Subgroups — D₃×#64969 #14439749

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#4015	C₂	1	2	yes	maximal
2	#11750781	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#64969	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748027	C₂	1	4	yes	maximal, complemented
6	D₃×#3949	C₂	1	4	yes	maximal
7	#11746948	C₂	1	2	yes	maximal, complemented
8	C₂×#64969	—	3	1	no	maximal, Hall
9	C₂×D₃×Dic₁₀	—	2	2	no	—
10	#91758	C₂²	1	2	yes	—
11	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
12	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×#4015	C₂²	1	2	yes	—
14	#91469	—	2	2	no	—
15	#91469	C₂²	1	4	yes	—
16	D₃×C₄⋊Dic₅	—	2	2	no	—
17	D₃×C₄⋊Dic₅	C₂²	1	4	yes	—
18	#91389	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#91101	C₂²	1	4	yes	—
21	C₅×C₈×Dic₃	C₂²	1	2	yes	—
22	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×#3949	C₂²	1	4	yes	—
24	#90987	C₂²	1	8	yes	—
25	D₃×#498	—	5	1	no	maximal, Hall
26	C₂×#4015	—	3	2	no	—
27	C₂×C₄×C₅×C₈	—	3	1	no	—
28	C₂×#3949	—	3	4	no	—
29	#64969	—	3	2	no	—
30	#64969	—	3	1	no	—
31	#64969	—	3	4	no	—
32	#64969	D₃	1	1	yes	characteristic, direct product factor
33	C₂×Dic₃₀	—	2	2	no	—
34	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₁₀	—	2	2	no	—
37	C₂×#1204	—	2	4	no	—
38	C₂×D₃×Dic₅	—	2	4	no	—
39	D₃×Dic₁₀	—	4	8	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	C₂³	1	4	yes	—
44	C₄×C₅×Dic₃	D₄	1	2	yes	—
45	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
47	C₅×D₃×C₈	Q₈	1	8	yes	complemented
48	C₃×C₄⋊Dic₅	—	2	2	no	—
49	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
50	Dic₃⋊Dic₅	—	2	4	no	—
51	Dic₃⋊Dic₅	—	2	4	no	—
52	D₃×C₄⋊Q₈	—	5	2	no	—
53	C₃×#498	—	5	1	no	—
54	#5080	—	5	2	no	—
55	D₃×#106	—	5	4	no	—
56	#4855	—	5	4	no	—
57	C₄×D₃×C₈	D₅	1	1	yes	characteristic
58	#4680	—	5	1	no	—
59	C₂²×Dic₁₀	—	6	2	no	—
60	C₂²×C₅×C₈	—	3	2	no	—
61	C₂×C₄²×C₅	—	3	1	no	—
62	C₂×C₄⋊Dic₅	—	3	4	no	—
63	C₂×C₄⋊Dic₅	—	6	2	no	—
64	#4015	—	3	2	no	—
65	#4015	—	3	4	no	—
66	#4015	—	6	2	no	—
67	#4015	—	6	2	no	—
68	#4015	C₂×D₃	1	2	yes	—
69	C₄×C₅×C₈	—	3	2	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#3949	—	3	4	no	—
73	#3949	—	3	8	no	—
74	#3949	C₂×D₃	1	4	yes	—
75	C₂×#498	—	15	1	no	Sylow
76	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₁₅	—	2	4	no	—
80	Dic₃₀	—	4	4	no	—
81	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
83	C₄×C₅×D₃	SD₁₆	1	8	yes	—
84	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
85	C₂×C₃×Dic₅	—	2	4	no	—
86	C₃×Dic₁₀	—	4	4	no	—
87	#1204	—	4	8	no	—
88	D₃×Dic₅	—	4	8	no	—
89	C₃×C₅×C₈	C₂×Q₈	1	4	yes	complemented
90	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	complemented
91	C₂×D₃×Q₈	—	10	2	no	—
92	C₃×C₄⋊Q₈	—	5	2	no	—
93	#944	—	10	2	no	—
94	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
95	D₃×C₄⋊C₄	—	5	4	no	—
96	D₃×C₄⋊C₄	—	10	2	no	—
97	#888	—	5	4	no	—
98	#888	—	10	2	no	—
99	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
100	#869	—	5	2	no	—
101	C₃×#106	—	5	4	no	—
102	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	#817	—	5	4	no	—
104	C₈×Dic₃	C₂×D₅	1	2	yes	—
105	#808	—	5	8	no	—
106	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₄×C₅	—	3	2	no	—
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×Dic₅	—	6	4	no	—
110	C₂×Dic₁₀	—	2	2	no	—
111	C₂×Dic₁₀	—	6	4	no	—
112	C₂×Dic₁₀	—	6	2	no	—
113	C₂×Dic₁₀	—	12	8	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	3	8	no	—
116	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
117	C₄²×C₅	—	3	2	no	—
118	C₄²×C₅	—	3	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	—	3	4	no	—
122	C₄⋊Dic₅	—	6	4	no	—
123	C₄⋊Dic₅	—	6	4	no	—
124	C₄⋊Dic₅	—	6	2	no	—
125	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
126	C₂×C₄⋊Q₈	—	15	2	no	—
127	#498	—	5	1	no	—
128	#498	—	15	4	no	—
129	#498	—	15	2	no	—
130	#498	—	15	1	no	—
131	C₂×#106	—	15	4	no	—
132	C₂×C₄×C₈	—	3	1	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
135	C₂×C₅×D₃	C₂×SD₁₆	1	4	yes	—
136	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
137	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
138	Dic₁₅	—	4	4	no	—
139	C₃×Dic₅	—	4	4	no	—
140	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
141	C₅×Dic₃	C₂×SD₁₆	1	4	yes	—
142	C₂×C₃×Q₈	—	10	2	no	—
143	D₃×Q₈	—	20	8	no	—
144	C₂×C₄×D₃	—	10	4	no	—
145	C₂×C₄×D₃	D₂₀	1	2	yes	—
146	C₂×C₄×D₃	C₂²×D₅	1	1	yes	characteristic
147	C₂×Dic₆	—	10	2	no	—
148	C₂×Dic₆	—	10	4	no	—
149	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
150	C₃×C₄⋊C₄	—	5	4	no	—
151	C₃×C₄⋊C₄	—	10	2	no	—
152	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
153	C₄⋊Dic₃	—	5	4	no	—
154	C₄⋊Dic₃	—	10	2	no	—
155	Dic₃⋊C₄	—	10	4	no	—
156	Dic₃⋊C₄	—	10	4	no	—
157	C₄×Dic₃	D₂₀	1	2	yes	—
158	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
160	D₃×C₈	Dic₁₀	1	8	yes	complemented
161	C₂³×C₅	—	3	1	no	—
162	C₂×C₄×C₅	—	3	1	no	—
163	C₂×C₄×C₅	—	3	2	no	—
164	C₂×C₄×C₅	—	3	8	no	—
165	C₂×C₄×C₅	—	3	4	no	—
166	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
168	C₂×Dic₅	—	2	4	no	—
169	C₂×Dic₅	—	6	4	no	—
170	C₂×Dic₅	—	12	8	no	—
171	Dic₁₀	—	4	4	no	—
172	Dic₁₀	—	12	8	no	—
173	Dic₁₀	—	12	4	no	—
174	C₅×C₈	—	3	4	no	—
175	C₅×C₈	D₃×Q₈	1	4	yes	complemented
176	C₂²×Q₈	—	30	2	no	—
177	C₂²×C₈	—	3	2	no	—
178	C₄⋊Q₈	—	5	2	no	—
179	C₄⋊Q₈	—	15	4	no	—
180	C₄⋊Q₈	—	15	2	no	—
181	C₄⋊Q₈	—	30	2	no	—
182	C₄⋊Q₈	—	30	2	no	—
183	C₂×C₄⋊C₄	—	15	4	no	—
184	C₂×C₄⋊C₄	—	30	2	no	—
185	C₂×C₄²	—	3	1	no	—
186	#106	—	5	4	no	—
187	#106	—	15	8	no	—
188	#106	—	15	4	no	—
189	C₄×C₈	—	3	2	no	—
190	C₄×C₈	—	3	1	no	—
191	C₄×C₈	D₃×D₅	1	1	yes	characteristic
192	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
194	C₅×D₃	#498	1	4	yes	complemented
195	C₂²×D₃	C₂×D₂₀	1	1	yes	characteristic
196	C₃×Q₈	—	20	4	no	—
197	C₂×C₃×C₄	—	10	4	no	—
198	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
199	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
200	C₂×Dic₃	—	10	4	no	—
201	C₂×Dic₃	C₂×D₂₀	1	1	yes	characteristic
202	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
203	C₄×D₃	—	20	8	no	—
204	C₄×D₃	#677	1	8	yes	—
205	C₄×D₃	C₂×Dic₁₀	1	4	yes	—
206	Dic₆	—	20	4	no	—
207	Dic₆	—	20	8	no	—
208	C₃×C₈	C₂×Dic₁₀	1	4	yes	complemented
209	C₃⋊C₈	C₂×Dic₁₀	1	4	yes	complemented
210	C₂²×C₅	—	3	2	no	—
211	C₂²×C₅	—	3	4	no	—
212	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
213	C₄×C₅	—	3	2	no	—
214	C₄×C₅	—	3	4	no	—
215	C₄×C₅	D₃×SD₁₆	1	4	yes	—
216	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
217	Dic₅	—	4	4	no	—
218	Dic₅	—	12	4	no	—
219	C₂×Q₈	—	10	2	no	—
220	C₂×Q₈	—	30	4	no	—
221	C₂×Q₈	—	30	2	no	—
222	C₂×Q₈	—	60	8	no	—
223	C₂²×C₄	—	3	2	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	30	4	no	—
226	C₂×C₈	—	3	2	no	—
227	C₂×C₈	—	3	8	no	—
228	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
229	C₄⋊C₄	—	5	4	no	—
230	C₄⋊C₄	—	10	2	no	—
231	C₄⋊C₄	—	15	4	no	—
232	C₄⋊C₄	—	30	4	no	—
233	C₄⋊C₄	—	30	4	no	—
234	C₄⋊C₄	—	30	2	no	—
235	C₄²	—	3	2	no	—
236	C₄²	—	3	1	no	—
237	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
238	C₃×C₅	C₂×#498	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
240	C₂×D₃	#4015	1	2	yes	—
241	C₂×D₃	C₂×#677	1	4	yes	—
242	C₃×C₄	—	20	4	no	—
243	C₃×C₄	C₂×#677	1	4	yes	—
244	C₃×C₄	C₂²×Dic₁₀	1	2	yes	—
245	Dic₃	—	20	4	no	—
246	Dic₃	#4015	1	2	yes	—
247	Dic₃	C₂×#677	1	4	yes	—
248	C₂×C₅	—	3	4	no	—
249	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
250	C₂×C₅	C₂×D₃×SD₁₆	1	2	yes	—
251	C₂³	—	3	1	no	—
252	Q₈	—	20	4	no	—
253	Q₈	—	60	8	no	—
254	Q₈	—	60	4	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	4	no	—
258	C₂×C₄	—	3	8	no	—
259	C₂×C₄	—	10	4	no	—
260	C₂×C₄	—	30	4	no	—
261	C₂×C₄	—	60	8	no	—
262	C₂×C₄	D₃×D₂₀	1	2	yes	—
263	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₈	—	3	4	no	—
265	C₈	D₃×Dic₁₀	1	4	yes	complemented
266	C₂×C₃	C₂×#4015	1	1	yes	characteristic
267	C₂×C₃	C₂²×#677	1	2	yes	—
268	D₃	#64969	1	4	yes	direct product factor
269	C₅	D₃×#498	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	3	2	no	—
271	C₂²	—	3	4	no	—
272	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
273	C₄	—	3	2	no	—
274	C₄	—	3	4	no	—
275	C₄	—	20	4	no	—
276	C₄	—	60	4	no	—
277	C₄	D₃×#677	1	4	yes	—
278	C₄	C₂×D₃×Dic₁₀	1	2	yes	—
279	C₃	C₂×#64969	1	1	yes	characteristic, complemented, Sylow
280	C₂	—	3	4	no	—
281	C₂	D₃×#4015	1	1	yes	characteristic
282	C₂	C₂×D₃×#677	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃×D₂₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#498
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11750781	C₂	semidirect
4. C₃×#64969	C₂	semidirect
5. #11748027	C₂	semidirect
7. #11746948	C₂	semidirect
32. #64969	D₃	direct (N × H)
47. C₅×D₃×C₈	Q₈	semidirect
89. C₃×C₅×C₈	C₂×Q₈	semidirect
90. C₅×C₃⋊C₈	C₂×Q₈	semidirect
160. D₃×C₈	Dic₁₀	semidirect
175. C₅×C₈	D₃×Q₈	semidirect
194. C₅×D₃	#498	semidirect
208. C₃×C₈	C₂×Dic₁₀	semidirect
209. C₃⋊C₈	C₂×Dic₁₀	semidirect
238. C₃×C₅	C₂×#498	semidirect
265. C₈	D₃×Dic₁₀	semidirect
268. D₃	#64969	direct (N × H)
269. C₅	D₃×#498	semidirect
279. C₃	C₂×#64969	semidirect

Maximal subgroups

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

subgroup	index
D₃×#4015	2
#11750781	2
C₄×C₅×D₃×C₈	2
C₃×#64969	2
#11748027	2
D₃×#3949	2
#11746948	2
C₂×#64969	3
D₃×#498	5

Subgroups — D3×#64969 #14439749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#64969 #14439749