Subgroups — #14315185

Order 1920.

292 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	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	#11751245	C₂	1	1	yes	maximal, characteristic
3	#11751002	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746382	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746336	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602245	—	3	1	no	maximal, Hall
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	#91839	C₄	1	1	yes	characteristic
11	#91790	C₂²	1	1	yes	characteristic
12	C₃×C₅×M₃₂	—	2	1	no	—
13	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₆	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	#91121	C₄	1	1	yes	characteristic
17	#91109	C₄	1	1	yes	characteristic
18	#91106	C₄	1	1	yes	characteristic
19	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
20	#91096	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
22	#91082	—	2	1	no	—
23	C₁₆×D₁₅	—	2	1	no	—
24	#71452	—	5	1	no	maximal, Hall
25	C₅×#504	D₃	1	1	yes	characteristic, complemented
26	#65433	—	3	1	no	—
27	#65190	—	3	1	no	—
28	#64781	—	3	1	no	—
29	#64735	—	3	1	no	—
30	#64731	—	3	1	no	—
31	C₁₆×Dic₅	—	3	1	no	—
32	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	#6817	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	#6718	C₈	1	1	yes	characteristic
39	#6717	C₈	1	1	yes	characteristic
40	#6716	C₈	1	1	yes	characteristic
41	#6715	C₈	1	1	yes	characteristic
42	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
43	#6710	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
45	#6705	—	2	1	no	—
46	C₈×D₁₅	—	2	1	no	—
47	C₃×C₅×C₁₆	—	2	1	no	—
48	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
49	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
50	C₃⋊#672	—	2	1	no	—
51	C₃⋊#672	—	2	1	no	—
52	C₃×#504	D₅	1	1	yes	characteristic, complemented
53	#5144	—	5	1	no	—
54	#4901	—	5	1	no	—
55	#4561	—	5	1	no	—
56	#4515	—	5	1	no	—
57	#4511	—	5	1	no	—
58	Dic₃×C₁₆	—	5	1	no	—
59	C₅×#131	C₂×D₃	1	1	yes	characteristic
60	#4096	—	3	1	no	—
61	#4047	—	3	1	no	—
62	C₅×M₃₂	—	2	1	no	—
63	C₅×M₃₂	C₂×D₃	1	1	yes	characteristic
64	C₂×C₅×C₁₆	—	2	1	no	—
65	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
66	#3969	—	3	1	no	—
67	#3957	—	3	1	no	—
68	#3954	—	3	1	no	—
69	C₈×Dic₅	—	3	1	no	—
70	#3944	—	3	1	no	—
71	C₂×#672	—	3	1	no	—
72	#3930	—	6	1	no	—
73	D₅×C₁₆	—	6	1	no	—
74	#2170	—	15	1	no	Sylow
75	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	#1220	—	2	1	no	—
81	C₂×Dic₁₅	—	2	1	no	—
82	D₆₀	C₂×C₈	1	1	yes	characteristic
83	C₄×D₁₅	—	2	1	no	—
84	Dic₃₀	C₂×C₈	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	#41	1	1	yes	characteristic
88	C₃⋊#168	—	2	1	no	—
89	C₃⋊#168	C₂×C₈	1	1	yes	characteristic
90	C₃⋊#168	C₂×C₈	1	1	yes	characteristic
91	C₃×#131	C₂×D₅	1	1	yes	characteristic
92	#950	—	5	1	no	—
93	#901	—	5	1	no	—
94	C₃×M₃₂	—	2	1	no	—
95	C₃×M₃₂	C₂×D₅	1	1	yes	characteristic
96	C₂×C₃×C₁₆	—	2	1	no	—
97	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
98	#837	—	5	1	no	—
99	#825	—	5	1	no	—
100	#822	—	5	1	no	—
101	C₈×Dic₃	—	5	1	no	—
102	#812	—	5	1	no	—
103	C₂×C₃⋊C₁₆	—	5	1	no	—
104	#798	—	10	1	no	—
105	D₃×C₁₆	—	10	1	no	—
106	C₅×#41	C₄×D₃	1	1	yes	characteristic
107	#709	—	3	1	no	—
108	C₅×M₁₆	—	2	1	no	—
109	C₅×M₁₆	C₄×D₃	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
112	#689	—	3	1	no	—
113	#688	—	3	1	no	—
114	#687	—	3	1	no	—
115	#686	—	3	1	no	—
116	C₄×Dic₅	—	3	1	no	—
117	#681	—	3	1	no	—
118	C₂×#168	—	3	1	no	—
119	#676	—	6	1	no	—
120	C₈×D₅	—	6	1	no	—
121	C₅×C₁₆	—	2	1	no	—
122	C₅×C₁₆	#67	1	1	yes	characteristic
123	C₅×C₁₆	#67	1	1	yes	characteristic
124	#672	—	6	1	no	—
125	#672	—	6	1	no	—
126	#504	—	15	1	no	—
127	#504	D₁₅	1	1	yes	characteristic, complemented
128	#443	—	15	1	no	—
129	#364	—	15	1	no	—
130	#350	—	15	1	no	—
131	#350	—	15	1	no	—
132	C₄×C₁₆	—	15	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#131	1	1	yes	characteristic
135	C₂×D₁₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	Dic₁₅	—	2	2	no	—
140	Dic₁₅	—	2	1	no	—
141	C₃×#41	C₄×D₅	1	1	yes	characteristic
142	#235	—	5	1	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	C₄×D₅	1	1	yes	characteristic
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
147	#216	—	5	1	no	—
148	#215	—	5	1	no	—
149	#214	—	5	1	no	—
150	#213	—	5	1	no	—
151	C₄×Dic₃	—	5	1	no	—
152	#208	—	5	1	no	—
153	C₂×C₃⋊C₈	—	5	1	no	—
154	ΓL(1,𝔽₂₅)	—	10	1	no	—
155	D₃×C₈	—	10	1	no	—
156	C₃×C₁₆	—	2	1	no	—
157	C₃×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
158	C₃×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
159	C₃⋊C₁₆	—	10	1	no	—
160	C₃⋊C₁₆	—	10	1	no	—
161	C₅×Q₈	D₃×C₈	1	1	yes	characteristic
162	C₅×D₄	—	2	1	no	—
163	C₅×D₄	D₃×C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂²⋊D₅	—	6	1	no	—
167	C₂×Dic₅	—	6	1	no	—
168	D₂₀	—	3	1	no	—
169	C₄×D₅	—	6	1	no	—
170	Dic₁₀	—	3	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	#235	1	1	yes	characteristic
173	C₅×C₈	C₂×#67	1	1	yes	characteristic
174	#168	—	3	1	no	—
175	#168	—	3	1	no	—
176	#168	—	6	1	no	—
177	#135	—	15	1	no	—
178	#131	—	15	1	no	—
179	#131	C₂×D₁₅	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	15	1	no	—
182	M₃₂	—	30	1	no	—
183	M₃₂	C₂×D₁₅	1	1	yes	characteristic
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	15	1	no	—
186	C₂×C₁₆	—	30	1	no	—
187	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
188	#108	—	15	1	no	—
189	#104	—	15	1	no	—
190	#104	—	15	1	no	—
191	C₄×C₈	—	15	1	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
195	D₁₅	—	4	1	no	—
196	C₃×Q₈	C₈×D₅	1	1	yes	characteristic
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₈×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
201	#67	—	10	1	no	—
202	C₂×Dic₃	—	10	1	no	—
203	D₁₂	—	5	1	no	—
204	C₄×D₃	—	10	1	no	—
205	Dic₆	—	5	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	#709	1	1	yes	characteristic
208	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
209	C₃⋊C₈	—	5	1	no	—
210	C₃⋊C₈	—	5	1	no	—
211	C₃⋊C₈	—	10	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	#901	1	1	yes	characteristic
214	C₂×D₅	—	6	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
217	C₄×C₅	C₄×#67	1	1	yes	characteristic
218	Dic₅	—	6	2	no	—
219	Dic₅	—	6	1	no	—
220	#41	—	15	1	no	—
221	#41	C₄×D₁₅	1	1	yes	characteristic
222	Q₁₆	—	15	1	no	—
223	SD₁₆	—	15	1	no	—
224	SD₁₆	—	15	1	no	—
225	D₈	—	15	1	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	15	1	no	—
228	M₁₆	—	30	1	no	—
229	M₁₆	C₄×D₁₅	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	15	1	no	—
232	C₂×C₈	—	30	1	no	—
233	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
234	C₄²	—	15	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	30	1	no	—
237	C₁₆	—	30	1	no	—
238	C₁₆	#1220	1	1	yes	characteristic
239	C₁₆	#1220	1	1	yes	characteristic
240	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	#4047	1	1	yes	characteristic
243	C₂×D₃	—	10	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
246	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
247	Dic₃	—	10	2	no	—
248	Dic₃	—	10	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	C₈×#67	1	1	yes	characteristic
252	D₅	—	12	1	no	—
253	Q₈	—	15	1	no	—
254	Q₈	C₈×D₁₅	1	1	yes	characteristic
255	D₄	—	2	1	no	—
256	D₄	—	15	1	no	—
257	D₄	—	30	1	no	—
258	D₄	C₈×D₁₅	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	30	1	no	—
261	C₂×C₄	—	30	1	no	—
262	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	15	1	no	—
265	C₈	—	15	1	no	—
266	C₈	—	30	1	no	—
267	C₈	#6817	1	1	yes	characteristic
268	C₈	C₂×#1220	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	C₈×C₂²⋊D₅	1	1	yes	characteristic
272	D₃	—	20	1	no	—
273	C₅	#71452	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	30	1	no	—
276	C₂²	#91790	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	30	2	no	—
279	C₄	—	30	1	no	—
280	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
281	C₄	C₄×#1220	1	1	yes	characteristic
282	C₃	#10602245	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	4	1	no	—
285	C₂	—	60	1	no	—
286	C₂	C₈×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#1220
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₃×C₅×#504	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	#91839	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#504	D₁₅
hypocenter	C₃×C₅	#2170
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#504	C₂	semidirect
3. #11751002	C₂	semidirect
4. #11746382	C₂	semidirect
5. #11746336	C₂	semidirect
6. #11746332	C₂	semidirect
7. C₁₆×Dic₁₅	C₂	semidirect
25. C₅×#504	D₃	semidirect
52. C₃×#504	D₅	semidirect
127. #504	D₁₅	semidirect
240. C₃×C₅	#2170	semidirect
273. C₅	#71452	semidirect
282. C₃	#10602245	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#504	2
#11751245	2
#11751002	2
#11746382	2
#11746336	2
#11746332	2
C₁₆×Dic₁₅	2
#10602245	3
#71452	5