Subgroups — #14439845

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756447	C₂	1	2	yes	maximal
2	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751166	C₂	1	2	yes	maximal, complemented
4	#11750981	C₂	1	2	yes	maximal
5	#11750975	C₂	1	2	yes	maximal, complemented
6	#11750961	C₂	1	2	yes	maximal
7	#11750956	C₂	1	2	yes	maximal, complemented
8	C₄×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
9	#11750784	C₂	1	1	yes	maximal, characteristic, complemented
10	#10613647	—	3	1	no	maximal, Hall
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
14	#91785	—	2	4	no	—
15	#91781	C₂²	1	4	yes	—
16	#91777	C₂²	1	4	yes	—
17	#91776	—	2	4	no	—
18	C₄²×D₁₅	C₂²	1	1	yes	characteristic
19	#91759	C₂²	1	2	yes	—
20	C₃×C₅×#107	C₂²	1	2	yes	—
21	C₃×C₅×#106	C₂²	1	2	yes	—
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	#91102	C₂²	1	2	yes	—
24	#91101	C₂²	1	2	yes	—
25	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
26	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
27	#91092	C₂²	1	4	yes	—
28	#91091	C₂²	1	4	yes	—
29	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
30	#81916	—	5	1	no	maximal, Hall
31	#66022	—	3	2	no	—
32	C₅×#499	D₃	1	1	yes	characteristic, complemented
33	#65353	—	3	2	no	—
34	#65169	—	3	2	no	—
35	#65163	—	3	2	no	—
36	#65149	—	3	2	no	—
37	#65144	—	3	2	no	—
38	C₄×C₈×D₅	—	3	1	no	—
39	#64972	—	3	1	no	—
40	C₂×C₄×D₁₅	D₄	1	2	yes	—
41	C₂×C₄×D₁₅	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₀	—	2	4	no	—
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	#6714	—	2	4	no	—
49	C₄⋊Dic₁₅	C₂³	1	4	yes	—
50	#6712	—	2	4	no	—
51	#6712	—	2	4	no	—
52	C₄×Dic₁₅	—	2	4	no	—
53	C₄×Dic₁₅	D₄	1	2	yes	—
54	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
55	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
56	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
57	C₈×D₁₅	Q₈	1	4	yes	complemented
58	C₈×D₁₅	Q₈	1	4	yes	complemented
59	#5688	—	5	2	no	—
60	C₃×#499	D₅	1	1	yes	characteristic, complemented
61	#5065	—	5	2	no	—
62	#4880	—	5	2	no	—
63	#4874	—	5	2	no	—
64	#4860	—	5	2	no	—
65	#4855	—	5	2	no	—
66	C₄×D₃×C₈	—	5	1	no	—
67	#4683	—	5	1	no	—
68	C₅×#125	C₂×D₃	1	2	yes	—
69	C₂×C₈×D₅	—	3	1	no	—
70	C₂×C₈×D₅	—	3	1	no	—
71	#4042	—	6	4	no	—
72	#4038	—	3	4	no	—
73	#4034	—	3	4	no	—
74	#4033	—	6	4	no	—
75	C₄²×D₅	—	3	1	no	—
76	#4016	—	3	2	no	—
77	C₅×#107	C₂×D₃	1	2	yes	—
78	C₅×#106	C₂×D₃	1	2	yes	—
79	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
80	#3950	—	3	2	no	—
81	#3949	—	3	2	no	—
82	C₈×Dic₅	—	3	1	no	—
83	C₈×Dic₅	—	3	1	no	—
84	#3940	—	3	4	no	—
85	#3939	—	3	4	no	—
86	C₄×#168	—	3	1	no	—
87	#3160	—	15	1	no	Sylow
88	C₂²×D₁₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₁₅	—	2	4	no	—
93	C₂×Dic₁₅	C₂×D₄	1	1	yes	characteristic
94	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
95	C₄×D₁₅	—	2	4	no	—
96	C₄×D₁₅	C₂×Q₈	1	2	yes	—
97	C₄×D₁₅	C₂×Q₈	1	2	yes	—
98	Dic₃₀	—	4	8	no	—
99	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
100	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₃⋊#168	C₂×Q₈	1	2	yes	—
102	C₃⋊#168	C₂×Q₈	1	2	yes	—
103	C₃×#125	C₂×D₅	1	2	yes	—
104	C₂×D₃×C₈	—	5	1	no	—
105	C₂×D₃×C₈	—	5	1	no	—
106	#896	—	10	4	no	—
107	#892	—	5	4	no	—
108	#888	—	5	4	no	—
109	#887	—	10	4	no	—
110	C₄²×D₃	—	5	1	no	—
111	#870	—	5	2	no	—
112	C₃×#107	C₂×D₅	1	2	yes	—
113	C₃×#106	C₂×D₅	1	2	yes	—
114	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	#818	—	5	2	no	—
116	#817	—	5	2	no	—
117	C₈×Dic₃	—	5	1	no	—
118	C₈×Dic₃	—	5	1	no	—
119	#808	—	5	4	no	—
120	#807	—	5	4	no	—
121	C₄×C₃⋊C₈	—	5	1	no	—
122	C₂×C₄×D₅	—	3	1	no	—
123	C₂×C₄×D₅	—	3	2	no	—
124	C₂×Dic₁₀	—	6	4	no	—
125	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
127	C₅×C₄⋊C₄	—	2	4	no	—
128	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	#685	—	6	4	no	—
131	C₄⋊Dic₅	—	3	4	no	—
132	#683	—	6	4	no	—
133	#683	—	6	4	no	—
134	C₄×Dic₅	—	3	2	no	—
135	C₄×Dic₅	—	3	1	no	—
136	C₄×Dic₅	—	6	4	no	—
137	C₂×#168	—	3	1	no	—
138	C₂×#168	—	3	1	no	—
139	C₈×D₅	—	3	4	no	—
140	C₈×D₅	—	3	4	no	—
141	#533	—	15	2	no	—
142	#500	—	15	2	no	—
143	#499	—	15	2	no	—
144	#499	—	15	1	no	—
145	#499	D₁₅	1	1	yes	characteristic, complemented
146	#498	—	15	2	no	—
147	#427	—	15	2	no	—
148	#427	—	15	2	no	—
149	C₂×C₄×C₈	—	15	1	no	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₅	—	2	2	no	—
152	C₂×D₁₅	C₄⋊Q₈	1	2	yes	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	—	4	4	no	—
155	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
157	Dic₁₅	—	4	4	no	—
158	Dic₁₅	C₄⋊Q₈	1	2	yes	—
159	Dic₁₅	#135	1	4	yes	—
160	C₂×C₄×D₃	—	5	1	no	—
161	C₂×C₄×D₃	—	5	2	no	—
162	C₂×Dic₆	—	10	4	no	—
163	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	2	4	no	—
166	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
167	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
168	(C₂×D₃)⋊C₄	—	10	4	no	—
169	C₄⋊Dic₃	—	5	4	no	—
170	Dic₃⋊C₄	—	10	4	no	—
171	Dic₃⋊C₄	—	10	4	no	—
172	C₄×Dic₃	—	5	2	no	—
173	C₄×Dic₃	—	5	1	no	—
174	C₄×Dic₃	—	10	4	no	—
175	C₂×C₃⋊C₈	—	5	1	no	—
176	C₂×C₃⋊C₈	—	5	1	no	—
177	D₃×C₈	—	5	4	no	—
178	D₃×C₈	—	5	4	no	—
179	C₂²×D₅	—	3	1	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
183	C₂×Dic₅	—	3	1	no	—
184	C₂×Dic₅	—	3	2	no	—
185	C₂×Dic₅	—	6	4	no	—
186	C₄×D₅	—	3	2	no	—
187	C₄×D₅	—	3	2	no	—
188	C₄×D₅	—	6	4	no	—
189	Dic₁₀	—	12	8	no	—
190	C₅×C₈	D₃×Q₈	1	2	yes	—
191	C₅×C₈	D₃×Q₈	1	2	yes	—
192	#168	—	3	2	no	—
193	#168	—	3	2	no	—
194	C₂²×C₈	—	15	1	no	—
195	C₂²×C₈	—	15	1	no	—
196	C₄⋊Q₈	—	15	4	no	—
197	#125	—	15	2	no	—
198	#125	C₂×D₁₅	1	2	yes	—
199	C₂²⋊Q₈	—	30	4	no	—
200	C₄×Q₈	—	30	4	no	—
201	#117	—	15	4	no	—
202	C₂×C₄²	—	15	1	no	—
203	#107	—	15	2	no	—
204	#107	—	15	4	no	—
205	#107	C₂×D₁₅	1	2	yes	—
206	#106	—	15	2	no	—
207	#106	—	15	4	no	—
208	#106	C₂×D₁₅	1	2	yes	—
209	C₄×C₈	—	15	1	no	—
210	C₄×C₈	—	15	1	no	—
211	C₄×C₈	—	15	1	no	—
212	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂×#135	1	2	yes	—
215	D₁₅	—	2	2	no	—
216	C₂²×D₃	—	5	1	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
219	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
220	C₂×Dic₃	—	5	1	no	—
221	C₂×Dic₃	—	5	2	no	—
222	C₂×Dic₃	—	10	4	no	—
223	C₄×D₃	—	5	2	no	—
224	C₄×D₃	—	5	2	no	—
225	C₄×D₃	—	10	4	no	—
226	Dic₆	—	20	8	no	—
227	C₃×C₈	Q₈×D₅	1	2	yes	—
228	C₃×C₈	Q₈×D₅	1	2	yes	—
229	C₃⋊C₈	—	5	2	no	—
230	C₃⋊C₈	—	5	2	no	—
231	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
232	C₂×D₅	—	3	2	no	—
233	C₂×D₅	—	6	2	no	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
237	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
238	Dic₅	—	3	2	no	—
239	Dic₅	—	3	4	no	—
240	Dic₅	—	12	4	no	—
241	C₂×Q₈	—	30	4	no	—
242	C₂²×C₄	—	15	1	no	—
243	C₂²×C₄	—	15	2	no	—
244	C₂×C₈	—	15	1	no	—
245	C₂×C₈	—	15	1	no	—
246	C₂×C₈	—	15	4	no	—
247	C₂×C₈	—	15	4	no	—
248	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	15	4	no	—
252	C₄⋊C₄	—	30	4	no	—
253	C₄⋊C₄	—	30	4	no	—
254	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
255	C₂²⋊C₄	—	30	4	no	—
256	C₄²	—	15	2	no	—
257	C₄²	—	15	1	no	—
258	C₄²	—	30	4	no	—
259	C₄²	C₂²×D₁₅	1	1	yes	characteristic
260	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
262	C₂×D₃	—	5	2	no	—
263	C₂×D₃	—	10	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
267	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
268	Dic₃	—	5	2	no	—
269	Dic₃	—	5	4	no	—
270	Dic₃	—	20	4	no	—
271	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
272	C₂×C₅	D₃×#135	1	2	yes	—
273	D₅	—	6	2	no	—
274	C₂³	—	15	1	no	—
275	Q₈	—	60	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	15	1	no	—
278	C₂×C₄	—	15	2	no	—
279	C₂×C₄	—	15	2	no	—
280	C₂×C₄	—	15	2	no	—
281	C₂×C₄	—	30	4	no	—
282	C₂×C₄	—	30	4	no	—
283	C₂×C₄	D₄×D₁₅	1	2	yes	—
284	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
285	C₈	—	15	2	no	—
286	C₈	—	15	2	no	—
287	C₈	Q₈×D₁₅	1	2	yes	—
288	C₈	Q₈×D₁₅	1	2	yes	—
289	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
290	C₂×C₃	D₅×#135	1	2	yes	—
291	D₃	—	10	2	no	—
292	C₅	#81916	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	15	2	no	—
294	C₂²	—	30	2	no	—
295	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	15	2	no	—
299	C₄	—	15	4	no	—
300	C₄	—	60	4	no	—
301	C₄	C₂×Q₈×D₁₅	1	1	yes	characteristic
302	C₄	C₂×Q₈×D₁₅	1	1	yes	characteristic
303	C₃	#10613647	1	1	yes	characteristic, complemented, Sylow
304	C₂	—	30	2	no	—
305	C₂	D₁₅×C₄⋊Q₈	1	1	yes	characteristic
306	C₂	D₁₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#499	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×D₁₅	C₂×D₄
hypercenter	#499	D₁₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#499	C₂	semidirect
3. #11751166	C₂	semidirect
5. #11750975	C₂	semidirect
7. #11750956	C₂	semidirect
9. #11750784	C₂	semidirect
32. C₅×#499	D₃	semidirect
57. C₈×D₁₅	Q₈	semidirect
58. C₈×D₁₅	Q₈	semidirect
60. C₃×#499	D₅	semidirect
145. #499	D₁₅	semidirect
260. C₃×C₅	#3160	semidirect
292. C₅	#81916	semidirect
303. C₃	#10613647	semidirect

Maximal subgroups

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

subgroup	index
#11756447	2
C₃×C₅×#499	2
#11751166	2
#11750981	2
#11750975	2
#11750961	2
#11750956	2
C₄×C₈×D₁₅	2
#11750784	2
#10613647	3
#81916	5