Subgroups — C₃×#10613622 #14439832

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	2	yes	maximal
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65367	C₂	1	2	yes	maximal, complemented
4	C₃×#65149	C₂	1	4	yes	maximal
5	C₃×#65147	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
7	C₃×#64969	C₂	1	1	yes	maximal, characteristic, complemented
8	#10613622	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×#4092	—	2	2	no	—
11	C₃×Q₈×Dic₅	—	2	2	no	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	C₃×#4043	—	2	2	no	—
14	C₃×#4038	C₂²	1	4	yes	—
15	C₃×#4036	C₂²	1	4	yes	—
16	C₃×#4033	—	2	2	no	—
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×#4015	C₂²	1	2	yes	—
19	C₃×C₅×#106	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×#3949	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×#3939	C₂²	1	8	yes	—
24	C₃×C₄×#168	C₂²	1	1	yes	characteristic
25	C₃×#3160	—	5	1	no	maximal, Hall
26	#66062	C₂×C₃	1	2	yes	—
27	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
28	#65367	C₂×C₃	1	2	yes	complemented
29	#65149	C₂×C₃	1	4	yes	—
30	#65147	C₂×C₃	1	4	yes	complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
32	#64969	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
35	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₁₀	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₃×#685	—	2	4	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	—	2	4	no	—
44	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
45	C₃×#683	—	2	4	no	—
46	C₃×C₄×Dic₅	—	2	4	no	—
47	C₃×C₄×Dic₅	D₄	1	2	yes	—
48	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×#168	C₂³	1	2	yes	—
50	C₃×C₈×D₅	Q₈	1	8	yes	complemented
51	C₃×#533	—	5	2	no	—
52	C₃×#500	—	5	2	no	—
53	C₃×#499	—	5	4	no	—
54	C₃×#498	—	5	1	no	—
55	C₃×#498	D₅	1	1	yes	characteristic, complemented
56	C₃×#427	—	5	4	no	—
57	C₂×C₃×C₄×C₈	—	5	1	no	—
58	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
59	#4092	—	2	2	no	—
60	Q₈×Dic₅	—	2	2	no	—
61	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
62	#4043	—	2	2	no	—
63	#4038	C₂²×C₃	1	4	yes	—
64	#4036	C₂²×C₃	1	4	yes	—
65	#4033	—	2	2	no	—
66	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
67	#4015	C₂²×C₃	1	2	yes	—
68	C₅×#106	C₂²×C₃	1	4	yes	—
69	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
70	#3949	C₂²×C₃	1	4	yes	—
71	C₈×Dic₅	C₂²×C₃	1	2	yes	—
72	#3939	C₂²×C₃	1	8	yes	—
73	C₄×#168	C₂²×C₃	1	1	yes	characteristic
74	#3160	—	5	1	no	Sylow
75	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	4	4	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₃×Dic₅	—	2	4	no	—
81	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
83	C₃×C₄×D₅	—	2	4	no	—
84	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
85	C₃×Dic₁₀	—	4	4	no	—
86	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
87	C₃×#168	C₂×Q₈	1	4	yes	—
88	C₂²×C₃×C₈	—	5	2	no	—
89	C₃×C₄⋊Q₈	—	5	2	no	—
90	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
91	C₃×#125	—	5	4	no	—
92	C₃×C₂²⋊Q₈	—	10	2	no	—
93	C₃×C₂²⋊Q₈	—	10	2	no	—
94	C₃×C₄×Q₈	—	10	2	no	—
95	C₃×C₄×Q₈	—	10	2	no	—
96	C₃×#117	—	5	4	no	—
97	C₂×C₃×C₄²	—	5	1	no	—
98	C₃×#107	—	5	8	no	—
99	C₃×#106	—	5	4	no	—
100	C₃×#106	C₂×D₅	1	4	yes	—
101	C₃×C₄×C₈	—	5	2	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₅×Q₈	—	2	2	no	—
105	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
106	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₁₀	—	2	2	no	—
108	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
111	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
112	#685	—	2	4	no	—
113	C₄⋊Dic₅	—	2	2	no	—
114	C₄⋊Dic₅	—	2	4	no	—
115	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
116	#683	—	2	4	no	—
117	C₄×Dic₅	—	2	4	no	—
118	C₄×Dic₅	C₃×D₄	1	2	yes	—
119	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
120	C₂×#168	C₂³×C₃	1	2	yes	—
121	C₈×D₅	C₃×Q₈	1	8	yes	complemented
122	#533	—	5	2	no	—
123	#500	—	5	2	no	—
124	#499	—	5	4	no	—
125	#498	—	5	1	no	—
126	#498	C₃×D₅	1	1	yes	characteristic, complemented
127	#427	—	5	4	no	—
128	C₂×C₄×C₈	—	5	1	no	—
129	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	2	no	—
131	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
132	C₃×C₄×C₅	—	4	4	no	—
133	C₃×C₄×C₅	#135	1	4	yes	—
134	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
135	C₃×Dic₅	—	2	2	no	—
136	C₃×Dic₅	—	4	4	no	—
137	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
138	C₂×C₃×Q₈	—	2	2	no	—
139	C₂×C₃×Q₈	—	10	2	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	—	5	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	8	no	—
144	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	5	4	no	—
147	C₃×C₄⋊C₄	—	10	4	no	—
148	C₃×C₄⋊C₄	—	10	4	no	—
149	C₃×C₄⋊C₄	—	10	2	no	—
150	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
151	C₃×C₂²⋊C₄	—	10	4	no	—
152	C₃×C₄²	—	5	2	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	10	4	no	—
155	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
156	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₅×Q₈	—	4	4	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
161	C₂×Dic₅	—	2	4	no	—
162	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
163	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₄×D₅	—	2	4	no	—
165	C₄×D₅	C₂×C₃×Q₈	1	4	yes	—
166	Dic₁₀	—	4	4	no	—
167	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
168	#168	C₂×C₃×Q₈	1	4	yes	—
169	C₂²×C₈	—	5	2	no	—
170	C₄⋊Q₈	—	5	2	no	—
171	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
172	#125	—	5	4	no	—
173	C₂²⋊Q₈	—	10	2	no	—
174	C₂²⋊Q₈	—	10	2	no	—
175	C₄×Q₈	—	10	2	no	—
176	C₄×Q₈	—	10	2	no	—
177	#117	—	5	4	no	—
178	C₂×C₄²	—	5	1	no	—
179	#107	—	5	8	no	—
180	#106	—	5	4	no	—
181	#106	C₂×C₃×D₅	1	4	yes	—
182	C₄×C₈	—	5	2	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂×#135	1	2	yes	—
187	C₃×D₅	—	2	2	no	—
188	C₂³×C₃	—	5	1	no	—
189	C₃×Q₈	—	4	4	no	—
190	C₃×Q₈	—	20	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	4	no	—
194	C₂×C₃×C₄	—	5	2	no	—
195	C₂×C₃×C₄	—	10	4	no	—
196	C₂×C₃×C₄	—	10	4	no	—
197	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
198	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
199	C₃×C₈	—	5	4	no	—
200	C₃×C₈	Q₈×D₅	1	4	yes	—
201	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
202	C₂×D₅	—	2	2	no	—
203	C₂×D₅	C₃×C₄⋊Q₈	1	2	yes	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	C₃×#135	1	4	yes	—
206	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
207	Dic₅	—	2	2	no	—
208	Dic₅	—	4	4	no	—
209	Dic₅	C₃×C₄⋊Q₈	1	2	yes	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	10	2	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	2	no	—
214	C₂×C₈	—	5	2	no	—
215	C₂×C₈	—	5	8	no	—
216	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	5	4	no	—
219	C₄⋊C₄	—	10	4	no	—
220	C₄⋊C₄	—	10	4	no	—
221	C₄⋊C₄	—	10	2	no	—
222	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
223	C₂²⋊C₄	—	10	4	no	—
224	C₄²	—	5	2	no	—
225	C₄²	—	5	1	no	—
226	C₄²	—	10	4	no	—
227	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
228	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	5	2	no	—
230	C₂²×C₃	—	10	2	no	—
231	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	—	5	2	no	—
234	C₃×C₄	—	10	2	no	—
235	C₃×C₄	—	20	4	no	—
236	C₃×C₄	#4062	1	4	yes	—
237	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
238	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
239	C₂×C₅	C₂×C₃×#135	1	2	yes	—
240	D₅	—	2	2	no	—
241	C₂³	—	5	1	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	20	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	4	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	10	4	no	—
249	C₂×C₄	—	10	4	no	—
250	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
251	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
252	C₈	—	5	4	no	—
253	C₈	C₃×Q₈×D₅	1	4	yes	—
254	C₂×C₃	—	10	2	no	—
255	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₃	C₂×#4062	1	2	yes	—
257	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
258	C₂²	—	5	2	no	—
259	C₂²	—	10	2	no	—
260	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
261	C₄	—	4	4	no	—
262	C₄	—	5	2	no	—
263	C₄	—	10	2	no	—
264	C₄	—	20	4	no	—
265	C₄	C₃×#4062	1	4	yes	—
266	C₄	C₂×C₃×Q₈×D₅	1	2	yes	—
267	C₃	#10613622	1	1	yes	characteristic, direct product factor, Sylow
268	C₂	—	10	2	no	—
269	C₂	C₃×D₅×C₄⋊Q₈	1	1	yes	characteristic
270	C₂	C₂×C₃×#4062	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#498	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₃×#498	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#498

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#498	C₂	semidirect
3. C₃×#65367	C₂	semidirect
5. C₃×#65147	C₂	semidirect
7. C₃×#64969	C₂	semidirect
8. #10613622	C₃	direct (N × H)
27. C₅×#498	C₂×C₃	semidirect
28. #65367	C₂×C₃	semidirect
30. #65147	C₂×C₃	semidirect
32. #64969	C₂×C₃	semidirect
50. C₃×C₈×D₅	Q₈	semidirect
55. C₃×#498	D₅	semidirect
121. C₈×D₅	C₃×Q₈	semidirect
126. #498	C₃×D₅	semidirect
228. C₃×C₅	#3160	semidirect
257. C₅	C₃×#3160	semidirect
267. C₃	#10613622	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×#66062	2
C₃×C₅×#498	2
C₃×#65367	2
C₃×#65149	2
C₃×#65147	2
C₃×C₄×C₈×D₅	2
C₃×#64969	2
#10613622	3
C₃×#3160	5

Subgroups — C3×#10613622 #14439832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613622 #14439832