Subgroups — #14280474

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6738	C₂	1	1	yes	maximal, characteristic
2	#11751797	C₂	1	1	yes	maximal, characteristic
3	#11747583	C₂	1	1	yes	maximal, characteristic
4	#11746751	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604010	—	3	1	no	maximal, Hall
9	C₂×#6738	C₄	1	2	yes	—
10	C₄×#1200	C₂²	1	1	yes	characteristic
11	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₃×C₄×#170	C₂²	1	1	yes	characteristic
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	Dic₃⋊#170	C₄	1	2	yes	—
16	#91171	C₄	1	2	yes	—
17	Dic₃×#170	C₄	1	2	yes	—
18	#91104	C₄	1	1	yes	characteristic
19	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
20	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
21	Dic₃⋊#168	C₄	1	1	yes	characteristic
22	#90957	C₄	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68047	—	5	1	no	maximal, Hall
25	#65680	—	3	1	no	—
26	C₄×D₅⋊C₈	—	3	1	no	—
27	#65122	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	—	3	1	no	—
30	C₈×#170	—	3	1	no	—
31	C₈×#170	D₃	1	1	yes	characteristic, complemented
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#1200	C₂×C₄	1	1	yes	characteristic
34	C₂×#1200	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	#6738	C₈	1	8	yes	complemented
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
47	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
48	#4685	—	5	1	no	—
49	#4685	—	5	1	no	—
50	C₄×D₃×C₈	—	5	1	no	—
51	C₃×C₈²	—	5	1	no	—
52	#4454	—	5	1	no	—
53	#4454	—	5	1	no	—
54	C₈×C₃⋊C₈	—	5	1	no	—
55	C₂×D₅⋊C₈	—	3	2	no	—
56	C₄²×D₅	—	3	1	no	—
57	#4003	—	3	2	no	—
58	C₄⋊#170	—	3	2	no	—
59	C₄×#170	—	3	2	no	—
60	C₄×#170	—	3	1	no	—
61	C₄×#170	—	3	1	no	—
62	C₄×#170	C₂×D₃	1	1	yes	characteristic
63	C₄×#170	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	3	1	no	—
65	#3952	—	3	1	no	—
66	#3952	—	3	1	no	—
67	C₈×Dic₅	—	3	1	no	—
68	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
69	C₄⋊#168	—	3	1	no	—
70	#1450	—	15	1	no	Sylow
71	C₂²×D₁₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
74	C₄×D₁₅	—	2	1	no	—
75	C₄×D₁₅	—	2	1	no	—
76	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
77	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
80	#1200	—	2	2	no	—
81	#1200	C₂×C₈	1	4	yes	—
82	C₃⋊#170	—	2	2	no	—
83	C₃⋊#170	C₂×C₈	1	4	yes	—
84	C₃×#170	—	2	2	no	—
85	C₃×#170	C₂×C₈	1	4	yes	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃⋊#168	—	2	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₅×C₃⋊C₈	—	2	1	no	—
90	C₂×D₃×C₈	—	5	2	no	—
91	C₄²×D₃	—	5	1	no	—
92	C₃×C₄×C₈	—	5	1	no	—
93	C₃×C₄×C₈	—	5	1	no	—
94	C₃×C₄×C₈	—	5	1	no	—
95	(C₂×D₃)⋊C₈	—	5	2	no	—
96	(C₂×D₃)⋊C₈	—	5	1	no	—
97	(C₂×D₃)⋊C₈	—	5	1	no	—
98	Dic₃⋊C₈	—	5	2	no	—
99	Dic₃⋊C₈	—	5	1	no	—
100	Dic₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₈×Dic₃	—	5	2	no	—
102	C₄×C₃⋊C₈	—	5	1	no	—
103	C₄×C₃⋊C₈	—	5	1	no	—
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₂×C₄×D₅	—	3	1	no	—
106	C₂×C₄×D₅	—	3	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×#170	—	3	2	no	—
109	C₂×#170	—	3	2	no	—
110	C₂×#170	C₄×D₃	1	2	yes	—
111	C₂×#170	C₄×D₃	1	2	yes	—
112	D₅⋊C₈	—	3	8	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
115	C₄²×C₅	—	3	1	no	—
116	C₄×Dic₅	—	3	1	no	—
117	C₄×Dic₅	—	3	1	no	—
118	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
119	C₂×#168	—	3	1	no	—
120	C₂×#168	C₄×D₃	1	1	yes	characteristic
121	#431	—	15	1	no	—
122	#431	—	15	1	no	—
123	C₂×C₄×C₈	—	15	1	no	—
124	#322	—	15	1	no	—
125	#322	—	15	1	no	—
126	C₈²	—	5	1	no	—
127	C₈²	—	15	1	no	—
128	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
129	C₂×D₁₅	—	2	1	no	—
130	C₂×D₁₅	—	2	1	no	—
131	C₂×D₁₅	C₄×C₈	1	2	yes	—
132	C₃×C₄×C₅	#131	1	1	yes	characteristic
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	Dic₁₅	—	2	1	no	—
135	C₃×Dic₅	C₂²×C₈	1	2	yes	—
136	C₃×Dic₅	#131	1	2	yes	—
137	C₅×Dic₃	—	2	1	no	—
138	C₅×Dic₃	C₄×C₈	1	2	yes	—
139	C₂×C₄×D₃	—	5	1	no	—
140	C₂×C₄×D₃	—	5	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₃×C₄²	—	5	1	no	—
147	C₄×Dic₃	—	5	1	no	—
148	C₄×Dic₃	—	5	1	no	—
149	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	5	2	no	—
151	C₂×C₃⋊C₈	—	5	2	no	—
152	C₂×C₃⋊C₈	—	5	1	no	—
153	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	D₃×C₈	—	5	8	no	—
155	C₂²×D₅	—	3	1	no	—
156	C₂×C₄×C₅	—	3	1	no	—
157	C₂×C₄×C₅	—	3	1	no	—
158	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
159	C₂×Dic₅	—	3	1	no	—
160	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₄×D₅	—	3	4	no	—
163	C₄×D₅	—	6	1	no	—
164	C₄×D₅	—	6	1	no	—
165	C₄×D₅	—	6	2	no	—
166	#170	—	2	2	no	—
167	#170	—	3	4	no	—
168	#170	—	6	2	no	—
169	#170	D₃×C₈	1	4	yes	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	6	1	no	—
172	#168	—	2	1	no	—
173	#168	—	6	1	no	—
174	C₂²×C₈	—	15	2	no	—
175	C₂×C₄²	—	15	1	no	—
176	C₄⋊C₈	—	3	1	no	—
177	C₄⋊C₈	—	15	1	no	—
178	C₄⋊C₈	—	15	2	no	—
179	C₂²⋊C₈	—	15	1	no	—
180	C₂²⋊C₈	—	15	2	no	—
181	C₂²⋊C₈	—	15	1	no	—
182	C₄×C₈	—	5	1	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	—	5	1	no	—
185	C₄×C₈	—	15	1	no	—
186	C₄×C₈	—	15	1	no	—
187	C₄×C₈	—	15	1	no	—
188	C₄×C₈	—	15	2	no	—
189	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
190	C₂×C₃×C₅	#405	1	1	yes	characteristic
191	C₂×C₃×C₅	#405	1	1	yes	characteristic
192	D₁₅	—	2	2	no	—
193	C₂²×D₃	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×Dic₃	—	5	1	no	—
198	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₄×D₃	—	5	4	no	—
201	C₄×D₃	—	10	2	no	—
202	C₄×D₃	—	10	1	no	—
203	C₄×D₃	—	10	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	5	4	no	—
206	C₃×C₈	—	10	2	no	—
207	C₃×C₈	—	10	1	no	—
208	C₃⋊C₈	—	2	1	no	—
209	C₃⋊C₈	—	5	4	no	—
210	C₃⋊C₈	—	10	2	no	—
211	C₃⋊C₈	—	10	1	no	—
212	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
213	C₂×D₅	—	3	2	no	—
214	C₂×D₅	—	6	1	no	—
215	C₂×D₅	—	6	1	no	—
216	C₄×C₅	—	3	2	no	—
217	C₄×C₅	—	6	1	no	—
218	C₄×C₅	#901	1	1	yes	characteristic
219	C₄×C₅	#901	1	1	yes	characteristic
220	Dic₅	—	6	1	no	—
221	Dic₅	C₂×D₃×C₈	1	2	yes	—
222	Dic₅	#901	1	2	yes	—
223	C₂²×C₄	—	15	1	no	—
224	C₂²×C₄	—	15	1	no	—
225	C₂²×C₄	—	15	1	no	—
226	C₂×C₈	—	3	1	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	5	2	no	—
229	C₂×C₈	—	5	1	no	—
230	C₂×C₈	—	15	8	no	—
231	C₂×C₈	—	15	2	no	—
232	C₂×C₈	—	15	2	no	—
233	C₂×C₈	—	15	1	no	—
234	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₄²	—	3	1	no	—
236	C₄²	—	5	1	no	—
237	C₄²	—	15	1	no	—
238	C₄²	—	15	1	no	—
239	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
241	C₂×D₃	—	5	2	no	—
242	C₂×D₃	—	10	1	no	—
243	C₂×D₃	—	10	1	no	—
244	C₃×C₄	—	5	2	no	—
245	C₃×C₄	—	5	2	no	—
246	C₃×C₄	#4131	1	1	yes	characteristic
247	C₃×C₄	#4133	1	1	yes	characteristic
248	Dic₃	—	2	1	no	—
249	Dic₃	—	10	1	no	—
250	Dic₃	C₈×AGL(1,𝔽₅)	1	2	yes	—
251	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
252	C₂×C₅	#4689	1	1	yes	characteristic
253	C₂×C₅	#4689	1	1	yes	characteristic
254	D₅	—	6	2	no	—
255	C₂³	—	15	1	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	15	4	no	—
261	C₂×C₄	—	15	1	no	—
262	C₂×C₄	—	30	2	no	—
263	C₂×C₄	—	30	1	no	—
264	C₂×C₄	—	30	1	no	—
265	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	5	4	no	—
268	C₈	—	6	1	no	—
269	C₈	—	10	2	no	—
270	C₈	—	10	1	no	—
271	C₈	—	15	4	no	—
272	C₈	—	30	2	no	—
273	C₈	—	30	1	no	—
274	C₂×C₃	#65701	1	1	yes	characteristic
275	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
276	C₂×C₃	#65728	1	1	yes	characteristic
277	D₃	—	10	2	no	—
278	C₅	#68047	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	15	2	no	—
280	C₂²	—	30	1	no	—
281	C₂²	—	30	1	no	—
282	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
283	C₄	—	3	2	no	—
284	C₄	—	5	2	no	—
285	C₄	—	5	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	30	1	no	—
288	C₄	#91914	1	1	yes	characteristic
289	C₄	#91927	1	1	yes	characteristic
290	C₃	#10604010	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	30	2	no	—
292	C₂	D₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
293	C₂	#11751671	1	1	yes	characteristic
294	C₂	#11751872	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×D₁₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1450
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11746751	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. #11746487	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
31. C₈×#170	D₃	semidirect
39. #6738	C₈	semidirect
239. C₃×C₅	#1450	semidirect
278. C₅	#68047	semidirect
290. C₃	#10604010	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6738	2
#11751797	2
#11747583	2
#11746751	2
C₃×C₈×#170	2
#11746487	2
C₃⋊C₈×#170	2
#10604010	3
#68047	5