Subgroups — C₃×C₅×#2184 #14275218

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 304 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, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#466	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×#2184	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	1	no	—
8	C₂×C₃×C₅×D₈	—	2	4	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#121	—	2	1	no	—
14	C₃×C₅×D₁₆	—	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×#102	—	2	1	no	—
18	C₃×#2184	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×D₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×D₁₆	C₂×C₃	1	2	yes	complemented
21	C₅×#466	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×D₄	—	2	4	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	1	no	—
28	C₂×C₃×C₅×D₄	—	4	2	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂×C₃×C₅×D₄	—	4	1	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×D₈	—	2	4	no	—
33	C₃×C₅×D₈	—	2	2	no	—
34	C₃×C₅×D₈	—	2	2	no	—
35	C₃×C₅×D₈	D₄	1	4	yes	complemented
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	—	2	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, complemented
43	C₂×C₃×D₁₆	C₂×C₅	1	2	yes	complemented
44	C₃×#466	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×D₈	—	2	4	no	—
49	C₂×C₅×D₈	—	2	1	no	—
50	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
51	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
52	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×#121	—	2	1	no	—
54	C₅×D₁₆	—	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×#102	—	2	1	no	—
58	#2184	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	—	4	1	no	—
62	C₂³×C₃×C₅	—	8	2	no	—
63	C₂³×C₃×C₅	—	8	2	no	—
64	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
65	C₃×C₅×D₄	—	2	4	no	—
66	C₃×C₅×D₄	—	4	4	no	—
67	C₃×C₅×D₄	—	4	1	no	—
68	C₃×C₅×D₄	—	4	2	no	—
69	C₃×C₅×D₄	—	4	1	no	—
70	C₃×C₅×D₄	—	8	2	no	—
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×C₈	—	2	1	no	—
77	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₂²×C₃×D₄	—	2	1	no	—
79	C₂×C₃×D₈	—	2	4	no	—
80	C₂×C₃×D₈	—	2	1	no	—
81	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
82	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
83	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₃×#121	—	2	1	no	—
85	C₃×D₁₆	—	2	4	no	—
86	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
87	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
88	C₃×#102	—	2	1	no	—
89	C₂⁴×C₅	—	4	1	no	—
90	C₂×C₅×D₄	—	2	4	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	2	1	no	—
93	C₂×C₅×D₄	—	4	2	no	—
94	C₂×C₅×D₄	—	4	1	no	—
95	C₂×C₅×D₄	—	4	1	no	—
96	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₅×D₈	—	2	4	no	—
98	C₅×D₈	—	2	2	no	—
99	C₅×D₈	—	2	2	no	—
100	C₅×D₈	C₃×D₄	1	4	yes	complemented
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₅×C₁₆	—	2	2	no	—
107	C₂²×D₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
108	C₂×D₁₆	C₂×C₃×C₅	1	2	yes	complemented
109	#466	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	#357	C₂×C₃×C₅	1	2	yes	complemented
111	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	4	4	no	—
115	C₂²×C₃×C₅	—	8	8	no	—
116	C₂²×C₃×C₅	—	8	2	no	—
117	C₂²×C₃×C₅	—	8	2	no	—
118	C₂²×C₃×C₅	—	8	1	no	—
119	C₂²×C₃×C₅	—	8	1	no	—
120	C₂²×C₃×C₅	—	8	1	no	—
121	C₂²×C₃×C₅	D₁₆	1	2	yes	complemented
122	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	8	1	no	—
125	C₃×C₄×C₅	#120	1	1	yes	characteristic
126	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂⁴×C₃	—	4	1	no	—
128	C₂×C₃×D₄	—	2	4	no	—
129	C₂×C₃×D₄	—	2	2	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	4	2	no	—
132	C₂×C₃×D₄	—	4	1	no	—
133	C₂×C₃×D₄	—	4	1	no	—
134	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
135	C₃×D₈	—	2	4	no	—
136	C₃×D₈	—	2	2	no	—
137	C₃×D₈	—	2	2	no	—
138	C₃×D₈	C₅×D₄	1	4	yes	complemented
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₂²⋊C₄	—	4	1	no	—
144	C₃×C₁₆	—	2	2	no	—
145	C₂³×C₅	—	4	4	no	—
146	C₂³×C₅	—	4	2	no	—
147	C₂³×C₅	—	4	1	no	—
148	C₂³×C₅	—	8	2	no	—
149	C₂³×C₅	—	8	2	no	—
150	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
151	C₅×D₄	—	2	4	no	—
152	C₅×D₄	—	4	4	no	—
153	C₅×D₄	—	4	1	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	1	no	—
156	C₅×D₄	—	8	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₅×C₈	—	2	1	no	—
163	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
164	C₂²×D₄	—	2	1	no	—
165	C₂×D₈	—	2	4	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
168	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
169	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	#121	—	2	1	no	—
171	D₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	complemented
173	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
174	#102	—	2	1	no	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	—	8	4	no	—
177	C₂×C₃×C₅	—	16	1	no	—
178	C₂×C₃×C₅	#447	1	1	yes	characteristic
179	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
180	C₂×C₃×C₅	#509	1	1	yes	characteristic
181	C₂³×C₃	—	4	4	no	—
182	C₂³×C₃	—	4	2	no	—
183	C₂³×C₃	—	4	1	no	—
184	C₂³×C₃	—	8	2	no	—
185	C₂³×C₃	—	8	2	no	—
186	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
187	C₃×D₄	—	2	4	no	—
188	C₃×D₄	—	4	4	no	—
189	C₃×D₄	—	4	1	no	—
190	C₃×D₄	—	4	2	no	—
191	C₃×D₄	—	4	1	no	—
192	C₃×D₄	—	8	2	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	4	1	no	—
196	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	4	4	no	—
203	C₂²×C₅	—	8	8	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	—	8	2	no	—
206	C₂²×C₅	—	8	1	no	—
207	C₂²×C₅	—	8	1	no	—
208	C₂²×C₅	—	8	1	no	—
209	C₂²×C₅	C₃×D₁₆	1	2	yes	complemented
210	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	8	1	no	—
213	C₄×C₅	C₃×#120	1	1	yes	characteristic
214	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂⁴	—	4	1	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
223	D₈	—	2	4	no	—
224	D₈	—	2	2	no	—
225	D₈	—	2	2	no	—
226	D₈	C₃×C₅×D₄	1	4	yes	complemented
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	4	1	no	—
232	C₁₆	—	2	2	no	—
233	C₃×C₅	#2184	1	1	yes	characteristic, direct product factor, Hall
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	4	4	no	—
237	C₂²×C₃	—	8	8	no	—
238	C₂²×C₃	—	8	2	no	—
239	C₂²×C₃	—	8	2	no	—
240	C₂²×C₃	—	8	1	no	—
241	C₂²×C₃	—	8	1	no	—
242	C₂²×C₃	—	8	1	no	—
243	C₂²×C₃	C₅×D₁₆	1	2	yes	complemented
244	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	8	1	no	—
247	C₃×C₄	C₅×#120	1	1	yes	characteristic
248	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	8	4	no	—
251	C₂×C₅	—	16	1	no	—
252	C₂×C₅	C₃×#447	1	1	yes	characteristic
253	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
254	C₂×C₅	C₃×#509	1	1	yes	characteristic
255	C₂³	—	4	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
261	D₄	—	2	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	1	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	1	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	8	4	no	—
276	C₂×C₃	—	16	1	no	—
277	C₂×C₃	C₅×#447	1	1	yes	characteristic
278	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
279	C₂×C₃	C₅×#509	1	1	yes	characteristic
280	C₅	C₃×#2184	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	8	8	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	1	no	—
288	C₂²	—	8	1	no	—
289	C₂²	—	8	1	no	—
290	C₂²	C₃×C₅×D₁₆	1	2	yes	complemented
291	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	8	1	no	—
294	C₄	C₃×C₅×#120	1	1	yes	characteristic
295	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₃	C₅×#2184	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	—	16	1	no	—
300	C₂	C₃×C₅×#447	1	1	yes	characteristic
301	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
302	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
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
1. C₂²×C₃×C₅×D₈	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₃×C₅×#466	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
6. C₅×#2184	C₃	direct (N × H)
11. C₂×C₃×C₅×D₈	C₂²	semidirect
15. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. C₃×C₅×#107	C₂²	semidirect
18. C₃×#2184	C₅	direct (N × H)
19. C₂²×C₅×D₈	C₂×C₃	semidirect
20. C₂×C₅×D₁₆	C₂×C₃	semidirect
21. C₅×#466	C₂×C₃	semidirect
22. C₅×#357	C₂×C₃	semidirect
23. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
35. C₃×C₅×D₈	D₄	semidirect
42. C₂²×C₃×D₈	C₂×C₅	semidirect
43. C₂×C₃×D₁₆	C₂×C₅	semidirect
44. C₃×#466	C₂×C₅	semidirect
45. C₃×#357	C₂×C₅	semidirect
46. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
51. C₂×C₅×D₈	C₂²×C₃	semidirect
55. C₂×C₅×C₁₆	C₂²×C₃	semidirect
56. C₅×#107	C₂²×C₃	semidirect
58. #2184	C₃×C₅	direct (N × H)
82. C₂×C₃×D₈	C₂²×C₅	semidirect
86. C₂×C₃×C₁₆	C₂²×C₅	semidirect
87. C₃×#107	C₂²×C₅	semidirect
100. C₅×D₈	C₃×D₄	semidirect
107. C₂²×D₈	C₂×C₃×C₅	semidirect
108. C₂×D₁₆	C₂×C₃×C₅	semidirect
109. #466	C₂×C₃×C₅	semidirect
110. #357	C₂×C₃×C₅	semidirect
111. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
121. C₂²×C₃×C₅	D₁₆	semidirect
138. C₃×D₈	C₅×D₄	semidirect
168. C₂×D₈	C₂²×C₃×C₅	semidirect
172. C₂×C₁₆	C₂²×C₃×C₅	semidirect
173. #107	C₂²×C₃×C₅	semidirect
209. C₂²×C₅	C₃×D₁₆	semidirect
226. D₈	C₃×C₅×D₄	semidirect
233. C₃×C₅	#2184	direct (N × H)
243. C₂²×C₃	C₅×D₁₆	semidirect
280. C₅	C₃×#2184	direct (N × H)
290. C₂²	C₃×C₅×D₁₆	semidirect
296. C₃	C₅×#2184	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×D₁₆	2
C₃×C₅×#466	2
C₃×C₅×#357	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2184	3
C₃×#2184	5

Subgroups — C3×C5×#2184 #14275218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2184 #14275218