Subgroups — C₃×#10599324 #14285754

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 312 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	1	yes	maximal, characteristic
2	C₃×#65124	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#676	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599324	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#676	—	2	1	no	—
10	C₂×C₃×#676	C₄	1	1	yes	characteristic
11	C₂×C₃×#676	C₄	1	1	yes	characteristic
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#3952	—	2	1	no	—
17	C₃×#3947	—	2	1	no	—
18	C₃×#3947	C₄	1	1	yes	characteristic
19	C₃×#3947	C₄	1	1	yes	characteristic
20	C₃×#3946	—	2	1	no	—
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	C₃×#1569	—	5	1	no	maximal, Hall
25	#65126	C₂×C₃	1	1	yes	characteristic
26	#65124	C₂×C₃	1	1	yes	characteristic
27	C₄×#676	C₂×C₃	1	1	yes	characteristic
28	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
29	#64700	C₂×C₃	1	1	yes	characteristic, complemented
30	#64699	C₂×C₃	1	1	yes	characteristic, complemented
31	#64678	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
43	C₂×C₃×#168	—	2	1	no	—
44	C₂×C₃×#168	—	2	1	no	—
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₃×#676	—	2	2	no	—
48	C₃×#676	—	2	2	no	—
49	C₃×#676	—	4	2	no	—
50	C₃×#432	—	5	1	no	—
51	C₃×#423	—	5	1	no	—
52	C₃×C₄×M₁₆	—	5	1	no	—
53	C₃×#335	—	5	1	no	—
54	C₃×#335	—	5	1	no	—
55	C₃×#334	—	5	1	no	—
56	C₃×#334	D₅	1	1	yes	characteristic, complemented
57	C₂×#676	—	2	1	no	—
58	C₂×#676	C₃×C₄	1	1	yes	characteristic
59	C₂×#676	C₃×C₄	1	1	yes	characteristic
60	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
64	#3952	—	2	1	no	—
65	#3947	—	2	1	no	—
66	#3947	C₃×C₄	1	1	yes	characteristic
67	#3947	C₃×C₄	1	1	yes	characteristic
68	#3946	—	2	1	no	—
69	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
70	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
71	C₄×#168	C₂²×C₃	1	1	yes	characteristic
72	#1569	—	5	1	no	Sylow
73	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₄×D₅	—	2	1	no	—
83	C₃×C₄×D₅	—	2	1	no	—
84	C₃×C₄×D₅	—	4	1	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
89	C₃×#168	—	2	1	no	—
90	C₃×#168	—	4	1	no	—
91	C₃×#168	—	4	1	no	—
92	C₃×#168	M₁₆	1	2	yes	complemented
93	C₂×C₃×M₁₆	—	5	1	no	—
94	C₂×C₃×M₁₆	—	5	1	no	—
95	C₂×C₃×M₁₆	—	10	1	no	—
96	C₂×C₃×C₄²	—	5	1	no	—
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	5	1	no	—
99	C₃×C₄⋊C₈	—	10	1	no	—
100	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
102	C₃×C₂²⋊C₈	—	10	1	no	—
103	C₃×#97	—	5	1	no	—
104	C₃×#97	—	5	1	no	—
105	C₃×#97	—	10	1	no	—
106	C₃×C₄×C₈	—	5	1	no	—
107	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
118	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
119	C₂×#168	—	2	1	no	—
120	C₂×#168	—	2	1	no	—
121	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
123	#676	—	2	2	no	—
124	#676	—	2	2	no	—
125	#676	—	4	2	no	—
126	#432	—	5	1	no	—
127	#423	—	5	1	no	—
128	C₄×M₁₆	—	5	1	no	—
129	#335	—	5	1	no	—
130	#335	—	5	1	no	—
131	#334	—	5	1	no	—
132	#334	C₃×D₅	1	1	yes	characteristic, complemented
133	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	1	no	—
135	C₂×C₃×D₅	—	2	1	no	—
136	C₂×C₃×D₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×C₄×C₅	#137	1	1	yes	characteristic
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	2	1	no	—
144	C₃×Dic₅	—	2	1	no	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₃×M₁₆	—	10	2	no	—
149	C₃×M₁₆	—	10	2	no	—
150	C₃×M₁₆	—	20	2	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	—	5	1	no	—
154	C₂×C₃×C₈	—	5	1	no	—
155	C₂×C₃×C₈	—	10	1	no	—
156	C₂×C₃×C₈	—	10	1	no	—
157	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
159	C₃×C₄²	—	5	1	no	—
160	C₃×C₄²	—	5	1	no	—
161	C₃×C₄²	—	5	1	no	—
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
168	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
169	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₄×D₅	—	2	1	no	—
171	C₄×D₅	—	2	1	no	—
172	C₄×D₅	—	2	1	no	—
173	C₄×D₅	—	2	1	no	—
174	C₄×D₅	—	4	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	4	1	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
179	#168	—	2	1	no	—
180	#168	—	4	1	no	—
181	#168	—	4	1	no	—
182	#168	C₃×M₁₆	1	2	yes	complemented
183	C₂×M₁₆	—	5	1	no	—
184	C₂×M₁₆	—	5	1	no	—
185	C₂×M₁₆	—	10	1	no	—
186	C₂×C₄²	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	10	1	no	—
190	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂²⋊C₈	—	10	1	no	—
193	#97	—	5	1	no	—
194	#97	—	5	1	no	—
195	#97	—	10	1	no	—
196	C₄×C₈	—	5	1	no	—
197	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
198	C₂×C₃×C₅	#423	1	1	yes	characteristic
199	C₂×C₃×C₅	#428	1	1	yes	characteristic
200	C₂×C₃×C₅	#430	1	1	yes	characteristic
201	C₃×D₅	—	4	1	no	—
202	C₂³×C₃	—	5	1	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	10	1	no	—
207	C₂×C₃×C₄	—	10	1	no	—
208	C₂×C₃×C₄	—	10	1	no	—
209	C₂×C₃×C₄	—	10	1	no	—
210	C₂×C₃×C₄	—	20	1	no	—
211	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
212	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	—	5	2	no	—
218	C₃×C₈	—	10	1	no	—
219	C₃×C₈	—	20	1	no	—
220	C₃×C₈	—	20	1	no	—
221	C₃×C₈	#676	1	2	yes	complemented
222	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
223	C₂×D₅	—	2	1	no	—
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
228	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
229	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₄×C₅	C₃×#137	1	1	yes	characteristic
231	Dic₅	—	2	1	no	—
232	Dic₅	—	2	1	no	—
233	Dic₅	—	2	1	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	M₁₆	—	10	2	no	—
238	M₁₆	—	10	2	no	—
239	M₁₆	—	20	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	10	1	no	—
245	C₂×C₈	—	10	1	no	—
246	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
247	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#1569	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	10	1	no	—
254	C₂²×C₃	—	10	1	no	—
255	C₂²×C₃	—	10	1	no	—
256	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	10	1	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	—	10	1	no	—
261	C₃×C₄	C₂×#676	1	1	yes	characteristic
262	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
263	C₃×C₄	#4060	1	1	yes	characteristic
264	C₃×C₄	#4061	1	1	yes	characteristic
265	C₂×C₅	C₃×#423	1	1	yes	characteristic
266	C₂×C₅	C₃×#428	1	1	yes	characteristic
267	C₂×C₅	C₃×#430	1	1	yes	characteristic
268	D₅	—	4	1	no	—
269	C₂³	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	10	1	no	—
274	C₂×C₄	—	10	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	1	no	—
277	C₂×C₄	—	20	1	no	—
278	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
279	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
280	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	5	2	no	—
285	C₈	—	10	1	no	—
286	C₈	—	20	1	no	—
287	C₈	—	20	1	no	—
288	C₈	C₃×#676	1	2	yes	complemented
289	C₂×C₃	—	20	1	no	—
290	C₂×C₃	#65126	1	1	yes	characteristic
291	C₂×C₃	#65150	1	1	yes	characteristic
292	C₂×C₃	#65182	1	1	yes	characteristic
293	C₅	C₃×#1569	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	10	1	no	—
295	C₂²	—	10	1	no	—
296	C₂²	—	10	1	no	—
297	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	10	1	no	—
300	C₄	—	10	1	no	—
301	C₄	—	10	1	no	—
302	C₄	C₂×C₃×#676	1	1	yes	characteristic
303	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
304	C₄	C₃×#4060	1	1	yes	characteristic
305	C₄	C₃×#4061	1	1	yes	characteristic
306	C₃	#10599324	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	20	1	no	—
308	C₂	C₃×#65126	1	1	yes	characteristic
309	C₂	C₃×#65150	1	1	yes	characteristic
310	C₂	C₃×#65182	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×#334	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#334	D₅
hypocenter	C₅	C₃×#1569
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#334	C₂	semidirect
5. C₃×#64700	C₂	semidirect
6. C₃×#64699	C₂	semidirect
7. C₃×#64678	C₂	semidirect
8. #10599324	C₃	direct (N × H)
28. C₅×#334	C₂×C₃	semidirect
29. #64700	C₂×C₃	semidirect
30. #64699	C₂×C₃	semidirect
31. #64678	C₂×C₃	semidirect
56. C₃×#334	D₅	semidirect
88. C₃×C₅×C₈	M₁₆	semidirect
92. C₃×#168	M₁₆	semidirect
132. #334	C₃×D₅	semidirect
178. C₅×C₈	C₃×M₁₆	semidirect
182. #168	C₃×M₁₆	semidirect
221. C₃×C₈	#676	semidirect
252. C₃×C₅	#1569	semidirect
288. C₈	C₃×#676	semidirect
293. C₅	C₃×#1569	semidirect
306. C₃	#10599324	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×#65124	2
C₃×C₄×#676	2
C₃×C₅×#334	2
C₃×#64700	2
C₃×#64699	2
C₃×#64678	2
#10599324	3
C₃×#1569	5

Subgroups — C3×#10599324 #14285754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599324 #14285754