Subgroups — C₃×#10615299 #14471519

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#65904	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65383	C₂	1	3	yes	maximal, complemented
4	C₃×#65382	C₂	1	3	yes	maximal, complemented
5	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65308	C₂	1	3	yes	maximal
7	C₃×C₄×#687	C₂	1	3	yes	maximal
8	#10615299	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₃×C₅×C₂²⋊Q₈	—	2	3	no	—
12	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₄	C₂²	1	3	yes	—
14	C₃×Q₈×Dic₅	—	2	1	no	—
15	C₂×C₃×#687	C₂²	1	3	yes	—
16	C₃×#4036	—	2	3	no	—
17	C₃×#4015	C₂²	1	3	yes	—
18	C₃×C₄×Dic₁₀	C₂²	1	3	yes	—
19	C₃×#3964	C₂²	1	1	yes	characteristic
20	C₃×#3942	C₂²	1	3	yes	—
21	C₃×#3942	C₂²	1	6	yes	—
22	C₃×#3940	C₂²	1	3	yes	—
23	C₃×C₄⋊#168	C₂²	1	3	yes	—
24	C₃×C₄×#168	C₂²	1	3	yes	—
25	C₃×#3359	—	5	1	no	maximal, Hall
26	C₅×D₄×Q₈	C₂×C₃	1	1	yes	characteristic
27	#65904	C₂×C₃	1	1	yes	characteristic, complemented
28	#65383	C₂×C₃	1	3	yes	complemented
29	#65382	C₂×C₃	1	3	yes	complemented
30	Q₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
31	#65308	C₂×C₃	1	3	yes	—
32	C₄×#687	C₂×C₃	1	3	yes	—
33	C₂×C₃×C₅×Q₈	—	2	3	no	—
34	C₂×C₃×C₅×Q₈	—	4	4	no	—
35	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	—	2	3	no	—
38	C₂×C₃×Dic₁₀	C₂³	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	3	no	—
40	C₃×C₅×C₄⋊C₄	D₄	1	3	yes	—
41	C₃×C₅×C₄⋊C₄	C₂³	1	3	yes	—
42	C₃×C₅×C₂²⋊C₄	—	2	3	no	—
43	C₃×C₄²×C₅	C₂³	1	3	yes	—
44	C₃×#687	—	2	6	no	—
45	C₃×C₄⋊Dic₅	—	2	3	no	—
46	C₃×C₄⋊Dic₅	—	2	3	no	—
47	C₃×C₄⋊Dic₅	C₂³	1	1	yes	characteristic
48	C₃×#683	—	2	3	no	—
49	C₃×C₄×Dic₅	—	2	3	no	—
50	C₂×C₃×#168	C₂³	1	3	yes	—
51	C₂×C₃×#168	C₂³	1	1	yes	characteristic
52	C₃×#557	—	5	1	no	—
53	C₃×D₄×Q₈	D₅	1	1	yes	characteristic
54	C₃×#487	—	5	3	no	—
55	C₃×#475	—	5	3	no	—
56	C₃×#461	—	5	3	no	—
57	C₃×C₈×Q₈	—	5	1	no	—
58	C₃×C₄×SD₁₆	—	5	3	no	—
59	C₂²×C₅×Q₈	—	2	1	no	—
60	C₅×C₄⋊Q₈	C₂²×C₃	1	3	yes	—
61	C₅×C₂²⋊Q₈	—	2	3	no	—
62	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₅×D₄	C₂²×C₃	1	3	yes	—
64	Q₈×Dic₅	—	2	1	no	—
65	C₂×#687	C₂²×C₃	1	3	yes	—
66	#4036	—	2	3	no	—
67	#4015	C₂²×C₃	1	3	yes	—
68	C₄×Dic₁₀	C₂²×C₃	1	3	yes	—
69	#3964	C₂²×C₃	1	1	yes	characteristic
70	#3942	C₂²×C₃	1	3	yes	—
71	#3942	C₂²×C₃	1	6	yes	—
72	#3940	C₂²×C₃	1	3	yes	—
73	C₄⋊#168	C₂²×C₃	1	3	yes	—
74	C₄×#168	C₂²×C₃	1	3	yes	—
75	#3359	—	5	1	no	Sylow
76	C₂³×C₃×C₅	—	2	1	no	—
77	C₃×C₅×Q₈	—	4	6	no	—
78	C₃×C₅×Q₈	SD₁₆	1	4	yes	complemented
79	C₃×C₅×D₄	—	2	1	no	—
80	C₃×C₅×D₄	#41	1	2	yes	—
81	C₂×C₃×C₄×C₅	—	2	3	no	—
82	C₂×C₃×C₄×C₅	—	4	6	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₃×Dic₅	—	2	1	no	—
87	C₂×C₃×Dic₅	—	2	3	no	—
88	C₃×Dic₁₀	—	2	3	no	—
89	C₃×Dic₁₀	—	2	3	no	—
90	C₃×#168	—	2	3	no	—
91	C₃×#168	—	2	1	no	—
92	C₂²×C₃×Q₈	—	2	1	no	—
93	C₂×C₃×SD₁₆	—	5	3	no	—
94	C₃×C₄⋊Q₈	—	5	3	no	—
95	C₃×C₄⋊Q₈	C₂×D₅	1	3	yes	—
96	C₃×#125	—	10	3	no	—
97	C₃×C₂²⋊Q₈	—	2	3	no	—
98	C₃×C₄×Q₈	—	5	3	no	—
99	C₃×C₄×Q₈	—	10	1	no	—
100	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄×D₄	C₂×D₅	1	3	yes	—
102	C₃×#106	—	5	3	no	—
103	C₃×C₄⋊C₈	—	5	3	no	—
104	C₃×#103	—	5	3	no	—
105	C₃×#103	—	5	6	no	—
106	C₃×#102	—	5	1	no	—
107	C₃×C₄×C₈	—	5	3	no	—
108	C₂×C₅×Q₈	—	2	3	no	—
109	C₂×C₅×Q₈	—	4	4	no	—
110	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
111	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	3	no	—
113	C₂×Dic₁₀	C₂³×C₃	1	3	yes	—
114	C₅×C₄⋊C₄	—	2	3	no	—
115	C₅×C₄⋊C₄	C₃×D₄	1	3	yes	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	3	yes	—
117	C₅×C₂²⋊C₄	—	2	3	no	—
118	C₄²×C₅	C₂³×C₃	1	3	yes	—
119	#687	—	2	6	no	—
120	C₄⋊Dic₅	—	2	3	no	—
121	C₄⋊Dic₅	—	2	3	no	—
122	C₄⋊Dic₅	C₂³×C₃	1	1	yes	characteristic
123	#683	—	2	3	no	—
124	C₄×Dic₅	—	2	3	no	—
125	C₂×#168	C₂³×C₃	1	1	yes	characteristic
126	C₂×#168	C₂³×C₃	1	3	yes	—
127	#557	—	5	1	no	—
128	D₄×Q₈	C₃×D₅	1	1	yes	characteristic
129	#487	—	5	3	no	—
130	#475	—	5	3	no	—
131	#461	—	5	3	no	—
132	C₈×Q₈	—	5	1	no	—
133	C₄×SD₁₆	—	5	3	no	—
134	C₂²×C₃×C₅	—	2	2	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₃×C₄×C₅	—	2	3	no	—
139	C₃×C₄×C₅	—	4	3	no	—
140	C₃×C₄×C₅	C₂×SD₁₆	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
143	C₃×Dic₅	—	4	3	no	—
144	C₃×Dic₅	—	4	1	no	—
145	C₂×C₃×Q₈	—	2	3	no	—
146	C₂×C₃×Q₈	—	4	4	no	—
147	C₂×C₃×Q₈	—	5	3	no	—
148	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₃×D₄	C₂²×D₅	1	1	yes	characteristic
150	C₂²×C₃×C₄	—	2	3	no	—
151	C₃×SD₁₆	—	10	6	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	5	3	no	—
154	C₃×C₄⋊C₄	—	2	3	no	—
155	C₃×C₄⋊C₄	—	5	1	no	—
156	C₃×C₄⋊C₄	—	10	3	no	—
157	C₃×C₄⋊C₄	—	10	3	no	—
158	C₃×C₄⋊C₄	—	10	3	no	—
159	C₃×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
160	C₃×C₄⋊C₄	C₂²×D₅	1	3	yes	—
161	C₃×C₂²⋊C₄	—	2	3	no	—
162	C₃×C₄²	—	10	3	no	—
163	C₃×C₄²	C₂²×D₅	1	3	yes	—
164	C₂³×C₅	—	2	1	no	—
165	C₅×Q₈	—	4	6	no	—
166	C₅×Q₈	C₃×SD₁₆	1	4	yes	complemented
167	C₅×D₄	—	2	1	no	—
168	C₅×D₄	C₃×#41	1	2	yes	—
169	C₂×C₄×C₅	—	2	3	no	—
170	C₂×C₄×C₅	—	4	6	no	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
172	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
173	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×Dic₅	—	2	1	no	—
175	C₂×Dic₅	—	2	3	no	—
176	Dic₁₀	—	2	3	no	—
177	Dic₁₀	—	2	3	no	—
178	#168	—	2	3	no	—
179	#168	—	2	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂×SD₁₆	—	5	3	no	—
182	C₄⋊Q₈	—	5	3	no	—
183	C₄⋊Q₈	C₂×C₃×D₅	1	3	yes	—
184	#125	—	10	3	no	—
185	C₂²⋊Q₈	—	2	3	no	—
186	C₄×Q₈	—	5	3	no	—
187	C₄×Q₈	—	10	1	no	—
188	C₄×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
189	C₄×D₄	C₂×C₃×D₅	1	3	yes	—
190	#106	—	5	3	no	—
191	C₄⋊C₈	—	5	3	no	—
192	#103	—	5	3	no	—
193	#103	—	5	6	no	—
194	#102	—	5	1	no	—
195	C₄×C₈	—	5	3	no	—
196	C₂×C₃×C₅	—	4	2	no	—
197	C₂×C₃×C₅	#548	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
199	C₂×C₃×C₅	#578	1	1	yes	characteristic
200	C₂³×C₃	—	2	1	no	—
201	C₃×Q₈	—	4	6	no	—
202	C₃×Q₈	—	10	3	no	—
203	C₃×Q₈	—	10	3	no	—
204	C₃×Q₈	#687	1	4	yes	complemented
205	C₃×D₄	—	2	1	no	—
206	C₃×D₄	#713	1	2	yes	—
207	C₂×C₃×C₄	—	2	3	no	—
208	C₂×C₃×C₄	—	4	6	no	—
209	C₂×C₃×C₄	—	10	1	no	—
210	C₂×C₃×C₄	—	10	3	no	—
211	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
212	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
213	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
214	C₃×C₈	—	10	3	no	—
215	C₃×C₈	—	10	1	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	4	1	no	—
218	C₂²×C₅	—	4	1	no	—
219	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₄×C₅	—	2	3	no	—
221	C₄×C₅	—	4	3	no	—
222	C₄×C₅	C₂×C₃×SD₁₆	1	6	yes	—
223	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
224	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
225	Dic₅	—	4	1	no	—
226	Dic₅	—	4	3	no	—
227	C₂×Q₈	—	2	3	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	5	3	no	—
230	C₂×Q₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
231	C₂×D₄	C₂²×C₃×D₅	1	1	yes	characteristic
232	C₂²×C₄	—	2	3	no	—
233	SD₁₆	—	10	6	no	—
234	C₂×C₈	—	5	1	no	—
235	C₂×C₈	—	5	3	no	—
236	C₄⋊C₄	—	2	3	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	10	3	no	—
239	C₄⋊C₄	—	10	3	no	—
240	C₄⋊C₄	—	10	3	no	—
241	C₄⋊C₄	C₃×C₂²⋊D₅	1	3	yes	—
242	C₄⋊C₄	C₂²×C₃×D₅	1	3	yes	—
243	C₂²⋊C₄	—	2	3	no	—
244	C₄²	—	10	3	no	—
245	C₄²	C₂²×C₃×D₅	1	3	yes	—
246	C₃×C₅	#3359	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
251	C₃×C₄	—	2	3	no	—
252	C₃×C₄	—	4	3	no	—
253	C₃×C₄	—	20	3	no	—
254	C₃×C₄	—	20	1	no	—
255	C₃×C₄	C₂×#687	1	6	yes	—
256	C₃×C₄	C₂×#713	1	1	yes	characteristic
257	C₃×C₄	#4147	1	1	yes	characteristic
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	C₃×#548	1	1	yes	characteristic
260	C₂×C₅	C₂²×C₃×SD₁₆	1	1	yes	characteristic
261	C₂×C₅	C₃×#578	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	Q₈	—	4	6	no	—
264	Q₈	—	10	3	no	—
265	Q₈	—	10	3	no	—
266	Q₈	C₃×#687	1	4	yes	complemented
267	D₄	—	2	1	no	—
268	D₄	C₃×#713	1	2	yes	—
269	C₂×C₄	—	2	3	no	—
270	C₂×C₄	—	4	6	no	—
271	C₂×C₄	—	10	1	no	—
272	C₂×C₄	—	10	3	no	—
273	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	3	yes	—
274	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	3	yes	—
275	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
276	C₈	—	10	3	no	—
277	C₈	—	10	1	no	—
278	C₂×C₃	—	4	2	no	—
279	C₂×C₃	C₂²×#687	1	1	yes	characteristic
280	C₂×C₃	#66151	1	1	yes	characteristic
281	C₂×C₃	#66172	1	1	yes	characteristic
282	C₅	C₃×#3359	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	2	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
287	C₄	—	2	3	no	—
288	C₄	—	4	3	no	—
289	C₄	—	20	3	no	—
290	C₄	—	20	1	no	—
291	C₄	C₂×C₃×#687	1	6	yes	—
292	C₄	C₂×C₃×#713	1	1	yes	characteristic
293	C₄	C₃×#4147	1	1	yes	characteristic
294	C₃	#10615299	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	4	2	no	—
296	C₂	C₂²×C₃×#687	1	1	yes	characteristic
297	C₂	C₃×#66151	1	1	yes	characteristic
298	C₂	C₃×#66172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×D₄×Q₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×C₅×D₄	C₂³
hypercenter	C₃×D₄×Q₈	D₅
hypocenter	C₅	C₃×#3359
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ C₃×D₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×#65904	C₂	semidirect
3. C₃×#65383	C₂	semidirect
4. C₃×#65382	C₂	semidirect
5. C₃×Q₈×#168	C₂	semidirect
8. #10615299	C₃	direct (N × H)
27. #65904	C₂×C₃	semidirect
28. #65383	C₂×C₃	semidirect
29. #65382	C₂×C₃	semidirect
30. Q₈×#168	C₂×C₃	semidirect
78. C₃×C₅×Q₈	SD₁₆	semidirect
166. C₅×Q₈	C₃×SD₁₆	semidirect
204. C₃×Q₈	#687	semidirect
246. C₃×C₅	#3359	semidirect
266. Q₈	C₃×#687	semidirect
282. C₅	C₃×#3359	semidirect
294. C₃	#10615299	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×C₅×D₄×Q₈	2
C₃×#65904	2
C₃×#65383	2
C₃×#65382	2
C₃×Q₈×#168	2
C₃×#65308	2
C₃×C₄×#687	2
#10615299	3
C₃×#3359	5

Subgroups — C3×#10615299 #14471519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615299 #14471519