Subgroups — C₃×#10610898 #14417337

Order 1920.

1056 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	C₂²×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₃×#65855	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65290	C₂	1	2	yes	maximal, complemented
6	C₃×#65161	C₂	1	4	yes	maximal, complemented
7	C₃×#65158	C₂	1	4	yes	maximal, complemented
8	#10610898	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₃×#4072	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄₀	C₄	1	8	yes	complemented
15	C₂×C₃×D₄₀	C₂²	1	2	yes	—
16	C₂×C₃×D₄₀	C₂²	1	2	yes	—
17	C₃×#4039	C₂²	1	8	yes	complemented
18	C₃×D₅×C₂²⋊C₄	—	2	4	no	—
19	C₃×C₄×D₂₀	—	2	4	no	—
20	C₃×C₅×#107	C₂²	1	2	yes	complemented
21	C₃×C₅×#106	C₂²	1	2	yes	complemented
22	C₃×#3947	C₂²	1	2	yes	complemented
23	C₃×C₈×Dic₅	C₂²	1	2	yes	complemented
24	C₃×#3941	C₂²	1	8	yes	complemented
25	C₃×#2948	—	5	1	no	maximal, Hall
26	C₂²×D₄₀	C₂×C₃	1	1	yes	characteristic
27	#65855	C₂×C₃	1	2	yes	complemented
28	C₅×#427	C₂×C₃	1	1	yes	characteristic, complemented
29	#65390	C₂×C₃	1	1	yes	characteristic, complemented
30	#65290	C₂×C₃	1	2	yes	complemented
31	#65161	C₂×C₃	1	4	yes	complemented
32	#65158	C₂×C₃	1	4	yes	complemented
33	C₂³×C₃×D₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₂₀	—	2	2	no	—
36	C₂×C₃×D₂₀	—	2	4	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	8	yes	complemented
38	C₂×C₃×D₂₀	C₂³	1	4	yes	—
39	C₂×C₃×C₄×D₅	—	2	8	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₃×C₄²×C₅	—	2	2	no	—
46	C₃×C₂²⋊Dic₅	D₄	1	2	yes	—
47	C₃×#685	—	2	8	no	—
48	C₃×C₄⋊Dic₅	D₄	1	2	yes	—
49	C₃×C₄×Dic₅	C₂³	1	2	yes	—
50	C₃×#681	—	2	2	no	—
51	C₂×C₃×#168	C₂³	1	2	yes	—
52	C₃×D₄₀	C₂×C₄	1	8	yes	complemented
53	C₃×D₄₀	C₂×C₄	1	8	yes	complemented
54	C₂²×C₃×D₈	—	5	1	no	—
55	C₃×#518	—	5	2	no	—
56	C₃×#442	—	5	4	no	—
57	C₃×C₄×D₈	—	5	4	no	—
58	C₃×#427	D₅	1	1	yes	characteristic, complemented
59	C₃×#418	—	5	2	no	—
60	C₃×#405	—	5	1	no	—
61	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₅×#117	C₂²×C₃	1	2	yes	—
64	#4072	C₂²×C₃	1	1	yes	characteristic
65	C₂×#681	C₂²×C₃	1	1	yes	characteristic
66	C₂×D₄₀	C₃×C₄	1	8	yes	complemented
67	C₂×D₄₀	C₂²×C₃	1	2	yes	—
68	C₂×D₄₀	C₂²×C₃	1	2	yes	—
69	#4039	C₂²×C₃	1	8	yes	complemented
70	D₅×C₂²⋊C₄	—	2	4	no	—
71	C₄×D₂₀	—	2	4	no	—
72	C₅×#107	C₂²×C₃	1	2	yes	complemented
73	C₅×#106	C₂²×C₃	1	2	yes	complemented
74	#3947	C₂²×C₃	1	2	yes	complemented
75	C₈×Dic₅	C₂²×C₃	1	2	yes	complemented
76	#3941	C₂²×C₃	1	8	yes	complemented
77	#2948	—	5	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₃×D₅	—	2	4	no	—
80	C₂²×C₃×D₅	—	2	8	no	—
81	C₂²×C₃×D₅	—	8	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×C₃×Dic₅	C₂×D₄	1	4	yes	—
88	C₃×D₂₀	—	2	8	no	—
89	C₃×D₂₀	—	2	4	no	—
90	C₃×D₂₀	C₂²×C₄	1	8	yes	—
91	C₃×C₄×D₅	—	8	8	no	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₃×#168	—	2	2	no	—
95	C₂²×C₃×D₄	—	5	2	no	—
96	C₂×C₃×D₈	—	5	8	no	—
97	C₂×C₃×D₈	—	5	2	no	—
98	C₂×C₃×D₈	—	5	2	no	—
99	C₂×C₃×M₁₆	—	5	1	no	—
100	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄×D₄	—	5	8	no	—
102	C₃×C₄×D₄	—	10	4	no	—
103	C₃×#117	—	5	1	no	—
104	C₃×#117	C₂×D₅	1	2	yes	—
105	C₂×C₃×C₂²⋊C₄	—	10	4	no	—
106	C₃×#107	C₂×D₅	1	2	yes	complemented
107	C₃×#106	C₂×D₅	1	2	yes	complemented
108	C₃×#102	—	5	8	no	—
109	C₃×#97	—	5	2	no	—
110	C₃×C₄×C₈	—	5	2	no	—
111	C₂³×D₅	—	2	2	no	—
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₂×D₂₀	—	2	2	no	—
114	C₂×D₂₀	—	2	4	no	—
115	C₂×D₂₀	C₂×C₃×C₄	1	8	yes	complemented
116	C₂×D₂₀	C₂³×C₃	1	4	yes	—
117	C₂×C₄×D₅	—	2	8	no	—
118	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	complemented
119	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
121	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₂²⋊Dic₅	C₃×D₄	1	2	yes	—
125	#685	—	2	8	no	—
126	C₄⋊Dic₅	C₃×D₄	1	2	yes	—
127	C₄×Dic₅	C₂³×C₃	1	2	yes	—
128	#681	—	2	2	no	—
129	C₂×#168	C₂³×C₃	1	2	yes	—
130	D₄₀	C₂×C₃×C₄	1	8	yes	complemented
131	D₄₀	C₂×C₃×C₄	1	8	yes	complemented
132	C₂²×D₈	—	5	1	no	—
133	#518	—	5	2	no	—
134	#442	—	5	4	no	—
135	C₄×D₈	—	5	4	no	—
136	#427	C₃×D₅	1	1	yes	characteristic, complemented
137	#418	—	5	2	no	—
138	#405	—	5	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₃×D₅	—	2	8	no	—
143	C₂×C₃×D₅	—	4	8	no	—
144	C₂×C₃×D₅	—	8	8	no	—
145	C₃×C₄×C₅	—	4	4	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₃×Dic₅	—	2	4	no	—
150	C₂⁴×C₃	—	10	2	no	—
151	C₂×C₃×D₄	—	5	8	no	—
152	C₂×C₃×D₄	—	5	4	no	—
153	C₂×C₃×D₄	—	10	2	no	—
154	C₂×C₃×D₄	—	10	4	no	—
155	C₂²×C₃×C₄	—	10	8	no	—
156	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₃×D₈	—	5	8	no	—
158	C₃×D₈	—	5	8	no	—
159	C₃×M₁₆	—	10	2	no	—
160	C₂×C₃×C₈	—	5	2	no	—
161	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
162	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	5	2	no	—
165	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
166	C₃×C₂²⋊C₄	—	2	2	no	—
167	C₃×C₂²⋊C₄	—	5	2	no	—
168	C₃×C₂²⋊C₄	—	10	8	no	—
169	C₃×C₄²	—	2	2	no	—
170	C₃×C₄²	—	5	2	no	—
171	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂²×D₅	—	2	4	no	—
173	C₂²×D₅	—	2	8	no	—
174	C₂²×D₅	—	8	4	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
177	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₃×#41	1	2	yes	—
179	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
180	C₂×Dic₅	C₂×C₃×D₄	1	4	yes	—
181	D₂₀	—	2	8	no	—
182	D₂₀	—	2	4	no	—
183	D₂₀	C₂²×C₃×C₄	1	8	yes	—
184	C₄×D₅	—	8	8	no	—
185	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
186	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
187	#168	—	2	2	no	—
188	C₂²×D₄	—	5	2	no	—
189	C₂×D₈	—	5	8	no	—
190	C₂×D₈	—	5	2	no	—
191	C₂×D₈	—	5	2	no	—
192	C₂×M₁₆	—	5	1	no	—
193	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
194	C₄×D₄	—	5	8	no	—
195	C₄×D₄	—	10	4	no	—
196	#117	—	5	1	no	—
197	#117	C₂×C₃×D₅	1	2	yes	—
198	C₂×C₂²⋊C₄	—	10	4	no	—
199	#107	C₂×C₃×D₅	1	2	yes	complemented
200	#106	C₂×C₃×D₅	1	2	yes	complemented
201	#102	—	5	8	no	—
202	#97	—	5	2	no	—
203	C₄×C₈	—	5	2	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×C₃×C₅	#576	1	2	yes	—
207	C₃×D₅	—	4	8	no	—
208	C₂³×C₃	—	10	4	no	—
209	C₂³×C₃	—	10	8	no	—
210	C₂³×C₃	—	40	4	no	—
211	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
212	C₃×D₄	—	5	8	no	—
213	C₃×D₄	—	10	4	no	—
214	C₃×D₄	—	10	8	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	5	4	no	—
217	C₂×C₃×C₄	—	40	8	no	—
218	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
219	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	#713	1	2	yes	—
221	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
222	C₃×C₈	—	10	2	no	—
223	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
224	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
227	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₂×D₅	—	2	8	no	—
229	C₂×D₅	—	4	8	no	—
230	C₂×D₅	—	8	8	no	—
231	C₄×C₅	—	4	4	no	—
232	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
233	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
234	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
235	Dic₅	—	2	4	no	—
236	C₂⁴	—	10	2	no	—
237	C₂×D₄	—	5	8	no	—
238	C₂×D₄	—	5	4	no	—
239	C₂×D₄	—	10	2	no	—
240	C₂×D₄	—	10	4	no	—
241	C₂²×C₄	—	10	8	no	—
242	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
243	D₈	—	5	8	no	—
244	D₈	—	5	8	no	—
245	M₁₆	—	10	2	no	—
246	C₂×C₈	—	5	2	no	—
247	C₂×C₈	C₃×C₄×D₅	1	4	yes	complemented
248	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	5	2	no	—
254	C₂²⋊C₄	—	10	8	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	5	2	no	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	10	8	no	—
260	C₂²×C₃	—	20	8	no	—
261	C₂²×C₃	—	40	8	no	—
262	C₂²×C₃	#4039	1	2	yes	—
263	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
264	C₃×C₄	—	4	4	no	—
265	C₃×C₄	—	10	4	no	—
266	C₃×C₄	#4039	1	2	yes	—
267	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
268	C₃×C₄	C₂×#713	1	1	yes	characteristic
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
271	C₂×C₅	C₃×#576	1	2	yes	—
272	D₅	—	4	8	no	—
273	C₂³	—	10	4	no	—
274	C₂³	—	10	8	no	—
275	C₂³	—	40	4	no	—
276	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
277	D₄	—	5	8	no	—
278	D₄	—	10	4	no	—
279	D₄	—	10	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	5	4	no	—
282	C₂×C₄	—	40	8	no	—
283	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
284	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
285	C₂×C₄	C₃×#713	1	2	yes	—
286	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
287	C₈	—	10	2	no	—
288	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
289	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	20	8	no	—
292	C₂×C₃	C₂×#4039	1	1	yes	characteristic
293	C₂×C₃	#66103	1	2	yes	—
294	C₅	C₃×#2948	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	10	8	no	—
297	C₂²	—	20	8	no	—
298	C₂²	—	40	8	no	—
299	C₂²	C₃×#4039	1	2	yes	—
300	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
301	C₄	—	4	4	no	—
302	C₄	—	10	4	no	—
303	C₄	C₃×#4039	1	2	yes	—
304	C₄	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
305	C₄	C₂×C₃×#713	1	1	yes	characteristic
306	C₃	#10610898	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	20	8	no	—
309	C₂	C₂×C₃×#4039	1	1	yes	characteristic
310	C₂	C₃×#66103	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₄₀	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#427	D₅
hypocenter	C₅	C₃×#2948
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#427

Complements

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

N	H ≅ G/N	type
2. C₃×#65855	C₂	semidirect
3. C₃×C₅×#427	C₂	semidirect
4. C₃×#65390	C₂	semidirect
5. C₃×#65290	C₂	semidirect
6. C₃×#65161	C₂	semidirect
7. C₃×#65158	C₂	semidirect
8. #10610898	C₃	direct (N × H)
14. C₂×C₃×D₄₀	C₄	semidirect
17. C₃×#4039	C₂²	semidirect
20. C₃×C₅×#107	C₂²	semidirect
21. C₃×C₅×#106	C₂²	semidirect
22. C₃×#3947	C₂²	semidirect
23. C₃×C₈×Dic₅	C₂²	semidirect
24. C₃×#3941	C₂²	semidirect
27. #65855	C₂×C₃	semidirect
28. C₅×#427	C₂×C₃	semidirect
29. #65390	C₂×C₃	semidirect
30. #65290	C₂×C₃	semidirect
31. #65161	C₂×C₃	semidirect
32. #65158	C₂×C₃	semidirect
37. C₂×C₃×D₂₀	C₂×C₄	semidirect
40. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
52. C₃×D₄₀	C₂×C₄	semidirect
53. C₃×D₄₀	C₂×C₄	semidirect
58. C₃×#427	D₅	semidirect
66. C₂×D₄₀	C₃×C₄	semidirect
69. #4039	C₂²×C₃	semidirect
72. C₅×#107	C₂²×C₃	semidirect
73. C₅×#106	C₂²×C₃	semidirect
74. #3947	C₂²×C₃	semidirect
75. C₈×Dic₅	C₂²×C₃	semidirect
76. #3941	C₂²×C₃	semidirect
106. C₃×#107	C₂×D₅	semidirect
107. C₃×#106	C₂×D₅	semidirect
115. C₂×D₂₀	C₂×C₃×C₄	semidirect
118. C₂×C₅×C₈	C₂×C₃×C₄	semidirect
130. D₄₀	C₂×C₃×C₄	semidirect
131. D₄₀	C₂×C₃×C₄	semidirect
136. #427	C₃×D₅	semidirect
161. C₂×C₃×C₈	C₄×D₅	semidirect
199. #107	C₂×C₃×D₅	semidirect
200. #106	C₂×C₃×D₅	semidirect
247. C₂×C₈	C₃×C₄×D₅	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	C₃×#2948	semidirect
306. C₃	#10610898	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×D₄₀	2
C₃×#65855	2
C₃×C₅×#427	2
C₃×#65390	2
C₃×#65290	2
C₃×#65161	2
C₃×#65158	2
#10610898	3
C₃×#2948	5

Subgroups — C3×#10610898 #14417337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610898 #14417337