Subgroups — #14285045

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	#11751027	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745862	C₂	1	2	yes	maximal, complemented
5	#11745853	C₂	1	2	yes	maximal, complemented
6	#10602600	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#930	—	2	1	no	—
10	C₅×#930	—	2	1	no	—
11	C₅×#875	C₄	1	2	yes	—
12	C₅×#874	C₂²	1	1	yes	characteristic
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	—	2	1	no	—
15	C₅×#869	C₂²	1	1	yes	characteristic
16	C₂²×C₃×#168	—	2	1	no	—
17	#91114	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
19	#91087	—	2	1	no	—
20	C₃×C₄⋊#168	—	2	1	no	—
21	C₃×C₄⋊#168	C₂²	1	2	yes	—
22	#70433	—	5	1	no	maximal, Hall
23	C₅×#532	—	3	1	no	—
24	#65215	—	3	1	no	—
25	C₂×C₄⋊#168	D₃	1	1	yes	characteristic, complemented
26	#64766	—	3	2	no	—
27	#64764	—	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×C₅×#67	—	2	2	no	—
32	C₅×#235	—	4	4	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₅×D₃	—	2	2	no	—
37	C₂×C₅×Dic₆	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	C₂×C₃⋊#168	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#168	—	4	2	no	—
49	#5477	D₅	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₈	—	5	1	no	—
51	#4926	—	5	1	no	—
52	#4485	—	5	2	no	—
53	#4481	—	5	2	no	—
54	C₂×C₅×#41	—	3	2	no	—
55	C₅×C₄⋊Q₈	—	3	1	no	—
56	C₅×#127	—	3	1	no	—
57	C₅×#124	—	3	2	no	—
58	C₅×#121	—	6	1	no	—
59	C₅×#121	—	6	1	no	—
60	C₄×C₅×D₄	—	6	1	no	—
61	C₄×C₅×D₄	—	6	1	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	C₂²×#168	—	2	1	no	—
64	C₂²⋊#168	—	6	1	no	—
65	C₄⋊#168	—	2	1	no	—
66	C₄⋊#168	—	3	2	no	—
67	C₄⋊#168	C₂×D₃	1	2	yes	—
68	#3935	—	6	1	no	—
69	#1551	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₅×D₃	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	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	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₅×#67	—	4	4	no	—
83	C₂×C₅×Dic₃	—	2	2	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×D₁₂	—	4	2	no	—
86	C₅×D₁₂	—	4	1	no	—
87	C₅×D₁₂	—	4	1	no	—
88	C₄×C₅×D₃	—	4	4	no	—
89	C₅×Dic₆	—	4	2	no	—
90	C₃⋊#168	—	4	2	no	—
91	C₃×#168	—	4	2	no	—
92	C₂×#235	Dic₅	1	2	yes	—
93	C₂²×C₃×C₈	—	10	1	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	#930	—	2	1	no	—
96	#930	—	2	1	no	—
97	#875	Dic₅	1	2	yes	—
98	#874	C₂×D₅	1	1	yes	characteristic
99	C₄×D₁₂	—	2	1	no	—
100	C₄×D₁₂	—	2	1	no	—
101	#869	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	—	5	2	no	—
103	C₃×C₄⋊C₈	—	10	1	no	—
104	#830	—	10	1	no	—
105	C₄⋊(C₃⋊C₈)	—	5	2	no	—
106	#803	—	10	1	no	—
107	C₅×#41	—	12	4	no	—
108	C₂×C₅×Q₈	—	3	2	no	—
109	C₂×C₅×D₄	—	3	2	no	—
110	C₂×C₅×D₄	—	6	2	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂×C₅×D₄	—	6	1	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	6	1	no	—
118	C₅×C₄⋊C₄	—	6	1	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₅×C₂²⋊C₄	—	6	2	no	—
121	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	D₁₂	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×#168	—	2	2	no	—
126	C₂×#168	—	4	2	no	—
127	C₂×#168	—	6	2	no	—
128	#532	—	3	1	no	—
129	#432	—	15	1	no	—
130	C₂×C₄⋊C₈	—	5	1	no	—
131	#333	—	15	2	no	—
132	#331	—	15	2	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	#100	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₅×D₃	—	4	2	no	—
138	C₂×C₅×D₃	—	4	2	no	—
139	C₂×C₅×D₃	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	#135	1	2	yes	—
145	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₃×C₄×C₅	#137	1	1	yes	characteristic
147	C₅×Dic₃	—	4	2	no	—
148	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
151	C₂×#67	—	2	2	no	—
152	#235	—	4	4	no	—
153	C₂×D₁₂	—	2	1	no	—
154	C₂×D₁₂	—	2	1	no	—
155	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
156	C₂×C₄×D₃	—	2	2	no	—
157	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
158	C₂×C₃×C₈	—	10	2	no	—
159	C₂×C₃×C₈	—	20	2	no	—
160	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
164	(C₂×D₃)⋊C₄	—	2	2	no	—
165	(C₂×D₃)⋊C₄	—	2	2	no	—
166	C₄⋊Dic₃	—	2	1	no	—
167	C₄⋊Dic₃	—	2	1	no	—
168	C₂×C₃⋊C₈	—	10	2	no	—
169	C₂³×C₅	—	6	2	no	—
170	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₅×Q₈	—	12	2	no	—
172	C₅×D₄	—	12	2	no	—
173	C₅×D₄	—	12	4	no	—
174	C₅×D₄	—	12	2	no	—
175	C₅×D₄	—	12	1	no	—
176	C₅×D₄	—	12	1	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	6	2	no	—
182	C₂×C₄×C₅	—	12	4	no	—
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×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
189	#168	—	4	2	no	—
190	#168	—	12	2	no	—
191	C₂×#41	—	3	2	no	—
192	C₂²×C₈	—	10	1	no	—
193	C₄⋊Q₈	—	3	1	no	—
194	#127	—	3	1	no	—
195	#124	—	3	2	no	—
196	#121	—	6	1	no	—
197	#121	—	6	1	no	—
198	C₄×D₄	—	6	1	no	—
199	C₄×D₄	—	6	1	no	—
200	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
201	C₄⋊C₈	—	5	2	no	—
202	C₄⋊C₈	—	10	1	no	—
203	C₄⋊C₈	—	15	2	no	—
204	C₂²⋊C₈	—	30	1	no	—
205	#97	—	30	1	no	—
206	C₂×C₃×C₅	—	2	2	no	—
207	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
208	C₂×C₃×C₅	#416	1	1	yes	characteristic
209	C₂×C₃×C₅	#417	1	1	yes	characteristic
210	C₅×D₃	—	8	2	no	—
211	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂²×D₃	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	#67	—	4	4	no	—
224	C₂×Dic₃	—	2	2	no	—
225	D₁₂	—	4	2	no	—
226	D₁₂	—	4	2	no	—
227	D₁₂	—	4	1	no	—
228	D₁₂	—	4	1	no	—
229	C₄×D₃	—	4	4	no	—
230	Dic₆	—	4	2	no	—
231	C₃×C₈	—	20	2	no	—
232	C₃⋊C₈	—	20	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	12	2	no	—
236	C₂²×C₅	—	12	2	no	—
237	C₂²×C₅	—	12	2	no	—
238	C₂²×C₅	#823	1	2	yes	—
239	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	12	2	no	—
245	C₄×C₅	#904	1	2	yes	—
246	C₄×C₅	#932	1	1	yes	characteristic
247	C₄×C₅	#942	1	1	yes	characteristic
248	#41	—	12	4	no	—
249	C₂×Q₈	—	3	2	no	—
250	C₂×D₄	—	3	2	no	—
251	C₂×D₄	—	6	2	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂×D₄	—	6	1	no	—
254	C₂²×C₄	—	6	2	no	—
255	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
256	C₂²×C₄	#1201	1	1	yes	characteristic
257	C₂²×C₄	#1203	1	1	yes	characteristic
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	20	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₄⋊C₄	—	6	1	no	—
262	C₄⋊C₄	—	6	1	no	—
263	C₂²⋊C₄	—	6	2	no	—
264	C₂²⋊C₄	—	6	2	no	—
265	C₄²	D₃×Dic₅	1	1	yes	characteristic
266	C₄²	#1201	1	1	yes	characteristic
267	C₄²	#1203	1	1	yes	characteristic
268	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
269	C₃×C₅	#1551	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	#3965	1	2	yes	—
273	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₂×D₃	—	4	2	no	—
275	C₂×D₃	—	4	2	no	—
276	C₂×D₃	—	4	2	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	#4095	1	1	yes	characteristic
282	C₃×C₄	#4096	1	2	yes	—
283	C₃×C₄	#4097	1	1	yes	characteristic
284	Dic₃	—	4	2	no	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	24	2	no	—
287	C₂×C₅	#4966	1	1	yes	characteristic
288	C₂×C₅	#5113	1	1	yes	characteristic
289	C₂×C₅	C₂×#823	1	1	yes	characteristic
290	C₂³	—	6	2	no	—
291	C₂³	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
292	Q₈	—	12	2	no	—
293	D₄	—	12	2	no	—
294	D₄	—	12	4	no	—
295	D₄	—	12	1	no	—
296	D₄	—	12	2	no	—
297	D₄	—	12	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	—	12	4	no	—
304	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
305	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
306	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
307	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
308	C₂×C₄	C₂×#1201	1	1	yes	characteristic
309	C₂×C₄	C₂×#1203	1	1	yes	characteristic
310	C₈	—	20	2	no	—
311	C₈	—	60	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₂×#3965	1	1	yes	characteristic
314	C₂×C₃	#65521	1	1	yes	characteristic
315	C₂×C₃	#65522	1	1	yes	characteristic
316	D₃	—	8	2	no	—
317	C₅	#70433	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	12	2	no	—
321	C₂²	—	12	2	no	—
322	C₂²	—	12	2	no	—
323	C₂²	#90959	1	2	yes	—
324	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	12	2	no	—
330	C₄	#91298	1	1	yes	characteristic
331	C₄	#91303	1	2	yes	—
332	C₄	#91309	1	1	yes	characteristic
333	C₃	#10602600	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	24	2	no	—
336	C₂	C₂×#90959	1	1	yes	characteristic
337	C₂	#11748384	1	1	yes	characteristic
338	C₂	#11748388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊Dic₅
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	C₅×#5477	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1551
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. #11751027	C₂	semidirect
3. C₂×C₃×C₄⋊#168	C₂	semidirect
4. #11745862	C₂	semidirect
5. #11745853	C₂	semidirect
25. C₂×C₄⋊#168	D₃	semidirect
269. C₃×C₅	#1551	semidirect
317. C₅	#70433	semidirect
333. C₃	#10602600	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
#11751027	2
C₂×C₃×C₄⋊#168	2
#11745862	2
#11745853	2
#10602600	3
#70433	5