Subgroups — #14295504

Order 1920.

568 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	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#11751341	C₂	1	1	yes	maximal, characteristic
3	#11751338	C₂	1	1	yes	maximal, characteristic
4	#11751202	C₂	1	2	yes	maximal, complemented
5	#11746377	C₂	1	2	yes	maximal, complemented
6	#10602804	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#131	—	2	2	no	—
9	C₃×C₅×#131	—	2	2	no	—
10	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	complemented
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₄	1	2	yes	complemented
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	#91815	C₂²	1	2	yes	—
15	C₂×#6710	C₂²	1	2	yes	—
16	#91120	C₄	1	2	yes	complemented
17	#91118	C₄	1	4	yes	—
18	#91117	C₄	1	2	yes	complemented
19	#91099	—	2	2	no	—
20	C₈×Dic₁₅	—	2	2	no	—
21	#72011	—	5	1	no	maximal, Hall
22	C₂×C₅×#131	D₃	1	1	yes	characteristic
23	#65529	—	3	1	no	—
24	#65526	—	3	1	no	—
25	#65390	—	3	2	no	—
26	#64776	—	3	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	2	no	—
35	C₃×C₅×M₁₆	—	2	4	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
42	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
43	#6719	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₅	—	2	2	no	—
45	C₄×Dic₁₅	—	2	2	no	—
46	#6710	—	2	2	no	—
47	#6710	C₂×C₄	1	4	yes	—
48	C₂×C₃⋊#168	—	2	2	no	—
49	C₂×C₃×#131	D₅	1	1	yes	characteristic
50	#5240	—	5	1	no	—
51	#5237	—	5	1	no	—
52	#5101	—	5	2	no	—
53	#4556	—	5	2	no	—
54	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
55	C₅×#131	—	2	2	no	—
56	C₅×#131	—	2	2	no	—
57	C₂×C₅×M₁₆	Dic₃	1	2	yes	complemented
58	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₅×C₈	Dic₃	1	2	yes	complemented
60	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
61	#4072	—	3	2	no	—
62	C₂×#681	—	3	2	no	—
63	#3968	—	3	2	no	—
64	#3966	—	3	4	no	—
65	#3965	—	3	2	no	—
66	#3947	—	6	2	no	—
67	C₈×Dic₅	—	6	2	no	—
68	#1757	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₄²	1	2	yes	—
70	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	2	no	—
72	C₃×C₅×D₄	—	2	4	no	—
73	C₃×C₅×D₄	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×Dic₁₅	—	2	4	no	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
86	C₃⋊#168	—	2	4	no	—
87	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
88	C₃×#131	—	2	2	no	—
89	C₃×#131	—	2	2	no	—
90	C₂×C₃×M₁₆	Dic₅	1	2	yes	complemented
91	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃×C₈	Dic₅	1	2	yes	complemented
93	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
94	#926	—	5	2	no	—
95	C₂×#208	—	5	2	no	—
96	#836	—	5	2	no	—
97	#834	—	5	4	no	—
98	#833	—	5	2	no	—
99	#815	—	10	2	no	—
100	C₈×Dic₃	—	10	2	no	—
101	C₅×#41	—	2	2	no	—
102	C₅×#41	—	2	2	no	—
103	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
104	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
105	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₅×M₁₆	—	2	2	no	—
109	C₅×M₁₆	—	2	4	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
113	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
115	C₂×C₅×C₈	#67	1	2	yes	—
116	C₂×C₅×C₈	#67	1	2	yes	—
117	C₂²⋊Dic₅	—	3	4	no	—
118	C₄⋊Dic₅	—	6	2	no	—
119	C₄×Dic₅	—	6	2	no	—
120	#681	—	3	4	no	—
121	#681	—	6	2	no	—
122	C₂×#168	—	6	2	no	—
123	C₂×#131	D₁₅	1	1	yes	characteristic
124	#413	—	15	1	no	—
125	#410	—	15	1	no	—
126	#405	—	15	2	no	—
127	#344	—	15	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	4	1	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#117	1	1	yes	characteristic
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	Dic₁₅	—	4	4	no	—
138	C₃×#41	—	2	2	no	—
139	C₃×#41	—	2	2	no	—
140	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
141	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
142	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₃×M₁₆	—	2	2	no	—
146	C₃×M₁₆	—	2	4	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
150	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
153	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
154	#217	—	5	4	no	—
155	C₄⋊Dic₃	—	10	2	no	—
156	C₄×Dic₃	—	10	2	no	—
157	#208	—	5	4	no	—
158	#208	—	10	2	no	—
159	C₂×C₃⋊C₈	—	10	2	no	—
160	C₂³×C₅	C₄×Dic₃	1	2	yes	—
161	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄×C₅	#235	1	2	yes	—
170	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×#67	1	2	yes	—
172	C₂×Dic₅	—	6	4	no	—
173	C₅×C₈	—	2	2	no	—
174	C₅×C₈	#217	1	1	yes	characteristic
175	C₅×C₈	#217	1	2	yes	—
176	C₅×C₈	#217	1	1	yes	characteristic
177	#168	—	6	4	no	—
178	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
179	#131	—	2	2	no	—
180	#131	—	2	2	no	—
181	C₂×M₁₆	—	15	2	no	—
182	C₂×M₁₆	Dic₁₅	1	2	yes	complemented
183	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
184	C₂²×C₈	Dic₁₅	1	2	yes	complemented
185	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
186	#117	—	15	2	no	—
187	#101	—	15	2	no	—
188	#100	—	15	2	no	—
189	#99	—	15	4	no	—
190	#97	—	30	2	no	—
191	C₄×C₈	—	30	2	no	—
192	C₂×C₃×C₅	—	2	2	no	—
193	C₂×C₃×C₅	—	2	1	no	—
194	C₂×C₃×C₅	—	4	2	no	—
195	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₃	C₄×Dic₅	1	2	yes	—
197	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
198	C₃×Q₈	—	2	2	no	—
199	C₃×D₄	—	2	4	no	—
200	C₃×D₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
205	C₂×C₃×C₄	#709	1	2	yes	—
206	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
208	C₂×Dic₃	—	10	4	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
212	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₃⋊C₈	—	10	4	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
218	C₂²×C₅	C₄×#67	1	2	yes	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	#926	1	1	yes	characteristic
221	C₄×C₅	C₄×#67	1	2	yes	—
222	C₄×C₅	C₂×#217	1	1	yes	characteristic
223	Dic₅	—	12	4	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	C₄×D₁₅	1	1	yes	characteristic
227	C₂×D₄	C₄×D₁₅	1	2	yes	—
228	C₂×D₄	C₄×D₁₅	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
230	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	15	4	no	—
234	M₁₆	—	30	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	30	2	no	—
238	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
239	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
240	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
241	C₂×C₈	#1220	1	2	yes	—
242	C₂×C₈	#1220	1	2	yes	—
243	C₄⋊C₄	—	30	2	no	—
244	C₂²⋊C₄	—	15	4	no	—
245	C₄²	—	30	2	no	—
246	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
251	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	#4072	1	1	yes	characteristic
254	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
255	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
256	Dic₃	—	20	4	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	C₄×#217	1	1	yes	characteristic
261	C₂³	C₄×Dic₁₅	1	2	yes	—
262	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	30	4	no	—
269	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
270	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
271	C₂×C₄	#6817	1	2	yes	—
272	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
273	C₂×C₄	C₂×#1220	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	30	4	no	—
276	C₈	#6719	1	1	yes	characteristic
277	C₈	#6719	1	2	yes	—
278	C₈	#6719	1	1	yes	characteristic
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
283	C₅	#72011	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
288	C₂²	C₄×#1220	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	60	4	no	—
291	C₄	#91815	1	1	yes	characteristic
292	C₄	C₄×#1220	1	2	yes	—
293	C₄	C₂×#6719	1	1	yes	characteristic
294	C₃	#10602804	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	4	2	no	—
298	C₂	C₄×#6719	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#6719
commutator	C₂²×C₃×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#131	D₁₅
hypocenter	C₃×C₅	#1757
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
4. #11751202	C₂	semidirect
5. #11746377	C₂	semidirect
10. C₂×C₃×C₅×M₁₆	C₄	semidirect
12. C₂²×C₃×C₅×C₈	C₄	semidirect
16. #91120	C₄	semidirect
18. #91117	C₄	semidirect
57. C₂×C₅×M₁₆	Dic₃	semidirect
59. C₂²×C₅×C₈	Dic₃	semidirect
90. C₂×C₃×M₁₆	Dic₅	semidirect
92. C₂²×C₃×C₈	Dic₅	semidirect
182. C₂×M₁₆	Dic₁₅	semidirect
184. C₂²×C₈	Dic₁₅	semidirect
246. C₃×C₅	#1757	semidirect
283. C₅	#72011	semidirect
294. C₃	#10602804	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#131	2
#11751341	2
#11751338	2
#11751202	2
#11746377	2
#10602804	3
#72011	5