Subgroups — #14438700

Order 1920.

784 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	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	#11748413	C₂	1	4	yes	maximal, complemented
5	#11746936	C₂	1	2	yes	maximal, complemented
6	#11746933	C₂	1	4	yes	maximal, complemented
7	#11746911	C₂	1	2	yes	maximal, complemented
8	#10610260	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91819	—	2	2	no	—
12	#91764	C₂²	1	2	yes	—
13	#91763	C₂²	1	1	yes	characteristic
14	C₄×D₆₀	—	2	2	no	—
15	#91758	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂×C₃×#683	—	2	2	no	—
19	C₃×#4016	—	2	2	no	—
20	C₃×#4016	C₂²	1	2	yes	—
21	C₃×#4016	C₂²	1	2	yes	—
22	C₅×#803	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
24	#90974	C₂²	1	8	yes	—
25	#90968	C₂²	1	8	yes	—
26	#83012	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₂×#4016	D₃	1	1	yes	characteristic, complemented
29	C₄×C₅×M₁₆	—	3	1	no	—
30	#65422	—	3	4	no	—
31	#65002	—	3	2	no	—
32	#64992	—	3	2	no	—
33	#64986	—	3	4	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×#1220	—	2	2	no	—
37	#6817	—	2	8	no	—
38	C₂×D₆₀	—	2	2	no	—
39	C₂×D₆₀	C₂³	1	2	yes	—
40	C₂×C₄×D₁₅	—	2	2	no	—
41	C₂×Dic₃₀	C₂³	1	2	yes	—
42	C₂²×C₃×Dic₅	—	2	2	no	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#6714	—	2	4	no	—
47	C₄⋊Dic₁₅	—	2	2	no	—
48	C₅×#208	—	2	4	no	—
49	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
50	C₃×C₄⋊Dic₅	—	2	2	no	—
51	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
52	C₃×#683	—	2	4	no	—
53	C₃×#683	—	2	4	no	—
54	C₂×C₃×#125	—	5	1	no	—
55	#5477	—	5	1	no	—
56	#5069	—	5	2	no	—
57	#5068	—	5	4	no	—
58	#5067	—	5	2	no	—
59	#4966	—	5	4	no	—
60	C₄×#208	D₅	1	1	yes	characteristic, complemented
61	C₂×#709	—	3	2	no	—
62	C₂×C₅×M₁₆	—	3	2	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	#4076	—	6	2	no	—
65	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
66	C₂×#683	—	2	2	no	—
67	#4021	—	3	2	no	—
68	#4020	—	3	1	no	—
69	C₄×D₂₀	—	6	2	no	—
70	#4016	—	2	2	no	—
71	#4016	C₂×D₃	1	2	yes	—
72	#4016	C₂×D₃	1	2	yes	—
73	#4015	—	3	1	no	—
74	C₅×#97	—	3	2	no	—
75	C₄×C₅×C₈	—	3	2	no	—
76	#3953	—	3	8	no	—
77	#3948	—	3	8	no	—
78	#3142	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×D₁₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	4	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	#1220	—	4	4	no	—
88	C₂×Dic₁₅	—	2	2	no	—
89	D₆₀	—	2	4	no	—
90	D₆₀	—	4	4	no	—
91	C₄×D₁₅	—	4	4	no	—
92	Dic₃₀	—	2	4	no	—
93	C₂×C₃×Dic₅	—	2	4	no	—
94	C₂×C₃×Dic₅	—	4	4	no	—
95	C₅×C₃⋊C₈	—	2	4	no	—
96	C₂×#235	—	5	2	no	—
97	C₃×#125	—	5	2	no	—
98	C₃×#125	—	5	2	no	—
99	C₃×#125	—	10	2	no	—
100	C₂×C₃×C₄⋊C₄	—	5	2	no	—
101	C₂×C₃×C₄⋊C₄	—	10	2	no	—
102	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	#930	—	10	2	no	—
104	C₂×#208	C₂×D₅	1	2	yes	—
105	#875	—	5	2	no	—
106	#874	—	5	1	no	—
107	C₄×D₁₂	—	10	2	no	—
108	#869	—	5	1	no	—
109	#810	—	5	8	no	—
110	#809	—	5	8	no	—
111	#803	C₂×D₅	1	2	yes	—
112	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
113	C₂²×C₄×C₅	#67	1	2	yes	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₂²⋊D₅	—	6	2	no	—
116	C₂²×Dic₅	—	2	2	no	—
117	#709	—	6	8	no	—
118	C₂×D₂₀	—	3	2	no	—
119	C₂×D₂₀	—	6	2	no	—
120	C₂×C₄×D₅	—	6	2	no	—
121	C₂×Dic₁₀	—	3	2	no	—
122	C₅×M₁₆	—	6	4	no	—
123	C₂×C₅×C₈	—	3	4	no	—
124	C₄²×C₅	#67	1	2	yes	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	#685	—	6	4	no	—
128	C₄⋊Dic₅	—	2	2	no	—
129	C₄⋊Dic₅	—	6	2	no	—
130	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
131	#683	—	2	4	no	—
132	#683	—	2	4	no	—
133	#532	—	15	1	no	—
134	C₂×#125	—	5	1	no	—
135	#491	—	15	2	no	—
136	#490	—	15	2	no	—
137	#488	—	15	4	no	—
138	#417	—	15	4	no	—
139	C₄×M₁₆	—	3	1	no	—
140	C₂²×C₃×C₅	—	2	2	no	—
141	C₂²×C₃×C₅	#124	1	2	yes	—
142	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₅	—	4	2	no	—
144	C₂×D₁₅	—	4	4	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	#124	1	2	yes	—
148	C₃×C₄×C₅	C₂×#41	1	2	yes	—
149	Dic₁₅	—	4	2	no	—
150	C₃×Dic₅	—	4	4	no	—
151	C₂²×C₃×C₄	—	10	2	no	—
152	C₂²×C₃×C₄	D₂₀	1	2	yes	—
153	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
154	C₂×#67	—	10	2	no	—
155	#235	—	10	8	no	—
156	C₂×D₁₂	—	5	2	no	—
157	C₂×D₁₂	—	10	2	no	—
158	C₂×C₄×D₃	—	10	2	no	—
159	C₂×Dic₆	—	5	2	no	—
160	C₃×C₄⋊C₄	—	5	4	no	—
161	C₃×C₄⋊C₄	—	10	4	no	—
162	C₃×C₄⋊C₄	—	10	4	no	—
163	C₃×C₄⋊C₄	—	10	2	no	—
164	C₃×C₄²	D₂₀	1	2	yes	—
165	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
167	(C₂×D₃)⋊C₄	—	10	4	no	—
168	C₄⋊Dic₃	—	10	2	no	—
169	#208	—	2	4	no	—
170	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
171	C₂³×C₅	C₂×#67	1	1	yes	characteristic
172	C₂²×D₅	—	6	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	#239	1	4	yes	—
175	C₂×C₄×C₅	C₂×#67	1	2	yes	—
176	C₂×C₄×C₅	C₂×#67	1	2	yes	—
177	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
179	C₂²⋊D₅	—	12	4	no	—
180	C₂×Dic₅	—	2	4	no	—
181	C₂×Dic₅	—	4	4	no	—
182	C₂×Dic₅	—	6	2	no	—
183	D₂₀	—	6	4	no	—
184	D₂₀	—	12	4	no	—
185	C₄×D₅	—	12	4	no	—
186	Dic₁₀	—	6	4	no	—
187	C₅×C₈	—	6	4	no	—
188	C₂×#41	—	15	2	no	—
189	C₂×M₁₆	—	3	2	no	—
190	C₄⋊Q₈	—	15	1	no	—
191	#127	—	15	1	no	—
192	#125	—	5	2	no	—
193	#125	—	5	2	no	—
194	#125	—	10	2	no	—
195	#124	—	15	2	no	—
196	#121	—	30	2	no	—
197	C₄×D₄	—	30	2	no	—
198	C₂×C₄⋊C₄	—	5	2	no	—
199	C₂×C₄⋊C₄	—	10	2	no	—
200	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
201	#103	—	15	8	no	—
202	#102	—	15	8	no	—
203	#97	—	3	2	no	—
204	C₄×C₈	—	3	2	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
207	C₂×C₃×C₅	#575	1	2	yes	—
208	D₁₅	—	8	2	no	—
209	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
210	C₂²×D₃	—	10	2	no	—
211	C₂×C₃×C₄	—	4	2	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	—	20	4	no	—
214	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
215	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
216	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
217	C₂×C₃×C₄	#709	1	4	yes	—
218	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
219	#67	—	20	4	no	—
220	C₂×Dic₃	—	10	2	no	—
221	D₁₂	—	10	4	no	—
222	D₁₂	—	20	4	no	—
223	C₄×D₃	—	20	4	no	—
224	Dic₆	—	10	4	no	—
225	C₃⋊C₈	—	2	4	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	#947	1	2	yes	—
228	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
229	C₂×D₅	—	12	2	no	—
230	C₂×D₅	—	12	4	no	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	#947	1	2	yes	—
234	C₄×C₅	C₂×#239	1	2	yes	—
235	Dic₅	—	4	4	no	—
236	Dic₅	—	12	2	no	—
237	#41	—	30	8	no	—
238	C₂×Q₈	—	15	2	no	—
239	C₂×D₄	—	15	2	no	—
240	C₂×D₄	—	30	2	no	—
241	C₂×D₄	—	30	2	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	—	30	2	no	—
244	C₂²×C₄	#1202	1	2	yes	—
245	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
246	M₁₆	—	6	4	no	—
247	C₂×C₈	—	3	4	no	—
248	C₄⋊C₄	—	5	4	no	—
249	C₄⋊C₄	—	10	2	no	—
250	C₄⋊C₄	—	10	4	no	—
251	C₄⋊C₄	—	10	4	no	—
252	C₄⋊C₄	—	30	2	no	—
253	C₂²⋊C₄	—	30	4	no	—
254	C₄²	#1202	1	2	yes	—
255	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
256	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
257	C₃×C₅	#3142	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	#4021	1	2	yes	—
260	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
261	C₂×D₃	—	20	2	no	—
262	C₂×D₃	—	20	4	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	20	4	no	—
266	C₃×C₄	#4021	1	2	yes	—
267	C₃×C₄	C₂×#709	1	2	yes	—
268	Dic₃	—	20	2	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₂×#947	1	1	yes	characteristic
271	C₂×C₅	#5822	1	2	yes	—
272	D₅	—	24	2	no	—
273	C₂³	—	30	2	no	—
274	C₂³	C₂×#1202	1	1	yes	characteristic
275	Q₈	—	30	4	no	—
276	D₄	—	30	4	no	—
277	D₄	—	60	4	no	—
278	D₄	—	60	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	10	4	no	—
281	C₂×C₄	—	20	4	no	—
282	C₂×C₄	—	30	2	no	—
283	C₂×C₄	—	60	4	no	—
284	C₂×C₄	#6769	1	4	yes	—
285	C₂×C₄	C₂×#1202	1	2	yes	—
286	C₂×C₄	C₂×#1202	1	2	yes	—
287	C₂×C₄	C₂×#1202	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
289	C₈	—	6	4	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	C₂×#4021	1	1	yes	characteristic
292	C₂×C₃	#66082	1	2	yes	—
293	D₃	—	40	2	no	—
294	C₅	#83012	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	60	2	no	—
297	C₂²	—	60	4	no	—
298	C₂²	#91374	1	2	yes	—
299	C₂²	C₂²×#1202	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	20	4	no	—
303	C₄	—	60	2	no	—
304	C₄	#91374	1	2	yes	—
305	C₄	C₂×#6769	1	2	yes	—
306	C₃	#10610260	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	120	2	no	—
309	C₂	#11754154	1	2	yes	—
310	C₂	C₂×#91374	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1202
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	#11756236	C₂
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#3142
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4016	C₂	semidirect
3. C₄×C₅×#208	C₂	semidirect
4. #11748413	C₂	semidirect
5. #11746936	C₂	semidirect
6. #11746933	C₂	semidirect
7. #11746911	C₂	semidirect
28. C₂×#4016	D₃	semidirect
60. C₄×#208	D₅	semidirect
257. C₃×C₅	#3142	semidirect
294. C₅	#83012	semidirect
306. C₃	#10610260	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₂×C₃×#4016	2
C₄×C₅×#208	2
#11748413	2
#11746936	2
#11746933	2
#11746911	2
#10610260	3
#83012	5