Subgroups — #14315269

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5140	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65433	C₂	1	1	yes	maximal, characteristic
3	#11747524	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746382	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745930	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745885	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745873	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602246	—	3	1	no	maximal, Hall
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	C₃×#4096	C₄	1	1	yes	characteristic
11	C₃×#4047	C₂²	1	1	yes	characteristic
12	#91096	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
14	C₅×#812	—	2	1	no	—
15	C₅×#812	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#3969	C₄	1	1	yes	characteristic
19	C₃×#3957	C₄	1	1	yes	characteristic
20	C₃×#3954	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	#90934	—	2	1	no	—
23	D₅×C₃⋊C₁₆	—	2	1	no	—
24	#70924	—	5	1	no	maximal, Hall
25	C₅×#504	—	3	1	no	—
26	#65433	D₃	1	1	yes	characteristic
27	#65190	—	3	1	no	—
28	#64781	—	3	1	no	—
29	#64735	—	3	1	no	—
30	#64731	—	3	1	no	—
31	#64722	—	3	1	no	—
32	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₃×#709	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×#689	C₈	1	1	yes	characteristic
39	C₃×#688	C₈	1	1	yes	characteristic
40	C₃×#687	C₈	1	1	yes	characteristic
41	C₃×#686	C₈	1	1	yes	characteristic
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×#681	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
45	C₃×#676	—	2	1	no	—
46	C₃×C₈×D₅	—	2	1	no	—
47	C₃⋊#672	—	2	1	no	—
48	C₃⋊#672	—	2	1	no	—
49	C₅×C₃⋊C₁₆	—	2	1	no	—
50	C₅×C₃⋊C₁₆	D₄	1	1	yes	characteristic
51	C₅×C₃⋊C₁₆	D₄	1	1	yes	characteristic
52	C₃×#443	—	5	1	no	—
53	#5140	—	5	1	no	—
54	#5140	D₅	1	1	yes	characteristic, complemented
55	#4561	—	5	1	no	—
56	#4561	—	5	1	no	—
57	#4463	—	5	1	no	—
58	#4461	—	5	1	no	—
59	C₅×#131	C₂×D₃	1	1	yes	characteristic
60	#4096	Dic₃	1	1	yes	characteristic
61	#4047	C₂×D₃	1	1	yes	characteristic
62	C₅×M₃₂	—	3	1	no	—
63	C₅×M₃₂	—	6	1	no	—
64	C₂×C₅×C₁₆	—	3	1	no	—
65	C₂×C₅×C₁₆	—	6	1	no	—
66	#3969	Dic₃	1	1	yes	characteristic
67	#3957	Dic₃	1	1	yes	characteristic
68	#3954	Dic₃	1	1	yes	characteristic
69	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
70	#3944	—	3	1	no	—
71	C₂×#672	—	3	1	no	—
72	#3930	—	6	1	no	—
73	D₅×C₁₆	—	6	1	no	—
74	#2171	—	15	1	no	Sylow
75	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₂²⋊D₅	—	2	1	no	—
81	C₂×C₃×Dic₅	—	2	1	no	—
82	C₃×D₂₀	C₂×C₈	1	1	yes	characteristic
83	C₃×C₄×D₅	—	2	1	no	—
84	C₃×Dic₁₀	C₂×C₈	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	#41	1	1	yes	characteristic
88	C₃×#168	—	2	1	no	—
89	C₃×#168	C₂×C₈	1	1	yes	characteristic
90	C₃×#168	C₂×C₈	1	1	yes	characteristic
91	C₃×#135	—	5	1	no	—
92	C₃×#131	—	5	1	no	—
93	C₃×#131	C₂×D₅	1	1	yes	characteristic
94	C₃×#108	—	5	1	no	—
95	C₃×#104	—	5	1	no	—
96	C₃×#104	—	5	1	no	—
97	C₃×C₄×C₈	—	5	1	no	—
98	#812	—	2	1	no	—
99	#812	—	5	1	no	—
100	#812	—	10	1	no	—
101	#812	C₂×D₅	1	1	yes	characteristic
102	C₂×C₃⋊C₁₆	—	2	1	no	—
103	C₂×C₃⋊C₁₆	—	5	1	no	—
104	C₂×C₃⋊C₁₆	—	10	1	no	—
105	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
106	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
107	#709	C₂×Dic₃	1	1	yes	characteristic
108	C₅×M₁₆	—	2	1	no	—
109	C₅×M₁₆	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
112	#689	C₃⋊C₈	1	1	yes	characteristic
113	#688	C₃⋊C₈	1	1	yes	characteristic
114	#687	C₃⋊C₈	1	1	yes	characteristic
115	#686	C₃⋊C₈	1	1	yes	characteristic
116	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
117	#681	C₂×Dic₃	1	1	yes	characteristic
118	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
119	#676	—	2	1	no	—
120	C₈×D₅	—	2	1	no	—
121	C₅×C₁₆	—	3	1	no	—
122	C₅×C₁₆	—	3	1	no	—
123	C₅×C₁₆	—	6	1	no	—
124	#672	—	6	1	no	—
125	#672	—	6	1	no	—
126	#504	—	3	1	no	—
127	#504	—	15	1	no	—
128	#443	—	5	1	no	—
129	#364	—	15	1	no	—
130	#350	—	15	1	no	—
131	#350	—	15	1	no	—
132	#346	—	15	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#131	1	1	yes	characteristic
135	C₂×C₃×D₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₃×Dic₅	—	2	1	no	—
140	C₃×Dic₅	—	4	1	no	—
141	C₃×#41	—	5	1	no	—
142	C₃×#41	C₄×D₅	1	1	yes	characteristic
143	C₃×Q₁₆	—	5	1	no	—
144	C₃×SD₁₆	—	5	1	no	—
145	C₃×SD₁₆	—	5	1	no	—
146	C₃×D₈	—	5	1	no	—
147	C₃×M₁₆	—	2	1	no	—
148	C₃×M₁₆	—	5	1	no	—
149	C₃×M₁₆	—	10	1	no	—
150	C₃×M₁₆	C₄×D₅	1	1	yes	characteristic
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	10	1	no	—
154	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
155	C₃×C₄²	—	5	1	no	—
156	C₃⋊C₁₆	—	2	1	no	—
157	C₃⋊C₁₆	—	10	1	no	—
158	C₃⋊C₁₆	—	10	1	no	—
159	C₃⋊C₁₆	C₂²⋊D₅	1	1	yes	characteristic
160	C₃⋊C₁₆	C₂²⋊D₅	1	1	yes	characteristic
161	C₅×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
162	C₅×D₄	—	2	1	no	—
163	C₅×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
166	C₂²⋊D₅	—	2	1	no	—
167	C₂×Dic₅	—	2	1	no	—
168	D₂₀	C₂×C₃⋊C₈	1	1	yes	characteristic
169	C₄×D₅	—	2	1	no	—
170	Dic₁₀	C₂×C₃⋊C₈	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	D₃×D₄	1	1	yes	characteristic
173	C₅×C₈	#237	1	1	yes	characteristic
174	#168	—	2	1	no	—
175	#168	C₂×C₃⋊C₈	1	1	yes	characteristic
176	#168	C₂×C₃⋊C₈	1	1	yes	characteristic
177	#135	—	5	1	no	—
178	#131	—	5	1	no	—
179	#131	D₃×D₅	1	1	yes	characteristic
180	M₃₂	—	3	1	no	—
181	M₃₂	—	6	1	no	—
182	M₃₂	—	15	1	no	—
183	M₃₂	—	30	1	no	—
184	C₂×C₁₆	—	3	1	no	—
185	C₂×C₁₆	—	6	1	no	—
186	C₂×C₁₆	—	15	1	no	—
187	C₂×C₁₆	—	30	1	no	—
188	#108	—	5	1	no	—
189	#104	—	5	1	no	—
190	#104	—	5	1	no	—
191	C₄×C₈	—	5	1	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
195	C₃×D₅	—	4	1	no	—
196	C₃×Q₈	—	5	1	no	—
197	C₃×Q₈	C₈×D₅	1	1	yes	characteristic
198	C₃×D₄	—	2	1	no	—
199	C₃×D₄	—	5	1	no	—
200	C₃×D₄	—	10	1	no	—
201	C₃×D₄	C₈×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	10	1	no	—
204	C₂×C₃×C₄	—	10	1	no	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	5	1	no	—
208	C₃×C₈	—	5	1	no	—
209	C₃×C₈	—	10	1	no	—
210	C₃×C₈	#709	1	1	yes	characteristic
211	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	#948	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
217	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
218	Dic₅	—	2	1	no	—
219	Dic₅	—	4	1	no	—
220	#41	—	5	1	no	—
221	#41	D₅×Dic₃	1	1	yes	characteristic
222	Q₁₆	—	5	1	no	—
223	SD₁₆	—	5	1	no	—
224	SD₁₆	—	5	1	no	—
225	D₈	—	5	1	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	5	1	no	—
228	M₁₆	—	10	1	no	—
229	M₁₆	D₅×Dic₃	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	5	1	no	—
232	C₂×C₈	—	10	1	no	—
233	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
234	C₄²	—	5	1	no	—
235	C₁₆	—	3	1	no	—
236	C₁₆	—	3	1	no	—
237	C₁₆	—	6	1	no	—
238	C₁₆	—	30	1	no	—
239	C₁₆	—	30	1	no	—
240	C₃×C₅	#2171	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	10	1	no	—
243	C₂²×C₃	#4047	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	10	1	no	—
246	C₃×C₄	—	20	1	no	—
247	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
248	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
252	D₅	—	4	1	no	—
253	Q₈	—	5	1	no	—
254	Q₈	D₅×C₃⋊C₈	1	1	yes	characteristic
255	D₄	—	2	1	no	—
256	D₄	—	5	1	no	—
257	D₄	—	10	1	no	—
258	D₄	D₅×C₃⋊C₈	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	10	1	no	—
262	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	5	1	no	—
265	C₈	—	5	1	no	—
266	C₈	—	10	1	no	—
267	C₈	#6779	1	1	yes	characteristic
268	C₈	D₃×C₂²⋊D₅	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	—	20	1	no	—
272	C₂×C₃	C₈×C₂²⋊D₅	1	1	yes	characteristic
273	C₅	#70924	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	10	1	no	—
276	C₂²	#91283	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	10	1	no	—
279	C₄	—	20	1	no	—
280	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
281	C₄	Dic₃×C₂²⋊D₅	1	1	yes	characteristic
282	C₃	#10602246	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	4	1	no	—
285	C₂	—	20	1	no	—
286	C₂	C₃⋊C₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₃×C₂²⋊D₅
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₃×C₅×#131	C₂²
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₃×#4096	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#131	D₃×D₅
hypocenter	C₃×C₅	#2171
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. C₅×#5140	C₂	semidirect
3. #11747524	C₂	semidirect
4. #11746382	C₂	semidirect
5. #11745930	C₂	semidirect
6. #11745885	C₂	semidirect
7. #11745873	C₂	semidirect
54. #5140	D₅	semidirect
240. C₃×C₅	#2171	semidirect
273. C₅	#70924	semidirect
282. C₃	#10602246	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5140	2
C₃×#65433	2
#11747524	2
#11746382	2
#11745930	2
#11745885	2
#11745873	2
#10602246	3
#70924	5