Subgroups — C₃×C₅×#2187 #14275221

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#467	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2187	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₃×C₅×D₈	—	2	4	no	—
10	C₂×C₃×C₅×D₈	—	2	1	no	—
11	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
14	C₃×C₅×SD₃₂	—	2	4	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
17	C₃×C₅×#103	—	2	1	no	—
18	C₃×#2187	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×D₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	complemented
21	C₅×#467	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#357	C₂×C₃	1	2	yes	complemented
23	C₅×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂⁴×C₃×C₅	—	4	1	no	—
25	C₂×C₃×C₅×Q₈	—	2	1	no	—
26	C₂×C₃×C₅×D₄	—	2	4	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	4	2	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₃×C₅×Q₁₆	—	2	2	no	—
32	C₃×C₅×D₈	—	2	4	no	—
33	C₃×C₅×D₈	—	2	2	no	—
34	C₃×C₅×D₈	D₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	4	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₃×C₅×C₁₆	—	2	2	no	—
42	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
43	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	complemented
44	C₃×#467	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×#357	C₂×C₅	1	2	yes	complemented
46	C₃×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂²×C₅×D₄	—	2	1	no	—
48	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
49	C₂×C₅×D₈	—	2	4	no	—
50	C₂×C₅×D₈	—	2	1	no	—
51	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
52	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₂²⋊Q₈	—	2	1	no	—
54	C₅×SD₃₂	—	2	4	no	—
55	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	complemented
56	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
57	C₅×#103	—	2	1	no	—
58	#2187	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₃×C₅	—	4	4	no	—
60	C₂³×C₃×C₅	—	4	2	no	—
61	C₂³×C₃×C₅	—	8	2	no	—
62	C₂³×C₃×C₅	—	8	2	no	—
63	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
64	C₃×C₅×Q₈	—	4	1	no	—
65	C₃×C₅×Q₈	—	4	1	no	—
66	C₃×C₅×D₄	—	2	4	no	—
67	C₃×C₅×D₄	—	4	4	no	—
68	C₃×C₅×D₄	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×C₈	—	2	1	no	—
76	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₂²×C₃×D₄	—	2	1	no	—
78	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
79	C₂×C₃×D₈	—	2	4	no	—
80	C₂×C₃×D₈	—	2	1	no	—
81	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
82	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₂²⋊Q₈	—	2	1	no	—
84	C₃×SD₃₂	—	2	4	no	—
85	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
86	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₃×#103	—	2	1	no	—
88	C₂⁴×C₅	—	4	1	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×D₄	—	2	4	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	1	no	—
94	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×D₈	—	2	4	no	—
97	C₅×D₈	—	2	2	no	—
98	C₅×D₈	C₃×D₄	1	4	yes	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	4	1	no	—
104	C₅×C₂²⋊C₄	—	4	1	no	—
105	C₅×C₁₆	—	2	2	no	—
106	C₂²×D₈	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	complemented
108	#467	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	#357	C₂×C₃×C₅	1	2	yes	complemented
110	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₅	—	2	1	no	—
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	4	4	no	—
114	C₂²×C₃×C₅	—	8	8	no	—
115	C₂²×C₃×C₅	—	8	2	no	—
116	C₂²×C₃×C₅	—	8	2	no	—
117	C₂²×C₃×C₅	SD₃₂	1	2	yes	complemented
118	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	8	1	no	—
121	C₃×C₄×C₅	—	8	1	no	—
122	C₃×C₄×C₅	#120	1	1	yes	characteristic
123	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂⁴×C₃	—	4	1	no	—
125	C₂×C₃×Q₈	—	2	1	no	—
126	C₂×C₃×D₄	—	2	4	no	—
127	C₂×C₃×D₄	—	2	2	no	—
128	C₂×C₃×D₄	—	4	2	no	—
129	C₂×C₃×D₄	—	4	1	no	—
130	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
131	C₃×Q₁₆	—	2	2	no	—
132	C₃×D₈	—	2	4	no	—
133	C₃×D₈	—	2	2	no	—
134	C₃×D₈	C₅×D₄	1	4	yes	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	4	1	no	—
140	C₃×C₂²⋊C₄	—	4	1	no	—
141	C₃×C₁₆	—	2	2	no	—
142	C₂³×C₅	—	4	4	no	—
143	C₂³×C₅	—	4	2	no	—
144	C₂³×C₅	—	8	2	no	—
145	C₂³×C₅	—	8	2	no	—
146	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
147	C₅×Q₈	—	4	1	no	—
148	C₅×Q₈	—	4	1	no	—
149	C₅×D₄	—	2	4	no	—
150	C₅×D₄	—	4	4	no	—
151	C₅×D₄	—	4	2	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₅×C₈	—	2	1	no	—
159	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
160	C₂²×D₄	—	2	1	no	—
161	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
165	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂²⋊Q₈	—	2	1	no	—
167	SD₃₂	—	2	4	no	—
168	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	complemented
169	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
170	#103	—	2	1	no	—
171	C₂×C₃×C₅	—	2	2	no	—
172	C₂×C₃×C₅	—	8	4	no	—
173	C₂×C₃×C₅	#447	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
175	C₂×C₃×C₅	#509	1	1	yes	characteristic
176	C₂³×C₃	—	4	4	no	—
177	C₂³×C₃	—	4	2	no	—
178	C₂³×C₃	—	8	2	no	—
179	C₂³×C₃	—	8	2	no	—
180	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
181	C₃×Q₈	—	4	1	no	—
182	C₃×Q₈	—	4	1	no	—
183	C₃×D₄	—	2	4	no	—
184	C₃×D₄	—	4	4	no	—
185	C₃×D₄	—	4	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	8	no	—
198	C₂²×C₅	—	8	2	no	—
199	C₂²×C₅	—	8	2	no	—
200	C₂²×C₅	C₃×SD₃₂	1	2	yes	complemented
201	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	8	1	no	—
204	C₄×C₅	—	8	1	no	—
205	C₄×C₅	C₃×#120	1	1	yes	characteristic
206	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂⁴	—	4	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
214	Q₁₆	—	2	2	no	—
215	D₈	—	2	4	no	—
216	D₈	—	2	2	no	—
217	D₈	C₃×C₅×D₄	1	4	yes	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₁₆	—	2	2	no	—
225	C₃×C₅	#2187	1	1	yes	characteristic, direct product factor, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	4	4	no	—
229	C₂²×C₃	—	8	8	no	—
230	C₂²×C₃	—	8	2	no	—
231	C₂²×C₃	—	8	2	no	—
232	C₂²×C₃	C₅×SD₃₂	1	2	yes	complemented
233	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	8	1	no	—
236	C₃×C₄	—	8	1	no	—
237	C₃×C₄	C₅×#120	1	1	yes	characteristic
238	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂×C₅	—	2	2	no	—
240	C₂×C₅	—	8	4	no	—
241	C₂×C₅	C₃×#447	1	1	yes	characteristic
242	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
243	C₂×C₅	C₃×#509	1	1	yes	characteristic
244	C₂³	—	4	4	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	—	8	2	no	—
248	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	2	4	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	8	4	no	—
264	C₂×C₃	C₅×#447	1	1	yes	characteristic
265	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
266	C₂×C₃	C₅×#509	1	1	yes	characteristic
267	C₅	C₃×#2187	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₃×C₅×SD₃₂	1	2	yes	complemented
275	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	C₃×C₅×#120	1	1	yes	characteristic
280	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₃	C₅×#2187	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	2	2	no	—
283	C₂	—	8	4	no	—
284	C₂	C₃×C₅×#447	1	1	yes	characteristic
285	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
286	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₈	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×SD₃₂	C₂	semidirect
3. C₃×C₅×#467	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
6. C₅×#2187	C₃	direct (N × H)
8. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
15. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. C₃×C₅×#107	C₂²	semidirect
18. C₃×#2187	C₅	direct (N × H)
20. C₂×C₅×SD₃₂	C₂×C₃	semidirect
21. C₅×#467	C₂×C₃	semidirect
22. C₅×#357	C₂×C₃	semidirect
23. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
43. C₂×C₃×SD₃₂	C₂×C₅	semidirect
44. C₃×#467	C₂×C₅	semidirect
45. C₃×#357	C₂×C₅	semidirect
46. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
48. C₂×C₅×Q₁₆	C₂²×C₃	semidirect
55. C₂×C₅×C₁₆	C₂²×C₃	semidirect
56. C₅×#107	C₂²×C₃	semidirect
58. #2187	C₃×C₅	direct (N × H)
78. C₂×C₃×Q₁₆	C₂²×C₅	semidirect
85. C₂×C₃×C₁₆	C₂²×C₅	semidirect
86. C₃×#107	C₂²×C₅	semidirect
107. C₂×SD₃₂	C₂×C₃×C₅	semidirect
108. #467	C₂×C₃×C₅	semidirect
109. #357	C₂×C₃×C₅	semidirect
110. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
117. C₂²×C₃×C₅	SD₃₂	semidirect
161. C₂×Q₁₆	C₂²×C₃×C₅	semidirect
168. C₂×C₁₆	C₂²×C₃×C₅	semidirect
169. #107	C₂²×C₃×C₅	semidirect
200. C₂²×C₅	C₃×SD₃₂	semidirect
225. C₃×C₅	#2187	direct (N × H)
232. C₂²×C₃	C₅×SD₃₂	semidirect
267. C₅	C₃×#2187	direct (N × H)
274. C₂²	C₃×C₅×SD₃₂	semidirect
281. C₃	C₅×#2187	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×C₅×D₈	2
C₂×C₃×C₅×SD₃₂	2
C₃×C₅×#467	2
C₃×C₅×#357	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2187	3
C₃×#2187	5

Subgroups — C3×C5×#2187 #14275221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2187 #14275221