Subgroups — #14284472

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745848	C₂	1	4	yes	maximal, complemented
5	#10601604	—	3	1	no	maximal, Hall
6	C₂×#6817	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	#91819	—	2	1	no	—
9	#91819	—	2	1	no	—
10	#91764	C₂²	1	2	yes	—
11	#91763	C₄	1	1	yes	characteristic
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	—	2	1	no	—
14	#91758	C₄	1	1	yes	characteristic
15	C₅×#830	—	2	1	no	—
16	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
17	C₅×#803	C₂²	1	2	yes	—
18	C₃×C₂²⋊#168	—	2	1	no	—
19	C₃×C₄⋊#168	—	2	1	no	—
20	C₃×#3935	C₂²	1	2	yes	—
21	#70811	—	5	1	no	maximal, Hall
22	#65811	—	3	1	no	—
23	C₅×#432	—	3	1	no	—
24	#65215	D₃	1	1	yes	characteristic, complemented
25	#64684	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×#1220	—	2	2	no	—
30	#6817	—	4	4	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₅	—	2	2	no	—
35	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#6714	—	2	2	no	—
41	#6714	—	2	2	no	—
42	C₄⋊Dic₁₅	—	2	1	no	—
43	C₄⋊Dic₁₅	—	2	1	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×#168	—	2	2	no	—
46	#5477	—	5	1	no	—
47	C₃×#432	—	5	1	no	—
48	#4926	D₅	1	1	yes	characteristic, complemented
49	#4464	—	5	4	no	—
50	C₂×#709	—	3	2	no	—
51	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
52	#4076	—	6	1	no	—
53	#4076	—	6	1	no	—
54	#4021	—	3	2	no	—
55	#4020	—	3	1	no	—
56	C₄×D₂₀	—	6	1	no	—
57	C₄×D₂₀	—	6	1	no	—
58	#4015	—	3	1	no	—
59	C₅×C₄⋊C₈	—	6	1	no	—
60	C₅×C₂²⋊C₈	—	6	1	no	—
61	C₅×#97	—	3	2	no	—
62	C₂²⋊#168	—	2	1	no	—
63	C₄⋊#168	—	2	1	no	—
64	#3935	C₂×D₃	1	2	yes	—
65	#1536	—	15	1	no	Sylow
66	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×D₁₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	#1220	—	4	4	no	—
78	C₂×Dic₁₅	—	2	2	no	—
79	D₆₀	—	4	2	no	—
80	D₆₀	—	4	2	no	—
81	D₆₀	—	4	2	no	—
82	C₄×D₁₅	—	4	4	no	—
83	Dic₃₀	—	4	2	no	—
84	C₃×#168	—	4	2	no	—
85	C₅×C₃⋊C₈	—	4	2	no	—
86	C₂×#235	—	5	2	no	—
87	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
88	#930	—	10	1	no	—
89	#930	—	10	1	no	—
90	#875	—	5	2	no	—
91	#874	—	5	1	no	—
92	C₄×D₁₂	—	10	1	no	—
93	C₄×D₁₂	—	10	1	no	—
94	#869	—	5	1	no	—
95	C₃×C₄⋊C₈	—	10	1	no	—
96	C₃×C₂²⋊C₈	—	10	1	no	—
97	C₃×#97	—	5	2	no	—
98	#830	—	2	1	no	—
99	C₄⋊(C₃⋊C₈)	—	2	1	no	—
100	#803	C₂×D₅	1	2	yes	—
101	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
102	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
103	C₂²×C₄×C₅	#67	1	1	yes	characteristic
104	C₂×C₂²⋊D₅	—	6	2	no	—
105	#709	—	12	4	no	—
106	C₂×D₂₀	—	3	2	no	—
107	C₂×D₂₀	—	6	1	no	—
108	C₂×D₂₀	—	6	1	no	—
109	C₂×C₄×D₅	—	6	2	no	—
110	C₂×Dic₁₀	—	3	2	no	—
111	C₂×C₅×C₈	—	6	2	no	—
112	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
113	C₄²×C₅	D₁₂	1	1	yes	characteristic
114	C₄²×C₅	#67	1	1	yes	characteristic
115	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
116	#685	—	6	2	no	—
117	#685	—	6	2	no	—
118	C₄⋊Dic₅	—	6	1	no	—
119	C₄⋊Dic₅	—	6	1	no	—
120	C₂×#168	—	2	2	no	—
121	#532	—	15	1	no	—
122	#432	—	3	1	no	—
123	#432	—	5	1	no	—
124	#329	—	15	4	no	—
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×D₁₅	—	4	2	no	—
130	C₂×D₁₅	—	4	2	no	—
131	C₂×D₁₅	—	4	2	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	#100	1	2	yes	—
138	Dic₁₅	—	4	2	no	—
139	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
142	C₂×#67	—	10	2	no	—
143	#235	—	20	4	no	—
144	C₂×D₁₂	—	5	2	no	—
145	C₂×D₁₂	—	10	1	no	—
146	C₂×D₁₂	—	10	1	no	—
147	C₂×C₄×D₃	—	10	2	no	—
148	C₂×Dic₆	—	5	2	no	—
149	C₂×C₃×C₈	—	10	2	no	—
150	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
151	C₃×C₄²	D₂₀	1	1	yes	characteristic
152	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
153	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	10	2	no	—
155	(C₂×D₃)⋊C₄	—	10	2	no	—
156	C₄⋊Dic₃	—	10	1	no	—
157	C₄⋊Dic₃	—	10	1	no	—
158	C₂×C₃⋊C₈	—	2	2	no	—
159	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₂²×D₅	—	6	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
170	C₂²⋊D₅	—	12	4	no	—
171	C₂×Dic₅	—	6	2	no	—
172	D₂₀	—	12	2	no	—
173	D₂₀	—	12	2	no	—
174	D₂₀	—	12	2	no	—
175	C₄×D₅	—	12	4	no	—
176	Dic₁₀	—	12	2	no	—
177	C₅×C₈	—	12	2	no	—
178	#168	—	4	2	no	—
179	C₂×#41	—	15	2	no	—
180	C₄⋊Q₈	—	15	1	no	—
181	#127	—	15	1	no	—
182	#124	—	15	2	no	—
183	#121	—	30	1	no	—
184	#121	—	30	1	no	—
185	C₄×D₄	—	30	1	no	—
186	C₄×D₄	—	30	1	no	—
187	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
188	C₄⋊C₈	—	6	1	no	—
189	C₄⋊C₈	—	10	1	no	—
190	C₂²⋊C₈	—	6	1	no	—
191	C₂²⋊C₈	—	10	1	no	—
192	#97	—	3	2	no	—
193	#97	—	5	2	no	—
194	C₂×C₃×C₅	—	4	1	no	—
195	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
196	C₂×C₃×C₅	#421	1	1	yes	characteristic
197	C₂×C₃×C₅	#421	1	1	yes	characteristic
198	D₁₅	—	8	2	no	—
199	C₂³×C₃	#685	1	1	yes	characteristic
200	C₂²×D₃	—	10	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	#685	1	1	yes	characteristic
205	C₂×C₃×C₄	#685	1	1	yes	characteristic
206	C₂×C₃×C₄	#685	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	#67	—	20	4	no	—
211	C₂×Dic₃	—	10	2	no	—
212	D₁₂	—	20	2	no	—
213	D₁₂	—	20	2	no	—
214	D₁₂	—	20	2	no	—
215	C₄×D₃	—	20	4	no	—
216	Dic₆	—	20	2	no	—
217	C₃×C₈	—	20	2	no	—
218	C₃⋊C₈	—	4	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₂×D₅	—	12	2	no	—
224	C₂×D₅	—	12	2	no	—
225	C₂×D₅	—	12	2	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	#823	1	2	yes	—
232	Dic₅	—	12	2	no	—
233	#41	—	60	4	no	—
234	C₂×Q₈	—	15	2	no	—
235	C₂×D₄	—	15	2	no	—
236	C₂×D₄	—	30	2	no	—
237	C₂×D₄	—	30	1	no	—
238	C₂×D₄	—	30	1	no	—
239	C₂²×C₄	—	30	2	no	—
240	C₂²×C₄	#1200	1	1	yes	characteristic
241	C₂²×C₄	#1202	1	1	yes	characteristic
242	C₂²×C₄	#1203	1	1	yes	characteristic
243	C₂×C₈	—	6	2	no	—
244	C₂×C₈	—	10	2	no	—
245	C₄⋊C₄	—	30	1	no	—
246	C₄⋊C₄	—	30	1	no	—
247	C₂²⋊C₄	—	30	2	no	—
248	C₂²⋊C₄	—	30	2	no	—
249	C₄²	#1200	1	1	yes	characteristic
250	C₄²	#1202	1	1	yes	characteristic
251	C₄²	#1203	1	1	yes	characteristic
252	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
253	C₃×C₅	#1536	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	C₂×#685	1	1	yes	characteristic
258	C₂×D₃	—	20	2	no	—
259	C₂×D₃	—	20	2	no	—
260	C₂×D₃	—	20	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	#3955	1	2	yes	—
267	Dic₃	—	20	2	no	—
268	C₂×C₅	—	4	1	no	—
269	C₂×C₅	#5005	1	1	yes	characteristic
270	C₂×C₅	C₂×#823	1	1	yes	characteristic
271	C₂×C₅	#5139	1	1	yes	characteristic
272	D₅	—	24	2	no	—
273	C₂³	—	30	2	no	—
274	C₂³	#6667	1	1	yes	characteristic
275	Q₈	—	60	2	no	—
276	D₄	—	60	2	no	—
277	D₄	—	60	2	no	—
278	D₄	—	60	2	no	—
279	D₄	—	60	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	30	2	no	—
284	C₂×C₄	—	60	4	no	—
285	C₂×C₄	#6667	1	1	yes	characteristic
286	C₂×C₄	#6667	1	1	yes	characteristic
287	C₂×C₄	#6667	1	1	yes	characteristic
288	C₂×C₄	C₂×#1200	1	1	yes	characteristic
289	C₂×C₄	C₂×#1202	1	1	yes	characteristic
290	C₂×C₄	C₂×#1203	1	1	yes	characteristic
291	C₈	—	12	2	no	—
292	C₈	—	20	2	no	—
293	C₂×C₃	—	4	1	no	—
294	C₂×C₃	#65294	1	1	yes	characteristic
295	C₂×C₃	C₂×#3955	1	1	yes	characteristic
296	C₂×C₃	#65428	1	1	yes	characteristic
297	D₃	—	40	2	no	—
298	C₅	#70811	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	60	2	no	—
303	C₂²	—	60	2	no	—
304	C₂²	—	60	2	no	—
305	C₂²	C₂×#6667	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	60	2	no	—
312	C₄	#90960	1	2	yes	—
313	C₃	#10601604	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	4	1	no	—
315	C₂	—	120	2	no	—
316	C₂	C₂×#90960	1	1	yes	characteristic
317	C₂	#11748428	1	1	yes	characteristic
318	C₂	#11748429	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1536
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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. C₅×#4926	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. #11745848	C₂	semidirect
24. #65215	D₃	semidirect
48. #4926	D₅	semidirect
253. C₃×C₅	#1536	semidirect
298. C₅	#70811	semidirect
313. C₃	#10601604	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₅×#4926	2
C₃×#65215	2
#11745848	2
#10601604	3
#70811	5