Subgroups — #14296657

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#91102	C₂	1	1	yes	maximal, characteristic
3	C₂×#91101	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Dic₁₅	C₂	1	2	yes	maximal
5	#11746372	C₂	1	2	yes	maximal
6	#10600607	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₅	—	2	2	no	—
13	C₃×C₅×#97	C₄	1	4	yes	complemented
14	#91115	—	2	2	no	—
15	#91102	C₄	1	4	yes	complemented
16	#91101	C₄	1	4	yes	complemented
17	#69814	—	5	1	no	maximal, Hall
18	C₂×C₅×#97	D₃	1	1	yes	characteristic
19	C₂×#3950	—	3	1	no	—
20	C₂×#3949	—	3	1	no	—
21	C₄×C₄⋊Dic₅	—	3	2	no	—
22	#64771	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₅	—	2	2	no	—
26	C₂²×Dic₁₅	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	Q₈	1	2	yes	—
35	C₄⋊Dic₁₅	—	2	2	no	—
36	C₄⋊Dic₁₅	—	2	2	no	—
37	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
38	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₅	—	4	4	no	—
40	C₂×C₃×#97	D₅	1	1	yes	characteristic
41	C₂×#818	—	5	1	no	—
42	C₂×#817	—	5	1	no	—
43	C₄×C₄⋊Dic₃	—	5	2	no	—
44	#4551	—	5	2	no	—
45	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅	—	3	2	no	—
49	C₂×C₄⋊Dic₅	—	3	2	no	—
50	C₂×C₄×Dic₅	—	6	2	no	—
51	C₅×#97	Dic₃	1	4	yes	complemented
52	#3963	—	6	2	no	—
53	#3950	—	3	4	no	—
54	#3949	—	3	4	no	—
55	#1776	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×Dic₁₅	—	2	4	no	—
67	C₂×Dic₁₅	—	2	4	no	—
68	C₂×Dic₁₅	—	4	2	no	—
69	C₂×Dic₁₅	—	4	2	no	—
70	C₂×Dic₁₅	—	4	2	no	—
71	C₂×Dic₁₅	—	4	2	no	—
72	C₃×C₅×C₈	—	2	2	no	—
73	C₃×C₅×C₈	C₄²	1	4	yes	complemented
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
77	C₂×C₄⋊Dic₃	—	5	2	no	—
78	C₂×C₄⋊Dic₃	—	5	2	no	—
79	C₂×C₄×Dic₃	—	10	2	no	—
80	C₃×#97	Dic₅	1	4	yes	complemented
81	C₃⋊#95	—	10	2	no	—
82	#818	—	5	4	no	—
83	#817	—	5	4	no	—
84	C₂²×C₄×C₅	D₁₂	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	6	2	no	—
87	C₂²×Dic₅	—	6	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
91	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
92	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
93	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
94	C₄²×C₅	Dic₆	1	2	yes	—
95	C₄²×C₅	C₂×Dic₃	1	2	yes	—
96	C₄⋊Dic₅	—	3	4	no	—
97	C₄⋊Dic₅	—	3	4	no	—
98	C₄⋊Dic₅	—	6	2	no	—
99	C₄⋊Dic₅	—	6	2	no	—
100	C₄×Dic₅	—	12	4	no	—
101	C₂×#107	—	15	1	no	—
102	C₂×#106	—	15	1	no	—
103	C₂×#97	D₁₅	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	15	2	no	—
105	#340	—	15	2	no	—
106	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₃×C₅	#137	1	2	yes	—
111	C₃×C₄×C₅	—	2	4	no	—
112	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₃×C₄×C₅	#117	1	1	yes	characteristic
114	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
116	Dic₁₅	—	4	4	no	—
117	Dic₁₅	—	4	4	no	—
118	C₂²×C₃×C₄	D₂₀	1	2	yes	—
119	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	10	2	no	—
121	C₂²×Dic₃	—	10	2	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
125	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
127	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
128	C₃×C₄²	Dic₁₀	1	2	yes	—
129	C₃×C₄²	C₂×Dic₅	1	2	yes	—
130	C₄⋊Dic₃	—	5	4	no	—
131	C₄⋊Dic₃	—	5	4	no	—
132	C₄⋊Dic₃	—	10	2	no	—
133	C₄⋊Dic₃	—	10	2	no	—
134	C₄×Dic₃	—	20	4	no	—
135	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
139	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	#235	1	1	yes	characteristic
143	C₂×C₄×C₅	#235	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
145	C₂×Dic₅	—	6	4	no	—
146	C₂×Dic₅	—	6	4	no	—
147	C₂×Dic₅	—	12	2	no	—
148	C₂×Dic₅	—	12	2	no	—
149	C₂×Dic₅	—	12	2	no	—
150	C₂×Dic₅	—	12	2	no	—
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	C₄×Dic₃	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	2	no	—
156	C₂×C₄⋊C₄	—	15	2	no	—
157	C₂×C₄²	—	30	2	no	—
158	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
159	#107	—	15	4	no	—
160	#106	—	15	4	no	—
161	#97	Dic₁₅	1	4	yes	complemented
162	#95	—	30	2	no	—
163	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₅	#428	1	2	yes	—
165	C₂×C₃×C₅	#440	1	2	yes	—
166	C₂×C₃×C₅	#441	1	1	yes	characteristic
167	C₂×C₃×C₅	#442	1	1	yes	characteristic
168	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
172	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	#709	1	1	yes	characteristic
176	C₂×C₃×C₄	#709	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
178	C₂×Dic₃	—	10	4	no	—
179	C₂×Dic₃	—	10	4	no	—
180	C₂×Dic₃	—	20	2	no	—
181	C₂×Dic₃	—	20	2	no	—
182	C₂×Dic₃	—	20	2	no	—
183	C₂×Dic₃	—	20	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	C₄×Dic₅	1	4	yes	complemented
186	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
187	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
188	C₂²×C₅	#908	1	2	yes	—
189	C₂²×C₅	#909	1	2	yes	—
190	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₄×C₅	—	2	4	no	—
192	C₄×C₅	C₄×Dic₆	1	1	yes	characteristic
193	C₄×C₅	C₄×Dic₆	1	1	yes	characteristic
194	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
195	C₄×C₅	#926	1	1	yes	characteristic
196	Dic₅	—	12	4	no	—
197	Dic₅	—	12	4	no	—
198	C₂²×C₄	—	30	2	no	—
199	C₂²×C₄	—	30	2	no	—
200	C₂²×C₄	D₆₀	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	C₄×D₁₅	1	2	yes	—
205	C₂×C₈	C₄×D₁₅	1	2	yes	—
206	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
207	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
208	C₄⋊C₄	—	15	4	no	—
209	C₄⋊C₄	—	15	4	no	—
210	C₄⋊C₄	—	30	2	no	—
211	C₄⋊C₄	—	30	2	no	—
212	C₄²	—	60	4	no	—
213	C₄²	Dic₃₀	1	2	yes	—
214	C₄²	C₂×Dic₁₅	1	2	yes	—
215	C₃×C₅	#1776	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
217	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
218	C₂²×C₃	#4054	1	2	yes	—
219	C₂²×C₃	#4055	1	2	yes	—
220	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
223	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
224	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
225	C₃×C₄	#4072	1	1	yes	characteristic
226	Dic₃	—	20	4	no	—
227	Dic₃	—	20	4	no	—
228	C₂×C₅	#4710	1	2	yes	—
229	C₂×C₅	#4711	1	1	yes	characteristic
230	C₂×C₅	#4716	1	1	yes	characteristic
231	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
232	C₂×C₅	#5121	1	2	yes	—
233	C₂³	C₂×D₆₀	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	30	4	no	—
237	C₂×C₄	—	30	4	no	—
238	C₂×C₄	—	60	2	no	—
239	C₂×C₄	—	60	2	no	—
240	C₂×C₄	—	60	2	no	—
241	C₂×C₄	—	60	2	no	—
242	C₂×C₄	C₄⋊Dic₁₅	1	4	yes	—
243	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
246	C₂×C₄	#6817	1	1	yes	characteristic
247	C₂×C₄	#6817	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄×Dic₁₅	1	4	yes	complemented
251	C₂×C₃	#64999	1	2	yes	—
252	C₂×C₃	#65000	1	1	yes	characteristic
253	C₂×C₃	#65005	1	1	yes	characteristic
254	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
255	C₂×C₃	#65410	1	2	yes	—
256	C₅	#69814	1	1	yes	characteristic, complemented, Sylow
257	C₂²	C₄×D₆₀	1	1	yes	characteristic
258	C₂²	C₄×D₆₀	1	1	yes	characteristic
259	C₂²	#91797	1	2	yes	—
260	C₂²	#91798	1	2	yes	—
261	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	60	4	no	—
264	C₄	—	60	4	no	—
265	C₄	C₄×Dic₃₀	1	1	yes	characteristic
266	C₄	C₄×Dic₃₀	1	1	yes	characteristic
267	C₄	C₂×C₄×Dic₁₅	1	1	yes	characteristic
268	C₄	#91815	1	1	yes	characteristic
269	C₃	#10600607	1	1	yes	characteristic, complemented, Sylow
270	C₂	#11750811	1	2	yes	—
271	C₂	#11750812	1	1	yes	characteristic
272	C₂	#11750817	1	1	yes	characteristic
273	C₂	C₄×C₄⋊Dic₁₅	1	1	yes	characteristic
274	C₂	#11751222	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#97	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#97	D₁₅
hypocenter	C₃×C₅	#1776
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
13. C₃×C₅×#97	C₄	semidirect
15. #91102	C₄	semidirect
16. #91101	C₄	semidirect
51. C₅×#97	Dic₃	semidirect
73. C₃×C₅×C₈	C₄²	semidirect
80. C₃×#97	Dic₅	semidirect
152. C₅×C₈	C₄×Dic₃	semidirect
161. #97	Dic₁₅	semidirect
185. C₃×C₈	C₄×Dic₅	semidirect
215. C₃×C₅	#1776	semidirect
250. C₈	C₄×Dic₁₅	semidirect
256. C₅	#69814	semidirect
269. C₃	#10600607	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#97	2
C₂×#91102	2
C₂×#91101	2
C₄×C₄⋊Dic₁₅	2
#11746372	2
#10600607	3
#69814	5