Subgroups — C₃×#10601283 #14285648

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₃×#65213	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₃×#64700	C₂	1	4	yes	maximal, complemented
5	#10601283	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×#681	—	2	1	no	—
9	C₂²×C₃×#168	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	—	2	1	no	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₄⋊#168	—	2	1	no	—
13	C₃×C₄⋊#168	C₂²	1	2	yes	—
14	C₃×C₄×#168	C₄	1	1	yes	characteristic
15	C₃×C₄×#168	C₄	1	1	yes	characteristic
16	C₃×C₄×#168	C₂²	1	2	yes	—
17	C₃×#1566	—	5	1	no	maximal, Hall
18	C₅×#423	C₂×C₃	1	1	yes	characteristic
19	#65213	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
21	#64700	C₂×C₃	1	4	yes	complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	4	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₃×#681	—	4	2	no	—
32	C₂×C₃×#168	—	2	4	no	—
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	C₂×C₄	1	2	yes	—
35	C₂×C₃×#168	C₂×C₄	1	2	yes	—
36	C₃×#423	—	5	1	no	—
37	C₃×#423	D₅	1	1	yes	characteristic
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₃×#334	—	5	4	no	—
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂×#681	—	2	1	no	—
43	C₂²×#168	C₃×C₄	1	2	yes	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
46	C₄⋊#168	—	2	1	no	—
47	C₄⋊#168	C₂²×C₃	1	2	yes	—
48	C₄×#168	C₃×C₄	1	1	yes	characteristic
49	C₄×#168	C₃×C₄	1	1	yes	characteristic
50	C₄×#168	C₂²×C₃	1	2	yes	—
51	#1566	—	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	C₄⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
59	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₅×C₈	—	4	2	no	—
64	C₃×#168	—	2	2	no	—
65	C₃×#168	—	4	2	no	—
66	C₃×#168	M₁₆	1	4	yes	complemented
67	C₂×C₃×M₁₆	—	2	1	no	—
68	C₂×C₃×M₁₆	—	10	1	no	—
69	C₂²×C₃×C₈	—	5	2	no	—
70	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
71	C₃×C₄⋊C₈	—	2	1	no	—
72	C₃×C₄⋊C₈	—	5	2	no	—
73	C₃×C₄⋊C₈	—	10	1	no	—
74	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
75	C₃×C₄×C₈	—	5	1	no	—
76	C₃×C₄×C₈	—	5	1	no	—
77	C₃×C₄×C₈	—	5	2	no	—
78	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
81	C₅×M₁₆	—	4	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
86	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
87	#681	—	4	2	no	—
88	C₂×#168	—	2	4	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	C₂×C₃×C₄	1	2	yes	—
91	C₂×#168	C₂×C₃×C₄	1	2	yes	—
92	#423	—	5	1	no	—
93	#423	C₃×D₅	1	1	yes	characteristic
94	C₂×C₄×C₈	—	5	1	no	—
95	#334	—	5	4	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	#106	1	2	yes	—
99	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	#106	1	2	yes	—
103	C₃×C₄×C₅	#108	1	2	yes	—
104	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
105	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
106	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₃×C₄	Dic₁₀	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
110	C₃×M₁₆	—	4	2	no	—
111	C₃×M₁₆	—	20	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	5	2	no	—
114	C₂×C₃×C₈	—	5	2	no	—
115	C₂×C₃×C₈	—	10	4	no	—
116	C₂×C₃×C₈	—	10	2	no	—
117	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
118	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
119	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
120	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
121	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
128	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
129	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₅×C₈	—	4	2	no	—
133	#168	—	2	2	no	—
134	#168	—	4	2	no	—
135	#168	C₃×M₁₆	1	4	yes	complemented
136	C₂×M₁₆	—	2	1	no	—
137	C₂×M₁₆	—	10	1	no	—
138	C₂²×C₈	—	5	2	no	—
139	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	5	2	no	—
142	C₄⋊C₈	—	10	1	no	—
143	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
144	C₄×C₈	—	5	1	no	—
145	C₄×C₈	—	5	1	no	—
146	C₄×C₈	—	5	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	#423	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
151	C₂³×C₃	#683	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	#683	1	1	yes	characteristic
155	C₂×C₃×C₄	#683	1	1	yes	characteristic
156	C₂×C₃×C₄	#683	1	1	yes	characteristic
157	C₂×C₃×C₄	#687	1	2	yes	—
158	C₂×C₃×C₄	#688	1	2	yes	—
159	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₈	—	4	2	no	—
163	C₃×C₈	—	5	4	no	—
164	C₃×C₈	—	10	2	no	—
165	C₃×C₈	—	20	2	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	C₃×#106	1	2	yes	—
169	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	C₃×#106	1	2	yes	—
173	C₄×C₅	C₃×#108	1	2	yes	—
174	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
175	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
176	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
177	C₂²×C₄	C₃×Dic₁₀	1	1	yes	characteristic
178	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
179	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	20	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	5	2	no	—
184	C₂×C₈	—	5	2	no	—
185	C₂×C₈	—	10	4	no	—
186	C₂×C₈	—	10	2	no	—
187	C₄²	C₃×Dic₁₀	1	1	yes	characteristic
188	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
189	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
190	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
191	C₃×C₅	#1566	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	#3940	1	2	yes	—
195	C₂²×C₃	C₂×#683	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	#3940	1	2	yes	—
199	C₃×C₄	#3954	1	2	yes	—
200	C₃×C₄	D₅×M₁₆	1	2	yes	—
201	C₃×C₄	C₂×#687	1	1	yes	characteristic
202	C₃×C₄	C₂×#688	1	1	yes	characteristic
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	C₃×#423	1	1	yes	characteristic
205	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
206	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
207	C₂³	C₃×#683	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	C₃×#683	1	1	yes	characteristic
211	C₂×C₄	C₃×#683	1	1	yes	characteristic
212	C₂×C₄	C₃×#683	1	1	yes	characteristic
213	C₂×C₄	C₃×#687	1	2	yes	—
214	C₂×C₄	C₃×#688	1	2	yes	—
215	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
218	C₈	—	4	2	no	—
219	C₈	—	5	4	no	—
220	C₈	—	10	2	no	—
221	C₈	—	20	2	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	C₂×#3940	1	1	yes	characteristic
224	C₂×C₃	#65407	1	1	yes	characteristic
225	C₂×C₃	C₂×#3954	1	1	yes	characteristic
226	C₅	C₃×#1566	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	C₃×#3940	1	2	yes	—
230	C₂²	C₂×C₃×#683	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	C₃×#3940	1	2	yes	—
234	C₄	C₃×#3954	1	2	yes	—
235	C₄	C₃×D₅×M₁₆	1	2	yes	—
236	C₄	C₂×C₃×#687	1	1	yes	characteristic
237	C₄	C₂×C₃×#688	1	1	yes	characteristic
238	C₃	#10601283	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	2	2	no	—
240	C₂	C₂×C₃×#3940	1	1	yes	characteristic
241	C₂	C₃×#65407	1	1	yes	characteristic
242	C₂	C₂×C₃×#3954	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#423	D₅
hypocenter	C₅	C₃×#1566
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
4. C₃×#64700	C₂	semidirect
5. #10601283	C₃	direct (N × H)
21. #64700	C₂×C₃	semidirect
66. C₃×#168	M₁₆	semidirect
135. #168	C₃×M₁₆	semidirect
191. C₃×C₅	#1566	semidirect
226. C₅	C₃×#1566	semidirect
238. C₃	#10601283	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×#423	2
C₃×#65213	2
C₂×C₃×C₄×#168	2
C₃×#64700	2
#10601283	3
C₃×#1566	5

Subgroups — C3×#10601283 #14285648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601283 #14285648