Subgroups — #14417548

Order 1920.

912 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	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic
2	#11756275	C₂	1	2	yes	maximal
3	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
4	#11751221	C₂	1	2	yes	maximal
5	#11751207	C₂	1	1	yes	maximal, characteristic
6	#11750817	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₆₀	C₂	1	4	yes	maximal, complemented
8	#10610304	—	3	1	no	maximal, Hall
9	C₂²×Dic₃₀	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	#91815	C₂²	1	2	yes	—
14	C₂×Dic₆₀	C₄	1	8	yes	complemented
15	C₂×Dic₆₀	C₂²	1	2	yes	—
16	C₂×Dic₆₀	C₂²	1	2	yes	—
17	#91776	—	2	4	no	—
18	#91766	—	2	4	no	—
19	C₄×Dic₃₀	C₂²	1	8	yes	—
20	C₃×C₅×#97	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
22	#91102	C₂²	1	2	yes	—
23	#91101	C₂²	1	2	yes	—
24	#91100	C₂²	1	8	yes	—
25	#78573	—	5	1	no	maximal, Hall
26	C₂²×Dic₂₀	—	3	1	no	—
27	#65850	—	3	2	no	—
28	C₅×#405	D₃	1	1	yes	characteristic
29	#65409	—	3	2	no	—
30	#65395	—	3	1	no	—
31	#65005	—	3	4	no	—
32	C₄×Dic₂₀	—	3	4	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₅	—	2	2	no	—
35	C₂×Dic₃₀	—	2	2	no	—
36	C₂×Dic₃₀	—	2	4	no	—
37	C₂×Dic₃₀	C₂×C₄	1	8	yes	—
38	C₂×Dic₃₀	C₂³	1	4	yes	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
45	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	#6719	—	2	2	no	—
48	C₄⋊Dic₁₅	C₂³	1	4	yes	—
49	#6712	—	2	8	no	—
50	C₄×Dic₁₅	—	2	8	no	—
51	C₄×Dic₁₅	—	2	2	no	—
52	Dic₆₀	C₂×C₄	1	8	yes	—
53	Dic₆₀	C₂×C₄	1	8	yes	—
54	C₂²×Dic₁₂	—	5	1	no	—
55	#5516	—	5	2	no	—
56	C₃×#405	D₅	1	1	yes	characteristic
57	#5120	—	5	2	no	—
58	#5106	—	5	1	no	—
59	#4716	—	5	4	no	—
60	C₄×Dic₁₂	—	5	4	no	—
61	C₂²×Dic₁₀	—	3	2	no	—
62	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	C₅×#117	C₂×D₃	1	1	yes	characteristic
65	#4072	—	3	2	no	—
66	C₂×Dic₂₀	—	3	8	no	—
67	C₂×Dic₂₀	—	3	2	no	—
68	C₂×Dic₂₀	—	3	2	no	—
69	#4033	—	6	4	no	—
70	#4023	—	6	4	no	—
71	C₄×Dic₁₀	—	3	8	no	—
72	C₅×#97	C₂×D₃	1	2	yes	—
73	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
74	#3950	—	3	2	no	—
75	#3949	—	3	2	no	—
76	#3948	—	3	8	no	—
77	#2950	—	15	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₁₅	—	2	8	no	—
85	C₂×Dic₁₅	—	2	4	no	—
86	C₂×Dic₁₅	—	2	4	no	—
87	Dic₃₀	—	2	8	no	—
88	Dic₃₀	—	2	4	no	—
89	Dic₃₀	C₂²×C₄	1	8	yes	—
90	C₃×C₅×C₈	—	2	2	no	—
91	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₆	—	5	2	no	—
94	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
96	C₃×#117	C₂×D₅	1	1	yes	characteristic
97	#926	—	5	2	no	—
98	C₂×Dic₁₂	—	5	8	no	—
99	C₂×Dic₁₂	—	5	2	no	—
100	C₂×Dic₁₂	—	5	2	no	—
101	#887	—	10	4	no	—
102	#877	—	10	4	no	—
103	C₄×Dic₆	—	5	8	no	—
104	C₃×#97	C₂×D₅	1	2	yes	—
105	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
106	#818	—	5	2	no	—
107	#817	—	5	2	no	—
108	#816	—	5	8	no	—
109	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₂²×Dic₅	—	6	2	no	—
111	C₂×Dic₁₀	—	3	8	no	—
112	C₂×Dic₁₀	—	3	4	no	—
113	C₂×Dic₁₀	—	6	2	no	—
114	C₂×Dic₁₀	—	6	4	no	—
115	C₅×M₁₆	—	2	2	no	—
116	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
117	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
118	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₅×C₄⋊C₄	D₁₂	1	2	yes	—
121	C₅×C₂²⋊C₄	D₁₂	1	2	yes	—
122	C₄²×C₅	C₂²×D₃	1	2	yes	—
123	C₂²⋊Dic₅	—	6	2	no	—
124	C₄⋊Dic₅	—	3	4	no	—
125	#683	—	6	8	no	—
126	C₄×Dic₅	—	6	8	no	—
127	C₄×Dic₅	—	6	2	no	—
128	Dic₂₀	—	3	8	no	—
129	Dic₂₀	—	3	8	no	—
130	C₂²×Q₁₆	—	15	1	no	—
131	#519	—	15	2	no	—
132	#441	—	15	4	no	—
133	C₄×Q₁₆	—	15	4	no	—
134	#427	—	15	1	no	—
135	#419	—	15	2	no	—
136	#405	D₁₅	1	1	yes	characteristic
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
139	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	4	no	—
141	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	Dic₁₅	—	2	8	no	—
145	Dic₁₅	—	4	4	no	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	10	2	no	—
148	C₂×Dic₆	—	5	8	no	—
149	C₂×Dic₆	—	5	4	no	—
150	C₂×Dic₆	—	10	2	no	—
151	C₂×Dic₆	—	10	4	no	—
152	C₃×M₁₆	—	2	2	no	—
153	C₂×C₃×C₈	C₄×D₅	1	4	yes	—
154	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
155	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₃×C₄⋊C₄	D₂₀	1	2	yes	—
158	C₃×C₂²⋊C₄	D₂₀	1	2	yes	—
159	C₃×C₄²	C₂²×D₅	1	2	yes	—
160	#217	—	10	2	no	—
161	C₄⋊Dic₃	—	5	4	no	—
162	Dic₃⋊C₄	—	10	8	no	—
163	C₄×Dic₃	—	10	8	no	—
164	C₄×Dic₃	—	10	2	no	—
165	Dic₁₂	—	5	8	no	—
166	Dic₁₂	—	5	8	no	—
167	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
169	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
170	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
171	C₂×C₄×C₅	#235	1	2	yes	—
172	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₂×Dic₅	—	6	8	no	—
174	C₂×Dic₅	—	6	4	no	—
175	C₂×Dic₅	—	6	4	no	—
176	Dic₁₀	—	3	8	no	—
177	Dic₁₀	—	6	8	no	—
178	Dic₁₀	—	6	4	no	—
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
181	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
182	C₂²×Q₈	—	15	2	no	—
183	C₂×Q₁₆	—	15	8	no	—
184	C₂×Q₁₆	—	15	2	no	—
185	C₂×Q₁₆	—	15	2	no	—
186	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
187	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
188	C₄×Q₈	—	15	8	no	—
189	C₄×Q₈	—	30	4	no	—
190	#117	—	15	2	no	—
191	#117	—	30	4	no	—
192	#117	C₂×D₁₅	1	1	yes	characteristic
193	#107	—	15	2	no	—
194	#106	—	15	2	no	—
195	#103	—	15	8	no	—
196	#97	C₂×D₁₅	1	2	yes	—
197	C₄×C₈	C₂×D₁₅	1	2	yes	—
198	C₂×C₃×C₅	—	2	2	no	—
199	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×C₃×C₅	#578	1	2	yes	—
201	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
203	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
204	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₃×C₄	#709	1	2	yes	—
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	—	10	4	no	—
208	C₂×Dic₃	—	10	8	no	—
209	C₂×Dic₃	—	10	4	no	—
210	Dic₆	—	5	8	no	—
211	Dic₆	—	10	4	no	—
212	Dic₆	—	10	8	no	—
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
215	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	C₄×D₁₂	1	2	yes	—
218	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	C₄×D₁₂	1	2	yes	—
221	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
222	C₄×C₅	C₂×#235	1	1	yes	characteristic
223	Dic₅	—	6	8	no	—
224	Dic₅	—	12	4	no	—
225	C₂×Q₈	—	15	8	no	—
226	C₂×Q₈	—	15	4	no	—
227	C₂×Q₈	—	30	2	no	—
228	C₂×Q₈	—	30	4	no	—
229	C₂²×C₄	—	30	2	no	—
230	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
231	Q₁₆	—	15	8	no	—
232	Q₁₆	—	15	8	no	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	C₄×D₁₅	1	4	yes	—
235	C₂×C₈	C₂²×D₁₅	1	2	yes	—
236	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
238	C₄⋊C₄	—	15	4	no	—
239	C₄⋊C₄	—	30	8	no	—
240	C₄⋊C₄	D₆₀	1	2	yes	—
241	C₂²⋊C₄	—	30	2	no	—
242	C₂²⋊C₄	D₆₀	1	2	yes	—
243	C₄²	—	30	2	no	—
244	C₄²	—	30	8	no	—
245	C₄²	C₂²×D₁₅	1	2	yes	—
246	C₃×C₅	#2950	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	C₄×D₂₀	1	2	yes	—
249	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	C₄×D₂₀	1	2	yes	—
252	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
253	C₃×C₄	C₂×#709	1	1	yes	characteristic
254	Dic₃	—	10	8	no	—
255	Dic₃	—	20	4	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
258	C₂×C₅	#5753	1	2	yes	—
259	C₂³	C₂×D₆₀	1	1	yes	characteristic
260	Q₈	—	15	8	no	—
261	Q₈	—	30	8	no	—
262	Q₈	—	30	4	no	—
263	C₂×C₄	—	30	8	no	—
264	C₂×C₄	—	30	4	no	—
265	C₂×C₄	—	30	4	no	—
266	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
267	C₂×C₄	C₂×D₆₀	1	4	yes	—
268	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
269	C₂×C₄	#6817	1	2	yes	—
270	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	C₂×C₄×D₁₅	1	2	yes	—
273	C₈	C₂×C₄×D₁₅	1	2	yes	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
276	C₂×C₃	#66087	1	2	yes	—
277	C₅	#78573	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	2	2	no	—
279	C₂²	C₄×D₆₀	1	2	yes	—
280	C₂²	C₂²×D₆₀	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	30	8	no	—
283	C₄	—	60	4	no	—
284	C₄	C₄×D₆₀	1	2	yes	—
285	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
286	C₄	C₂×#6817	1	1	yes	characteristic
287	C₃	#10610304	1	1	yes	characteristic, complemented, Sylow
288	C₂	—	2	2	no	—
289	C₂	C₂×C₄×D₆₀	1	1	yes	characteristic
290	C₂	#11756512	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₆₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#405	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	#405	D₁₅
hypocenter	C₃×C₅	#2950
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
6. #11750817	C₂	semidirect
7. C₄×Dic₆₀	C₂	semidirect
14. C₂×Dic₆₀	C₄	semidirect
246. C₃×C₅	#2950	semidirect
277. C₅	#78573	semidirect
287. C₃	#10610304	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₀	2
#11756275	2
C₃×C₅×#405	2
#11751221	2
#11751207	2
#11750817	2
C₄×Dic₆₀	2
#10610304	3
#78573	5