Subgroups — C₃×#10615544 #14412162

Order 1920.

1512 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4149	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65421	C₂	1	6	yes	maximal, complemented
5	C₂×C₃×#3955	C₂	1	6	yes	maximal, complemented
6	#10615544	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×#4149	—	2	4	no	—
9	C₃×#4149	—	2	4	no	—
10	C₂×C₃×#713	C₄	1	2	yes	—
11	C₂×C₃×D₄×D₅	C₄	1	6	yes	—
12	C₂×C₃×#709	C₂²	1	3	yes	—
13	C₂²×C₃×D₂₀	C₂²	1	3	yes	—
14	C₃×C₅×#131	—	2	4	no	—
15	C₂×C₃×C₅×M₁₆	C₂²	1	6	yes	complemented
16	C₃×#4094	—	2	4	no	—
17	C₂×C₃×#681	C₂²	1	6	yes	complemented
18	C₃×#3956	C₂²	1	4	yes	complemented
19	C₃×#3955	C₂²	1	12	yes	complemented
20	C₃×#2888	—	5	1	no	maximal, Hall
21	C₂×#4149	C₂×C₃	1	1	yes	characteristic
22	C₅×#568	C₂×C₃	1	1	yes	characteristic, complemented
23	#66153	C₂×C₃	1	1	yes	characteristic, complemented
24	#65421	C₂×C₃	1	6	yes	complemented
25	C₂×#3955	C₂×C₃	1	6	yes	complemented
26	C₂³×C₃×D₅	C₂×C₄	1	6	yes	—
27	C₃×C₅×#41	D₄	1	4	yes	—
28	C₃×C₅×#41	D₄	1	4	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	3	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
32	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	6	yes	—
33	C₃×#713	—	2	8	no	—
34	C₃×D₄×D₅	—	2	24	no	—
35	C₃×#709	—	2	6	no	—
36	C₃×#709	—	2	6	no	—
37	C₂×C₃×D₂₀	—	2	6	no	—
38	C₂×C₃×D₂₀	—	2	6	no	—
39	C₂×C₃×D₂₀	C₂×C₄	1	6	yes	—
40	C₂×C₃×D₂₀	C₂³	1	3	yes	—
41	C₂×C₃×C₄×D₅	C₂×C₄	1	6	yes	—
42	C₂×C₃×Dic₁₀	C₂³	1	1	yes	characteristic
43	C₃×C₅×M₁₆	—	2	6	no	—
44	C₃×C₅×M₁₆	C₂³	1	4	yes	—
45	C₂×C₃×C₅×C₈	—	2	6	no	—
46	C₃×#681	—	2	6	no	—
47	C₃×#681	C₂³	1	4	yes	—
48	C₂×C₃×#168	—	2	6	no	—
49	C₂×C₃×H(5,𝔽₂)	—	5	1	no	—
50	C₃×#568	—	5	1	no	—
51	C₃×#568	D₅	1	1	yes	characteristic, complemented
52	C₃×#413	—	5	6	no	—
53	C₂×C₃×#100	—	5	6	no	—
54	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
55	#4149	—	2	4	no	—
56	#4149	—	2	4	no	—
57	C₂×#713	C₃×C₄	1	2	yes	—
58	C₂×D₄×D₅	C₃×C₄	1	6	yes	—
59	C₂×#709	C₂²×C₃	1	3	yes	—
60	C₂²×D₂₀	C₂²×C₃	1	3	yes	—
61	C₅×#131	—	2	4	no	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	6	yes	complemented
63	#4094	—	2	4	no	—
64	C₂×#681	C₂²×C₃	1	6	yes	complemented
65	#3956	C₂²×C₃	1	4	yes	complemented
66	#3955	C₂²×C₃	1	12	yes	complemented
67	#2888	—	5	1	no	Sylow
68	C₂³×C₃×C₅	C₂²×C₄	1	3	yes	—
69	C₂²×C₃×D₅	—	2	12	no	—
70	C₂²×C₃×D₅	—	8	6	no	—
71	C₂²×C₃×D₅	C₂²×C₄	1	6	yes	—
72	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
73	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₃×C₂²⋊D₅	—	2	12	no	—
79	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
80	C₃×D₂₀	—	2	12	no	—
81	C₃×D₂₀	—	2	3	no	—
82	C₃×D₂₀	—	2	3	no	—
83	C₃×C₄×D₅	—	2	12	no	—
84	C₃×Dic₁₀	—	2	1	no	—
85	C₃×Dic₁₀	—	2	1	no	—
86	C₃×C₅×C₈	—	2	4	no	—
87	C₃×#168	—	2	4	no	—
88	C₃×H(5,𝔽₂)	—	10	4	no	—
89	C₃×H(5,𝔽₂)	—	10	4	no	—
90	C₂×C₃×#41	—	5	2	no	—
91	C₂×C₃×#41	—	5	3	no	—
92	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₃×D₄	—	5	6	no	—
94	C₂²×C₃×D₄	—	5	3	no	—
95	C₃×#131	—	2	4	no	—
96	C₃×#131	—	10	4	no	—
97	C₂×C₃×M₁₆	—	5	6	no	—
98	C₂×C₃×M₁₆	C₂×D₅	1	6	yes	complemented
99	C₃×#101	—	5	4	no	—
100	C₃×#100	—	5	12	no	—
101	C₂³×D₅	C₂×C₃×C₄	1	6	yes	—
102	C₅×#41	C₃×D₄	1	4	yes	—
103	C₅×#41	C₃×D₄	1	4	yes	—
104	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₅×D₄	C₂×C₃×C₄	1	3	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
107	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	6	yes	—
108	#713	—	2	8	no	—
109	D₄×D₅	—	2	24	no	—
110	#709	—	2	6	no	—
111	#709	—	2	6	no	—
112	C₂×D₂₀	—	2	6	no	—
113	C₂×D₂₀	—	2	6	no	—
114	C₂×D₂₀	C₂×C₃×C₄	1	6	yes	—
115	C₂×D₂₀	C₂³×C₃	1	3	yes	—
116	C₂×C₄×D₅	C₂×C₃×C₄	1	6	yes	—
117	C₂×Dic₁₀	C₂³×C₃	1	1	yes	characteristic
118	C₅×M₁₆	—	2	6	no	—
119	C₅×M₁₆	C₂³×C₃	1	4	yes	—
120	C₂×C₅×C₈	—	2	6	no	—
121	#681	—	2	6	no	—
122	#681	C₂³×C₃	1	4	yes	—
123	C₂×#168	—	2	6	no	—
124	C₂×H(5,𝔽₂)	—	5	1	no	—
125	#568	—	5	1	no	—
126	#568	C₃×D₅	1	1	yes	characteristic, complemented
127	#413	—	5	6	no	—
128	C₂×#100	—	5	6	no	—
129	C₂²×C₃×C₅	—	4	3	no	—
130	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
131	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₃×D₅	—	2	6	no	—
133	C₂×C₃×D₅	—	4	6	no	—
134	C₂×C₃×D₅	—	8	12	no	—
135	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
136	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₃×Dic₅	—	2	2	no	—
139	C₂⁴×C₃	—	5	6	no	—
140	C₃×#41	—	10	8	no	—
141	C₃×#41	—	10	6	no	—
142	C₃×#41	—	10	6	no	—
143	C₃×#41	D₂₀	1	4	yes	—
144	C₃×#41	C₂²⋊D₅	1	4	yes	—
145	C₂×C₃×Q₈	—	5	1	no	—
146	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
147	C₂×C₃×D₄	—	5	6	no	—
148	C₂×C₃×D₄	—	5	6	no	—
149	C₂×C₃×D₄	—	5	3	no	—
150	C₂×C₃×D₄	—	10	24	no	—
151	C₂×C₃×D₄	—	10	6	no	—
152	C₂×C₃×D₄	—	10	6	no	—
153	C₂×C₃×D₄	C₄×D₅	1	3	yes	—
154	C₂²×C₃×C₄	—	5	6	no	—
155	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
156	C₃×M₁₆	—	2	6	no	—
157	C₃×M₁₆	—	5	4	no	—
158	C₃×M₁₆	—	10	6	no	—
159	C₃×M₁₆	C₂²×D₅	1	4	yes	—
160	C₂×C₃×C₈	—	2	6	no	—
161	C₂×C₃×C₈	—	10	6	no	—
162	C₂³×C₅	C₂²×C₃×C₄	1	3	yes	—
163	C₂²×D₅	—	2	12	no	—
164	C₂²×D₅	—	8	6	no	—
165	C₂²×D₅	C₂²×C₃×C₄	1	6	yes	—
166	C₅×Q₈	C₃×C₂²⋊C₄	1	4	yes	—
167	C₅×D₄	C₃×C₂²⋊C₄	1	12	yes	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
169	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
171	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
172	C₂²⋊D₅	—	2	12	no	—
173	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
174	D₂₀	—	2	12	no	—
175	D₂₀	—	2	3	no	—
176	D₂₀	—	2	3	no	—
177	C₄×D₅	—	2	12	no	—
178	Dic₁₀	—	2	1	no	—
179	Dic₁₀	—	2	1	no	—
180	C₅×C₈	—	2	4	no	—
181	#168	—	2	4	no	—
182	H(5,𝔽₂)	—	10	4	no	—
183	H(5,𝔽₂)	—	10	4	no	—
184	C₂×#41	—	5	2	no	—
185	C₂×#41	—	5	3	no	—
186	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
187	C₂²×D₄	—	5	6	no	—
188	C₂²×D₄	—	5	3	no	—
189	#131	—	2	4	no	—
190	#131	—	10	4	no	—
191	C₂×M₁₆	—	5	6	no	—
192	C₂×M₁₆	C₂×C₃×D₅	1	6	yes	complemented
193	#101	—	5	4	no	—
194	#100	—	5	12	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	2	6	no	—
197	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₃×D₅	—	4	6	no	—
199	C₂³×C₃	—	5	6	no	—
200	C₂³×C₃	—	10	12	no	—
201	C₂³×C₃	—	40	6	no	—
202	C₂³×C₃	C₂×C₄×D₅	1	3	yes	—
203	C₃×Q₈	—	10	1	no	—
204	C₃×Q₈	—	10	1	no	—
205	C₃×Q₈	#685	1	4	yes	—
206	C₃×D₄	—	10	12	no	—
207	C₃×D₄	—	10	3	no	—
208	C₃×D₄	—	10	3	no	—
209	C₃×D₄	—	10	12	no	—
210	C₃×D₄	#685	1	12	yes	—
211	C₂×C₃×C₄	—	5	2	no	—
212	C₂×C₃×C₄	—	10	12	no	—
213	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
214	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
215	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
216	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
217	C₃×C₈	—	2	4	no	—
218	C₃×C₈	—	10	4	no	—
219	C₂²×C₅	—	4	3	no	—
220	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
221	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
222	C₂×D₅	—	2	6	no	—
223	C₂×D₅	—	4	6	no	—
224	C₂×D₅	—	8	12	no	—
225	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
226	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
228	Dic₅	—	2	2	no	—
229	C₂⁴	—	5	6	no	—
230	#41	—	10	8	no	—
231	#41	—	10	6	no	—
232	#41	—	10	6	no	—
233	#41	C₃×D₂₀	1	4	yes	—
234	#41	C₃×C₂²⋊D₅	1	4	yes	—
235	C₂×Q₈	—	5	1	no	—
236	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
237	C₂×D₄	—	5	6	no	—
238	C₂×D₄	—	5	6	no	—
239	C₂×D₄	—	5	3	no	—
240	C₂×D₄	—	10	24	no	—
241	C₂×D₄	—	10	6	no	—
242	C₂×D₄	—	10	6	no	—
243	C₂×D₄	C₃×C₄×D₅	1	3	yes	—
244	C₂²×C₄	—	5	6	no	—
245	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
246	M₁₆	—	2	6	no	—
247	M₁₆	—	5	4	no	—
248	M₁₆	—	10	6	no	—
249	M₁₆	C₂²×C₃×D₅	1	4	yes	—
250	C₂×C₈	—	2	6	no	—
251	C₂×C₈	—	10	6	no	—
252	C₃×C₅	#2888	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	4	3	no	—
254	C₂²×C₃	—	10	6	no	—
255	C₂²×C₃	—	20	6	no	—
256	C₂²×C₃	—	40	12	no	—
257	C₂²×C₃	C₂×#685	1	6	yes	—
258	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
259	C₃×C₄	—	10	2	no	—
260	C₃×C₄	C₂×#685	1	6	yes	—
261	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
262	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	2	6	no	—
265	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
266	D₅	—	4	6	no	—
267	C₂³	—	5	6	no	—
268	C₂³	—	10	12	no	—
269	C₂³	—	40	6	no	—
270	C₂³	C₂×C₃×C₄×D₅	1	3	yes	—
271	Q₈	—	10	1	no	—
272	Q₈	—	10	1	no	—
273	Q₈	C₃×#685	1	4	yes	—
274	D₄	—	10	12	no	—
275	D₄	—	10	3	no	—
276	D₄	—	10	3	no	—
277	D₄	—	10	12	no	—
278	D₄	C₃×#685	1	12	yes	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	10	12	no	—
281	C₂×C₄	C₂×C₃×C₄×D₅	1	3	yes	—
282	C₂×C₄	C₂×C₃×D₂₀	1	6	yes	—
283	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	6	yes	—
284	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	10	4	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	6	no	—
289	C₂×C₃	—	20	6	no	—
290	C₂×C₃	C₂²×#685	1	1	yes	characteristic
291	C₅	C₃×#2888	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	4	3	no	—
293	C₂²	—	10	6	no	—
294	C₂²	—	20	6	no	—
295	C₂²	—	40	12	no	—
296	C₂²	C₂×C₃×#685	1	6	yes	—
297	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
298	C₄	—	10	2	no	—
299	C₄	C₂×C₃×#685	1	6	yes	—
300	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
301	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
302	C₃	#10615544	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	2	1	no	—
304	C₂	—	2	6	no	—
305	C₂	—	20	6	no	—
306	C₂	C₂²×C₃×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×#685
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₃×#4149	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#568	D₅
hypocenter	C₅	C₃×#2888
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×#41 ◁ C₃×#568

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#568	C₂	semidirect
3. C₃×#66153	C₂	semidirect
4. C₃×#65421	C₂	semidirect
5. C₂×C₃×#3955	C₂	semidirect
6. #10615544	C₃	direct (N × H)
15. C₂×C₃×C₅×M₁₆	C₂²	semidirect
17. C₂×C₃×#681	C₂²	semidirect
18. C₃×#3956	C₂²	semidirect
19. C₃×#3955	C₂²	semidirect
22. C₅×#568	C₂×C₃	semidirect
23. #66153	C₂×C₃	semidirect
24. #65421	C₂×C₃	semidirect
25. C₂×#3955	C₂×C₃	semidirect
51. C₃×#568	D₅	semidirect
62. C₂×C₅×M₁₆	C₂²×C₃	semidirect
64. C₂×#681	C₂²×C₃	semidirect
65. #3956	C₂²×C₃	semidirect
66. #3955	C₂²×C₃	semidirect
98. C₂×C₃×M₁₆	C₂×D₅	semidirect
126. #568	C₃×D₅	semidirect
192. C₂×M₁₆	C₂×C₃×D₅	semidirect
252. C₃×C₅	#2888	semidirect
291. C₅	C₃×#2888	semidirect
302. C₃	#10615544	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×#4149	2
C₃×C₅×#568	2
C₃×#66153	2
C₃×#65421	2
C₂×C₃×#3955	2
#10615544	3
C₃×#2888	5

Subgroups — C3×#10615544 #14412162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615544 #14412162