Subgroups — #14285436

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 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×#4924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745863	C₂	1	2	yes	maximal, complemented
5	#11745854	C₂	1	2	yes	maximal, complemented
6	#10601298	—	3	1	no	maximal, Hall
7	C₂×#6817	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	—	2	1	no	—
10	#91819	—	2	1	no	—
11	#91764	C₄	1	2	yes	—
12	#91763	C₂²	1	1	yes	characteristic
13	C₄×D₆₀	—	2	1	no	—
14	C₄×D₆₀	—	2	1	no	—
15	#91758	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	—	2	1	no	—
17	C₂×C₃×#681	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₃×C₄⋊#168	—	2	1	no	—
21	C₃×C₄⋊#168	C₂²	1	2	yes	—
22	#70505	—	5	1	no	maximal, Hall
23	#65811	—	3	1	no	—
24	C₅×#423	—	3	1	no	—
25	#65213	D₃	1	1	yes	characteristic, complemented
26	#64705	—	3	2	no	—
27	#64701	—	3	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×#1220	—	2	2	no	—
32	#6817	—	4	4	no	—
33	C₂×D₆₀	—	2	1	no	—
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₅	—	2	2	no	—
37	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	#6714	—	2	2	no	—
43	#6714	—	2	2	no	—
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₅×#208	—	4	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₃×#681	—	4	2	no	—
49	C₂×C₃×#168	—	2	2	no	—
50	#5477	—	5	1	no	—
51	C₃×#423	—	5	1	no	—
52	#4924	D₅	1	1	yes	characteristic, complemented
53	#4485	—	5	2	no	—
54	#4481	—	5	2	no	—
55	C₂×#709	—	3	2	no	—
56	C₂×C₅×M₁₆	—	6	1	no	—
57	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
58	#4076	—	6	1	no	—
59	#4076	—	6	1	no	—
60	C₂×#681	—	2	1	no	—
61	#4021	—	3	2	no	—
62	#4020	—	3	1	no	—
63	C₄×D₂₀	—	6	1	no	—
64	C₄×D₂₀	—	6	1	no	—
65	#4015	—	3	1	no	—
66	C₅×C₄⋊C₈	—	3	2	no	—
67	C₅×C₄⋊C₈	—	6	1	no	—
68	C₄⋊#168	—	2	1	no	—
69	C₄⋊#168	C₂×D₃	1	2	yes	—
70	#1560	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	#1220	—	4	4	no	—
85	C₂×Dic₁₅	—	2	2	no	—
86	D₆₀	—	4	2	no	—
87	D₆₀	—	4	1	no	—
88	D₆₀	—	4	1	no	—
89	D₆₀	—	4	1	no	—
90	D₆₀	—	4	1	no	—
91	C₄×D₁₅	—	4	4	no	—
92	Dic₃₀	—	4	2	no	—
93	C₃×#168	—	4	2	no	—
94	C₅×C₃⋊C₈	—	4	2	no	—
95	C₂×#235	—	5	2	no	—
96	C₂×C₃×M₁₆	—	10	1	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	#930	—	10	1	no	—
99	#930	—	10	1	no	—
100	C₂×#208	—	2	1	no	—
101	#875	—	5	2	no	—
102	#874	—	5	1	no	—
103	C₄×D₁₂	—	10	1	no	—
104	C₄×D₁₂	—	10	1	no	—
105	#869	—	5	1	no	—
106	C₃×C₄⋊C₈	—	5	2	no	—
107	C₃×C₄⋊C₈	—	10	1	no	—
108	C₄⋊(C₃⋊C₈)	—	2	1	no	—
109	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
110	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	—	6	2	no	—
114	#709	—	12	4	no	—
115	C₂×D₂₀	—	3	2	no	—
116	C₂×D₂₀	—	6	1	no	—
117	C₂×D₂₀	—	6	1	no	—
118	C₂×C₄×D₅	—	6	2	no	—
119	C₂×Dic₁₀	—	3	2	no	—
120	C₅×M₁₆	—	12	2	no	—
121	C₂×C₅×C₈	—	6	2	no	—
122	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
123	C₄²×C₅	D₁₂	1	1	yes	characteristic
124	C₄²×C₅	#67	1	1	yes	characteristic
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#685	—	6	2	no	—
127	#685	—	6	2	no	—
128	C₄⋊Dic₅	—	6	1	no	—
129	C₄⋊Dic₅	—	6	1	no	—
130	#681	—	4	2	no	—
131	C₂×#168	—	2	2	no	—
132	#532	—	15	1	no	—
133	#423	—	3	1	no	—
134	#423	—	5	1	no	—
135	#333	—	15	2	no	—
136	#331	—	15	2	no	—
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	—	2	1	no	—
139	C₂²×C₃×C₅	—	2	1	no	—
140	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×D₁₅	—	4	2	no	—
142	C₂×D₁₅	—	4	2	no	—
143	C₂×D₁₅	—	4	2	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	2	1	no	—
147	C₃×C₄×C₅	#100	1	2	yes	—
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₃×C₄×C₅	#137	1	1	yes	characteristic
151	C₃×C₄×C₅	#137	1	1	yes	characteristic
152	Dic₁₅	—	4	2	no	—
153	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
155	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
156	C₂×#67	—	10	2	no	—
157	#235	—	20	4	no	—
158	C₂×D₁₂	—	5	2	no	—
159	C₂×D₁₂	—	10	1	no	—
160	C₂×D₁₂	—	10	1	no	—
161	C₂×C₄×D₃	—	10	2	no	—
162	C₂×Dic₆	—	5	2	no	—
163	C₃×M₁₆	—	20	2	no	—
164	C₂×C₃×C₈	—	10	2	no	—
165	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
166	C₃×C₄²	D₂₀	1	1	yes	characteristic
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
169	(C₂×D₃)⋊C₄	—	10	2	no	—
170	(C₂×D₃)⋊C₄	—	10	2	no	—
171	C₄⋊Dic₃	—	10	1	no	—
172	C₄⋊Dic₃	—	10	1	no	—
173	#208	—	4	2	no	—
174	C₂×C₃⋊C₈	—	2	2	no	—
175	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₂²×D₅	—	6	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₂²⋊D₅	—	12	4	no	—
189	C₂×Dic₅	—	6	2	no	—
190	D₂₀	—	12	2	no	—
191	D₂₀	—	12	1	no	—
192	D₂₀	—	12	1	no	—
193	D₂₀	—	12	1	no	—
194	D₂₀	—	12	1	no	—
195	C₄×D₅	—	12	4	no	—
196	Dic₁₀	—	12	2	no	—
197	C₅×C₈	—	12	2	no	—
198	#168	—	4	2	no	—
199	C₂×#41	—	15	2	no	—
200	C₂×M₁₆	—	6	1	no	—
201	C₂×M₁₆	—	10	1	no	—
202	C₄⋊Q₈	—	15	1	no	—
203	#127	—	15	1	no	—
204	#124	—	15	2	no	—
205	#121	—	30	1	no	—
206	#121	—	30	1	no	—
207	C₄×D₄	—	30	1	no	—
208	C₄×D₄	—	30	1	no	—
209	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
210	C₄⋊C₈	—	3	2	no	—
211	C₄⋊C₈	—	5	2	no	—
212	C₄⋊C₈	—	6	1	no	—
213	C₄⋊C₈	—	10	1	no	—
214	C₂×C₃×C₅	—	4	1	no	—
215	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
216	C₂×C₃×C₅	#417	1	1	yes	characteristic
217	C₂×C₃×C₅	#417	1	1	yes	characteristic
218	D₁₅	—	8	2	no	—
219	C₂³×C₃	#685	1	1	yes	characteristic
220	C₂²×D₃	—	10	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	#685	1	1	yes	characteristic
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	#685	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	#67	—	20	4	no	—
233	C₂×Dic₃	—	10	2	no	—
234	D₁₂	—	20	2	no	—
235	D₁₂	—	20	1	no	—
236	D₁₂	—	20	1	no	—
237	D₁₂	—	20	1	no	—
238	D₁₂	—	20	1	no	—
239	C₄×D₃	—	20	4	no	—
240	Dic₆	—	20	2	no	—
241	C₃×C₈	—	20	2	no	—
242	C₃⋊C₈	—	4	2	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	C₂×D₅	—	12	2	no	—
248	C₂×D₅	—	12	2	no	—
249	C₂×D₅	—	12	2	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	#823	1	2	yes	—
254	C₄×C₅	#908	1	1	yes	characteristic
255	C₄×C₅	#909	1	1	yes	characteristic
256	C₄×C₅	#932	1	1	yes	characteristic
257	C₄×C₅	#942	1	1	yes	characteristic
258	Dic₅	—	12	2	no	—
259	#41	—	60	4	no	—
260	C₂×Q₈	—	15	2	no	—
261	C₂×D₄	—	15	2	no	—
262	C₂×D₄	—	30	2	no	—
263	C₂×D₄	—	30	1	no	—
264	C₂×D₄	—	30	1	no	—
265	C₂²×C₄	—	30	2	no	—
266	C₂²×C₄	#1200	1	1	yes	characteristic
267	C₂²×C₄	#1202	1	1	yes	characteristic
268	C₂²×C₄	#1203	1	1	yes	characteristic
269	M₁₆	—	12	2	no	—
270	M₁₆	—	20	2	no	—
271	C₂×C₈	—	6	2	no	—
272	C₂×C₈	—	10	2	no	—
273	C₄⋊C₄	—	30	1	no	—
274	C₄⋊C₄	—	30	1	no	—
275	C₂²⋊C₄	—	30	2	no	—
276	C₂²⋊C₄	—	30	2	no	—
277	C₄²	#1200	1	1	yes	characteristic
278	C₄²	#1202	1	1	yes	characteristic
279	C₄²	#1203	1	1	yes	characteristic
280	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
281	C₃×C₅	#1560	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	C₂×#685	1	1	yes	characteristic
286	C₂×D₃	—	20	2	no	—
287	C₂×D₃	—	20	2	no	—
288	C₂×D₃	—	20	2	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	#3955	1	2	yes	—
293	C₃×C₄	#4054	1	1	yes	characteristic
294	C₃×C₄	#4055	1	1	yes	characteristic
295	C₃×C₄	#4078	1	1	yes	characteristic
296	C₃×C₄	#4088	1	1	yes	characteristic
297	Dic₃	—	20	2	no	—
298	C₂×C₅	—	4	1	no	—
299	C₂×C₅	#4966	1	1	yes	characteristic
300	C₂×C₅	C₂×#823	1	1	yes	characteristic
301	C₂×C₅	#5133	1	1	yes	characteristic
302	D₅	—	24	2	no	—
303	C₂³	—	30	2	no	—
304	C₂³	#6667	1	1	yes	characteristic
305	Q₈	—	60	2	no	—
306	D₄	—	60	2	no	—
307	D₄	—	60	4	no	—
308	D₄	—	60	1	no	—
309	D₄	—	60	1	no	—
310	D₄	—	60	1	no	—
311	D₄	—	60	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	30	2	no	—
318	C₂×C₄	—	60	4	no	—
319	C₂×C₄	#6667	1	1	yes	characteristic
320	C₂×C₄	#6667	1	1	yes	characteristic
321	C₂×C₄	#6667	1	1	yes	characteristic
322	C₂×C₄	C₂×#1200	1	1	yes	characteristic
323	C₂×C₄	C₂×#1202	1	1	yes	characteristic
324	C₂×C₄	C₂×#1203	1	1	yes	characteristic
325	C₈	—	12	2	no	—
326	C₈	—	20	2	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	#65255	1	1	yes	characteristic
329	C₂×C₃	C₂×#3955	1	1	yes	characteristic
330	C₂×C₃	#65422	1	1	yes	characteristic
331	D₃	—	40	2	no	—
332	C₅	#70505	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	60	2	no	—
337	C₂²	—	60	2	no	—
338	C₂²	—	60	2	no	—
339	C₂²	C₂×#6667	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	60	2	no	—
344	C₄	#90960	1	2	yes	—
345	C₄	#91304	1	1	yes	characteristic
346	C₄	#91312	1	1	yes	characteristic
347	C₄	#91315	1	1	yes	characteristic
348	C₄	#91321	1	1	yes	characteristic
349	C₃	#10601298	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	4	1	no	—
351	C₂	—	120	2	no	—
352	C₂	C₂×#90960	1	1	yes	characteristic
353	C₂	#11748394	1	1	yes	characteristic
354	C₂	#11748413	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	#1560
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×#4924	C₂	semidirect
3. C₃×#65213	C₂	semidirect
4. #11745863	C₂	semidirect
5. #11745854	C₂	semidirect
25. #65213	D₃	semidirect
52. #4924	D₅	semidirect
281. C₃×C₅	#1560	semidirect
332. C₅	#70505	semidirect
349. C₃	#10601298	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×#4924	2
C₃×#65213	2
#11745863	2
#11745854	2
#10601298	3
#70505	5