Subgroups — C₃×#10613759 #14478614

Order 1920.

700 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×#66100	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65468	C₂	1	2	yes	maximal, complemented
4	C₃×#65208	C₂	1	2	yes	maximal, complemented
5	C₃×#65207	C₂	1	4	yes	maximal, complemented
6	C₃×D₅×Q₃₂	C₂	1	2	yes	maximal, complemented
7	C₃×#65193	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#65190	C₂	1	1	yes	maximal, characteristic, complemented
9	#10613759	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×#4147	—	2	2	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₃×#4088	—	2	2	no	—
13	C₃×#4065	C₂²	1	4	yes	—
14	C₃×#4064	—	2	4	no	—
15	C₃×D₅×Q₁₆	C₂²	1	4	yes	—
16	C₃×#4050	C₂²	1	2	yes	—
17	C₃×#4047	C₂²	1	1	yes	characteristic
18	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
19	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₃×#3961	C₂²	1	4	yes	—
22	C₃×#3960	C₂²	1	4	yes	—
23	C₃×#3944	C₂²	1	1	yes	characteristic
24	C₃×Dic₄₀	C₂²	1	1	yes	characteristic
25	C₃×#3932	C₂²	1	2	yes	—
26	C₃×D₈₀	C₂²	1	1	yes	characteristic
27	C₃×#3930	C₂²	1	2	yes	—
28	C₃×D₅×C₁₆	C₂²	1	2	yes	—
29	C₃×#3417	—	5	1	no	maximal, Hall
30	#66100	C₂×C₃	1	2	yes	complemented
31	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
32	#65468	C₂×C₃	1	2	yes	complemented
33	#65208	C₂×C₃	1	2	yes	complemented
34	#65207	C₂×C₃	1	4	yes	complemented
35	D₅×Q₃₂	C₂×C₃	1	2	yes	complemented
36	#65193	C₂×C₃	1	1	yes	characteristic, complemented
37	#65190	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₅×Q₈	—	2	2	no	—
39	C₃×#713	—	2	4	no	—
40	C₃×#713	—	4	2	no	—
41	C₃×Q₈×D₅	—	2	4	no	—
42	C₃×Q₈×D₅	—	4	2	no	—
43	C₃×#709	—	2	2	no	—
44	C₃×#709	—	4	4	no	—
45	C₃×#709	D₄	1	1	yes	characteristic
46	C₃×C₅×Q₁₆	—	2	2	no	—
47	C₃×C₅×Q₁₆	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₃×#689	—	2	4	no	—
50	C₃×#688	—	2	4	no	—
51	C₃×#681	D₄	1	1	yes	characteristic
52	C₃×Dic₂₀	C₂³	1	2	yes	—
53	C₃×D₄₀	C₂³	1	2	yes	—
54	C₃×#677	—	2	2	no	—
55	C₃×#676	D₄	1	2	yes	—
56	C₃×C₈×D₅	C₂³	1	2	yes	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₃×#672	C₂³	1	2	yes	—
60	C₃×#578	—	5	2	no	—
61	C₃×#510	—	5	2	no	—
62	C₃×#510	—	5	4	no	—
63	C₃×#508	—	5	2	no	—
64	C₃×#508	—	5	1	no	—
65	C₂×C₃×Q₃₂	—	5	2	no	—
66	C₂×C₃×Q₃₂	D₅	1	1	yes	characteristic, complemented
67	C₃×#504	—	5	1	no	—
68	#4147	—	2	2	no	—
69	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
70	#4088	—	2	2	no	—
71	#4065	C₂²×C₃	1	4	yes	—
72	#4064	—	2	4	no	—
73	D₅×Q₁₆	C₂²×C₃	1	4	yes	—
74	#4050	C₂²×C₃	1	2	yes	—
75	#4047	C₂²×C₃	1	1	yes	characteristic
76	C₅×Q₃₂	C₂²×C₃	1	2	yes	—
77	C₅×Q₃₂	C₂²×C₃	1	2	yes	—
78	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
79	#3961	C₂²×C₃	1	4	yes	—
80	#3960	C₂²×C₃	1	4	yes	—
81	#3944	C₂²×C₃	1	1	yes	characteristic
82	Dic₄₀	C₂²×C₃	1	1	yes	characteristic
83	#3932	C₂²×C₃	1	2	yes	—
84	D₈₀	C₂²×C₃	1	1	yes	characteristic
85	#3930	C₂²×C₃	1	2	yes	—
86	D₅×C₁₆	C₂²×C₃	1	2	yes	—
87	#3417	—	5	1	no	Sylow
88	C₃×C₅×Q₈	—	2	4	no	—
89	C₃×C₅×Q₈	—	4	2	no	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₃×C₂²⋊D₅	—	4	2	no	—
93	C₃×C₂²⋊D₅	D₈	1	2	yes	—
94	C₃×D₂₀	—	2	2	no	—
95	C₃×D₂₀	—	4	4	no	—
96	C₃×D₂₀	D₈	1	1	yes	characteristic
97	C₃×C₄×D₅	—	4	4	no	—
98	C₃×C₄×D₅	—	4	2	no	—
99	C₃×C₄×D₅	—	4	4	no	—
100	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
101	C₃×Dic₁₀	—	2	2	no	—
102	C₃×Dic₁₀	—	4	4	no	—
103	C₃×Dic₁₀	D₈	1	1	yes	characteristic
104	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₃×#168	C₂×D₄	1	2	yes	—
107	C₃×2₋¹⁺⁴	—	10	2	no	—
108	C₃×#137	—	10	4	no	—
109	C₃×#137	—	10	2	no	—
110	C₃×#135	—	5	4	no	—
111	C₃×#135	—	5	2	no	—
112	C₂×C₃×Q₁₆	—	5	4	no	—
113	C₂×C₃×Q₁₆	C₂×D₅	1	2	yes	—
114	C₃×#131	—	5	1	no	—
115	C₃×Q₃₂	—	5	4	no	—
116	C₃×Q₃₂	—	5	1	no	—
117	C₃×Q₃₂	C₂×D₅	1	2	yes	—
118	C₃×Q₃₂	C₂×D₅	1	2	yes	—
119	C₃×SD₃₂	—	5	4	no	—
120	C₃×SD₃₂	—	5	2	no	—
121	C₃×D₁₆	—	5	1	no	—
122	C₃×M₃₂	—	5	1	no	—
123	C₃×M₃₂	—	5	2	no	—
124	C₂×C₃×C₁₆	—	5	2	no	—
125	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
126	C₂×C₅×Q₈	—	2	2	no	—
127	#713	—	2	4	no	—
128	#713	—	4	2	no	—
129	Q₈×D₅	—	2	4	no	—
130	Q₈×D₅	—	4	2	no	—
131	#709	—	2	2	no	—
132	#709	—	4	4	no	—
133	#709	C₃×D₄	1	1	yes	characteristic
134	C₅×Q₁₆	—	2	2	no	—
135	C₅×Q₁₆	C₂³×C₃	1	4	yes	—
136	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
137	#689	—	2	4	no	—
138	#688	—	2	4	no	—
139	#681	C₃×D₄	1	1	yes	characteristic
140	Dic₂₀	C₂³×C₃	1	2	yes	—
141	D₄₀	C₂³×C₃	1	2	yes	—
142	#677	—	2	2	no	—
143	#676	C₃×D₄	1	2	yes	—
144	C₈×D₅	C₂³×C₃	1	2	yes	—
145	C₅×C₁₆	C₂³×C₃	1	1	yes	characteristic
146	C₅×C₁₆	C₂³×C₃	1	1	yes	characteristic
147	#672	C₂³×C₃	1	2	yes	—
148	#578	—	5	2	no	—
149	#510	—	5	2	no	—
150	#510	—	5	4	no	—
151	#508	—	5	2	no	—
152	#508	—	5	1	no	—
153	C₂×Q₃₂	—	5	2	no	—
154	C₂×Q₃₂	C₃×D₅	1	1	yes	characteristic, complemented
155	#504	—	5	1	no	—
156	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×C₃×D₅	—	4	2	no	—
158	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
159	C₃×C₄×C₅	—	4	4	no	—
160	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₃×Dic₅	—	4	2	no	—
163	C₃×Dic₅	C₂×D₈	1	2	yes	—
164	C₃×#41	—	5	1	no	—
165	C₃×#41	—	10	4	no	—
166	C₃×#41	—	10	2	no	—
167	C₃×#41	—	20	4	no	—
168	C₃×#41	—	20	2	no	—
169	C₂×C₃×Q₈	—	2	2	no	—
170	C₂×C₃×Q₈	—	10	4	no	—
171	C₂×C₃×Q₈	—	20	2	no	—
172	C₃×Q₁₆	—	2	2	no	—
173	C₃×Q₁₆	—	5	2	no	—
174	C₃×Q₁₆	—	10	4	no	—
175	C₃×Q₁₆	C₂²×D₅	1	4	yes	—
176	C₃×SD₁₆	—	10	4	no	—
177	C₃×SD₁₆	—	10	2	no	—
178	C₃×D₈	—	5	2	no	—
179	C₃×M₁₆	—	5	1	no	—
180	C₃×M₁₆	—	5	2	no	—
181	C₂×C₃×C₈	—	5	2	no	—
182	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
183	C₃×C₁₆	—	5	2	no	—
184	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
185	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
186	C₅×Q₈	—	2	4	no	—
187	C₅×Q₈	—	4	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂²⋊D₅	—	4	2	no	—
191	C₂²⋊D₅	C₃×D₈	1	2	yes	—
192	D₂₀	—	2	2	no	—
193	D₂₀	—	4	4	no	—
194	D₂₀	C₃×D₈	1	1	yes	characteristic
195	C₄×D₅	—	4	4	no	—
196	C₄×D₅	—	4	2	no	—
197	C₄×D₅	—	4	4	no	—
198	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
199	Dic₁₀	—	2	2	no	—
200	Dic₁₀	—	4	4	no	—
201	Dic₁₀	C₃×D₈	1	1	yes	characteristic
202	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
203	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
204	#168	C₂×C₃×D₄	1	2	yes	—
205	2₋¹⁺⁴	—	10	2	no	—
206	#137	—	10	4	no	—
207	#137	—	10	2	no	—
208	#135	—	5	4	no	—
209	#135	—	5	2	no	—
210	C₂×Q₁₆	—	5	4	no	—
211	C₂×Q₁₆	C₂×C₃×D₅	1	2	yes	—
212	#131	—	5	1	no	—
213	Q₃₂	—	5	4	no	—
214	Q₃₂	—	5	1	no	—
215	Q₃₂	C₂×C₃×D₅	1	2	yes	—
216	Q₃₂	C₂×C₃×D₅	1	2	yes	—
217	SD₃₂	—	5	4	no	—
218	SD₃₂	—	5	2	no	—
219	D₁₆	—	5	1	no	—
220	M₃₂	—	5	1	no	—
221	M₃₂	—	5	2	no	—
222	C₂×C₁₆	—	5	2	no	—
223	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
226	C₃×D₅	—	2	2	no	—
227	C₃×D₅	—	8	2	no	—
228	C₃×Q₈	—	2	4	no	—
229	C₃×Q₈	—	4	2	no	—
230	C₃×Q₈	—	5	1	no	—
231	C₃×Q₈	—	10	2	no	—
232	C₃×Q₈	—	20	4	no	—
233	C₃×D₄	—	5	2	no	—
234	C₃×D₄	—	5	1	no	—
235	C₃×D₄	—	10	2	no	—
236	C₃×D₄	—	20	4	no	—
237	C₃×D₄	—	20	2	no	—
238	C₂×C₃×C₄	—	4	2	no	—
239	C₂×C₃×C₄	—	5	2	no	—
240	C₂×C₃×C₄	—	20	4	no	—
241	C₂×C₃×C₄	—	20	4	no	—
242	C₂×C₃×C₄	—	20	2	no	—
243	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
244	C₃×C₈	—	5	2	no	—
245	C₃×C₈	D₄×D₅	1	1	yes	characteristic
246	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
247	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
248	C₂×D₅	—	4	2	no	—
249	C₂×D₅	C₂×C₃×D₈	1	2	yes	—
250	C₄×C₅	—	4	4	no	—
251	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
252	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
253	Dic₅	—	4	2	no	—
254	Dic₅	C₂×C₃×D₈	1	2	yes	—
255	#41	—	5	1	no	—
256	#41	—	10	4	no	—
257	#41	—	10	2	no	—
258	#41	—	20	4	no	—
259	#41	—	20	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	10	4	no	—
262	C₂×Q₈	—	20	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	5	2	no	—
265	Q₁₆	—	10	4	no	—
266	Q₁₆	C₂²×C₃×D₅	1	4	yes	—
267	SD₁₆	—	10	4	no	—
268	SD₁₆	—	10	2	no	—
269	D₈	—	5	2	no	—
270	M₁₆	—	5	1	no	—
271	M₁₆	—	5	2	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
274	C₁₆	—	5	2	no	—
275	C₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
276	C₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
277	C₃×C₅	#3417	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	5	2	no	—
279	C₂²×C₃	—	20	2	no	—
280	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	—	5	2	no	—
283	C₃×C₄	—	20	2	no	—
284	C₃×C₄	D₅×D₈	1	1	yes	characteristic
285	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
288	D₅	—	2	2	no	—
289	D₅	—	8	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	5	1	no	—
293	Q₈	—	10	2	no	—
294	Q₈	—	20	4	no	—
295	D₄	—	5	2	no	—
296	D₄	—	5	1	no	—
297	D₄	—	10	2	no	—
298	D₄	—	20	4	no	—
299	D₄	—	20	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	20	4	no	—
303	C₂×C₄	—	20	4	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
306	C₈	—	5	2	no	—
307	C₈	C₃×D₄×D₅	1	1	yes	characteristic
308	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	10	2	no	—
311	C₂×C₃	—	40	2	no	—
312	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
313	C₅	C₃×#3417	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	5	2	no	—
315	C₂²	—	20	2	no	—
316	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	5	2	no	—
319	C₄	—	20	2	no	—
320	C₄	C₃×D₅×D₈	1	1	yes	characteristic
321	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
322	C₃	#10613759	1	1	yes	characteristic, direct product factor, Sylow
323	C₂	—	2	1	no	—
324	C₂	—	10	2	no	—
325	C₂	—	40	2	no	—
326	C₂	C₂×C₃×D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₅×D₈
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×Q₃₂	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×C₃×Q₃₂	D₅
hypocenter	C₅	C₃×#3417
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×Q₃₂

Complements

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

N	H ≅ G/N	type
1. C₃×#66100	C₂	semidirect
2. C₂×C₃×C₅×Q₃₂	C₂	semidirect
3. C₃×#65468	C₂	semidirect
4. C₃×#65208	C₂	semidirect
5. C₃×#65207	C₂	semidirect
6. C₃×D₅×Q₃₂	C₂	semidirect
7. C₃×#65193	C₂	semidirect
8. C₃×#65190	C₂	semidirect
9. #10613759	C₃	direct (N × H)
30. #66100	C₂×C₃	semidirect
31. C₂×C₅×Q₃₂	C₂×C₃	semidirect
32. #65468	C₂×C₃	semidirect
33. #65208	C₂×C₃	semidirect
34. #65207	C₂×C₃	semidirect
35. D₅×Q₃₂	C₂×C₃	semidirect
36. #65193	C₂×C₃	semidirect
37. #65190	C₂×C₃	semidirect
66. C₂×C₃×Q₃₂	D₅	semidirect
154. C₂×Q₃₂	C₃×D₅	semidirect
277. C₃×C₅	#3417	semidirect
313. C₅	C₃×#3417	semidirect
322. C₃	#10613759	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×#66100	2
C₂×C₃×C₅×Q₃₂	2
C₃×#65468	2
C₃×#65208	2
C₃×#65207	2
C₃×D₅×Q₃₂	2
C₃×#65193	2
C₃×#65190	2
#10613759	3
C₃×#3417	5

Subgroups — C3×#10613759 #14478614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613759 #14478614