Subgroups — #14417338

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	#11756280	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751202	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751102	C₂	1	2	yes	maximal, complemented
6	#11750973	C₂	1	4	yes	maximal, complemented
7	#11750970	C₂	1	4	yes	maximal, complemented
8	#10610898	—	3	1	no	maximal, Hall
9	C₂²×D₆₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	#91815	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	1	yes	characteristic
14	C₂×D₁₂₀	C₄	1	8	yes	complemented
15	C₂×D₁₂₀	C₂²	1	2	yes	—
16	C₂×D₁₂₀	C₂²	1	2	yes	—
17	#91782	C₂²	1	8	yes	complemented
18	C₂²⋊C₄×D₁₅	—	2	4	no	—
19	C₄×D₆₀	—	2	4	no	—
20	C₃×C₅×#107	C₂²	1	2	yes	complemented
21	C₃×C₅×#106	C₂²	1	2	yes	complemented
22	#91099	C₂²	1	2	yes	complemented
23	C₈×Dic₁₅	C₂²	1	2	yes	complemented
24	#91093	C₂²	1	8	yes	complemented
25	#79167	—	5	1	no	maximal, Hall
26	C₂²×D₄₀	—	3	1	no	—
27	#65855	—	3	2	no	—
28	C₅×#427	D₃	1	1	yes	characteristic, complemented
29	#65390	—	3	1	no	—
30	#65290	—	3	2	no	—
31	#65161	—	3	4	no	—
32	#65158	—	3	4	no	—
33	C₂³×D₁₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×D₆₀	—	2	2	no	—
36	C₂×D₆₀	—	2	4	no	—
37	C₂×D₆₀	C₂×C₄	1	8	yes	complemented
38	C₂×D₆₀	C₂³	1	4	yes	—
39	C₂×C₄×D₁₅	—	2	8	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₃×C₄²×C₅	—	2	2	no	—
46	#6719	D₄	1	2	yes	—
47	#6714	—	2	8	no	—
48	C₄⋊Dic₁₅	D₄	1	2	yes	—
49	C₄×Dic₁₅	C₂³	1	2	yes	—
50	#6710	—	2	2	no	—
51	C₂×C₃⋊#168	C₂³	1	2	yes	—
52	D₁₂₀	C₂×C₄	1	8	yes	complemented
53	D₁₂₀	C₂×C₄	1	8	yes	complemented
54	C₂²×D₂₄	—	5	1	no	—
55	#5521	—	5	2	no	—
56	C₃×#427	D₅	1	1	yes	characteristic, complemented
57	#5101	—	5	1	no	—
58	#5001	—	5	2	no	—
59	#4872	—	5	4	no	—
60	#4869	—	5	4	no	—
61	C₂²×D₂₀	—	3	2	no	—
62	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₅×#117	C₂×D₃	1	2	yes	—
64	#4072	—	3	1	no	—
65	C₂×#681	—	3	1	no	—
66	C₂×D₄₀	—	3	8	no	—
67	C₂×D₄₀	—	3	2	no	—
68	C₂×D₄₀	—	3	2	no	—
69	#4039	—	3	8	no	—
70	D₅×C₂²⋊C₄	—	6	4	no	—
71	C₄×D₂₀	—	6	4	no	—
72	C₅×#107	C₂×D₃	1	2	yes	complemented
73	C₅×#106	C₂×D₃	1	2	yes	complemented
74	#3947	—	3	2	no	—
75	C₈×Dic₅	—	3	2	no	—
76	#3941	—	3	8	no	—
77	#2948	—	15	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₁₅	—	2	4	no	—
80	C₂²×D₁₅	—	2	8	no	—
81	C₂²×D₁₅	—	8	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
88	D₆₀	—	2	8	no	—
89	D₆₀	—	2	4	no	—
90	D₆₀	C₂²×C₄	1	8	yes	—
91	C₄×D₁₅	—	8	8	no	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₃⋊#168	—	2	2	no	—
95	C₂²×D₁₂	—	5	2	no	—
96	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
97	C₃×#117	C₂×D₅	1	2	yes	—
98	#926	—	5	1	no	—
99	C₂×#208	—	5	1	no	—
100	C₂×D₂₄	—	5	8	no	—
101	C₂×D₂₄	—	5	2	no	—
102	C₂×D₂₄	—	5	2	no	—
103	#893	—	5	8	no	—
104	D₃×C₂²⋊C₄	—	10	4	no	—
105	C₄×D₁₂	—	10	4	no	—
106	C₃×#107	C₂×D₅	1	2	yes	complemented
107	C₃×#106	C₂×D₅	1	2	yes	complemented
108	#815	—	5	2	no	—
109	C₈×Dic₃	—	5	2	no	—
110	#809	—	5	8	no	—
111	C₂³×D₅	—	6	2	no	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×D₂₀	—	3	8	no	—
114	C₂×D₂₀	—	3	4	no	—
115	C₂×D₂₀	—	6	2	no	—
116	C₂×D₂₀	—	6	4	no	—
117	C₂×C₄×D₅	—	6	8	no	—
118	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
119	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₂²⋊Dic₅	—	3	2	no	—
125	#685	—	6	8	no	—
126	C₄⋊Dic₅	—	3	2	no	—
127	C₄×Dic₅	—	3	2	no	—
128	#681	—	6	2	no	—
129	C₂×#168	—	3	2	no	—
130	D₄₀	—	3	8	no	—
131	D₄₀	—	3	8	no	—
132	C₂²×D₈	—	15	1	no	—
133	#518	—	15	2	no	—
134	#442	—	15	4	no	—
135	C₄×D₈	—	15	4	no	—
136	#427	D₁₅	1	1	yes	characteristic, complemented
137	#418	—	15	2	no	—
138	#405	—	15	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₅	—	2	8	no	—
143	C₂×D₁₅	—	4	8	no	—
144	C₂×D₁₅	—	8	8	no	—
145	C₃×C₄×C₅	—	4	4	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	Dic₁₅	—	2	4	no	—
150	C₂³×D₃	—	10	2	no	—
151	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₂×D₁₂	—	5	8	no	—
153	C₂×D₁₂	—	5	4	no	—
154	C₂×D₁₂	—	10	2	no	—
155	C₂×D₁₂	—	10	4	no	—
156	C₂×C₄×D₃	—	10	8	no	—
157	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
158	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
159	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
160	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₄²	—	2	2	no	—
163	#217	—	5	2	no	—
164	(C₂×D₃)⋊C₄	—	10	8	no	—
165	C₄⋊Dic₃	—	5	2	no	—
166	C₄×Dic₃	—	5	2	no	—
167	#208	—	10	2	no	—
168	C₂×C₃⋊C₈	—	5	2	no	—
169	D₂₄	—	5	8	no	—
170	D₂₄	—	5	8	no	—
171	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
172	C₂²×D₅	—	6	4	no	—
173	C₂²×D₅	—	6	8	no	—
174	C₂²×D₅	—	24	4	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
177	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	#239	1	2	yes	—
179	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
180	C₂×Dic₅	—	3	4	no	—
181	D₂₀	—	3	8	no	—
182	D₂₀	—	6	8	no	—
183	D₂₀	—	6	4	no	—
184	C₄×D₅	—	24	8	no	—
185	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
186	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
187	#168	—	6	2	no	—
188	C₂²×D₄	—	15	2	no	—
189	C₂×D₈	—	15	8	no	—
190	C₂×D₈	—	15	2	no	—
191	C₂×D₈	—	15	2	no	—
192	C₂×M₁₆	—	15	1	no	—
193	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
194	C₄×D₄	—	15	8	no	—
195	C₄×D₄	—	30	4	no	—
196	#117	—	15	1	no	—
197	#117	C₂×D₁₅	1	2	yes	—
198	C₂×C₂²⋊C₄	—	30	4	no	—
199	#107	C₂×D₁₅	1	2	yes	complemented
200	#106	C₂×D₁₅	1	2	yes	complemented
201	#102	—	15	8	no	—
202	#97	—	15	2	no	—
203	C₄×C₈	—	15	2	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×C₃×C₅	#576	1	2	yes	—
207	D₁₅	—	4	8	no	—
208	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
209	C₂²×D₃	—	10	4	no	—
210	C₂²×D₃	—	10	8	no	—
211	C₂²×D₃	—	40	4	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
214	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	#713	1	2	yes	—
216	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
217	C₂×Dic₃	—	5	4	no	—
218	D₁₂	—	5	8	no	—
219	D₁₂	—	10	4	no	—
220	D₁₂	—	10	8	no	—
221	C₄×D₃	—	40	8	no	—
222	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
223	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
224	C₃⋊C₈	—	10	2	no	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	#893	1	2	yes	—
227	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
228	C₂×D₅	—	6	8	no	—
229	C₂×D₅	—	12	8	no	—
230	C₂×D₅	—	24	8	no	—
231	C₄×C₅	—	4	4	no	—
232	C₄×C₅	#893	1	2	yes	—
233	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
234	C₄×C₅	C₂×#239	1	1	yes	characteristic
235	Dic₅	—	6	4	no	—
236	C₂⁴	—	30	2	no	—
237	C₂×D₄	—	15	8	no	—
238	C₂×D₄	—	15	4	no	—
239	C₂×D₄	—	30	2	no	—
240	C₂×D₄	—	30	4	no	—
241	C₂²×C₄	—	30	8	no	—
242	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
243	D₈	—	15	8	no	—
244	D₈	—	15	8	no	—
245	M₁₆	—	30	2	no	—
246	C₂×C₈	—	15	2	no	—
247	C₂×C₈	C₄×D₁₅	1	4	yes	complemented
248	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
250	C₄⋊C₄	—	15	2	no	—
251	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	15	2	no	—
254	C₂²⋊C₄	—	30	8	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	15	2	no	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	#4039	1	2	yes	—
260	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
261	C₂×D₃	—	10	8	no	—
262	C₂×D₃	—	20	8	no	—
263	C₂×D₃	—	40	8	no	—
264	C₃×C₄	—	4	4	no	—
265	C₃×C₄	#4039	1	2	yes	—
266	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
267	C₃×C₄	C₂×#713	1	1	yes	characteristic
268	Dic₃	—	10	4	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₂×#893	1	1	yes	characteristic
271	C₂×C₅	#5769	1	2	yes	—
272	D₅	—	12	8	no	—
273	C₂³	—	30	4	no	—
274	C₂³	—	30	8	no	—
275	C₂³	—	120	4	no	—
276	C₂³	D₄×D₁₅	1	1	yes	characteristic
277	D₄	—	15	8	no	—
278	D₄	—	30	8	no	—
279	D₄	—	30	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	15	4	no	—
282	C₂×C₄	—	120	8	no	—
283	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
284	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
285	C₂×C₄	#6821	1	2	yes	—
286	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
287	C₈	—	30	2	no	—
288	C₈	C₂×C₄×D₁₅	1	2	yes	—
289	C₈	C₂×C₄×D₁₅	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	C₂×#4039	1	1	yes	characteristic
292	C₂×C₃	#66103	1	2	yes	—
293	D₃	—	20	8	no	—
294	C₅	#79167	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	30	8	no	—
297	C₂²	—	60	8	no	—
298	C₂²	—	120	8	no	—
299	C₂²	#91782	1	2	yes	—
300	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
301	C₄	—	4	4	no	—
302	C₄	—	30	4	no	—
303	C₄	#91782	1	2	yes	—
304	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
305	C₄	C₂×#6821	1	1	yes	characteristic
306	C₃	#10610898	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	60	8	no	—
309	C₂	C₂×#91782	1	1	yes	characteristic
310	C₂	#11756528	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	#427	D₁₅
hypocenter	C₃×C₅	#2948
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
2. #11756280	C₂	semidirect
3. C₃×C₅×#427	C₂	semidirect
4. #11751202	C₂	semidirect
5. #11751102	C₂	semidirect
6. #11750973	C₂	semidirect
7. #11750970	C₂	semidirect
14. C₂×D₁₂₀	C₄	semidirect
17. #91782	C₂²	semidirect
20. C₃×C₅×#107	C₂²	semidirect
21. C₃×C₅×#106	C₂²	semidirect
22. #91099	C₂²	semidirect
23. C₈×Dic₁₅	C₂²	semidirect
24. #91093	C₂²	semidirect
28. C₅×#427	D₃	semidirect
37. C₂×D₆₀	C₂×C₄	semidirect
40. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
52. D₁₂₀	C₂×C₄	semidirect
53. D₁₂₀	C₂×C₄	semidirect
56. C₃×#427	D₅	semidirect
72. C₅×#107	C₂×D₃	semidirect
73. C₅×#106	C₂×D₃	semidirect
106. C₃×#107	C₂×D₅	semidirect
107. C₃×#106	C₂×D₅	semidirect
118. C₂×C₅×C₈	C₄×D₃	semidirect
136. #427	D₁₅	semidirect
157. C₂×C₃×C₈	C₄×D₅	semidirect
199. #107	C₂×D₁₅	semidirect
200. #106	C₂×D₁₅	semidirect
247. C₂×C₈	C₄×D₁₅	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	#79167	semidirect
306. C₃	#10610898	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₀	2
#11756280	2
C₃×C₅×#427	2
#11751202	2
#11751102	2
#11750973	2
#11750970	2
#10610898	3
#79167	5