Subgroups — #27541846

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287705	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276953	C₂	1	2	yes	maximal
4	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276863	C₂	1	1	yes	maximal, characteristic
6	#20276855	C₂	1	2	yes	maximal, complemented
7	#20276779	C₂	1	2	yes	maximal
8	#20276620	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276586	C₂	1	2	yes	maximal, complemented
10	C₄×#13133160	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₀₆	—	2	1	no	—
13	#16811347	—	2	1	no	—
14	C₂×#13133160	C₂²	1	1	yes	characteristic
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811027	C₂²	1	2	yes	—
17	#16811008	C₂²	1	2	yes	—
18	#16810918	—	2	1	no	—
19	#16810916	—	2	2	no	—
20	#16810847	—	2	2	no	—
21	#16810825	—	2	1	no	—
22	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
24	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
25	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
26	#16810451	C₂²	1	2	yes	—
27	#16810450	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810427	C₂²	1	1	yes	characteristic
31	#16810427	C₂²	1	2	yes	—
32	#16810425	C₂²	1	2	yes	—
33	#16810423	C₂²	1	1	yes	characteristic
34	#16810398	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#494	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₃×#358	C₂²	1	2	yes	—
41	C₂×#11719924	—	2	2	no	—
42	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
43	#13133217	—	2	1	no	—
44	#13133217	—	4	1	no	—
45	#13133216	—	2	2	no	—
46	#13133216	—	4	2	no	—
47	#13133212	—	4	2	no	—
48	#13133206	—	2	2	no	—
49	#13133197	—	4	1	no	—
50	Q₈×Dic₅₃	—	2	2	no	—
51	Q₈×Dic₅₃	—	2	2	no	—
52	C₄×Dic₁₀₆	—	4	1	no	—
53	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
54	C₄×D₂₁₂	—	2	1	no	—
55	#13133175	—	4	1	no	—
56	#13133160	—	2	2	no	—
57	C₂×#11719908	C₂³	1	1	yes	characteristic
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	#13133151	—	2	1	no	—
60	#13133151	C₂³	1	2	yes	—
61	#13133151	C₂³	1	1	yes	characteristic
62	#13133150	—	2	2	no	—
63	C₄⋊#88322	D₄	1	1	yes	characteristic
64	#13133146	D₄	1	2	yes	—
65	#13133141	—	2	2	no	—
66	#13133141	—	2	2	no	—
67	#13133139	—	2	1	no	—
68	C₈×Dic₅₃	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₅₃	—	2	1	no	—
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₅₃	—	2	1	no	—
73	C₄×Q₈×C₅₃	—	2	1	no	—
74	Q₃₂×C₅₃	—	2	4	no	—
75	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#103×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×D₂₁₂	—	2	1	no	—
82	C₂×C₄×D₅₃	—	4	1	no	—
83	C₂×Dic₁₀₆	—	4	1	no	—
84	C₂×Dic₁₀₆	—	4	2	no	—
85	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
86	Q₁₆×C₅₃	—	2	2	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	8	1	no	—
90	Q₁₆×C₅₃	#41	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	—	4	1	no	—
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	#11719924	—	8	2	no	—
100	#11719921	—	4	1	no	—
101	#11719921	—	4	2	no	—
102	#11719921	—	4	2	no	—
103	C₄⋊Dic₅₃	—	2	1	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	4	1	no	—
106	C₄⋊Dic₅₃	—	4	2	no	—
107	C₄⋊Dic₅₃	—	4	2	no	—
108	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
109	#11719919	—	4	2	no	—
110	#11719919	D₈	1	2	yes	—
111	C₄×Dic₅₃	—	4	2	no	—
112	C₄×Dic₅₃	—	4	2	no	—
113	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
114	C₂×#88322	C₂×D₄	1	2	yes	—
115	D₄₂₄	—	2	1	no	—
116	D₄₂₄	—	2	1	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
134	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
135	D₂₁₂	—	4	1	no	—
136	D₂₁₂	—	4	1	no	—
137	C₄×D₅₃	—	8	2	no	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	—	16	2	no	—
140	Dic₁₀₆	—	16	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#88322	—	2	2	no	—
145	#33936	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₃	—	8	1	no	—
148	C₂×D₅₃	—	8	1	no	—
149	C₂×D₅₃	—	8	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	—	8	1	no	—
154	C₄×C₅₃	#510	1	2	yes	—
155	C₄×C₅₃	#548	1	1	yes	characteristic
156	C₄×C₅₃	#577	1	1	yes	characteristic
157	Dic₅₃	—	2	2	no	—
158	Dic₅₃	—	2	2	no	—
159	Dic₅₃	—	8	1	no	—
160	Dic₅₃	—	8	2	no	—
161	#3393	—	53	1	no	—
162	Q₈×Q₁₆	—	53	1	no	—
163	#2243	—	53	2	no	—
164	#2241	—	53	1	no	—
165	#2226	—	53	1	no	—
166	#2225	—	53	2	no	—
167	#2211	—	53	2	no	—
168	#2209	D₅₃	1	1	yes	characteristic, complemented
169	#2183	—	53	1	no	—
170	#2176	—	53	2	no	—
171	C₄×SD₃₂	—	53	1	no	—
172	C₂×C₅₃	#3358	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
174	C₂×C₅₃	#3416	1	1	yes	characteristic
175	D₅₃	—	16	1	no	—
176	Q₈²	—	106	1	no	—
177	#556	—	106	1	no	—
178	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
179	C₂×SD₃₂	—	53	1	no	—
180	#500	—	53	2	no	—
181	#500	—	53	1	no	—
182	#495	—	53	2	no	—
183	#494	C₂×D₅₃	1	1	yes	characteristic
184	#477	—	106	2	no	—
185	#477	—	106	1	no	—
186	#464	—	106	2	no	—
187	#461	—	106	1	no	—
188	C₈×Q₈	—	53	1	no	—
189	C₄×Q₁₆	—	53	2	no	—
190	C₄×Q₁₆	—	53	2	no	—
191	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
192	C₄×D₈	—	53	1	no	—
193	#367	—	53	2	no	—
194	#367	—	53	1	no	—
195	C₄⋊C₁₆	—	53	2	no	—
196	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
197	#358	—	53	1	no	—
198	#358	—	53	2	no	—
199	#358	—	53	2	no	—
200	#358	C₂×D₅₃	1	2	yes	—
201	#357	—	53	2	no	—
202	#357	—	53	1	no	—
203	#346	—	53	2	no	—
204	C₄×C₁₆	—	53	1	no	—
205	C₅₃	#33936	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
208	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	106	2	no	—
210	C₂×D₈	—	53	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	106	2	no	—
213	C₄⋊Q₈	—	106	1	no	—
214	C₄⋊Q₈	—	212	1	no	—
215	C₄⋊Q₈	—	212	2	no	—
216	#126	—	212	2	no	—
217	#124	—	106	2	no	—
218	C₂²⋊Q₈	—	212	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	53	1	no	—
221	C₄×Q₈	—	106	2	no	—
222	C₄×Q₈	—	106	2	no	—
223	C₄×Q₈	—	212	1	no	—
224	C₄×D₄	—	106	1	no	—
225	#117	—	212	1	no	—
226	Q₃₂	—	2	4	no	—
227	SD₃₂	—	106	2	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
231	#107	—	53	2	no	—
232	#107	—	53	1	no	—
233	#107	—	106	2	no	—
234	#107	—	106	1	no	—
235	#107	C₂²×D₅₃	1	1	yes	characteristic
236	C₄⋊C₈	—	53	2	no	—
237	C₄⋊C₈	—	53	1	no	—
238	#103	—	2	1	no	—
239	#103	—	106	2	no	—
240	#103	—	106	2	no	—
241	#102	—	106	1	no	—
242	C₄×C₈	—	53	2	no	—
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	53	1	no	—
247	C₂×Q₈	—	212	2	no	—
248	C₂×Q₈	—	212	1	no	—
249	C₂×D₄	—	106	1	no	—
250	C₂²×C₄	—	212	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	8	1	no	—
255	Q₁₆	#11719947	1	2	yes	—
256	SD₁₆	—	424	2	no	—
257	D₈	—	106	1	no	—
258	D₈	—	106	1	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	53	1	no	—
267	C₄⋊C₄	—	53	2	no	—
268	C₄⋊C₄	—	106	1	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	—	212	2	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	2	no	—
273	C₄⋊C₄	—	212	1	no	—
274	C₂²⋊C₄	—	212	2	no	—
275	C₂²⋊C₄	—	212	2	no	—
276	C₂²⋊C₄	—	212	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	53	2	no	—
279	C₄²	—	212	2	no	—
280	C₄²	—	212	2	no	—
281	C₄²	D₄×D₅₃	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	106	1	no	—
284	C₁₆	—	106	1	no	—
285	C₂³	—	212	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	848	1	no	—
291	Q₈	—	848	2	no	—
292	Q₈	—	106	2	no	—
293	D₄	—	212	1	no	—
294	D₄	—	212	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	424	2	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	106	2	no	—
308	C₈	C₂×#11719947	1	1	yes	characteristic
309	C₈	#13133264	1	1	yes	characteristic
310	C₂²	—	424	1	no	—
311	C₂²	—	424	1	no	—
312	C₂²	—	424	1	no	—
313	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	106	2	no	—
319	C₄	—	106	2	no	—
320	C₄	—	424	1	no	—
321	C₄	—	424	2	no	—
322	C₄	#16811065	1	2	yes	—
323	C₄	#16811279	1	1	yes	characteristic
324	C₄	#16811433	1	1	yes	characteristic
325	C₂	—	848	1	no	—
326	C₂	#20287345	1	1	yes	characteristic
327	C₂	C₂×#16811065	1	1	yes	characteristic
328	C₂	#20287860	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2209	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	#2209	D₅₃
hypocenter	C₅₃	#33936
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₀₆	C₂	semidirect
4. #20276942	C₂	semidirect
6. #20276855	C₂	semidirect
8. #20276620	C₂	semidirect
9. #20276586	C₂	semidirect
11. C₅₃×#2209	C₂	semidirect
168. #2209	D₅₃	semidirect
205. C₅₃	#33936	semidirect

Maximal subgroups

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

subgroup	index
#20287705	2
Q₁₆×Dic₁₀₆	2
#20276953	2
#20276942	2
#20276863	2
#20276855	2
#20276779	2
#20276620	2
#20276586	2
C₄×#13133160	2
C₅₃×#2209	2
#33936	53