Subgroups — C₃×#10603906 #14315136

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65014	C₂	1	1	yes	maximal, characteristic
4	C₃×C₂²⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64888	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64887	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64885	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603906	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₃×#3998	—	2	2	no	—
13	C₂×C₃×#674	C₂²	1	1	yes	characteristic
14	C₂×C₃×#674	C₂²	1	1	yes	characteristic
15	C₂×C₃×#674	C₂²	1	1	yes	characteristic
16	C₂×C₃×#674	C₂²	1	1	yes	characteristic
17	C₃×#3990	—	2	2	no	—
18	C₃×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
19	C₃×#3952	C₄	1	1	yes	characteristic
20	C₃×#3946	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₃×#2169	—	5	1	no	maximal, Hall
23	C₂×#3998	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3990	C₂×C₃	1	1	yes	characteristic, complemented
25	#65014	C₂×C₃	1	1	yes	characteristic
26	C₂²⋊#674	C₂×C₃	1	1	yes	characteristic, complemented
27	#64888	C₂×C₃	1	1	yes	characteristic, complemented
28	#64887	C₂×C₃	1	1	yes	characteristic, complemented
29	#64885	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₂²⋊Dic₅	C₈	1	1	yes	characteristic
36	C₃×#685	C₈	1	1	yes	characteristic
37	C₃×#683	C₈	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	—	2	1	no	—
40	C₂×C₃×#168	—	2	1	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂³	1	1	yes	characteristic
43	C₃×C₈×D₅	—	2	1	no	—
44	C₃×C₈×D₅	—	2	1	no	—
45	C₃×#674	—	2	1	no	—
46	C₃×#674	—	2	1	no	—
47	C₃×#674	—	2	1	no	—
48	C₃×#674	D₄	1	2	yes	complemented
49	C₂×C₃×M₃₂	—	5	1	no	—
50	C₂×C₃×M₃₂	—	5	1	no	—
51	C₃×C₈×D₄	—	5	1	no	—
52	C₃×C₄⋊C₁₆	—	5	1	no	—
53	C₃×C₂²⋊C₁₆	—	5	1	no	—
54	C₃×C₂²⋊C₁₆	—	5	1	no	—
55	C₃×C₄×C₁₆	—	5	1	no	—
56	C₄×C₂²⋊D₅	C₃×C₄	1	1	yes	characteristic
57	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
59	#3998	—	2	2	no	—
60	C₂×#674	C₂²×C₃	1	1	yes	characteristic
61	C₂×#674	C₂²×C₃	1	1	yes	characteristic
62	C₂×#674	C₂²×C₃	1	1	yes	characteristic
63	C₂×#674	C₂²×C₃	1	1	yes	characteristic
64	#3990	—	2	2	no	—
65	C₅×C₂²⋊C₈	C₃×C₄	1	1	yes	characteristic
66	#3952	C₃×C₄	1	1	yes	characteristic
67	#3946	C₃×C₄	1	1	yes	characteristic
68	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	#2169	—	5	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×#168	—	2	1	no	—
85	C₃×#168	C₂×D₄	1	1	yes	characteristic
86	C₃×#168	#41	1	1	yes	characteristic
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂²×C₃×C₈	—	5	1	no	—
89	C₃×C₄×D₄	—	5	1	no	—
90	C₃×M₃₂	—	10	2	no	—
91	C₃×M₃₂	—	10	2	no	—
92	C₂×C₃×C₁₆	—	5	1	no	—
93	C₂×C₃×C₁₆	—	5	1	no	—
94	C₂×C₃×C₁₆	—	5	1	no	—
95	C₂×C₃×C₁₆	—	5	1	no	—
96	C₃×C₄⋊C₈	—	5	1	no	—
97	C₃×C₂²⋊C₈	—	5	1	no	—
98	C₃×C₂²⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₃×C₄×C₈	—	5	1	no	—
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₂²⋊D₅	C₃×C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂²⋊Dic₅	C₃×C₈	1	1	yes	characteristic
106	#685	C₃×C₈	1	1	yes	characteristic
107	#683	C₃×C₈	1	1	yes	characteristic
108	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×#168	—	2	1	no	—
110	C₂×#168	—	2	1	no	—
111	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×#168	C₂³×C₃	1	1	yes	characteristic
113	C₈×D₅	—	2	1	no	—
114	C₈×D₅	—	2	1	no	—
115	#674	—	2	1	no	—
116	#674	—	2	1	no	—
117	#674	—	2	1	no	—
118	#674	C₃×D₄	1	2	yes	complemented
119	C₂×M₃₂	—	5	1	no	—
120	C₂×M₃₂	—	5	1	no	—
121	C₈×D₄	—	5	1	no	—
122	C₄⋊C₁₆	—	5	1	no	—
123	C₂²⋊C₁₆	—	5	1	no	—
124	C₂²⋊C₁₆	—	5	1	no	—
125	C₄×C₁₆	—	5	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	1	no	—
131	C₂×C₃×D₅	—	2	1	no	—
132	C₂×C₃×D₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	—	2	1	no	—
138	C₃×Dic₅	M₃₂	1	2	yes	complemented
139	C₂×C₃×D₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	—	10	1	no	—
145	C₂×C₃×C₈	—	10	1	no	—
146	C₂×C₃×C₈	—	10	1	no	—
147	C₂×C₃×C₈	—	10	1	no	—
148	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	5	1	no	—
151	C₃×C₂²⋊C₄	—	5	1	no	—
152	C₃×C₂²⋊C₄	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₁₆	—	5	2	no	—
155	C₃×C₁₆	—	10	1	no	—
156	C₃×C₁₆	—	10	1	no	—
157	C₃×C₁₆	—	10	1	no	—
158	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
159	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂²⋊D₅	—	2	2	no	—
165	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
166	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
167	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	#168	—	2	1	no	—
173	#168	C₂×C₃×D₄	1	1	yes	characteristic
174	#168	C₃×#41	1	1	yes	characteristic
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	5	1	no	—
177	C₄×D₄	—	5	1	no	—
178	M₃₂	—	10	2	no	—
179	M₃₂	—	10	2	no	—
180	C₂×C₁₆	—	5	1	no	—
181	C₂×C₁₆	—	5	1	no	—
182	C₂×C₁₆	—	5	1	no	—
183	C₂×C₁₆	—	5	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₂²⋊C₈	—	5	1	no	—
186	C₂²⋊C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₄×C₈	—	5	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	C₃×D₅	—	4	1	no	—
193	C₂³×C₃	—	5	1	no	—
194	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
195	C₃×D₄	—	10	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	10	1	no	—
202	C₂×C₃×C₄	—	10	1	no	—
203	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	5	1	no	—
208	C₃×C₈	—	5	1	no	—
209	C₃×C₈	—	10	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
219	C₄×C₅	C₃×#131	1	1	yes	characteristic
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	C₃×M₃₂	1	2	yes	complemented
223	C₂×D₄	—	5	1	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	10	1	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	—	10	1	no	—
231	C₂×C₈	—	10	1	no	—
232	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₄⋊C₄	—	5	1	no	—
235	C₂²⋊C₄	—	5	1	no	—
236	C₂²⋊C₄	—	5	1	no	—
237	C₄²	—	5	1	no	—
238	C₁₆	—	5	2	no	—
239	C₁₆	—	10	1	no	—
240	C₁₆	—	10	1	no	—
241	C₁₆	—	10	1	no	—
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	10	1	no	—
247	C₂²×C₃	—	10	1	no	—
248	C₂²×C₃	—	10	1	no	—
249	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	5	2	no	—
252	C₃×C₄	—	10	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	#4131	1	1	yes	characteristic
255	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
258	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
259	C₂×C₅	C₃×#504	1	1	yes	characteristic
260	D₅	—	4	1	no	—
261	C₂³	—	5	1	no	—
262	C₂³	C₃×D₅⋊C₈	1	1	yes	characteristic
263	D₄	—	10	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	10	1	no	—
271	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	5	1	no	—
276	C₈	—	5	1	no	—
277	C₈	—	10	1	no	—
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	—	20	1	no	—
280	C₂×C₃	#65692	1	1	yes	characteristic
281	C₂×C₃	D₅⋊M₃₂	1	1	yes	characteristic
282	C₂×C₃	#65725	1	1	yes	characteristic
283	C₅	C₃×#2169	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	10	1	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	5	2	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	1	no	—
295	C₄	C₃×#4131	1	1	yes	characteristic
296	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₃	#10603906	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	20	1	no	—
300	C₂	C₃×#65692	1	1	yes	characteristic
301	C₂	C₃×D₅⋊M₃₂	1	1	yes	characteristic
302	C₂	C₃×#65725	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₂²⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×C₂²⋊D₅	C₈
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₃×C₂²⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2169
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3998	C₂	semidirect
2. C₂×C₃×#3990	C₂	semidirect
4. C₃×C₂²⋊#674	C₂	semidirect
5. C₃×#64888	C₂	semidirect
6. C₃×#64887	C₂	semidirect
7. C₃×#64885	C₂	semidirect
8. #10603906	C₃	direct (N × H)
23. C₂×#3998	C₂×C₃	semidirect
24. C₂×#3990	C₂×C₃	semidirect
26. C₂²⋊#674	C₂×C₃	semidirect
27. #64888	C₂×C₃	semidirect
28. #64887	C₂×C₃	semidirect
29. #64885	C₂×C₃	semidirect
48. C₃×#674	D₄	semidirect
118. #674	C₃×D₄	semidirect
138. C₃×Dic₅	M₃₂	semidirect
222. Dic₅	C₃×M₃₂	semidirect
242. C₃×C₅	#2169	semidirect
283. C₅	C₃×#2169	semidirect
297. C₃	#10603906	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×#3998	2
C₂×C₃×#3990	2
C₃×#65014	2
C₃×C₂²⋊#674	2
C₃×#64888	2
C₃×#64887	2
C₃×#64885	2
#10603906	3
C₃×#2169	5

Subgroups — C3×#10603906 #14315136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603906 #14315136