Subgroups — #27540980

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287109	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20276943	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276919	C₂	1	2	yes	maximal, complemented
5	#20276867	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276805	C₂	1	2	yes	maximal, complemented
7	#20276765	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276647	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic
10	#358×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	#16811333	—	2	1	no	—
14	Q₈×D₂₁₂	—	2	1	no	—
15	C₂×#13133163	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	#16811028	C₂²	1	1	yes	characteristic
18	#16811003	C₂²	1	1	yes	characteristic
19	#16810957	—	2	2	no	—
20	#16810918	—	2	1	no	—
21	#16810865	C₂²	1	2	yes	—
22	#16810861	C₂²	1	2	yes	—
23	#16810833	—	2	1	no	—
24	#16810780	—	2	2	no	—
25	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
26	#107×D₅₃	C₂²	1	2	yes	—
27	#16810448	C₂²	1	2	yes	—
28	C₄⋊#11719908	C₂²	1	1	yes	characteristic
29	#16810428	C₂²	1	2	yes	—
30	#16810428	C₂²	1	2	yes	—
31	#16810427	C₂²	1	2	yes	—
32	#16810426	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	1	yes	characteristic
39	C₅₃×#358	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₅₃	—	2	2	no	—
42	C₂×Dic₂₁₂	—	2	1	no	—
43	C₂×Dic₂₁₂	C₂³	1	2	yes	—
44	C₂×#11719925	—	2	2	no	—
45	Q₁₆×D₅₃	—	2	4	no	—
46	Q₁₆×D₅₃	—	4	2	no	—
47	C₂×C₈×D₅₃	C₂³	1	2	yes	—
48	#13133217	—	2	1	no	—
49	#13133217	—	2	1	no	—
50	#13133217	—	4	1	no	—
51	#13133209	—	4	1	no	—
52	#13133197	—	2	2	no	—
53	#13133197	—	4	2	no	—
54	#13133195	—	4	2	no	—
55	#13133193	—	2	2	no	—
56	C₄×D₂₁₂	—	4	1	no	—
57	C₄×D₂₁₂	—	4	1	no	—
58	C₄×D₂₁₂	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	#13133163	—	4	2	no	—
61	C₂×#11719908	C₂³	1	2	yes	—
62	C₁₆×D₅₃	—	4	2	no	—
63	#13133151	—	2	1	no	—
64	#13133151	C₂³	1	2	yes	—
65	#13133150	—	2	2	no	—
66	C₄⋊#88322	D₄	1	1	yes	characteristic
67	#13133141	—	2	2	no	—
68	#13133140	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
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	2	no	—
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#107×C₅₃	C₂³	1	1	yes	characteristic
79	#103×C₅₃	—	2	1	no	—
80	#103×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	1	no	—
84	Q₈×D₅₃	—	4	4	no	—
85	Q₈×D₅₃	—	8	2	no	—
86	C₂×D₂₁₂	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	—	4	2	no	—
89	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₀₆	—	2	2	no	—
91	C₂×Dic₁₀₆	—	4	2	no	—
92	C₂×Dic₁₀₆	—	4	1	no	—
93	Q₁₆×C₅₃	—	2	1	no	—
94	Q₁₆×C₅₃	—	2	1	no	—
95	Q₁₆×C₅₃	—	2	1	no	—
96	Q₁₆×C₅₃	#41	1	2	yes	—
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	—	4	1	no	—
104	C₄²×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	#11719925	—	4	4	no	—
107	#11719921	—	4	2	no	—
108	#11719921	—	4	2	no	—
109	#11719921	D₈	1	2	yes	—
110	C₄⋊Dic₅₃	—	2	2	no	—
111	C₄⋊Dic₅₃	—	4	1	no	—
112	C₄⋊Dic₅₃	—	4	1	no	—
113	C₄⋊Dic₅₃	—	4	1	no	—
114	C₄⋊Dic₅₃	—	4	2	no	—
115	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
116	#11719919	—	4	2	no	—
117	#11719919	—	4	2	no	—
118	C₂×#88322	C₂×D₄	1	2	yes	—
119	Dic₂₁₂	—	2	2	no	—
120	Dic₂₁₂	—	2	2	no	—
121	Dic₂₁₂	—	8	1	no	—
122	C₈×D₅₃	—	2	2	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₁₆×C₅₃	—	2	1	no	—
125	C₁₆×C₅₃	—	2	1	no	—
126	#11719908	—	2	2	no	—
127	C₂²×D₅₃	C₂×D₈	1	2	yes	—
128	Q₈×C₅₃	—	2	2	no	—
129	Q₈×C₅₃	—	4	1	no	—
130	Q₈×C₅₃	—	4	1	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
142	D₂₁₂	Q₃₂	1	4	yes	complemented
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	2	no	—
145	C₄×D₅₃	—	8	4	no	—
146	C₄×D₅₃	—	16	2	no	—
147	Dic₁₀₆	—	4	2	no	—
148	Dic₁₀₆	—	4	2	no	—
149	Dic₁₀₆	—	8	4	no	—
150	Dic₁₀₆	—	16	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
154	#88322	—	2	2	no	—
155	#33866	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	C₂×Q₃₂	1	4	yes	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	4	2	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	—	8	1	no	—
164	C₄×C₅₃	—	8	1	no	—
165	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
166	C₄×C₅₃	#546	1	1	yes	characteristic
167	C₄×C₅₃	#577	1	1	yes	characteristic
168	Dic₅₃	—	2	2	no	—
169	Dic₅₃	—	8	2	no	—
170	Dic₅₃	—	8	2	no	—
171	#3338	—	53	1	no	—
172	D₄×Q₁₆	—	53	1	no	—
173	#2241	—	53	1	no	—
174	#2236	—	53	2	no	—
175	#2227	—	53	1	no	—
176	#2216	—	53	2	no	—
177	#2209	—	53	1	no	—
178	#2188	—	53	2	no	—
179	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
180	D₄×C₁₆	—	53	1	no	—
181	C₂×#358	—	53	2	no	—
182	C₂×C₅₃	#3294	1	1	yes	characteristic
183	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
184	C₂×C₅₃	#3416	1	1	yes	characteristic
185	D₅₃	—	2	4	no	—
186	C₂²×Q₁₆	—	53	2	no	—
187	#554	—	106	1	no	—
188	D₄×Q₈	—	106	1	no	—
189	C₂×Q₃₂	—	53	2	no	—
190	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
191	C₂²×C₁₆	—	53	2	no	—
192	#500	—	53	1	no	—
193	#494	—	53	1	no	—
194	#484	—	106	2	no	—
195	#477	—	106	1	no	—
196	#467	—	53	2	no	—
197	#467	—	53	2	no	—
198	#462	—	106	1	no	—
199	#451	—	106	2	no	—
200	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
201	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
202	C₈×D₄	—	53	1	no	—
203	C₂×#107	—	53	2	no	—
204	#366	—	53	2	no	—
205	#366	C₂×D₅₃	1	1	yes	characteristic
206	C₄⋊C₁₆	—	53	1	no	—
207	#358	—	53	2	no	—
208	#358	—	53	2	no	—
209	#358	—	53	2	no	—
210	#358	—	53	2	no	—
211	#358	C₂×D₅₃	1	1	yes	characteristic
212	#358	C₂×D₅₃	1	1	yes	characteristic
213	C₂²⋊C₁₆	—	53	2	no	—
214	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
215	C₅₃	#33866	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	106	2	no	—
217	C₂×Q₁₆	—	53	2	no	—
218	C₂×Q₁₆	—	106	2	no	—
219	C₂×Q₁₆	—	106	4	no	—
220	C₂×Q₁₆	—	106	1	no	—
221	C₂×Q₁₆	—	212	2	no	—
222	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
224	C₂²×C₈	—	53	2	no	—
225	C₄⋊Q₈	—	106	1	no	—
226	C₄⋊Q₈	—	106	1	no	—
227	C₄⋊Q₈	—	212	1	no	—
228	#125	—	212	1	no	—
229	C₂²⋊Q₈	—	106	2	no	—
230	C₂²⋊Q₈	—	106	2	no	—
231	C₂²⋊Q₈	—	212	2	no	—
232	C₂²⋊Q₈	—	212	2	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	53	1	no	—
236	C₄×D₄	—	212	1	no	—
237	C₄×D₄	—	212	1	no	—
238	C₂×C₄⋊C₄	—	106	2	no	—
239	Q₃₂	—	2	2	no	—
240	Q₃₂	—	212	2	no	—
241	C₂×C₁₆	—	53	2	no	—
242	C₂×C₁₆	—	212	2	no	—
243	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
245	#107	—	53	2	no	—
246	#107	—	106	2	no	—
247	#107	—	106	1	no	—
248	#107	C₂²×D₅₃	1	1	yes	characteristic
249	#107	C₂²×D₅₃	1	1	yes	characteristic
250	C₄⋊C₈	—	53	1	no	—
251	#103	—	2	1	no	—
252	#103	—	2	1	no	—
253	#103	—	106	2	no	—
254	#103	—	106	2	no	—
255	C₂²⋊C₈	—	53	2	no	—
256	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	106	2	no	—
260	C₂×Q₈	—	424	2	no	—
261	C₂×Q₈	—	212	2	no	—
262	C₂×Q₈	—	212	4	no	—
263	C₂×Q₈	—	212	1	no	—
264	C₂×D₄	—	53	1	no	—
265	C₂²×C₄	—	53	2	no	—
266	C₂²×C₄	—	212	2	no	—
267	C₂²×C₄	—	212	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	106	2	no	—
272	Q₁₆	—	106	2	no	—
273	Q₁₆	—	424	1	no	—
274	Q₁₆	—	212	4	no	—
275	Q₁₆	#11719945	1	2	yes	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	53	1	no	—
286	C₄⋊C₄	—	106	2	no	—
287	C₄⋊C₄	—	212	2	no	—
288	C₄⋊C₄	—	212	2	no	—
289	C₄⋊C₄	—	212	2	no	—
290	C₄⋊C₄	—	212	1	no	—
291	C₄⋊C₄	—	212	1	no	—
292	C₄⋊C₄	—	212	1	no	—
293	C₂²⋊C₄	—	53	2	no	—
294	C₂²⋊C₄	—	212	2	no	—
295	C₂²⋊C₄	—	212	2	no	—
296	C₄²	—	4	1	no	—
297	C₄²	—	4	1	no	—
298	C₄²	D₄×D₅₃	1	1	yes	characteristic
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	106	2	no	—
302	C₂³	—	53	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	848	1	no	—
308	Q₈	—	424	4	no	—
309	Q₈	—	212	2	no	—
310	Q₈	—	212	2	no	—
311	D₄	—	53	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	848	2	no	—
318	C₂×C₄	—	106	2	no	—
319	C₂×C₄	—	106	2	no	—
320	C₂×C₄	—	424	4	no	—
321	C₂×C₄	—	212	2	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
324	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	106	2	no	—
328	C₈	C₂×#11719945	1	1	yes	characteristic
329	C₈	#13133263	1	1	yes	characteristic
330	C₂²	—	53	4	no	—
331	C₂²	—	106	2	no	—
332	C₂²	—	106	2	no	—
333	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	106	2	no	—
340	C₄	—	424	2	no	—
341	C₄	—	424	2	no	—
342	C₄	Q₃₂×D₅₃	1	2	yes	—
343	C₄	#16811269	1	1	yes	characteristic
344	C₄	#16811430	1	1	yes	characteristic
345	C₂	—	106	4	no	—
346	C₂	#20286530	1	1	yes	characteristic
347	C₂	C₂×Q₃₂×D₅₃	1	1	yes	characteristic
348	C₂	#20287858	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₂₁₂	D₈
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33866
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #20276943	C₂	semidirect
4. #20276919	C₂	semidirect
5. #20276867	C₂	semidirect
6. #20276805	C₂	semidirect
7. #20276765	C₂	semidirect
8. #20276647	C₂	semidirect
10. #358×D₅₃	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
142. D₂₁₂	Q₃₂	semidirect
179. C₄×Q₃₂	D₅₃	semidirect
215. C₅₃	#33866	semidirect

Maximal subgroups

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

subgroup	index
#20287109	2
Q₁₆×D₂₁₂	2
#20276943	2
#20276919	2
#20276867	2
#20276805	2
#20276765	2
#20276647	2
C₁₆×D₂₁₂	2
#358×D₅₃	2
C₄×Q₃₂×C₅₃	2
#33866	53