Subgroups — C₃×#10599362 #14285676

Order 1920.

368 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×#65126	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64699	C₂	1	2	yes	maximal, complemented
5	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599362	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#676	—	2	2	no	—
8	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×#3946	—	2	2	no	—
14	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
16	C₃×C₄⋊#168	C₂²	1	2	yes	—
17	C₃×C₄×#168	C₂²	1	1	yes	characteristic
18	C₃×#1567	—	5	1	no	maximal, Hall
19	#65126	C₂×C₃	1	2	yes	—
20	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
21	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
22	#64699	C₂×C₃	1	2	yes	complemented
23	#64678	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
37	C₃×#676	—	4	4	no	—
38	C₃×C₈×D₅	—	2	2	no	—
39	C₃×C₈×D₅	—	2	2	no	—
40	C₃×#423	—	5	2	no	—
41	C₂×C₃×C₄×C₈	—	5	1	no	—
42	C₃×#335	—	5	2	no	—
43	C₃×#335	—	5	1	no	—
44	C₃×#335	D₅	1	1	yes	characteristic, complemented
45	C₂×#676	—	2	2	no	—
46	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
48	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
50	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	#3946	—	2	2	no	—
52	C₈×Dic₅	C₃×C₄	1	1	yes	characteristic
53	C₈×Dic₅	C₃×C₄	1	1	yes	characteristic
54	C₄⋊#168	C₂²×C₃	1	2	yes	—
55	C₄×#168	C₂²×C₃	1	1	yes	characteristic
56	#1567	—	5	1	no	Sylow
57	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₄×D₅	—	2	1	no	—
65	C₃×C₄×D₅	—	2	1	no	—
66	C₃×C₄×D₅	—	2	2	no	—
67	C₃×C₄×D₅	D₈	1	2	yes	—
68	C₃×C₄×D₅	Q₁₆	1	2	yes	—
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
72	C₃×#168	—	2	1	no	—
73	C₃×#168	—	4	2	no	—
74	C₃×#168	M₁₆	1	2	yes	complemented
75	C₂×C₃×M₁₆	—	10	2	no	—
76	C₂²×C₃×C₈	—	5	1	no	—
77	C₂²×C₃×C₈	—	5	1	no	—
78	C₂×C₃×C₄²	—	5	1	no	—
79	C₃×C₄⋊C₈	—	5	2	no	—
80	C₃×C₄⋊C₈	—	10	2	no	—
81	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
82	C₃×C₄×C₈	—	5	1	no	—
83	C₃×C₄×C₈	—	5	1	no	—
84	C₃×C₄×C₈	—	5	1	no	—
85	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
88	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
95	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
96	C₂×#168	—	2	2	no	—
97	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
99	#676	—	4	4	no	—
100	C₈×D₅	—	2	2	no	—
101	C₈×D₅	—	2	2	no	—
102	#423	—	5	2	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	#335	—	5	2	no	—
105	#335	—	5	1	no	—
106	#335	C₃×D₅	1	1	yes	characteristic, complemented
107	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	1	no	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	#107	1	2	yes	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	#107	1	2	yes	—
118	C₃×Dic₅	#108	1	2	yes	—
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₃×M₁₆	—	20	4	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	10	2	no	—
127	C₂×C₃×C₈	—	10	2	no	—
128	C₂×C₃×C₈	—	10	2	no	—
129	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
130	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
131	C₃×C₄²	—	5	1	no	—
132	C₃×C₄²	—	5	1	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
135	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
142	C₄×D₅	—	2	1	no	—
143	C₄×D₅	—	2	1	no	—
144	C₄×D₅	—	2	2	no	—
145	C₄×D₅	C₃×D₈	1	2	yes	—
146	C₄×D₅	C₃×Q₁₆	1	2	yes	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	4	2	no	—
149	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
150	#168	—	2	1	no	—
151	#168	—	4	2	no	—
152	#168	C₃×M₁₆	1	2	yes	complemented
153	C₂×M₁₆	—	10	2	no	—
154	C₂²×C₈	—	5	1	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂×C₄²	—	5	1	no	—
157	C₄⋊C₈	—	5	2	no	—
158	C₄⋊C₈	—	10	2	no	—
159	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
160	C₄×C₈	—	5	1	no	—
161	C₄×C₈	—	5	1	no	—
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
164	C₂×C₃×C₅	#423	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
167	C₃×D₅	—	2	2	no	—
168	C₂³×C₃	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	2	no	—
172	C₂×C₃×C₄	—	5	2	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	10	2	no	—
175	C₂×C₃×C₄	—	10	1	no	—
176	C₂×C₃×C₄	—	10	1	no	—
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	4	2	no	—
182	C₃×C₈	—	5	2	no	—
183	C₃×C₈	—	10	1	no	—
184	C₃×C₈	—	20	2	no	—
185	C₃×C₈	#676	1	2	yes	complemented
186	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂×D₅	—	2	1	no	—
188	C₂×D₅	—	2	1	no	—
189	C₂×D₅	C₃×#107	1	2	yes	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
192	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
193	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
194	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
195	Dic₅	—	2	1	no	—
196	Dic₅	C₃×#107	1	2	yes	—
197	Dic₅	C₃×#108	1	2	yes	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	—	5	1	no	—
201	M₁₆	—	20	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	5	1	no	—
204	C₂×C₈	—	5	1	no	—
205	C₂×C₈	—	10	2	no	—
206	C₂×C₈	—	10	2	no	—
207	C₂×C₈	—	10	2	no	—
208	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
209	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
210	C₄²	—	5	1	no	—
211	C₄²	—	5	1	no	—
212	C₄²	—	5	1	no	—
213	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
214	C₃×C₅	#1567	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	5	2	no	—
216	C₂²×C₃	—	10	1	no	—
217	C₂²×C₃	—	10	1	no	—
218	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	5	2	no	—
221	C₃×C₄	—	5	2	no	—
222	C₃×C₄	—	10	1	no	—
223	C₃×C₄	C₂×#676	1	1	yes	characteristic
224	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
225	C₃×C₄	D₅×D₈	1	1	yes	characteristic
226	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
227	C₂×C₅	C₃×#423	1	1	yes	characteristic
228	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
229	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
230	D₅	—	2	2	no	—
231	C₂³	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	1	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	5	2	no	—
236	C₂×C₄	—	5	1	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	—	10	1	no	—
239	C₂×C₄	—	10	1	no	—
240	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
241	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
242	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	5	2	no	—
246	C₈	—	10	1	no	—
247	C₈	—	20	2	no	—
248	C₈	C₃×#676	1	2	yes	complemented
249	C₂×C₃	—	10	2	no	—
250	C₂×C₃	#65126	1	1	yes	characteristic
251	C₂×C₃	D₅×#107	1	1	yes	characteristic
252	C₂×C₃	D₅×#108	1	1	yes	characteristic
253	C₅	C₃×#1567	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	5	2	no	—
255	C₂²	—	10	1	no	—
256	C₂²	—	10	1	no	—
257	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	5	2	no	—
260	C₄	—	5	2	no	—
261	C₄	—	10	1	no	—
262	C₄	C₂×C₃×#676	1	1	yes	characteristic
263	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
264	C₄	C₃×D₅×D₈	1	1	yes	characteristic
265	C₄	C₃×D₅×Q₁₆	1	1	yes	characteristic
266	C₃	#10599362	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	10	2	no	—
268	C₂	C₃×#65126	1	1	yes	characteristic
269	C₂	C₃×D₅×#107	1	1	yes	characteristic
270	C₂	C₃×D₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#335	D₅
hypocenter	C₅	C₃×#1567
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#335

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#335	C₂	semidirect
4. C₃×#64699	C₂	semidirect
5. C₃×#64678	C₂	semidirect
6. #10599362	C₃	direct (N × H)
21. C₅×#335	C₂×C₃	semidirect
22. #64699	C₂×C₃	semidirect
23. #64678	C₂×C₃	semidirect
44. C₃×#335	D₅	semidirect
71. C₃×C₅×C₈	M₁₆	semidirect
74. C₃×#168	M₁₆	semidirect
106. #335	C₃×D₅	semidirect
149. C₅×C₈	C₃×M₁₆	semidirect
152. #168	C₃×M₁₆	semidirect
185. C₃×C₈	#676	semidirect
214. C₃×C₅	#1567	semidirect
248. C₈	C₃×#676	semidirect
253. C₅	C₃×#1567	semidirect
266. C₃	#10599362	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×#65126	2
C₃×C₄×C₈×D₅	2
C₃×C₅×#335	2
C₃×#64699	2
C₃×#64678	2
#10599362	3
C₃×#1567	5

Subgroups — C3×#10599362 #14285676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599362 #14285676