Subgroups — #14280696

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752407	C₂	1	1	yes	maximal, characteristic
2	C₃×#65754	C₂	1	1	yes	maximal, characteristic
3	#11748316	C₂	1	1	yes	maximal, characteristic
4	#11746487	C₂	1	1	yes	maximal, characteristic, complemented
5	#170⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603896	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	#91241	C₄	1	2	yes	—
11	C₄⋊(C₃⋊#170)	C₄	1	2	yes	—
12	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₃×C₂²⋊#170	C₄	1	2	yes	—
15	C₃×#4004	C₄	1	2	yes	—
16	C₃×C₄×#170	C₂²	1	1	yes	characteristic
17	C₃×C₄×#170	C₂²	1	1	yes	characteristic
18	#91114	C₄	1	1	yes	characteristic
19	C₅×#830	C₄	1	1	yes	characteristic
20	#90982	C₄	1	1	yes	characteristic
21	#90982	C₄	1	1	yes	characteristic
22	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#70744	—	5	1	no	maximal, Hall
25	#65754	D₃	1	1	yes	characteristic
26	#65748	—	3	1	no	—
27	#65007	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	—	3	1	no	—
30	#64879	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	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	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
47	C₃×#431	—	5	1	no	—
48	#4925	—	5	1	no	—
49	#4925	—	5	1	no	—
50	#4455	—	5	1	no	—
51	#4454	—	5	1	no	—
52	#4454	—	5	1	no	—
53	C₈×C₃⋊C₈	—	5	1	no	—
54	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
55	C₂²⋊#170	—	3	2	no	—
56	C₂²⋊#170	Dic₃	1	2	yes	—
57	#4004	Dic₃	1	2	yes	—
58	C₄⋊#170	—	3	2	no	—
59	C₄×#170	—	3	1	no	—
60	C₄×#170	—	3	1	no	—
61	C₄×#170	C₂×D₃	1	1	yes	characteristic
62	C₄×#170	C₂×D₃	1	1	yes	characteristic
63	C₅×C₂²⋊C₈	—	3	1	no	—
64	C₂²⋊#168	—	3	1	no	—
65	#3946	—	3	1	no	—
66	#3946	—	3	1	no	—
67	C₈×Dic₅	—	3	1	no	—
68	C₈×Dic₅	—	3	1	no	—
69	#1454	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×Dic₅	—	2	2	no	—
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
81	C₃⋊#170	—	2	2	no	—
82	C₃⋊#170	—	2	2	no	—
83	C₃×#170	—	2	2	no	—
84	C₃×#170	—	2	2	no	—
85	C₃⋊#168	—	2	1	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₅×C₃⋊C₈	—	2	1	no	—
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₂×C₃×C₄²	—	5	1	no	—
90	C₃×C₄⋊C₈	—	5	2	no	—
91	C₃×C₂²⋊C₈	—	5	2	no	—
92	C₃×C₄×C₈	—	5	1	no	—
93	C₃×C₄×C₈	—	5	1	no	—
94	#830	—	5	2	no	—
95	#830	—	5	1	no	—
96	#830	AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₄⋊(C₃⋊C₈)	—	5	2	no	—
98	C₄⋊(C₃⋊C₈)	—	5	1	no	—
99	C₄⋊(C₃⋊C₈)	—	5	1	no	—
100	C₄×C₃⋊C₈	—	5	1	no	—
101	C₄×C₃⋊C₈	—	5	1	no	—
102	C₄×C₃⋊C₈	—	5	1	no	—
103	C₄×C₃⋊C₈	—	5	1	no	—
104	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
105	C₂²×Dic₅	C₄×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
107	C₂×#170	—	3	2	no	—
108	C₂×#170	—	3	2	no	—
109	C₂×#170	C₂×Dic₃	1	2	yes	—
110	C₂×#170	C₂×Dic₃	1	2	yes	—
111	C₂×C₅×C₈	—	3	1	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
114	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
115	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
116	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
117	C₂×#168	—	3	1	no	—
118	C₂×#168	—	3	1	no	—
119	#431	—	5	1	no	—
120	#431	—	15	1	no	—
121	#431	—	15	1	no	—
122	#322	—	15	1	no	—
123	#322	—	15	1	no	—
124	#322	—	15	1	no	—
125	C₈²	—	15	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	#131	1	1	yes	characteristic
132	C₃×C₄×C₅	#131	1	1	yes	characteristic
133	C₃×Dic₅	—	2	1	no	—
134	C₃×Dic₅	—	2	1	no	—
135	C₃×Dic₅	#131	1	2	yes	—
136	C₃×Dic₅	#131	1	2	yes	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×C₃×C₈	—	5	2	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₃×C₄²	—	5	1	no	—
143	C₃×C₄²	—	5	1	no	—
144	C₃×C₄²	—	5	1	no	—
145	C₃×C₄²	—	5	1	no	—
146	C₂×C₃⋊C₈	—	5	2	no	—
147	C₂×C₃⋊C₈	—	5	2	no	—
148	C₂×C₃⋊C₈	—	5	1	no	—
149	C₂×C₃⋊C₈	—	5	1	no	—
150	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×Dic₅	—	2	2	no	—
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
160	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
161	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
163	#170	—	2	2	no	—
164	#170	—	2	2	no	—
165	#170	—	6	2	no	—
166	#170	—	6	2	no	—
167	C₅×C₈	—	6	1	no	—
168	C₅×C₈	—	6	1	no	—
169	#168	—	6	1	no	—
170	#168	—	6	1	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₄⋊C₈	—	5	2	no	—
173	C₄⋊C₈	—	15	1	no	—
174	C₄⋊C₈	—	15	1	no	—
175	C₄⋊C₈	—	15	2	no	—
176	C₂²⋊C₈	—	3	1	no	—
177	C₂²⋊C₈	—	5	2	no	—
178	C₂²⋊C₈	—	15	2	no	—
179	C₂²⋊C₈	—	15	1	no	—
180	C₄×C₈	—	5	1	no	—
181	C₄×C₈	—	5	1	no	—
182	C₄×C₈	—	15	1	no	—
183	C₄×C₈	—	15	1	no	—
184	C₄×C₈	—	15	1	no	—
185	C₄×C₈	—	15	1	no	—
186	C₂×C₃×C₅	—	4	1	no	—
187	C₂×C₃×C₅	#405	1	1	yes	characteristic
188	C₂×C₃×C₅	#405	1	1	yes	characteristic
189	C₂×C₃×C₅	#405	1	1	yes	characteristic
190	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	10	2	no	—
198	C₂×C₃×C₄	—	10	2	no	—
199	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₃×C₈	—	10	2	no	—
202	C₃×C₈	—	10	2	no	—
203	C₃⋊C₈	—	2	1	no	—
204	C₃⋊C₈	—	2	1	no	—
205	C₃⋊C₈	—	10	2	no	—
206	C₃⋊C₈	—	10	2	no	—
207	C₃⋊C₈	—	10	1	no	—
208	C₃⋊C₈	—	10	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	#907	1	1	yes	characteristic
215	C₄×C₅	#907	1	1	yes	characteristic
216	Dic₅	—	2	1	no	—
217	Dic₅	—	2	1	no	—
218	Dic₅	#901	1	2	yes	—
219	Dic₅	#948	1	2	yes	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	1	no	—
222	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	15	2	no	—
228	C₂×C₈	—	15	2	no	—
229	C₂×C₈	—	15	1	no	—
230	C₂×C₈	—	15	1	no	—
231	C₄²	—	5	1	no	—
232	C₄²	—	5	1	no	—
233	C₄²	—	5	1	no	—
234	C₄²	—	5	1	no	—
235	C₃×C₅	#1454	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	5	2	no	—
242	C₃×C₄	—	5	2	no	—
243	C₃×C₄	—	10	1	no	—
244	C₃×C₄	—	10	1	no	—
245	C₃×C₄	#4131	1	1	yes	characteristic
246	C₃×C₄	#4131	1	1	yes	characteristic
247	C₂×C₅	—	4	1	no	—
248	C₂×C₅	#4997	1	1	yes	characteristic
249	C₂×C₅	#5101	1	1	yes	characteristic
250	C₂×C₅	#5122	1	1	yes	characteristic
251	C₂³	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
261	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₈	—	6	1	no	—
263	C₈	—	6	1	no	—
264	C₈	—	10	2	no	—
265	C₈	—	10	2	no	—
266	C₈	—	30	2	no	—
267	C₈	—	30	2	no	—
268	C₈	—	30	1	no	—
269	C₈	—	30	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	#65691	1	1	yes	characteristic
272	C₂×C₃	#65718	1	1	yes	characteristic
273	C₂×C₃	#65728	1	1	yes	characteristic
274	C₅	#70744	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	5	2	no	—
281	C₄	—	5	2	no	—
282	C₄	—	10	1	no	—
283	C₄	—	10	1	no	—
284	C₄	#91929	1	1	yes	characteristic
285	C₄	#91929	1	1	yes	characteristic
286	C₃	#10603896	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	4	1	no	—
288	C₂	#11751694	1	1	yes	characteristic
289	C₂	#11751695	1	1	yes	characteristic
290	C₂	#11751995	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1454
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11746487	C₂	semidirect
5. #170⋊(C₃⋊C₈)	C₂	semidirect
6. #11746485	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
235. C₃×C₅	#1454	semidirect
274. C₅	#70744	semidirect
286. C₃	#10603896	semidirect

Maximal subgroups

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

subgroup	index
#11752407	2
C₃×#65754	2
#11748316	2
#11746487	2
#170⋊(C₃⋊C₈)	2
#11746485	2
C₃⋊C₈×#170	2
#10603896	3
#70744	5