Subgroups — #14295542

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×#4094	C₂	1	1	yes	maximal, characteristic
2	#11751338	C₂	1	1	yes	maximal, characteristic
3	C₅×#5240	C₂	1	1	yes	maximal, characteristic
4	#11748323	C₂	1	2	yes	maximal, complemented
5	#11745988	C₂	1	2	yes	maximal, complemented
6	#10600833	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂×#6710	C₂²	1	2	yes	—
9	C₅×#926	C₂²	1	2	yes	—
10	C₃×#4094	—	2	2	no	—
11	C₃×#4094	—	2	2	no	—
12	C₂×C₃×#681	C₄	1	2	yes	complemented
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₂²×C₃×#168	C₄	1	2	yes	complemented
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	#91120	C₄	1	2	yes	complemented
17	#91117	C₄	1	2	yes	complemented
18	C₅×#834	C₄	1	4	yes	—
19	#90953	—	2	2	no	—
20	Dic₃×#168	—	2	2	no	—
21	#72011	—	5	1	no	maximal, Hall
22	C₂×#4094	D₃	1	1	yes	characteristic
23	C₅×#410	—	3	1	no	—
24	#65526	—	3	1	no	—
25	#65286	—	3	2	no	—
26	#64753	—	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	#6710	—	2	2	no	—
35	#6710	C₂×C₄	1	4	yes	—
36	C₂×C₃⋊#168	—	2	2	no	—
37	C₅×#217	C₂×C₄	1	4	yes	—
38	C₅×C₄⋊Dic₃	—	2	2	no	—
39	C₄×C₅×Dic₃	—	2	2	no	—
40	C₃×#681	—	2	2	no	—
41	C₃×#681	—	2	4	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	—	2	2	no	—
44	C₂×C₃×#168	C₂×C₄	1	2	yes	—
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	D₄	1	2	yes	—
48	C₂×C₃×#168	D₄	1	2	yes	—
49	C₂×C₃×#131	—	5	1	no	—
50	#5240	D₅	1	1	yes	characteristic
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×#117	—	3	2	no	—
56	#4094	—	2	2	no	—
57	#4094	—	2	2	no	—
58	C₂×#681	—	3	2	no	—
59	C₂×#681	Dic₃	1	2	yes	complemented
60	C₂×#681	C₂×D₃	1	1	yes	characteristic
61	C₂²×#168	Dic₃	1	2	yes	complemented
62	C₂²×#168	C₂×D₃	1	1	yes	characteristic
63	C₅×#99	—	3	4	no	—
64	#3968	—	3	2	no	—
65	#3965	—	3	2	no	—
66	#3935	—	6	2	no	—
67	C₄×#168	—	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×C₅×Dic₃	—	2	4	no	—
82	C₃⋊#168	—	2	4	no	—
83	C₃×#168	—	2	2	no	—
84	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×#168	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
88	C₃×#131	—	10	2	no	—
89	C₃×#131	—	10	2	no	—
90	C₂×C₃×M₁₆	—	5	2	no	—
91	C₂×C₃×M₁₆	—	5	1	no	—
92	C₂²×C₃×C₈	—	5	1	no	—
93	C₂²×C₃×C₈	—	5	2	no	—
94	#926	C₂×D₅	1	2	yes	—
95	C₂×#208	—	5	2	no	—
96	#836	—	5	2	no	—
97	#834	Dic₅	1	4	yes	—
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×C₄⋊C₄	—	6	2	no	—
109	C₅×C₂²⋊C₄	—	3	4	no	—
110	C₄²×C₅	—	6	2	no	—
111	#681	—	2	2	no	—
112	#681	—	2	4	no	—
113	#681	—	3	4	no	—
114	#681	—	6	2	no	—
115	C₂×#168	—	2	2	no	—
116	C₂×#168	—	2	2	no	—
117	C₂×#168	—	6	2	no	—
118	C₂×#168	C₂×Dic₃	1	2	yes	—
119	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
120	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
121	C₂×#168	#67	1	2	yes	—
122	C₂×#168	#67	1	2	yes	—
123	C₂×#131	—	5	1	no	—
124	#413	—	15	1	no	—
125	#410	—	3	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	C₅×Dic₃	—	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	1	yes	characteristic
142	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
143	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₃×M₁₆	—	10	4	no	—
146	C₃×M₁₆	—	10	2	no	—
147	C₂×C₃×C₈	—	5	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	5	2	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	C₂×C₃×C₈	—	10	2	no	—
153	C₂×C₃×C₈	—	10	2	no	—
154	#217	C₂×Dic₅	1	4	yes	—
155	C₄⋊Dic₃	—	2	2	no	—
156	C₄×Dic₃	—	2	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	—	6	4	no	—
168	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
169	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₄×C₅	#235	1	2	yes	—
171	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×#67	1	2	yes	—
173	#168	—	2	2	no	—
174	#168	—	6	4	no	—
175	#168	#217	1	1	yes	characteristic
176	#168	#217	1	1	yes	characteristic
177	#168	#217	1	2	yes	—
178	C₂×#41	D₃×D₅	1	1	yes	characteristic
179	#131	—	10	2	no	—
180	#131	—	10	2	no	—
181	C₂×M₁₆	—	5	2	no	—
182	C₂×M₁₆	—	5	1	no	—
183	C₂×M₁₆	—	15	2	no	—
184	C₂²×C₈	—	5	1	no	—
185	C₂²×C₈	—	5	2	no	—
186	#117	—	3	2	no	—
187	#101	—	15	2	no	—
188	#100	—	15	2	no	—
189	#99	—	3	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	C₂²×Dic₅	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	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
207	C₂×C₃×C₄	#711	1	2	yes	—
208	C₂×Dic₃	—	2	4	no	—
209	C₃×C₈	—	5	1	no	—
210	C₃×C₈	—	5	1	no	—
211	C₃×C₈	—	5	2	no	—
212	C₃×C₈	—	10	2	no	—
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	—	12	4	no	—
221	C₄×C₅	#926	1	1	yes	characteristic
222	C₄×C₅	C₄×#67	1	2	yes	—
223	C₄×C₅	C₂×#217	1	1	yes	characteristic
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	#1200	1	1	yes	characteristic
227	C₂×D₄	D₃×Dic₅	1	2	yes	—
228	C₂×D₄	#1200	1	1	yes	characteristic
229	C₂²×C₄	D₅×Dic₃	1	2	yes	—
230	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
231	M₁₆	—	10	4	no	—
232	M₁₆	—	10	2	no	—
233	M₁₆	—	15	4	no	—
234	M₁₆	—	30	2	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	5	2	no	—
239	C₂×C₈	—	5	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	—	30	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₂²⋊C₄	—	3	4	no	—
245	C₄²	—	6	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	D₄×Dic₅	1	2	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	#4023	1	1	yes	characteristic
254	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₃×C₄	#4027	1	2	yes	—
256	Dic₃	—	4	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	Dic₃×Dic₅	1	2	yes	—
262	C₂³	C₂×D₃×Dic₅	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	—	6	4	no	—
269	C₂×C₄	Dic₃×Dic₅	1	2	yes	—
270	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
271	C₂×C₄	#6782	1	2	yes	—
272	C₂×C₄	C₂×#1200	1	1	yes	characteristic
273	C₂×C₄	D₅×#67	1	2	yes	—
274	C₈	—	5	1	no	—
275	C₈	—	5	1	no	—
276	C₈	—	5	2	no	—
277	C₈	—	10	2	no	—
278	C₈	—	30	4	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₂²⋊C₄×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×Dic₃×Dic₅	1	1	yes	characteristic
288	C₂²	Dic₅×#67	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	12	4	no	—
291	C₄	#91528	1	1	yes	characteristic
292	C₄	D₅×#217	1	1	yes	characteristic
293	C₄	#91552	1	2	yes	—
294	C₃	#10600833	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	4	2	no	—
298	C₂	Dic₅×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅×#217
commutator	C₂²×C₃×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#41	C₂²
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#41	D₃×D₅
hypocenter	C₃×C₅	#1757
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
4. #11748323	C₂	semidirect
5. #11745988	C₂	semidirect
12. C₂×C₃×#681	C₄	semidirect
14. C₂²×C₃×#168	C₄	semidirect
16. #91120	C₄	semidirect
17. #91117	C₄	semidirect
59. C₂×#681	Dic₃	semidirect
61. C₂²×#168	Dic₃	semidirect
246. C₃×C₅	#1757	semidirect
283. C₅	#72011	semidirect
294. C₃	#10600833	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4094	2
#11751338	2
C₅×#5240	2
#11748323	2
#11745988	2
#10600833	3
#72011	5