Subgroups — #27541043

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287147	C₂	1	1	yes	maximal, characteristic, complemented
2	#20286474	C₂	1	1	yes	maximal, characteristic
3	#20276939	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276909	C₂	1	2	yes	maximal, complemented
5	#20276857	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276814	C₂	1	2	yes	maximal, complemented
7	#20276755	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276654	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276421	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133240	C₂²	1	2	yes	—
13	#16811346	—	2	1	no	—
14	#16811295	—	2	1	no	—
15	C₂×#13133160	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	#16811028	C₂²	1	1	yes	characteristic
18	#16811000	C₂²	1	1	yes	characteristic
19	#16810963	—	2	2	no	—
20	#16810908	—	2	1	no	—
21	#16810861	C₂²	1	2	yes	—
22	#16810860	C₂²	1	2	yes	—
23	#16810823	—	2	1	no	—
24	#16810771	—	2	2	no	—
25	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
26	#16810660	C₂²	1	2	yes	—
27	#16810450	C₂²	1	2	yes	—
28	C₄⋊#11719908	C₂²	1	1	yes	characteristic
29	#16810427	C₂²	1	2	yes	—
30	#16810425	C₂²	1	2	yes	—
31	#16810425	C₂²	1	2	yes	—
32	#16810423	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×#11719949	—	2	2	no	—
42	#13133240	—	2	4	no	—
43	#13133240	—	4	2	no	—
44	C₂×#11719924	—	2	2	no	—
45	C₂×D₄₂₄	—	2	1	no	—
46	C₂×D₄₂₄	C₂³	1	2	yes	—
47	C₂×C₈×D₅₃	C₂³	1	2	yes	—
48	#13133217	—	2	1	no	—
49	#13133214	—	2	1	no	—
50	#13133214	—	4	1	no	—
51	#13133212	—	4	2	no	—
52	#13133205	—	4	1	no	—
53	#13133193	—	2	2	no	—
54	#13133192	—	2	2	no	—
55	#13133192	—	4	2	no	—
56	C₄×Dic₁₀₆	—	4	1	no	—
57	C₄×D₂₁₂	—	4	1	no	—
58	C₄×D₂₁₂	D₄	1	1	yes	characteristic
59	#13133175	—	2	2	no	—
60	#13133160	—	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	#13133148	—	2	2	no	—
66	C₄⋊#88322	D₄	1	1	yes	characteristic
67	#13133141	—	2	2	no	—
68	#13133137	—	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	#11719949	—	4	4	no	—
85	#11719949	—	8	2	no	—
86	C₂×D₂₁₂	—	2	2	no	—
87	C₂×D₂₁₂	—	4	2	no	—
88	C₂×D₂₁₂	—	4	1	no	—
89	C₂×D₂₁₂	—	4	2	no	—
90	C₂×D₂₁₂	—	4	1	no	—
91	C₂×D₂₁₂	D₈	1	1	yes	characteristic
92	C₂×C₄×D₅₃	—	4	2	no	—
93	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₁₀₆	—	4	1	no	—
95	Q₁₆×C₅₃	—	2	1	no	—
96	Q₁₆×C₅₃	—	2	1	no	—
97	Q₁₆×C₅₃	—	2	1	no	—
98	Q₁₆×C₅₃	#41	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	—	4	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	#11719924	—	4	4	no	—
109	#11719921	—	4	2	no	—
110	#11719921	—	4	2	no	—
111	#11719921	—	4	2	no	—
112	#11719921	D₈	1	2	yes	—
113	C₄⋊Dic₅₃	—	2	2	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
116	#11719919	—	4	2	no	—
117	C₄×Dic₅₃	—	4	2	no	—
118	C₂×#88322	C₂×D₄	1	2	yes	—
119	D₄₂₄	—	2	2	no	—
120	D₄₂₄	—	2	2	no	—
121	D₄₂₄	—	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	—	4	2	no	—
128	C₂²×D₅₃	C₂×D₈	1	2	yes	—
129	Q₈×C₅₃	—	2	2	no	—
130	Q₈×C₅₃	—	4	1	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	Q₈×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	—	4	1	no	—
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₅₃	—	4	2	no	—
141	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
142	D₂₁₂	—	2	2	no	—
143	D₂₁₂	—	4	2	no	—
144	D₂₁₂	—	4	2	no	—
145	D₂₁₂	—	8	4	no	—
146	D₂₁₂	—	8	2	no	—
147	D₂₁₂	—	16	1	no	—
148	D₂₁₂	—	16	2	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	2	2	no	—
151	C₄×D₅₃	—	8	4	no	—
152	Dic₁₀₆	—	8	2	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
156	#88322	—	2	2	no	—
157	#33871	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₅₃	—	2	2	no	—
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	8	2	no	—
163	C₂×D₅₃	—	8	2	no	—
164	C₂×D₅₃	—	8	2	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	4	2	no	—
167	C₄×C₅₃	—	8	1	no	—
168	C₄×C₅₃	—	8	1	no	—
169	C₄×C₅₃	—	8	1	no	—
170	C₄×C₅₃	#508	1	2	yes	—
171	C₄×C₅₃	#546	1	1	yes	characteristic
172	C₄×C₅₃	#577	1	1	yes	characteristic
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	8	2	no	—
175	#3342	—	53	1	no	—
176	#3287	—	53	1	no	—
177	#2240	—	53	1	no	—
178	#2234	—	53	2	no	—
179	#2225	—	53	1	no	—
180	#2217	—	53	2	no	—
181	#2207	—	53	1	no	—
182	#2189	—	53	2	no	—
183	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
184	D₄×C₁₆	—	53	1	no	—
185	#2138	—	53	2	no	—
186	C₂×C₅₃	#3294	1	1	yes	characteristic
187	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
188	C₂×C₅₃	#3416	1	1	yes	characteristic
189	D₅₃	—	4	2	no	—
190	D₅₃	—	16	2	no	—
191	C₂×#135	—	53	2	no	—
192	#556	—	106	1	no	—
193	#550	—	106	1	no	—
194	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
195	C₂×SD₃₂	—	53	2	no	—
196	C₂²×C₁₆	—	53	2	no	—
197	#500	—	53	1	no	—
198	#493	—	53	1	no	—
199	#485	—	106	2	no	—
200	#475	—	106	1	no	—
201	#467	—	53	2	no	—
202	#466	—	53	2	no	—
203	#460	—	106	1	no	—
204	#449	—	106	2	no	—
205	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
206	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₈×D₄	—	53	1	no	—
208	#427	—	53	2	no	—
209	#367	—	53	2	no	—
210	#366	C₂×D₅₃	1	1	yes	characteristic
211	C₄⋊C₁₆	—	53	1	no	—
212	#358	—	53	2	no	—
213	#358	C₂×D₅₃	1	1	yes	characteristic
214	#358	C₂×D₅₃	1	1	yes	characteristic
215	#357	—	53	2	no	—
216	#357	—	53	2	no	—
217	#357	—	53	2	no	—
218	C₂²⋊C₁₆	—	53	2	no	—
219	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
220	C₅₃	#33871	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	106	2	no	—
222	#135	—	106	4	no	—
223	#135	—	212	2	no	—
224	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×SD₁₆	—	106	2	no	—
227	C₂×D₈	—	53	2	no	—
228	C₂×D₈	—	106	1	no	—
229	C₂²×C₈	—	53	2	no	—
230	C₄⋊Q₈	—	106	1	no	—
231	#127	—	106	1	no	—
232	#127	—	212	1	no	—
233	#126	—	212	2	no	—
234	#124	—	212	1	no	—
235	C₂²⋊Q₈	—	106	2	no	—
236	#121	—	106	2	no	—
237	#121	—	212	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	212	1	no	—
241	C₄×D₄	—	53	1	no	—
242	C₄×D₄	—	212	1	no	—
243	#117	—	106	2	no	—
244	Q₃₂	—	2	2	no	—
245	SD₃₂	—	212	2	no	—
246	C₂×C₁₆	—	53	2	no	—
247	C₂×C₁₆	—	212	2	no	—
248	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
250	#107	—	53	2	no	—
251	#107	—	106	1	no	—
252	#107	C₂²×D₅₃	1	1	yes	characteristic
253	#107	C₂²×D₅₃	1	1	yes	characteristic
254	#106	—	106	2	no	—
255	C₄⋊C₈	—	53	1	no	—
256	#103	—	2	1	no	—
257	#103	—	2	1	no	—
258	#103	—	106	2	no	—
259	#102	—	106	2	no	—
260	C₂²⋊C₈	—	53	2	no	—
261	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	#41	—	424	2	no	—
263	#41	—	212	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	212	1	no	—
267	C₂×D₄	—	53	1	no	—
268	C₂×D₄	—	106	2	no	—
269	C₂×D₄	—	212	2	no	—
270	C₂×D₄	—	212	2	no	—
271	C₂×D₄	—	212	1	no	—
272	C₂×D₄	—	212	1	no	—
273	C₂²×C₄	—	53	2	no	—
274	C₂²×C₄	—	212	2	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	#11719945	1	2	yes	—
279	SD₁₆	—	212	4	no	—
280	D₈	—	106	2	no	—
281	D₈	—	106	2	no	—
282	D₈	—	424	1	no	—
283	C₂×C₈	—	53	2	no	—
284	C₂×C₈	—	106	2	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
287	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	53	1	no	—
293	C₄⋊C₄	—	106	2	no	—
294	C₄⋊C₄	—	212	2	no	—
295	C₄⋊C₄	—	212	1	no	—
296	C₂²⋊C₄	—	53	2	no	—
297	C₂²⋊C₄	—	212	2	no	—
298	C₂²⋊C₄	—	212	2	no	—
299	C₂²⋊C₄	—	212	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	212	2	no	—
303	C₄²	D₄×D₅₃	1	1	yes	characteristic
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	106	2	no	—
307	C₂³	—	53	2	no	—
308	C₂³	—	212	2	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	424	2	no	—
314	D₄	—	848	1	no	—
315	D₄	—	848	2	no	—
316	D₄	—	106	2	no	—
317	D₄	—	424	4	no	—
318	D₄	—	424	2	no	—
319	D₄	—	212	2	no	—
320	D₄	—	212	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	424	4	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
331	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	106	2	no	—
335	C₈	C₂×#11719945	1	1	yes	characteristic
336	C₈	#13133263	1	1	yes	characteristic
337	C₂²	—	106	2	no	—
338	C₂²	—	106	2	no	—
339	C₂²	—	106	2	no	—
340	C₂²	—	424	2	no	—
341	C₂²	—	424	2	no	—
342	C₂²	—	424	2	no	—
343	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	106	2	no	—
350	C₄	—	424	2	no	—
351	C₄	#16811056	1	2	yes	—
352	C₄	#16811269	1	1	yes	characteristic
353	C₄	#16811430	1	1	yes	characteristic
354	C₂	—	848	2	no	—
355	C₂	—	212	2	no	—
356	C₂	#20286530	1	1	yes	characteristic
357	C₂	C₂×#16811056	1	1	yes	characteristic
358	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	#20286474	C₂
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33871
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
1. #20287147	C₂	semidirect
3. #20276939	C₂	semidirect
4. #20276909	C₂	semidirect
5. #20276857	C₂	semidirect
6. #20276814	C₂	semidirect
7. #20276755	C₂	semidirect
8. #20276654	C₂	semidirect
9. C₁₆×D₂₁₂	C₂	semidirect
10. #20276421	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
183. C₄×Q₃₂	D₅₃	semidirect
220. C₅₃	#33871	semidirect

Maximal subgroups

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

subgroup	index
#20287147	2
#20286474	2
#20276939	2
#20276909	2
#20276857	2
#20276814	2
#20276755	2
#20276654	2
C₁₆×D₂₁₂	2
#20276421	2
C₄×Q₃₂×C₅₃	2
#33871	53