Subgroups — #14439839

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	#11756487	C₂	1	2	yes	maximal
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751181	C₂	1	2	yes	maximal, complemented
4	#11750981	C₂	1	4	yes	maximal
5	#11750979	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
7	#11750783	C₂	1	1	yes	maximal, characteristic, complemented
8	#10613669	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	#91835	—	2	2	no	—
11	Q₈×Dic₁₅	—	2	2	no	—
12	C₂×C₈×D₁₅	C₂²	1	2	yes	—
13	#91786	—	2	2	no	—
14	#91781	C₂²	1	4	yes	—
15	#91779	C₂²	1	4	yes	—
16	#91776	—	2	2	no	—
17	C₄²×D₁₅	C₂²	1	1	yes	characteristic
18	#91758	C₂²	1	2	yes	—
19	C₃×C₅×#107	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#91102	C₂²	1	4	yes	—
22	C₈×Dic₁₅	C₂²	1	2	yes	—
23	#91092	C₂²	1	8	yes	—
24	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
25	#81938	—	5	1	no	maximal, Hall
26	#66062	—	3	2	no	—
27	C₅×#500	D₃	1	1	yes	characteristic, complemented
28	#65368	—	3	2	no	—
29	#65169	—	3	4	no	—
30	#65167	—	3	4	no	—
31	C₄×C₈×D₅	—	3	1	no	—
32	#64971	—	3	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
35	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
36	C₂×C₄×D₁₅	C₂³	1	1	yes	characteristic
37	C₂×Dic₃₀	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	#6714	—	2	4	no	—
43	C₄⋊Dic₁₅	—	2	2	no	—
44	C₄⋊Dic₁₅	—	2	4	no	—
45	C₄⋊Dic₁₅	C₂³	1	4	yes	—
46	#6712	—	2	4	no	—
47	C₄×Dic₁₅	—	2	4	no	—
48	C₄×Dic₁₅	D₄	1	1	yes	characteristic
49	C₄×Dic₁₅	D₄	1	1	yes	characteristic
50	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
51	C₂×C₃⋊#168	C₂³	1	2	yes	—
52	C₈×D₁₅	Q₈	1	8	yes	complemented
53	#5728	—	5	2	no	—
54	C₃×#500	D₅	1	1	yes	characteristic, complemented
55	#5080	—	5	2	no	—
56	#4880	—	5	4	no	—
57	#4878	—	5	4	no	—
58	C₄×D₃×C₈	—	5	1	no	—
59	#4682	—	5	1	no	—
60	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
61	#4092	—	6	2	no	—
62	Q₈×Dic₅	—	6	2	no	—
63	C₂×C₈×D₅	—	3	2	no	—
64	#4043	—	6	2	no	—
65	#4038	—	3	4	no	—
66	#4036	—	3	4	no	—
67	#4033	—	6	2	no	—
68	C₄²×D₅	—	3	1	no	—
69	#4015	—	3	2	no	—
70	C₅×#107	C₂×D₃	1	4	yes	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#3950	—	3	4	no	—
73	C₈×Dic₅	—	3	2	no	—
74	#3940	—	3	8	no	—
75	C₄×#168	—	3	1	no	—
76	#3160	—	15	1	no	Sylow
77	C₂²×D₁₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×Q₈	—	4	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₅	—	2	4	no	—
84	C₂×Dic₁₅	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₅	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₅	C₂×D₄	1	1	yes	characteristic
87	C₄×D₁₅	—	2	2	no	—
88	C₄×D₁₅	—	2	2	no	—
89	C₄×D₁₅	C₂×Q₈	1	4	yes	—
90	Dic₃₀	—	4	4	no	—
91	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
92	C₃⋊#168	C₂×Q₈	1	4	yes	—
93	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
94	#946	—	10	2	no	—
95	Q₈×Dic₃	—	10	2	no	—
96	C₂×D₃×C₈	—	5	2	no	—
97	#897	—	10	2	no	—
98	#892	—	5	4	no	—
99	#890	—	5	4	no	—
100	#887	—	10	2	no	—
101	C₄²×D₃	—	5	1	no	—
102	#869	—	5	2	no	—
103	C₃×#107	C₂×D₅	1	4	yes	—
104	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	#818	—	5	4	no	—
106	C₈×Dic₃	—	5	2	no	—
107	#808	—	5	8	no	—
108	C₄×C₃⋊C₈	—	5	1	no	—
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₄×D₅	—	3	1	no	—
111	C₂×C₄×D₅	—	3	1	no	—
112	C₂×C₄×D₅	—	3	1	no	—
113	C₂×Dic₁₀	—	6	2	no	—
114	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
117	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
118	#685	—	6	4	no	—
119	C₄⋊Dic₅	—	3	4	no	—
120	C₄⋊Dic₅	—	6	4	no	—
121	C₄⋊Dic₅	—	6	2	no	—
122	#683	—	6	4	no	—
123	C₄×Dic₅	—	3	1	no	—
124	C₄×Dic₅	—	3	1	no	—
125	C₄×Dic₅	—	3	1	no	—
126	C₄×Dic₅	—	6	4	no	—
127	C₂×#168	—	3	2	no	—
128	C₈×D₅	—	3	8	no	—
129	#533	—	15	2	no	—
130	#500	—	15	1	no	—
131	#500	D₁₅	1	1	yes	characteristic, complemented
132	#499	—	15	4	no	—
133	#498	—	15	2	no	—
134	#427	—	15	4	no	—
135	C₂×C₄×C₈	—	15	1	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₅	—	2	1	no	—
138	C₂×D₁₅	—	2	1	no	—
139	C₂×D₁₅	C₄⋊Q₈	1	2	yes	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	C₃×C₄×C₅	#135	1	2	yes	—
143	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
144	Dic₁₅	—	2	1	no	—
145	Dic₁₅	—	2	1	no	—
146	Dic₁₅	—	4	4	no	—
147	Dic₁₅	C₄⋊Q₈	1	2	yes	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₄×D₃	—	5	1	no	—
150	C₂×C₄×D₃	—	5	1	no	—
151	C₂×C₄×D₃	—	5	1	no	—
152	C₂×Dic₆	—	10	2	no	—
153	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
156	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	—	10	4	no	—
158	C₄⋊Dic₃	—	5	4	no	—
159	C₄⋊Dic₃	—	10	4	no	—
160	C₄⋊Dic₃	—	10	2	no	—
161	Dic₃⋊C₄	—	10	4	no	—
162	C₄×Dic₃	—	5	1	no	—
163	C₄×Dic₃	—	5	1	no	—
164	C₄×Dic₃	—	5	1	no	—
165	C₄×Dic₃	—	10	4	no	—
166	C₂×C₃⋊C₈	—	5	2	no	—
167	D₃×C₈	—	5	8	no	—
168	C₂²×D₅	—	3	1	no	—
169	C₅×Q₈	—	4	4	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₂×Dic₅	—	3	1	no	—
175	C₂×Dic₅	—	3	1	no	—
176	C₂×Dic₅	—	3	1	no	—
177	C₂×Dic₅	—	6	4	no	—
178	C₄×D₅	—	3	4	no	—
179	C₄×D₅	—	6	2	no	—
180	C₄×D₅	—	6	2	no	—
181	Dic₁₀	—	12	4	no	—
182	C₅×C₈	D₃×Q₈	1	4	yes	—
183	#168	—	3	4	no	—
184	C₂²×C₈	—	15	2	no	—
185	C₄⋊Q₈	—	15	2	no	—
186	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
187	#125	—	15	4	no	—
188	C₂²⋊Q₈	—	30	2	no	—
189	C₂²⋊Q₈	—	30	2	no	—
190	C₄×Q₈	—	30	2	no	—
191	C₄×Q₈	—	30	2	no	—
192	#117	—	15	4	no	—
193	C₂×C₄²	—	15	1	no	—
194	#107	—	15	4	no	—
195	#107	C₂×D₁₅	1	4	yes	—
196	#106	—	15	8	no	—
197	C₄×C₈	—	15	2	no	—
198	C₄×C₈	—	15	1	no	—
199	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
203	D₁₅	—	2	2	no	—
204	C₂²×D₃	—	5	1	no	—
205	C₃×Q₈	—	4	4	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
210	C₂×Dic₃	—	5	1	no	—
211	C₂×Dic₃	—	5	1	no	—
212	C₂×Dic₃	—	5	1	no	—
213	C₂×Dic₃	—	10	4	no	—
214	C₄×D₃	—	5	4	no	—
215	C₄×D₃	—	10	2	no	—
216	C₄×D₃	—	10	2	no	—
217	Dic₆	—	20	4	no	—
218	C₃×C₈	Q₈×D₅	1	4	yes	—
219	C₃⋊C₈	—	5	4	no	—
220	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
221	C₂×D₅	—	3	2	no	—
222	C₂×D₅	—	6	1	no	—
223	C₂×D₅	—	6	1	no	—
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	#912	1	2	yes	—
226	C₄×C₅	#919	1	2	yes	—
227	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
228	Dic₅	—	3	2	no	—
229	Dic₅	—	6	1	no	—
230	Dic₅	—	6	1	no	—
231	Dic₅	—	12	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	30	2	no	—
234	C₂²×C₄	—	15	1	no	—
235	C₂²×C₄	—	15	1	no	—
236	C₂²×C₄	—	15	1	no	—
237	C₂×C₈	—	15	2	no	—
238	C₂×C₈	—	15	8	no	—
239	C₂×C₈	C₂²×D₁₅	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	15	4	no	—
242	C₄⋊C₄	—	30	4	no	—
243	C₄⋊C₄	—	30	4	no	—
244	C₄⋊C₄	—	30	2	no	—
245	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
246	C₂²⋊C₄	—	30	4	no	—
247	C₄²	—	15	1	no	—
248	C₄²	—	15	1	no	—
249	C₄²	—	15	1	no	—
250	C₄²	—	30	4	no	—
251	C₄²	C₂²×D₁₅	1	1	yes	characteristic
252	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
254	C₂×D₃	—	5	2	no	—
255	C₂×D₃	—	10	1	no	—
256	C₂×D₃	—	10	1	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	#4058	1	2	yes	—
259	C₃×C₄	#4065	1	2	yes	—
260	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
261	Dic₃	—	5	2	no	—
262	Dic₃	—	10	1	no	—
263	Dic₃	—	10	1	no	—
264	Dic₃	—	20	4	no	—
265	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
266	C₂×C₅	C₂×#912	1	1	yes	characteristic
267	C₂×C₅	C₂×#919	1	1	yes	characteristic
268	D₅	—	6	2	no	—
269	C₂³	—	15	1	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	60	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	15	1	no	—
274	C₂×C₄	—	15	4	no	—
275	C₂×C₄	—	15	1	no	—
276	C₂×C₄	—	15	1	no	—
277	C₂×C₄	—	30	2	no	—
278	C₂×C₄	—	30	2	no	—
279	C₂×C₄	—	30	4	no	—
280	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
281	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
282	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
283	C₈	—	15	4	no	—
284	C₈	Q₈×D₁₅	1	4	yes	—
285	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
286	C₂×C₃	C₂×#4058	1	1	yes	characteristic
287	C₂×C₃	C₂×#4065	1	1	yes	characteristic
288	D₃	—	10	2	no	—
289	C₅	#81938	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	15	2	no	—
291	C₂²	—	30	1	no	—
292	C₂²	—	30	1	no	—
293	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
294	C₄	—	4	4	no	—
295	C₄	—	15	2	no	—
296	C₄	—	30	1	no	—
297	C₄	—	30	1	no	—
298	C₄	—	60	4	no	—
299	C₄	#91801	1	2	yes	—
300	C₄	#91808	1	2	yes	—
301	C₄	C₂×Q₈×D₁₅	1	2	yes	—
302	C₃	#10613669	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	30	2	no	—
304	C₂	D₁₅×C₄⋊Q₈	1	1	yes	characteristic
305	C₂	C₂×#91801	1	1	yes	characteristic
306	C₂	C₂×#91808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#500	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×D₁₅	C₂×D₄
hypercenter	#500	D₁₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#500	C₂	semidirect
3. #11751181	C₂	semidirect
5. #11750979	C₂	semidirect
7. #11750783	C₂	semidirect
27. C₅×#500	D₃	semidirect
52. C₈×D₁₅	Q₈	semidirect
54. C₃×#500	D₅	semidirect
131. #500	D₁₅	semidirect
252. C₃×C₅	#3160	semidirect
289. C₅	#81938	semidirect
302. C₃	#10613669	semidirect

Maximal subgroups

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

subgroup	index
#11756487	2
C₃×C₅×#500	2
#11751181	2
#11750981	2
#11750979	2
C₄×C₈×D₁₅	2
#11750783	2
#10613669	3
#81938	5