Subgroups — C₃×#10600833 #14295509

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4094	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
3	C₃×#65526	C₂	1	1	yes	maximal, characteristic
4	C₃×#65286	C₂	1	2	yes	maximal, complemented
5	C₃×#64753	C₂	1	2	yes	maximal, complemented
6	#10600833	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	C₃×#4094	—	2	2	no	—
10	C₃×#4094	—	2	2	no	—
11	C₂×C₃×#681	C₄	1	2	yes	complemented
12	C₂×C₃×#681	C₂²	1	2	yes	—
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₂²×C₃×#168	C₄	1	2	yes	complemented
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₃×C₅×#99	C₄	1	4	yes	—
17	C₃×#3968	C₄	1	2	yes	complemented
18	C₃×#3965	C₄	1	2	yes	complemented
19	C₃×#3935	—	2	2	no	—
20	C₃×C₄×#168	—	2	2	no	—
21	C₃×#1757	—	5	1	no	maximal, Hall
22	C₂×#4094	C₂×C₃	1	1	yes	characteristic
23	C₅×#410	C₂×C₃	1	1	yes	characteristic
24	#65526	C₂×C₃	1	1	yes	characteristic
25	#65286	C₂×C₃	1	2	yes	complemented
26	#64753	C₂×C₃	1	2	yes	complemented
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
36	C₃×C₄²×C₅	—	2	2	no	—
37	C₃×#681	—	2	2	no	—
38	C₃×#681	—	2	2	no	—
39	C₃×#681	—	2	4	no	—
40	C₃×#681	C₂×C₄	1	4	yes	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	—	2	2	no	—
44	C₂×C₃×#168	C₂×C₄	1	2	yes	—
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	D₄	1	2	yes	—
48	C₂×C₃×#168	D₄	1	2	yes	—
49	C₂×C₃×#131	—	5	1	no	—
50	C₃×#413	—	5	1	no	—
51	C₃×#410	D₅	1	1	yes	characteristic
52	C₃×#405	—	5	2	no	—
53	C₃×#344	—	5	2	no	—
54	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
55	C₅×#117	C₂²×C₃	1	2	yes	—
56	#4094	—	2	2	no	—
57	#4094	—	2	2	no	—
58	C₂×#681	C₃×C₄	1	2	yes	complemented
59	C₂×#681	C₂²×C₃	1	2	yes	—
60	C₂×#681	C₂²×C₃	1	1	yes	characteristic
61	C₂²×#168	C₃×C₄	1	2	yes	complemented
62	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
63	C₅×#99	C₃×C₄	1	4	yes	—
64	#3968	C₃×C₄	1	2	yes	complemented
65	#3965	C₃×C₄	1	2	yes	complemented
66	#3935	—	2	2	no	—
67	C₄×#168	—	2	2	no	—
68	#1757	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₄²	1	2	yes	—
70	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	2	no	—
72	C₃×C₅×D₄	—	2	4	no	—
73	C₃×C₅×D₄	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	#41	1	2	yes	—
82	C₃×#168	—	2	2	no	—
83	C₃×#168	—	2	4	no	—
84	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×#168	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
88	C₃×#131	—	10	2	no	—
89	C₃×#131	—	10	2	no	—
90	C₂×C₃×M₁₆	—	5	2	no	—
91	C₂×C₃×M₁₆	—	5	1	no	—
92	C₂×C₃×M₁₆	—	5	2	no	—
93	C₂²×C₃×C₈	—	5	1	no	—
94	C₂²×C₃×C₈	—	5	2	no	—
95	C₃×#117	C₂×D₅	1	2	yes	—
96	C₃×#101	—	5	2	no	—
97	C₃×#100	—	5	2	no	—
98	C₃×#99	Dic₅	1	4	yes	—
99	C₃×#97	—	10	2	no	—
100	C₃×C₄×C₈	—	10	2	no	—
101	C₅×#41	—	2	2	no	—
102	C₅×#41	—	2	2	no	—
103	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
110	C₄²×C₅	—	2	2	no	—
111	#681	—	2	2	no	—
112	#681	—	2	2	no	—
113	#681	—	2	4	no	—
114	#681	C₂×C₃×C₄	1	4	yes	—
115	C₂×#168	—	2	2	no	—
116	C₂×#168	—	2	2	no	—
117	C₂×#168	—	2	2	no	—
118	C₂×#168	C₂×C₃×C₄	1	2	yes	—
119	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×#168	C₃×D₄	1	2	yes	—
122	C₂×#168	C₃×D₄	1	2	yes	—
123	C₂×#131	—	5	1	no	—
124	#413	—	5	1	no	—
125	#410	C₃×D₅	1	1	yes	characteristic
126	#405	—	5	2	no	—
127	#344	—	5	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	4	1	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	—	4	4	no	—
135	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₃×C₄×C₅	#117	1	1	yes	characteristic
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×#41	—	2	2	no	—
139	C₃×#41	—	2	2	no	—
140	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
141	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
142	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
143	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₃×M₁₆	—	5	4	no	—
146	C₃×M₁₆	—	10	4	no	—
147	C₃×M₁₆	—	10	2	no	—
148	C₃×M₁₆	—	10	2	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	C₂×C₃×C₈	—	5	2	no	—
153	C₂×C₃×C₈	—	5	2	no	—
154	C₂×C₃×C₈	—	10	2	no	—
155	C₂×C₃×C₈	—	10	2	no	—
156	C₂×C₃×C₈	—	10	2	no	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
159	C₃×C₄²	—	2	2	no	—
160	C₂³×C₅	C₃×C₄²	1	2	yes	—
161	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₃×#41	1	2	yes	—
173	#168	—	2	2	no	—
174	#168	—	2	4	no	—
175	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	#168	C₃×C₂²⋊C₄	1	2	yes	—
178	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
179	#131	—	10	2	no	—
180	#131	—	10	2	no	—
181	C₂×M₁₆	—	5	2	no	—
182	C₂×M₁₆	—	5	1	no	—
183	C₂×M₁₆	—	5	2	no	—
184	C₂²×C₈	—	5	1	no	—
185	C₂²×C₈	—	5	2	no	—
186	#117	C₂×C₃×D₅	1	2	yes	—
187	#101	—	5	2	no	—
188	#100	—	5	2	no	—
189	#99	C₃×Dic₅	1	4	yes	—
190	#97	—	10	2	no	—
191	C₄×C₈	—	10	2	no	—
192	C₂×C₃×C₅	—	2	2	no	—
193	C₂×C₃×C₅	—	2	1	no	—
194	C₂×C₃×C₅	—	4	2	no	—
195	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₃	C₄×Dic₅	1	2	yes	—
197	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
198	C₃×Q₈	—	2	2	no	—
199	C₃×D₄	—	2	4	no	—
200	C₃×D₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
208	C₂×C₃×C₄	#711	1	2	yes	—
209	C₃×C₈	—	5	1	no	—
210	C₃×C₈	—	5	1	no	—
211	C₃×C₈	—	5	2	no	—
212	C₃×C₈	—	10	2	no	—
213	C₃×C₈	—	10	4	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
218	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	4	4	no	—
221	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₄×C₅	C₃×#117	1	1	yes	characteristic
223	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
227	C₂×D₄	C₃×C₄×D₅	1	1	yes	characteristic
228	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
229	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
230	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
231	M₁₆	—	5	4	no	—
232	M₁₆	—	10	4	no	—
233	M₁₆	—	10	2	no	—
234	M₁₆	—	10	2	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	5	2	no	—
239	C₂×C₈	—	5	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	—	10	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	C₂×C₃×Dic₅	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
251	C₂²×C₃	D₄×Dic₅	1	2	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	4	4	no	—
254	C₃×C₄	#4023	1	1	yes	characteristic
255	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₃×C₄	#4027	1	2	yes	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
261	C₂³	C₃×C₄×Dic₅	1	2	yes	—
262	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
270	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
272	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
273	C₂×C₄	C₃×#711	1	2	yes	—
274	C₈	—	5	1	no	—
275	C₈	—	5	1	no	—
276	C₈	—	5	2	no	—
277	C₈	—	10	2	no	—
278	C₈	—	10	4	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
283	C₅	C₃×#1757	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
288	C₂²	C₃×D₄×Dic₅	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	C₃×#4023	1	1	yes	characteristic
292	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
293	C₄	C₃×#4027	1	2	yes	—
294	C₃	#10600833	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	4	2	no	—
298	C₂	C₃×C₂²⋊C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₅×C₂²⋊C₄
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#410	D₅
hypocenter	C₅	C₃×#1757
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#410

Complements

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

N	H ≅ G/N	type
4. C₃×#65286	C₂	semidirect
5. C₃×#64753	C₂	semidirect
6. #10600833	C₃	direct (N × H)
11. C₂×C₃×#681	C₄	semidirect
14. C₂²×C₃×#168	C₄	semidirect
17. C₃×#3968	C₄	semidirect
18. C₃×#3965	C₄	semidirect
25. #65286	C₂×C₃	semidirect
26. #64753	C₂×C₃	semidirect
58. C₂×#681	C₃×C₄	semidirect
61. C₂²×#168	C₃×C₄	semidirect
64. #3968	C₃×C₄	semidirect
65. #3965	C₃×C₄	semidirect
246. C₃×C₅	#1757	semidirect
283. C₅	C₃×#1757	semidirect
294. C₃	#10600833	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×#4094	2
C₃×C₅×#410	2
C₃×#65526	2
C₃×#65286	2
C₃×#64753	2
#10600833	3
C₃×#1757	5

Subgroups — C3×#10600833 #14295509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600833 #14295509