Subgroups — C₃×#10600898 #14302930

Order 1920.

628 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66150	C₂	1	1	yes	maximal, characteristic
2	C₃×#66146	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65428	C₂	1	2	yes	maximal
5	C₃×#64777	C₂	1	2	yes	maximal, complemented
6	#10600898	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#709	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×#4093	C₄	1	2	yes	complemented
11	C₃×#4092	—	2	2	no	—
12	C₃×Q₈×Dic₅	—	2	4	no	—
13	C₃×#4090	C₄	1	2	yes	complemented
14	C₃×#4075	—	2	2	no	—
15	C₃×#4072	—	2	2	no	—
16	C₃×#4072	C₂²	1	1	yes	characteristic
17	C₃×#4072	C₂²	1	2	yes	—
18	C₃×C₅×#101	—	2	2	no	—
19	C₃×#3957	—	2	4	no	—
20	C₃×#1912	—	5	1	no	maximal, Hall
21	#66150	C₂×C₃	1	1	yes	characteristic
22	#66146	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₅×#101	C₂×C₃	1	1	yes	characteristic, complemented
24	#65428	C₂×C₃	1	2	yes	—
25	#64777	C₂×C₃	1	2	yes	complemented
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
33	C₃×#709	—	2	2	no	—
34	C₃×#709	—	2	2	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	—	2	1	no	—
37	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₃×C₂²⋊Dic₅	—	2	2	no	—
42	C₃×C₂²⋊Dic₅	—	2	1	no	—
43	C₃×#685	—	2	4	no	—
44	C₃×C₄⋊Dic₅	—	2	4	no	—
45	C₃×C₄⋊Dic₅	—	2	2	no	—
46	C₃×C₄⋊Dic₅	—	2	1	no	—
47	C₃×#683	—	2	4	no	—
48	C₃×C₄×Dic₅	—	2	4	no	—
49	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
50	C₃×C₄×Dic₅	D₄	1	4	yes	—
51	C₃×#536	—	5	1	no	—
52	C₃×#519	—	5	1	no	—
53	C₃×#421	—	5	2	no	—
54	C₂×C₃×#101	D₅	1	1	yes	characteristic, complemented
55	C₃×H(3,ℤ/4)	—	5	2	no	—
56	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×#709	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	#4093	C₃×C₄	1	2	yes	complemented
60	#4092	—	2	2	no	—
61	Q₈×Dic₅	—	2	4	no	—
62	#4090	C₃×C₄	1	2	yes	complemented
63	#4075	—	2	2	no	—
64	#4072	—	2	2	no	—
65	#4072	C₂²×C₃	1	1	yes	characteristic
66	#4072	C₂²×C₃	1	2	yes	—
67	C₅×#101	—	2	2	no	—
68	#3957	—	2	4	no	—
69	#1912	—	5	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×D₅	—	2	1	no	—
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₃×C₂²⋊D₅	—	4	1	no	—
82	C₃×C₂²⋊D₅	—	4	1	no	—
83	C₂×C₃×Dic₅	—	2	2	no	—
84	C₂×C₃×Dic₅	—	2	1	no	—
85	C₂×C₃×Dic₅	—	2	4	no	—
86	C₃×D₂₀	—	2	2	no	—
87	C₃×C₄×D₅	—	4	2	no	—
88	C₃×Dic₁₀	—	2	2	no	—
89	C₃×C₅×C₈	—	4	2	no	—
90	C₂×C₃×#41	—	5	1	no	—
91	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
92	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
93	C₃×C₄⋊Q₈	—	5	2	no	—
94	C₃×#124	—	5	2	no	—
95	C₃×#123	—	10	2	no	—
96	C₃×C₂²⋊Q₈	—	10	2	no	—
97	C₃×C₄×Q₈	—	10	4	no	—
98	C₃×#117	—	5	2	no	—
99	C₃×#117	—	5	1	no	—
100	C₃×#117	—	10	2	no	—
101	C₃×#104	—	10	4	no	—
102	C₃×#101	—	2	2	no	—
103	C₂×C₅×Q₈	—	2	2	no	—
104	C₂×C₅×Q₈	—	2	2	no	—
105	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
106	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
107	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	2	1	no	—
110	#709	—	2	2	no	—
111	#709	—	2	2	no	—
112	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅	—	2	1	no	—
114	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂²⋊Dic₅	—	2	2	no	—
119	C₂²⋊Dic₅	—	2	1	no	—
120	#685	—	2	4	no	—
121	C₄⋊Dic₅	—	2	4	no	—
122	C₄⋊Dic₅	—	2	2	no	—
123	C₄⋊Dic₅	—	2	1	no	—
124	#683	—	2	4	no	—
125	C₄×Dic₅	—	2	4	no	—
126	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
127	C₄×Dic₅	C₃×D₄	1	4	yes	—
128	#536	—	5	1	no	—
129	#519	—	5	1	no	—
130	#421	—	5	2	no	—
131	C₂×#101	C₃×D₅	1	1	yes	characteristic, complemented
132	H(3,ℤ/4)	—	5	2	no	—
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₃×C₅	#120	1	1	yes	characteristic
136	C₂²×C₃×C₅	#123	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	2	no	—
138	C₂×C₃×D₅	—	4	1	no	—
139	C₃×C₄×C₅	—	2	4	no	—
140	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₃×C₄×C₅	#121	1	2	yes	—
142	C₃×Dic₅	—	4	4	no	—
143	C₃×Dic₅	—	4	2	no	—
144	C₃×Dic₅	—	4	1	no	—
145	C₃×#41	—	10	2	no	—
146	C₃×#41	—	10	2	no	—
147	C₂×C₃×Q₈	—	2	2	no	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	—	5	1	no	—
150	C₂×C₃×Q₈	C₄×D₅	1	2	yes	—
151	C₂×C₃×Q₈	C₂²⋊D₅	1	2	yes	—
152	C₂×C₃×D₄	—	5	1	no	—
153	C₂×C₃×D₄	—	10	1	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₂²×C₃×C₄	D₂₀	1	2	yes	—
156	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	—	2	2	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₃×C₄⋊C₄	—	10	4	no	—
161	C₃×C₄⋊C₄	—	10	4	no	—
162	C₃×C₄⋊C₄	—	10	2	no	—
163	C₃×C₄⋊C₄	—	10	1	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	10	1	no	—
166	C₃×C₂²⋊C₄	—	10	4	no	—
167	C₃×C₄²	—	5	2	no	—
168	C₃×C₄²	—	5	4	no	—
169	C₃×C₄²	—	10	4	no	—
170	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂²×D₅	—	2	1	no	—
172	C₅×Q₈	—	2	4	no	—
173	C₅×Q₈	—	2	4	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₃×#41	1	2	yes	—
181	C₂²⋊D₅	—	4	1	no	—
182	C₂²⋊D₅	—	4	1	no	—
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	1	no	—
185	C₂×Dic₅	—	2	4	no	—
186	D₂₀	—	2	2	no	—
187	C₄×D₅	—	4	2	no	—
188	Dic₁₀	—	2	2	no	—
189	C₅×C₈	—	4	2	no	—
190	C₂×#41	—	5	1	no	—
191	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
193	C₄⋊Q₈	—	5	2	no	—
194	#124	—	5	2	no	—
195	#123	—	10	2	no	—
196	C₂²⋊Q₈	—	10	2	no	—
197	C₄×Q₈	—	10	4	no	—
198	#117	—	5	2	no	—
199	#117	—	5	1	no	—
200	#117	—	10	2	no	—
201	#104	—	10	4	no	—
202	#101	—	2	2	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	2	1	no	—
206	C₂×C₃×C₅	#386	1	1	yes	characteristic
207	C₃×D₅	—	8	1	no	—
208	C₂³×C₃	—	10	1	no	—
209	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	10	2	no	—
213	C₃×D₄	—	10	2	no	—
214	C₃×D₄	—	20	1	no	—
215	C₃×D₄	—	20	1	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	10	4	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	—	10	1	no	—
221	C₂×C₃×C₄	—	20	2	no	—
222	C₂×C₃×C₄	#685	1	4	yes	—
223	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
225	C₂×C₃×C₄	#711	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₃×C₈	—	4	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₅	C₃×#120	1	1	yes	characteristic
231	C₂²×C₅	C₃×#123	1	1	yes	characteristic
232	C₂×D₅	—	4	2	no	—
233	C₂×D₅	—	4	1	no	—
234	C₄×C₅	—	2	4	no	—
235	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
236	C₄×C₅	C₃×#121	1	2	yes	—
237	Dic₅	—	4	4	no	—
238	Dic₅	—	4	2	no	—
239	Dic₅	—	4	1	no	—
240	#41	—	10	2	no	—
241	#41	—	10	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	5	1	no	—
245	C₂×Q₈	C₃×C₄×D₅	1	2	yes	—
246	C₂×Q₈	C₃×C₂²⋊D₅	1	2	yes	—
247	C₂×D₄	—	5	1	no	—
248	C₂×D₄	—	10	1	no	—
249	C₂²×C₄	—	10	1	no	—
250	C₂²×C₄	C₃×D₂₀	1	2	yes	—
251	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₄⋊C₄	—	10	4	no	—
256	C₄⋊C₄	—	10	4	no	—
257	C₄⋊C₄	—	10	2	no	—
258	C₄⋊C₄	—	10	1	no	—
259	C₂²⋊C₄	—	10	2	no	—
260	C₂²⋊C₄	—	10	1	no	—
261	C₂²⋊C₄	—	10	4	no	—
262	C₄²	—	5	2	no	—
263	C₄²	—	5	4	no	—
264	C₄²	—	10	4	no	—
265	C₃×C₅	#1912	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	20	2	no	—
268	C₂²×C₃	—	20	1	no	—
269	C₂²×C₃	#4028	1	1	yes	characteristic
270	C₂²×C₃	#4032	1	1	yes	characteristic
271	C₂²×C₃	C₂×#685	1	1	yes	characteristic
272	C₃×C₄	—	2	4	no	—
273	C₃×C₄	—	20	4	no	—
274	C₃×C₄	—	20	2	no	—
275	C₃×C₄	—	20	1	no	—
276	C₃×C₄	#4027	1	2	yes	—
277	C₃×C₄	#4085	1	2	yes	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	C₃×#386	1	1	yes	characteristic
282	D₅	—	8	1	no	—
283	C₂³	—	10	1	no	—
284	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	10	2	no	—
288	D₄	—	10	2	no	—
289	D₄	—	20	1	no	—
290	D₄	—	20	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	10	4	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	1	no	—
296	C₂×C₄	—	20	2	no	—
297	C₂×C₄	C₃×#685	1	4	yes	—
298	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
299	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
300	C₂×C₄	C₃×#711	1	2	yes	—
301	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
302	C₈	—	4	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	40	1	no	—
307	C₂×C₃	#65247	1	1	yes	characteristic
308	C₅	C₃×#1912	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	4	1	no	—
310	C₂²	—	20	2	no	—
311	C₂²	—	20	1	no	—
312	C₂²	C₃×#4028	1	1	yes	characteristic
313	C₂²	C₃×#4032	1	1	yes	characteristic
314	C₂²	C₂×C₃×#685	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	20	4	no	—
317	C₄	—	20	2	no	—
318	C₄	—	20	1	no	—
319	C₄	C₃×#4027	1	2	yes	—
320	C₄	C₃×#4085	1	2	yes	—
321	C₃	#10600898	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	40	1	no	—
326	C₂	C₃×#65247	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65247
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#101	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#101	D₅
hypocenter	C₅	C₃×#1912
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#101

Complements

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

N	H ≅ G/N	type
2. C₃×#66146	C₂	semidirect
3. C₂×C₃×C₅×#101	C₂	semidirect
5. C₃×#64777	C₂	semidirect
6. #10600898	C₃	direct (N × H)
10. C₃×#4093	C₄	semidirect
13. C₃×#4090	C₄	semidirect
22. #66146	C₂×C₃	semidirect
23. C₂×C₅×#101	C₂×C₃	semidirect
25. #64777	C₂×C₃	semidirect
54. C₂×C₃×#101	D₅	semidirect
59. #4093	C₃×C₄	semidirect
62. #4090	C₃×C₄	semidirect
131. C₂×#101	C₃×D₅	semidirect
265. C₃×C₅	#1912	semidirect
308. C₅	C₃×#1912	semidirect
321. C₃	#10600898	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×#66150	2
C₃×#66146	2
C₂×C₃×C₅×#101	2
C₃×#65428	2
C₃×#64777	2
#10600898	3
C₃×#1912	5

Subgroups — C3×#10600898 #14302930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600898 #14302930