Subgroups — #14297910

Order 1920.

672 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	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#11751313	C₂	1	2	yes	maximal
3	C₂×#91102	C₂	1	2	yes	maximal, complemented
4	#11746372	C₂	1	2	yes	maximal, complemented
5	#10602764	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	—	2	4	no	—
8	C₂²×C₃×C₅×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#6719	—	2	2	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	4	yes	—
13	#91115	—	2	2	no	—
14	#91102	—	2	4	no	—
15	#71971	—	5	1	no	maximal, Hall
16	C₂³×C₅×C₈	D₃	1	1	yes	characteristic
17	#65501	—	3	2	no	—
18	C₂×#3950	—	3	2	no	—
19	#64771	—	3	2	no	—
20	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₅	—	2	4	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	8	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
33	#6719	—	4	4	no	—
34	C₄⋊Dic₁₅	—	2	4	no	—
35	C₄⋊Dic₁₅	—	2	4	no	—
36	C₂³×C₃×C₈	D₅	1	1	yes	characteristic
37	#5212	—	5	2	no	—
38	C₂×#818	—	5	2	no	—
39	#4551	—	5	2	no	—
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	2	4	no	—
42	C₂²×C₅×C₈	Dic₃	1	4	yes	complemented
43	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₂²⋊Dic₅	—	6	2	no	—
46	C₂×C₄⋊Dic₅	—	3	4	no	—
47	#3963	—	6	2	no	—
48	#3950	—	6	4	no	—
49	#1808	—	15	1	no	Sylow
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃×C₅	D₈	1	2	yes	—
54	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
55	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	C₂×Dic₁₅	—	4	4	no	—
66	C₂×Dic₁₅	—	4	4	no	—
67	C₂×Dic₁₅	—	4	4	no	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	complemented
70	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₃×C₈	—	2	4	no	—
72	C₂²×C₃×C₈	Dic₅	1	4	yes	complemented
73	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂×#217	—	10	2	no	—
76	C₂×C₄⋊Dic₃	—	5	4	no	—
77	C₃⋊#95	—	10	2	no	—
78	#818	—	10	4	no	—
79	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	Dic₆	1	2	yes	—
83	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	6	4	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	8	no	—
89	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
90	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
91	C₂×C₅×C₈	#67	1	4	yes	—
92	C₂²⋊Dic₅	—	12	4	no	—
93	C₄⋊Dic₅	—	6	4	no	—
94	C₄⋊Dic₅	—	6	4	no	—
95	C₂³×C₈	D₁₅	1	1	yes	characteristic
96	C₂×#107	—	15	2	no	—
97	#380	—	15	2	no	—
98	#340	—	15	2	no	—
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	4	no	—
102	C₂²×C₃×C₅	—	2	4	no	—
103	C₂²×C₃×C₅	#107	1	4	yes	complemented
104	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	#121	1	2	yes	—
106	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
108	C₂²×C₃×C₅	#135	1	2	yes	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	#117	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
113	Dic₁₅	—	8	4	no	—
114	C₂⁴×C₃	D₂₀	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
118	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂²×Dic₃	—	10	4	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	8	no	—
124	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
125	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
126	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
127	#217	—	20	4	no	—
128	C₄⋊Dic₃	—	10	4	no	—
129	C₄⋊Dic₃	—	10	4	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	D₂₄	1	2	yes	—
133	C₂³×C₅	Dic₁₂	1	2	yes	—
134	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
135	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
141	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₄×C₅	#235	1	2	yes	—
143	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#67	1	2	yes	—
145	C₂×Dic₅	—	12	4	no	—
146	C₂×Dic₅	—	12	4	no	—
147	C₂×Dic₅	—	12	4	no	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	#217	1	4	yes	complemented
150	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	Dic₁₅	1	4	yes	complemented
153	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
154	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	15	4	no	—
156	C₂×C₂²⋊C₄	—	30	2	no	—
157	#107	—	30	4	no	—
158	#95	—	30	2	no	—
159	C₂×C₃×C₅	—	2	4	no	—
160	C₂×C₃×C₅	#380	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
162	C₂×C₃×C₅	#427	1	1	yes	characteristic
163	C₂×C₃×C₅	#466	1	2	yes	—
164	C₂×C₃×C₅	#467	1	2	yes	—
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	—	2	2	no	—
167	C₂³×C₃	D₄₀	1	2	yes	—
168	C₂³×C₃	Dic₂₀	1	2	yes	—
169	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
170	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
176	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
177	C₂×C₃×C₄	#709	1	2	yes	—
178	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
180	C₂×Dic₃	—	20	4	no	—
181	C₂×Dic₃	—	20	4	no	—
182	C₂×Dic₃	—	20	4	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	C₂²⋊Dic₅	1	4	yes	complemented
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	#818	1	4	yes	complemented
190	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
191	C₂²×C₅	#904	1	2	yes	—
192	C₂²×C₅	C₂×Dic₁₂	1	1	yes	characteristic
193	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
194	C₂²×C₅	#930	1	2	yes	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	#924	1	2	yes	—
197	C₄×C₅	#926	1	1	yes	characteristic
198	C₄×C₅	C₂×#217	1	1	yes	characteristic
199	Dic₅	—	24	4	no	—
200	C₂⁴	D₆₀	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	30	4	no	—
204	C₂²×C₄	Dic₃₀	1	2	yes	—
205	C₂²×C₄	D₆₀	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
211	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
212	C₂×C₈	#1220	1	4	yes	—
213	C₄⋊C₄	—	30	4	no	—
214	C₄⋊C₄	—	30	4	no	—
215	C₂²⋊C₄	—	60	4	no	—
216	C₃×C₅	#1808	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	#3950	1	4	yes	complemented
222	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
223	C₂²×C₃	#4050	1	2	yes	—
224	C₂²×C₃	C₂×Dic₂₀	1	1	yes	characteristic
225	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
226	C₂²×C₃	#4076	1	2	yes	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	#4070	1	2	yes	—
229	C₃×C₄	#4072	1	1	yes	characteristic
230	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
231	Dic₃	—	40	4	no	—
232	C₂×C₅	—	2	4	no	—
233	C₂×C₅	C₂×#818	1	1	yes	characteristic
234	C₂×C₅	#5106	1	1	yes	characteristic
235	C₂×C₅	#5115	1	2	yes	—
236	C₂×C₅	#5116	1	2	yes	—
237	C₂×C₅	#5212	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	D₁₂₀	1	2	yes	—
241	C₂³	Dic₆₀	1	2	yes	—
242	C₂³	C₄⋊Dic₁₅	1	2	yes	—
243	C₂³	C₂×D₆₀	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	60	4	no	—
249	C₂×C₄	—	60	4	no	—
250	C₂×C₄	—	60	4	no	—
251	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
252	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
253	C₂×C₄	#6817	1	2	yes	—
254	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
255	C₂×C₄	C₂×#1220	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	#6719	1	4	yes	complemented
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	C₂×#3950	1	1	yes	characteristic
260	C₂×C₃	#65395	1	1	yes	characteristic
261	C₂×C₃	#65404	1	2	yes	—
262	C₂×C₃	#65405	1	2	yes	—
263	C₂×C₃	#65501	1	1	yes	characteristic
264	C₅	#71971	1	1	yes	characteristic, complemented, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	#91102	1	4	yes	complemented
270	C₂²	C₂×D₁₂₀	1	1	yes	characteristic
271	C₂²	#91793	1	2	yes	—
272	C₂²	C₂×Dic₆₀	1	1	yes	characteristic
273	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
274	C₂²	#91819	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	120	4	no	—
277	C₄	#91813	1	2	yes	—
278	C₄	#91815	1	1	yes	characteristic
279	C₄	C₂×#6719	1	1	yes	characteristic
280	C₃	#10602764	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	2	4	no	—
282	C₂	C₂×#91102	1	1	yes	characteristic
283	C₂	#11751207	1	1	yes	characteristic
284	C₂	#11751216	1	2	yes	—
285	C₂	#11751217	1	2	yes	—
286	C₂	#11751313	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂³×C₃×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₈	D₁₅
hypocenter	C₃×C₅	#1808
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#91102	C₂	semidirect
4. #11746372	C₂	semidirect
8. C₂²×C₃×C₅×C₈	C₄	semidirect
42. C₂²×C₅×C₈	Dic₃	semidirect
69. C₃×C₅×C₈	C₂²⋊C₄	semidirect
72. C₂²×C₃×C₈	Dic₅	semidirect
103. C₂²×C₃×C₅	#107	semidirect
149. C₅×C₈	#217	semidirect
152. C₂²×C₈	Dic₁₅	semidirect
184. C₃×C₈	C₂²⋊Dic₅	semidirect
189. C₂²×C₅	#818	semidirect
216. C₃×C₅	#1808	semidirect
221. C₂²×C₃	#3950	semidirect
257. C₈	#6719	semidirect
264. C₅	#71971	semidirect
269. C₂²	#91102	semidirect
280. C₃	#10602764	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₈	2
#11751313	2
C₂×#91102	2
#11746372	2
#10602764	3
#71971	5