dic₅	1	2	yes	complemented
151	C₄⋊C₈	—	10	4	no	—
152	C₄×C₈	—	10	2	no	—
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	#389	1	1	yes	characteristic
156	C₂×C₃×C₅	#443	1	2	yes	—
157	C₂³×C₃	#711	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	10	4	no	—
160	C₂×C₃×C₄	—	10	2	no	—
161	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
163	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
164	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
165	C₂×C₃×C₄	#713	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
167	C₃×C₈	—	10	2	no	—
168	C₃×C₈	—	20	2	no	—
169	C₃×C₈	C₄⋊Dic₅	1	2	yes	—
170	C₃×C₈	C₄⋊Dic₅	1	2	yes	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂²×C₅	C₃×#125	1	1	yes	characteristic
175	C₄×C₅	—	4	4	no	—
176	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
178	C₄×C₅	C₃×#121	1	1	yes	characteristic
179	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
180	Dic₅	—	2	4	no	—
181	C₂²×C₄	—	10	1	no	—
182	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
183	M₁₆	—	10	2	no	—
184	M₁₆	—	10	2	no	—
185	C₂×C₈	—	5	2	no	—
186	C₂×C₈	—	10	1	no	—
187	C₂×C₈	—	10	1	no	—
188	C₂×C₈	—	10	2	no	—
189	C₂×C₈	C₃×Dic₁₀	1	2	yes	—
190	C₂×C₈	C₃×D₂₀	1	2	yes	—
191	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
192	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
193	C₄⋊C₄	C₂×C₃×Dic₅	1	4	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	5	2	no	—
197	C₄²	—	10	1	no	—
198	C₃×C₅	#1945	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	#4036	1	1	yes	characteristic
201	C₂²×C₃	#4043	1	1	yes	characteristic
202	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
203	C₃×C₄	—	4	4	no	—
204	C₃×C₄	—	10	4	no	—
205	C₃×C₄	#4034	1	1	yes	characteristic
206	C₃×C₄	#4041	1	1	yes	characteristic
207	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
208	C₃×C₄	Q₈×Dic₅	1	1	yes	characteristic
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	C₃×#389	1	1	yes	characteristic
212	C₂×C₅	C₃×#443	1	2	yes	—
213	C₂³	C₃×#711	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	10	4	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
219	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
220	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
221	C₂×C₄	C₃×#713	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
223	C₈	—	10	2	no	—
224	C₈	—	20	2	no	—
225	C₈	C₃×C₄⋊Dic₅	1	2	yes	—
226	C₈	C₃×C₄⋊Dic₅	1	2	yes	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	#65274	1	1	yes	characteristic
230	C₂×C₃	#65485	1	2	yes	—
231	C₅	C₃×#1945	1	1	yes	characteristic, complemented, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	C₃×#4036	1	1	yes	characteristic
234	C₂²	C₃×#4043	1	1	yes	characteristic
235	C₂²	C₃×D₄×Dic₅	1	1	yes	characteristic
236	C₄	—	4	4	no	—
237	C₄	—	10	4	no	—
238	C₄	C₃×#4034	1	1	yes	characteristic
239	C₄	C₃×#4041	1	1	yes	characteristic
240	C₄	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
241	C₄	C₃×Q₈×Dic₅	1	1	yes	characteristic
242	C₃	#10601455	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	2	1	no	—
244	C₂	—	2	1	no	—
245	C₂	C₃×#65274	1	1	yes	characteristic
246	C₂	C₃×#65485	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#427	D₅
hypocenter	C₅	C₃×#1945
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#427

Complements

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

N	H ≅ G/N	type
6. #10601455	C₃	direct (N × H)
12. C₃×C₅×#107	C₄	semidirect
13. C₃×C₅×#106	C₄	semidirect
50. C₅×#107	C₃×C₄	semidirect
51. C₅×#106	C₃×C₄	semidirect
76. C₃×#107	Dic₅	semidirect
77. C₃×#106	Dic₅	semidirect
149. #107	C₃×Dic₅	semidirect
150. #106	C₃×Dic₅	semidirect
198. C₃×C₅	#1945	semidirect
231. C₅	C₃×#1945	semidirect
242. C₃	#10601455	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×#427	2
C₂×C₃×#3951	2
C₂×C₃×C₈×Dic₅	2
C₃×#65288	2
C₃×#64752	2
#10601455	3
C₃×#1945	5

Subgroups — C3×#10601455 #14305230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601455 #14305230