Subgroups — C₃×#10614287 #14412013

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4147	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66146	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65295	C₂	1	12	yes	maximal, complemented
5	#10614287	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₃×#4147	—	2	4	no	—
8	C₃×#4147	—	2	4	no	—
9	C₂×C₃×#713	C₄	1	4	yes	complemented
10	C₂×C₃×Q₈×D₅	C₄	1	4	yes	complemented
11	C₂×C₃×#709	C₂²	1	6	yes	—
12	C₃×C₄×C₅×Q₈	—	2	4	no	—
13	C₃×C₅×#117	C₂²	1	6	yes	—
14	C₃×Q₈×Dic₅	—	2	4	no	—
15	C₃×#4072	C₂²	1	6	yes	—
16	C₃×#3938	C₂²	1	16	yes	—
17	C₃×#2884	—	5	1	no	maximal, Hall
18	C₂×#4147	C₂×C₃	1	1	yes	characteristic
19	C₅×#519	C₂×C₃	1	1	yes	characteristic, complemented
20	#66146	C₂×C₃	1	1	yes	characteristic, complemented
21	#65295	C₂×C₃	1	12	yes	complemented
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
24	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
26	C₂×C₃×C₂²⋊D₅	C₂³	1	4	yes	—
27	C₃×#713	—	2	16	no	—
28	C₃×Q₈×D₅	—	2	16	no	—
29	C₃×#709	—	2	12	no	—
30	C₃×#709	—	2	12	no	—
31	C₂×C₃×D₂₀	C₂×C₄	1	6	yes	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	12	yes	—
33	C₂×C₃×Dic₁₀	C₂×C₄	1	6	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	6	no	—
35	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
36	C₃×C₄²×C₅	—	2	6	no	—
37	C₃×C₂²⋊Dic₅	C₂³	1	4	yes	—
38	C₃×C₄⋊Dic₅	—	2	6	no	—
39	C₃×C₄×Dic₅	—	2	6	no	—
40	C₂×C₃×2₋¹⁺⁴	—	5	1	no	—
41	C₃×#519	—	5	1	no	—
42	C₃×#519	D₅	1	1	yes	characteristic, complemented
43	C₃×#410	—	5	12	no	—
44	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
45	#4147	—	2	4	no	—
46	#4147	—	2	4	no	—
47	C₂×#713	C₃×C₄	1	4	yes	complemented
48	C₂×Q₈×D₅	C₃×C₄	1	4	yes	complemented
49	C₂×#709	C₂²×C₃	1	6	yes	—
50	C₄×C₅×Q₈	—	2	4	no	—
51	C₅×#117	C₂²×C₃	1	6	yes	—
52	Q₈×Dic₅	—	2	4	no	—
53	#4072	C₂²×C₃	1	6	yes	—
54	#3938	C₂²×C₃	1	16	yes	—
55	#2884	—	5	1	no	Sylow
56	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
57	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
58	C₃×C₅×Q₈	C₂²⋊C₄	1	16	yes	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
63	C₃×C₂²⋊D₅	—	2	4	no	—
64	C₃×C₂²⋊D₅	—	2	4	no	—
65	C₂×C₃×Dic₅	—	2	4	no	—
66	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
67	C₃×D₂₀	—	2	12	no	—
68	C₃×C₄×D₅	—	2	24	no	—
69	C₃×Dic₁₀	—	2	12	no	—
70	C₃×2₋¹⁺⁴	—	10	4	no	—
71	C₃×2₋¹⁺⁴	—	10	4	no	—
72	C₂×C₃×#41	—	5	4	no	—
73	C₂×C₃×#41	—	5	6	no	—
74	C₂²×C₃×Q₈	—	5	4	no	—
75	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
76	C₃×C₄×Q₈	—	2	4	no	—
77	C₃×C₄×Q₈	—	10	4	no	—
78	C₃×#117	—	5	6	no	—
79	C₃×#117	C₂×D₅	1	6	yes	—
80	C₃×#99	—	5	16	no	—
81	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
82	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
83	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
85	C₂×C₂²⋊D₅	C₂³×C₃	1	4	yes	—
86	#713	—	2	16	no	—
87	Q₈×D₅	—	2	16	no	—
88	#709	—	2	12	no	—
89	#709	—	2	12	no	—
90	C₂×D₂₀	C₂×C₃×C₄	1	6	yes	—
91	C₂×C₄×D₅	C₂×C₃×C₄	1	12	yes	—
92	C₂×Dic₁₀	C₂×C₃×C₄	1	6	yes	—
93	C₅×C₄⋊C₄	—	2	6	no	—
94	C₅×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
95	C₄²×C₅	—	2	6	no	—
96	C₂²⋊Dic₅	C₂³×C₃	1	4	yes	—
97	C₄⋊Dic₅	—	2	6	no	—
98	C₄×Dic₅	—	2	6	no	—
99	C₂×2₋¹⁺⁴	—	5	1	no	—
100	#519	—	5	1	no	—
101	#519	C₃×D₅	1	1	yes	characteristic, complemented
102	#410	—	5	12	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₃×D₅	—	2	4	no	—
108	C₂×C₃×D₅	—	4	4	no	—
109	C₃×C₄×C₅	—	4	4	no	—
110	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
111	C₃×Dic₅	—	2	4	no	—
112	C₃×Dic₅	—	4	4	no	—
113	C₃×#41	—	10	16	no	—
114	C₃×#41	—	10	12	no	—
115	C₃×#41	—	10	12	no	—
116	C₂×C₃×Q₈	—	5	6	no	—
117	C₂×C₃×Q₈	—	10	16	no	—
118	C₂×C₃×Q₈	C₄×D₅	1	4	yes	—
119	C₂×C₃×Q₈	D₂₀	1	4	yes	—
120	C₂×C₃×Q₈	C₂²⋊D₅	1	4	yes	—
121	C₂×C₃×D₄	—	5	4	no	—
122	C₂×C₃×D₄	—	5	6	no	—
123	C₂²×C₃×C₄	—	5	12	no	—
124	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
125	C₃×C₄⋊C₄	—	2	6	no	—
126	C₃×C₄⋊C₄	—	10	6	no	—
127	C₃×C₂²⋊C₄	—	5	4	no	—
128	C₃×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
129	C₃×C₄²	—	2	6	no	—
130	C₃×C₄²	—	10	6	no	—
131	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
132	C₂²×D₅	C₂²×C₃×C₄	1	4	yes	—
133	C₅×Q₈	C₃×C₂²⋊C₄	1	16	yes	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
136	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
137	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
138	C₂²⋊D₅	—	2	4	no	—
139	C₂²⋊D₅	—	2	4	no	—
140	C₂×Dic₅	—	2	4	no	—
141	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
142	D₂₀	—	2	12	no	—
143	C₄×D₅	—	2	24	no	—
144	Dic₁₀	—	2	12	no	—
145	2₋¹⁺⁴	—	10	4	no	—
146	2₋¹⁺⁴	—	10	4	no	—
147	C₂×#41	—	5	4	no	—
148	C₂×#41	—	5	6	no	—
149	C₂²×Q₈	—	5	4	no	—
150	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	—	10	4	no	—
153	#117	—	5	6	no	—
154	#117	C₂×C₃×D₅	1	6	yes	—
155	#99	—	5	16	no	—
156	C₂×C₃×C₅	—	2	1	no	—
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₃×D₅	—	4	4	no	—
161	C₂³×C₃	—	5	4	no	—
162	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
163	C₃×Q₈	—	10	12	no	—
164	C₃×Q₈	#685	1	16	yes	—
165	C₃×D₄	—	10	4	no	—
166	C₃×D₄	—	10	4	no	—
167	C₃×D₄	—	10	12	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	5	4	no	—
170	C₂×C₃×C₄	—	10	24	no	—
171	C₂×C₃×C₄	—	10	4	no	—
172	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
173	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
177	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
178	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
179	C₂×D₅	—	2	4	no	—
180	C₂×D₅	—	4	4	no	—
181	C₄×C₅	—	4	4	no	—
182	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	12	yes	—
183	Dic₅	—	2	4	no	—
184	Dic₅	—	4	4	no	—
185	#41	—	10	16	no	—
186	#41	—	10	12	no	—
187	#41	—	10	12	no	—
188	C₂×Q₈	—	5	6	no	—
189	C₂×Q₈	—	10	16	no	—
190	C₂×Q₈	C₃×C₄×D₅	1	4	yes	—
191	C₂×Q₈	C₃×D₂₀	1	4	yes	—
192	C₂×Q₈	C₃×C₂²⋊D₅	1	4	yes	—
193	C₂×D₄	—	5	4	no	—
194	C₂×D₄	—	5	6	no	—
195	C₂²×C₄	—	5	12	no	—
196	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
197	C₄⋊C₄	—	2	6	no	—
198	C₄⋊C₄	—	10	6	no	—
199	C₂²⋊C₄	—	5	4	no	—
200	C₂²⋊C₄	C₂²×C₃×D₅	1	4	yes	—
201	C₄²	—	2	6	no	—
202	C₄²	—	10	6	no	—
203	C₃×C₅	#2884	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	10	4	no	—
206	C₂²×C₃	—	20	4	no	—
207	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
208	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
209	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
210	C₃×C₄	—	4	4	no	—
211	C₃×C₄	—	10	4	no	—
212	C₃×C₄	—	20	4	no	—
213	C₃×C₄	C₂×#685	1	12	yes	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	1	no	—
217	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
218	D₅	—	4	4	no	—
219	C₂³	—	5	4	no	—
220	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
221	Q₈	—	10	12	no	—
222	Q₈	C₃×#685	1	16	yes	—
223	D₄	—	10	4	no	—
224	D₄	—	10	4	no	—
225	D₄	—	10	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	10	24	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
231	C₂×C₄	C₂×C₃×D₂₀	1	6	yes	—
232	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	6	yes	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	20	4	no	—
237	C₂×C₃	C₂²×#685	1	1	yes	characteristic
238	C₅	C₃×#2884	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	4	1	no	—
240	C₂²	—	10	4	no	—
241	C₂²	—	20	4	no	—
242	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
243	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
244	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
245	C₄	—	4	4	no	—
246	C₄	—	10	4	no	—
247	C₄	—	20	4	no	—
248	C₄	C₂×C₃×#685	1	12	yes	—
249	C₃	#10614287	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	20	4	no	—
254	C₂	C₂²×C₃×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×#685
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#519	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₃×#4147	C₂
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×#519	D₅
hypocenter	C₅	C₃×#2884
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×Q₈ ◁ C₃×#519

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#519	C₂	semidirect
3. C₃×#66146	C₂	semidirect
4. C₃×#65295	C₂	semidirect
5. #10614287	C₃	direct (N × H)
9. C₂×C₃×#713	C₄	semidirect
10. C₂×C₃×Q₈×D₅	C₄	semidirect
19. C₅×#519	C₂×C₃	semidirect
20. #66146	C₂×C₃	semidirect
21. #65295	C₂×C₃	semidirect
42. C₃×#519	D₅	semidirect
47. C₂×#713	C₃×C₄	semidirect
48. C₂×Q₈×D₅	C₃×C₄	semidirect
101. #519	C₃×D₅	semidirect
203. C₃×C₅	#2884	semidirect
238. C₅	C₃×#2884	semidirect
249. C₃	#10614287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#4147	2
C₃×C₅×#519	2
C₃×#66146	2
C₃×#65295	2
#10614287	3
C₃×#2884	5

Subgroups — C3×#10614287 #14412013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10614287 #14412013