Subgroups — C₃×#10615434 #14471916

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65911	C₂	1	1	yes	maximal, characteristic
3	C₃×#65377	C₂	1	3	yes	maximal, complemented
4	C₃×#65373	C₂	1	3	yes	maximal, complemented
5	C₃×#65316	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×#688	C₂	1	3	yes	maximal, complemented
7	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615434	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#713	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	—	2	3	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₃×C₄×C₅×Q₈	—	2	1	no	—
13	C₃×C₄×C₅×Q₈	C₂²	1	3	yes	—
14	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×#688	C₂²	1	3	yes	—
16	C₃×#4041	—	2	3	no	—
17	C₃×#4020	C₂²	1	3	yes	—
18	C₃×C₄×D₂₀	C₂²	1	3	yes	—
19	C₃×#3967	C₂²	1	1	yes	characteristic
20	C₃×#3941	C₂²	1	3	yes	—
21	C₃×#3941	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×#3362	—	5	1	no	maximal, Hall
26	C₅×Q₈²	C₂×C₃	1	1	yes	characteristic, complemented
27	#65911	C₂×C₃	1	1	yes	characteristic
28	#65377	C₂×C₃	1	3	yes	complemented
29	#65373	C₂×C₃	1	3	yes	complemented
30	#65316	C₂×C₃	1	3	yes	complemented
31	C₄×#688	C₂×C₃	1	3	yes	complemented
32	Q₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₃×C₅×Q₈	—	2	3	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
36	C₃×#713	—	4	4	no	—
37	C₂×C₃×D₂₀	—	2	3	no	—
38	C₂×C₃×D₂₀	—	2	3	no	—
39	C₂×C₃×D₂₀	C₂³	1	3	yes	—
40	C₂×C₃×C₄×D₅	—	2	3	no	—
41	C₃×C₅×C₄⋊C₄	—	2	3	no	—
42	C₃×C₅×C₄⋊C₄	—	2	3	no	—
43	C₃×C₅×C₄⋊C₄	D₄	1	3	yes	—
44	C₃×C₅×C₄⋊C₄	C₂³	1	3	yes	—
45	C₃×C₄²×C₅	—	2	3	no	—
46	C₃×C₄²×C₅	C₂³	1	3	yes	—
47	C₃×#688	—	2	6	no	—
48	C₃×#685	—	2	3	no	—
49	C₃×C₄⋊Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×#168	C₂³	1	3	yes	—
51	C₂×C₃×#168	C₂³	1	1	yes	characteristic
52	C₃×Q₈²	D₅	1	1	yes	characteristic, complemented
53	C₃×#550	—	5	1	no	—
54	C₃×#486	—	5	3	no	—
55	C₃×#476	—	5	3	no	—
56	C₃×#460	—	5	3	no	—
57	C₃×C₈×Q₈	—	5	1	no	—
58	C₃×C₄×SD₁₆	—	5	3	no	—
59	C₂×#713	—	2	1	no	—
60	C₅×C₄⋊Q₈	—	2	3	no	—
61	C₅×C₄⋊Q₈	C₂²×C₃	1	3	yes	—
62	C₄×C₅×Q₈	—	2	1	no	—
63	C₄×C₅×Q₈	C₂²×C₃	1	3	yes	—
64	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×#688	C₂²×C₃	1	3	yes	—
66	#4041	—	2	3	no	—
67	#4020	C₂²×C₃	1	3	yes	—
68	C₄×D₂₀	C₂²×C₃	1	3	yes	—
69	#3967	C₂²×C₃	1	1	yes	characteristic
70	#3941	C₂²×C₃	1	3	yes	—
71	#3941	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	#3362	—	5	1	no	Sylow
76	C₂²×C₃×D₅	—	2	3	no	—
77	C₃×C₅×Q₈	—	2	1	no	—
78	C₃×C₅×Q₈	—	8	3	no	—
79	C₃×C₅×Q₈	SD₁₆	1	4	yes	complemented
80	C₃×C₅×Q₈	#41	1	2	yes	complemented
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	3	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	C₃×D₂₀	—	2	3	no	—
88	C₃×D₂₀	—	2	3	no	—
89	C₃×D₂₀	—	4	6	no	—
90	C₃×D₂₀	—	8	3	no	—
91	C₃×C₄×D₅	—	4	6	no	—
92	C₃×#168	—	2	3	no	—
93	C₃×#168	—	2	1	no	—
94	C₂×C₃×#41	—	10	1	no	—
95	C₂×C₃×SD₁₆	—	5	3	no	—
96	C₃×C₄⋊Q₈	—	2	3	no	—
97	C₃×C₄⋊Q₈	C₂×D₅	1	3	yes	—
98	C₃×#127	—	5	3	no	—
99	C₃×#121	—	10	3	no	—
100	C₃×C₄×Q₈	—	2	1	no	—
101	C₃×C₄×Q₈	C₂×D₅	1	3	yes	—
102	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
103	C₃×C₄×D₄	—	5	3	no	—
104	C₃×#106	—	5	3	no	—
105	C₃×C₄⋊C₈	—	5	3	no	—
106	C₃×#103	—	5	1	no	—
107	C₃×#102	—	5	3	no	—
108	C₃×#102	—	5	6	no	—
109	C₃×C₄×C₈	—	5	3	no	—
110	C₂×C₅×Q₈	—	2	3	no	—
111	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
113	#713	—	4	4	no	—
114	C₂×D₂₀	—	2	3	no	—
115	C₂×D₂₀	—	2	3	no	—
116	C₂×D₂₀	C₂³×C₃	1	3	yes	—
117	C₂×C₄×D₅	—	2	3	no	—
118	C₅×C₄⋊C₄	—	2	3	no	—
119	C₅×C₄⋊C₄	—	2	3	no	—
120	C₅×C₄⋊C₄	C₃×D₄	1	3	yes	—
121	C₅×C₄⋊C₄	C₂³×C₃	1	3	yes	—
122	C₄²×C₅	—	2	3	no	—
123	C₄²×C₅	C₂³×C₃	1	3	yes	—
124	#688	—	2	6	no	—
125	#685	—	2	3	no	—
126	C₄⋊Dic₅	C₂³×C₃	1	1	yes	characteristic
127	C₂×#168	C₂³×C₃	1	1	yes	characteristic
128	C₂×#168	C₂³×C₃	1	3	yes	—
129	Q₈²	C₃×D₅	1	1	yes	characteristic, complemented
130	#550	—	5	1	no	—
131	#486	—	5	3	no	—
132	#476	—	5	3	no	—
133	#460	—	5	3	no	—
134	C₈×Q₈	—	5	1	no	—
135	C₄×SD₁₆	—	5	3	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	3	no	—
138	C₂×C₃×D₅	—	4	3	no	—
139	C₂×C₃×D₅	—	4	3	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	3	no	—
142	C₃×C₄×C₅	—	4	3	no	—
143	C₃×C₄×C₅	C₂×SD₁₆	1	6	yes	—
144	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
146	C₃×Dic₅	—	4	1	no	—
147	C₃×#41	—	20	4	no	—
148	C₂×C₃×Q₈	—	2	3	no	—
149	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
150	C₂×C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₃×D₄	—	5	3	no	—
152	C₂×C₃×D₄	—	10	3	no	—
153	C₂×C₃×D₄	—	10	3	no	—
154	C₂²×C₃×C₄	—	10	3	no	—
155	C₃×SD₁₆	—	10	6	no	—
156	C₂×C₃×C₈	—	5	1	no	—
157	C₂×C₃×C₈	—	5	3	no	—
158	C₃×C₄⋊C₄	—	2	3	no	—
159	C₃×C₄⋊C₄	—	2	3	no	—
160	C₃×C₄⋊C₄	—	5	1	no	—
161	C₃×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
162	C₃×C₄⋊C₄	C₂²×D₅	1	3	yes	—
163	C₃×C₂²⋊C₄	—	10	3	no	—
164	C₃×C₄²	—	2	3	no	—
165	C₃×C₄²	C₂²×D₅	1	3	yes	—
166	C₂²×D₅	—	2	3	no	—
167	C₅×Q₈	—	2	1	no	—
168	C₅×Q₈	—	8	3	no	—
169	C₅×Q₈	C₃×SD₁₆	1	4	yes	complemented
170	C₅×Q₈	C₃×#41	1	2	yes	complemented
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	3	no	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
174	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
175	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
176	C₂×Dic₅	—	2	1	no	—
177	D₂₀	—	2	3	no	—
178	D₂₀	—	2	3	no	—
179	D₂₀	—	4	6	no	—
180	D₂₀	—	8	3	no	—
181	C₄×D₅	—	4	6	no	—
182	#168	—	2	3	no	—
183	#168	—	2	1	no	—
184	C₂×#41	—	10	1	no	—
185	C₂×SD₁₆	—	5	3	no	—
186	C₄⋊Q₈	—	2	3	no	—
187	C₄⋊Q₈	C₂×C₃×D₅	1	3	yes	—
188	#127	—	5	3	no	—
189	#121	—	10	3	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	C₂×C₃×D₅	1	3	yes	—
192	C₄×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
193	C₄×D₄	—	5	3	no	—
194	#106	—	5	3	no	—
195	C₄⋊C₈	—	5	3	no	—
196	#103	—	5	1	no	—
197	#102	—	5	3	no	—
198	#102	—	5	6	no	—
199	C₄×C₈	—	5	3	no	—
200	C₂×C₃×C₅	#548	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
202	C₂×C₃×C₅	#576	1	1	yes	characteristic
203	C₃×D₅	—	8	3	no	—
204	C₂³×C₃	—	10	3	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	8	3	no	—
207	C₃×Q₈	#688	1	4	yes	complemented
208	C₃×Q₈	#713	1	2	yes	complemented
209	C₃×D₄	—	10	3	no	—
210	C₃×D₄	—	10	3	no	—
211	C₃×D₄	—	20	6	no	—
212	C₃×D₄	—	40	3	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	3	no	—
215	C₂×C₃×C₄	—	10	1	no	—
216	C₂×C₃×C₄	—	20	6	no	—
217	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
218	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
219	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
220	C₃×C₈	—	10	3	no	—
221	C₃×C₈	—	10	1	no	—
222	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
223	C₂×D₅	—	4	3	no	—
224	C₂×D₅	—	4	3	no	—
225	C₂×D₅	—	4	3	no	—
226	C₄×C₅	—	2	3	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	4	3	no	—
229	C₄×C₅	C₂×C₃×SD₁₆	1	6	yes	—
230	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
231	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
232	Dic₅	—	4	1	no	—
233	#41	—	20	4	no	—
234	C₂×Q₈	—	2	3	no	—
235	C₂×Q₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×Q₈	C₂²×C₃×D₅	1	1	yes	characteristic
237	C₂×D₄	—	5	3	no	—
238	C₂×D₄	—	10	3	no	—
239	C₂×D₄	—	10	3	no	—
240	C₂²×C₄	—	10	3	no	—
241	SD₁₆	—	10	6	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	5	3	no	—
244	C₄⋊C₄	—	2	3	no	—
245	C₄⋊C₄	—	2	3	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	C₃×C₂²⋊D₅	1	3	yes	—
248	C₄⋊C₄	C₂²×C₃×D₅	1	3	yes	—
249	C₂²⋊C₄	—	10	3	no	—
250	C₄²	—	2	3	no	—
251	C₄²	C₂²×C₃×D₅	1	3	yes	—
252	C₃×C₅	#3362	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	20	3	no	—
254	C₂²×C₃	—	20	3	no	—
255	C₂²×C₃	—	20	3	no	—
256	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	3	no	—
259	C₃×C₄	—	4	3	no	—
260	C₃×C₄	—	20	1	no	—
261	C₃×C₄	C₂×#688	1	6	yes	—
262	C₃×C₄	C₂×#713	1	1	yes	characteristic
263	C₃×C₄	#4147	1	1	yes	characteristic
264	C₂×C₅	C₃×#548	1	1	yes	characteristic
265	C₂×C₅	C₂²×C₃×SD₁₆	1	1	yes	characteristic
266	C₂×C₅	C₃×#576	1	1	yes	characteristic
267	D₅	—	8	3	no	—
268	C₂³	—	10	3	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	8	3	no	—
271	Q₈	C₃×#688	1	4	yes	complemented
272	Q₈	C₃×#713	1	2	yes	complemented
273	D₄	—	10	3	no	—
274	D₄	—	10	3	no	—
275	D₄	—	20	6	no	—
276	D₄	—	40	3	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	3	no	—
279	C₂×C₄	—	10	1	no	—
280	C₂×C₄	—	20	6	no	—
281	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	3	yes	—
282	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	3	yes	—
283	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
284	C₈	—	10	3	no	—
285	C₈	—	10	1	no	—
286	C₂×C₃	—	40	3	no	—
287	C₂×C₃	C₂²×#688	1	1	yes	characteristic
288	C₂×C₃	#66151	1	1	yes	characteristic
289	C₂×C₃	#66169	1	1	yes	characteristic
290	C₅	C₃×#3362	1	1	yes	characteristic, complemented, Sylow
291	C₂²	—	20	3	no	—
292	C₂²	—	20	3	no	—
293	C₂²	—	20	3	no	—
294	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	3	no	—
297	C₄	—	4	3	no	—
298	C₄	—	20	1	no	—
299	C₄	C₂×C₃×#688	1	6	yes	—
300	C₄	C₂×C₃×#713	1	1	yes	characteristic
301	C₄	C₃×#4147	1	1	yes	characteristic
302	C₃	#10615434	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	40	3	no	—
304	C₂	C₂²×C₃×#688	1	1	yes	characteristic
305	C₂	C₃×#66151	1	1	yes	characteristic
306	C₂	C₃×#66169	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×Q₈²	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65911	C₂
hypercenter	C₃×Q₈²	D₅
hypocenter	C₅	C₃×#3362
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ C₃×Q₈²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×Q₈²	C₂	semidirect
3. C₃×#65377	C₂	semidirect
4. C₃×#65373	C₂	semidirect
5. C₃×#65316	C₂	semidirect
6. C₃×C₄×#688	C₂	semidirect
7. C₃×Q₈×#168	C₂	semidirect
8. #10615434	C₃	direct (N × H)
26. C₅×Q₈²	C₂×C₃	semidirect
28. #65377	C₂×C₃	semidirect
29. #65373	C₂×C₃	semidirect
30. #65316	C₂×C₃	semidirect
31. C₄×#688	C₂×C₃	semidirect
32. Q₈×#168	C₂×C₃	semidirect
52. C₃×Q₈²	D₅	semidirect
79. C₃×C₅×Q₈	SD₁₆	semidirect
80. C₃×C₅×Q₈	#41	semidirect
129. Q₈²	C₃×D₅	semidirect
169. C₅×Q₈	C₃×SD₁₆	semidirect
170. C₅×Q₈	C₃×#41	semidirect
207. C₃×Q₈	#688	semidirect
208. C₃×Q₈	#713	semidirect
252. C₃×C₅	#3362	semidirect
271. Q₈	C₃×#688	semidirect
272. Q₈	C₃×#713	semidirect
290. C₅	C₃×#3362	semidirect
302. C₃	#10615434	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×Q₈²	2
C₃×#65911	2
C₃×#65377	2
C₃×#65373	2
C₃×#65316	2
C₃×C₄×#688	2
C₃×Q₈×#168	2
#10615434	3
C₃×#3362	5

Subgroups — C3×#10615434 #14471916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615434 #14471916