Subgroups — #27540880

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287863	C₂	1	2	yes	maximal, complemented
2	#20286502	C₂	1	2	yes	maximal, complemented
3	#20276976	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276845	C₂	1	2	yes	maximal, complemented
5	#20276785	C₂	1	2	yes	maximal, complemented
6	#20276695	C₂	1	2	yes	maximal, complemented
7	#20276674	C₂	1	2	yes	maximal, complemented
8	#20276597	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811440	C₂²	1	2	yes	—
11	#16811430	—	2	2	no	—
12	#16811428	C₂²	1	2	yes	complemented
13	#16811066	—	2	2	no	—
14	#16811066	C₂²	1	2	yes	complemented
15	#16811061	—	2	2	no	—
16	#16811061	C₂²	1	2	yes	—
17	#16811055	—	2	2	no	—
18	#16811055	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13133162	—	2	2	no	—
21	C₂×#13133162	C₂²	1	2	yes	complemented
22	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
23	#16811007	C₂²	1	2	yes	complemented
24	#16810898	C₂²	1	2	yes	complemented
25	#16810893	C₂²	1	2	yes	—
26	#16810887	—	2	2	no	—
27	#16810802	—	2	2	no	—
28	#16810797	—	2	2	no	—
29	#16810741	C₂²	1	2	yes	complemented
30	#16810453	C₂²	1	1	yes	characteristic, complemented
31	#16810440	C₂²	1	2	yes	complemented
32	#16810437	C₂²	1	2	yes	complemented
33	#16810431	C₂²	1	2	yes	complemented
34	C₅₃×#504	C₂²	1	2	yes	complemented
35	C₅₃×#443	C₂²	1	1	yes	characteristic
36	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#350	C₂²	1	2	yes	complemented
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	#13133264	—	2	2	no	—
40	#13133263	—	2	2	no	—
41	#13133250	—	2	2	no	—
42	#13133250	—	4	2	no	—
43	#13133249	—	4	2	no	—
44	#13133247	—	2	2	no	—
45	#13133245	—	2	2	no	—
46	#13133245	—	4	2	no	—
47	#13133242	—	2	2	no	—
48	#13133241	—	4	2	no	—
49	#13133239	—	2	2	no	—
50	#13133239	—	2	2	no	—
51	#13133239	C₂³	1	2	yes	—
52	#13133238	—	4	2	no	—
53	C₂×Dic₂₁₂	—	2	2	no	—
54	C₂×Dic₂₁₂	C₂³	1	2	yes	—
55	C₂×#11719913	—	2	2	no	—
56	SD₁₆×D₅₃	—	4	2	no	—
57	C₂×D₄₂₄	—	2	2	no	—
58	C₂×D₄₂₄	C₂³	1	2	yes	—
59	#13133223	—	2	2	no	—
60	#13133205	—	2	2	no	—
61	Dic₄₂₄	—	2	2	no	—
62	Dic₄₂₄	—	4	1	no	—
63	Dic₄₂₄	D₄	1	1	yes	characteristic
64	#13133162	—	2	2	no	—
65	#13133162	—	2	2	no	—
66	#13133162	—	2	2	no	—
67	#13133162	—	2	2	no	—
68	#13133162	D₄	1	2	yes	—
69	D₈₄₈	—	2	2	no	—
70	D₈₄₈	—	4	1	no	—
71	D₈₄₈	D₄	1	1	yes	characteristic
72	#13133153	C₂³	1	2	yes	—
73	#13133145	—	2	2	no	—
74	#13133135	—	2	2	no	—
75	#13133133	—	2	2	no	—
76	#135×C₅₃	D₄	1	1	yes	characteristic
77	#131×C₅₃	C₂³	1	2	yes	—
78	M₃₂×C₅₃	—	2	2	no	—
79	M₃₂×C₅₃	C₂³	1	2	yes	—
80	C₂×C₁₆×C₅₃	—	2	2	no	—
81	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
83	#108×C₅₃	D₄	1	1	yes	characteristic
84	#104×C₅₃	D₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	#41×C₅₃	C₂×D₄	1	2	yes	—
87	#11719949	—	4	2	no	—
88	Q₈×D₅₃	—	4	2	no	—
89	#11719947	—	4	2	no	—
90	#11719947	—	8	2	no	—
91	D₄×D₅₃	—	4	2	no	—
92	D₄×D₅₃	—	8	2	no	—
93	#11719945	—	2	2	no	—
94	#11719945	—	4	2	no	—
95	#11719945	—	4	2	no	—
96	C₂×D₂₁₂	—	2	2	no	—
97	C₂×D₂₁₂	—	4	2	no	—
98	C₂×Dic₁₀₆	—	2	2	no	—
99	C₂×Dic₁₀₆	—	4	2	no	—
100	Q₁₆×C₅₃	D₈	1	1	yes	characteristic
101	SD₁₆×C₅₃	D₈	1	2	yes	—
102	D₈×C₅₃	D₈	1	1	yes	characteristic
103	M₁₆×C₅₃	—	2	2	no	—
104	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	#11719925	—	4	2	no	—
110	#11719924	—	4	2	no	—
111	#11719923	—	4	2	no	—
112	#11719922	—	4	2	no	—
113	#11719921	—	8	2	no	—
114	#11719917	—	2	2	no	—
115	Dic₂₁₂	—	2	2	no	—
116	Dic₂₁₂	—	2	2	no	—
117	Dic₂₁₂	—	4	2	no	—
118	Dic₂₁₂	C₂×D₄	1	2	yes	—
119	D₄₂₄	—	2	2	no	—
120	D₄₂₄	—	2	2	no	—
121	D₄₂₄	—	4	2	no	—
122	D₄₂₄	C₂×D₄	1	2	yes	—
123	#11719913	—	2	2	no	—
124	#11719913	—	4	2	no	—
125	#11719913	—	4	2	no	—
126	#11719913	—	4	2	no	—
127	#11719913	—	4	2	no	—
128	#11719912	—	4	2	no	—
129	C₈×D₅₃	—	4	2	no	—
130	C₁₆×C₅₃	—	2	2	no	—
131	C₁₆×C₅₃	—	2	1	no	—
132	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×D₅₃	—	4	2	no	—
135	C₂²×D₅₃	—	8	2	no	—
136	Q₈×C₅₃	C₂×D₈	1	2	yes	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₅₃	—	4	2	no	—
143	C₂²⋊D₅₃	—	8	2	no	—
144	C₂²⋊D₅₃	—	8	2	no	—
145	C₂×Dic₅₃	—	4	2	no	—
146	C₂×Dic₅₃	—	8	2	no	—
147	D₂₁₂	—	2	2	no	—
148	D₂₁₂	—	4	2	no	—
149	D₂₁₂	—	4	2	no	—
150	D₂₁₂	—	8	2	no	—
151	C₄×D₅₃	—	4	2	no	—
152	C₄×D₅₃	—	8	2	no	—
153	C₄×D₅₃	—	8	2	no	—
154	Dic₁₀₆	—	2	2	no	—
155	Dic₁₀₆	—	4	2	no	—
156	Dic₁₀₆	—	4	2	no	—
157	Dic₁₀₆	—	8	2	no	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
160	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
161	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	#88322	—	4	2	no	—
164	#33857	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	#576	1	1	yes	characteristic
167	C₂×D₅₃	—	4	2	no	—
168	C₂×D₅₃	—	8	2	no	—
169	C₂×D₅₃	—	16	2	no	—
170	C₂×D₅₃	—	32	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	4	1	no	—
173	C₄×C₅₃	D₄²	1	1	yes	characteristic
174	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
175	Dic₅₃	—	4	2	no	—
176	Dic₅₃	—	8	2	no	—
177	#3416	—	53	2	no	—
178	#3290	—	53	2	no	—
179	#2249	—	53	1	no	—
180	#2223	—	53	2	no	—
181	#2212	—	53	2	no	—
182	#2195	—	53	2	no	—
183	#2192	—	53	2	no	—
184	#2178	—	53	1	no	—
185	#2170	D₅₃	1	1	yes	characteristic, complemented
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	4	2	no	—
188	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
189	D₅₃	—	8	2	no	—
190	D₅₃	—	16	2	no	—
191	#578	—	53	2	no	—
192	#577	—	106	2	no	—
193	#576	—	53	2	no	—
194	#510	—	53	2	no	—
195	#510	—	106	2	no	—
196	#509	—	53	2	no	—
197	#509	—	106	2	no	—
198	#508	—	53	1	no	—
199	#508	—	106	2	no	—
200	C₂×Q₃₂	—	53	1	no	—
201	C₂×SD₃₂	—	53	2	no	—
202	C₂×SD₃₂	—	106	2	no	—
203	C₂×D₁₆	—	53	1	no	—
204	#504	C₂×D₅₃	1	2	yes	complemented
205	#495	—	53	2	no	—
206	#473	—	53	2	no	—
207	#472	—	53	2	no	—
208	#471	—	106	2	no	—
209	#455	—	106	2	no	—
210	#454	—	106	2	no	—
211	#443	—	53	2	no	—
212	#443	C₂×D₅₃	1	1	yes	characteristic
213	#368	—	53	1	no	—
214	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
215	#362	—	53	2	no	—
216	#361	—	53	2	no	—
217	#359	—	53	2	no	—
218	#350	C₂×D₅₃	1	2	yes	complemented
219	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
220	C₅₃	#33857	1	1	yes	characteristic, complemented, Sylow
221	2₋¹⁺⁴	—	106	2	no	—
222	H(5,𝔽₂)	—	106	2	no	—
223	#137	—	106	2	no	—
224	#137	—	106	2	no	—
225	#137	—	212	2	no	—
226	#137	—	212	2	no	—
227	AGL(1,ℤ/8)	—	106	2	no	—
228	AGL(1,ℤ/8)	—	106	2	no	—
229	AGL(1,ℤ/8)	—	212	2	no	—
230	AGL(1,ℤ/8)	—	212	2	no	—
231	#135	—	53	2	no	—
232	#135	—	106	2	no	—
233	#135	—	106	2	no	—
234	#135	—	212	2	no	—
235	#135	D₂₁₂	1	1	yes	characteristic
236	C₂×Q₁₆	—	53	2	no	—
237	C₂×Q₁₆	—	106	2	no	—
238	C₂×SD₁₆	—	106	2	no	—
239	C₂×SD₁₆	—	212	2	no	—
240	C₂×D₈	—	53	2	no	—
241	C₂×D₈	—	106	2	no	—
242	#131	—	106	2	no	—
243	#131	C₂²×D₅₃	1	2	yes	—
244	#124	—	106	2	no	—
245	Q₃₂	—	53	1	no	—
246	Q₃₂	—	106	2	no	—
247	Q₃₂	—	212	1	no	—
248	SD₃₂	—	53	2	no	—
249	SD₃₂	—	106	2	no	—
250	SD₃₂	—	106	2	no	—
251	SD₃₂	—	106	2	no	—
252	SD₃₂	—	106	2	no	—
253	D₁₆	—	53	1	no	—
254	D₁₆	—	106	2	no	—
255	D₁₆	—	212	1	no	—
256	M₃₂	—	2	2	no	—
257	M₃₂	C₂²×D₅₃	1	2	yes	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
261	#108	—	53	2	no	—
262	#108	D₂₁₂	1	1	yes	characteristic
263	#104	—	106	2	no	—
264	#104	D₂₁₂	1	2	yes	—
265	#101	—	106	2	no	—
266	#100	—	106	2	no	—
267	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
268	#41	—	106	2	no	—
269	#41	—	424	2	no	—
270	#41	—	212	2	no	—
271	#41	—	212	2	no	—
272	#41	—	212	2	no	—
273	#41	—	212	2	no	—
274	#41	C₂×D₂₁₂	1	2	yes	—
275	C₂×Q₈	—	106	2	no	—
276	C₂×Q₈	—	212	2	no	—
277	C₂×Q₈	—	212	2	no	—
278	C₂×D₄	—	106	2	no	—
279	C₂×D₄	—	424	2	no	—
280	C₂×D₄	—	212	2	no	—
281	C₂×D₄	—	212	2	no	—
282	Q₁₆	—	53	2	no	—
283	Q₁₆	—	106	2	no	—
284	Q₁₆	—	106	2	no	—
285	Q₁₆	—	212	2	no	—
286	Q₁₆	—	212	2	no	—
287	Q₁₆	D₄₂₄	1	1	yes	characteristic
288	SD₁₆	—	106	2	no	—
289	SD₁₆	—	212	2	no	—
290	SD₁₆	—	212	2	no	—
291	SD₁₆	—	212	2	no	—
292	SD₁₆	—	212	2	no	—
293	SD₁₆	—	212	2	no	—
294	SD₁₆	—	212	2	no	—
295	SD₁₆	D₄₂₄	1	2	yes	—
296	D₈	—	53	2	no	—
297	D₈	—	106	2	no	—
298	D₈	—	106	2	no	—
299	D₈	—	212	2	no	—
300	D₈	—	212	2	no	—
301	D₈	D₄₂₄	1	1	yes	characteristic
302	M₁₆	—	2	2	no	—
303	M₁₆	—	106	2	no	—
304	M₁₆	—	212	2	no	—
305	M₁₆	C₂×D₂₁₂	1	2	yes	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	212	2	no	—
308	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
310	C₂²⋊C₄	—	424	2	no	—
311	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	D₄×D₅₃	1	1	yes	characteristic
315	C₁₆	D₄×D₅₃	1	1	yes	characteristic
316	C₂³	—	424	2	no	—
317	C₂³	—	212	2	no	—
318	Q₈	—	106	2	no	—
319	Q₈	—	424	2	no	—
320	Q₈	—	212	2	no	—
321	Q₈	—	212	2	no	—
322	Q₈	C₂×D₄₂₄	1	2	yes	—
323	D₄	—	2	2	no	—
324	D₄	—	106	2	no	—
325	D₄	—	424	2	no	—
326	D₄	—	424	2	no	—
327	D₄	—	424	2	no	—
328	D₄	—	212	2	no	—
329	D₄	—	212	2	no	—
330	D₄	—	212	2	no	—
331	D₄	C₂×D₄₂₄	1	2	yes	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	424	2	no	—
335	C₂×C₄	—	424	2	no	—
336	C₂×C₄	—	424	2	no	—
337	C₂×C₄	—	212	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	212	2	no	—
342	C₈	C₂×D₄₂₄	1	1	yes	characteristic
343	C₈	C₂×D₄₂₄	1	1	yes	characteristic
344	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
345	C₈	#13133263	1	1	yes	characteristic
346	C₂²	—	2	2	no	—
347	C₂²	—	848	2	no	—
348	C₂²	—	1696	2	no	—
349	C₂²	—	424	2	no	—
350	C₂²	—	212	2	no	—
351	C₂²	#16811428	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	424	2	no	—
355	C₄	—	212	2	no	—
356	C₄	D₄×D₂₁₂	1	1	yes	characteristic
357	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	4	2	no	—
360	C₂	—	848	2	no	—
361	C₂	—	424	2	no	—
362	C₂	D₄×D₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2170	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2170	D₅₃
hypocenter	C₅₃	#33857
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #20287863	C₂	semidirect
2. #20286502	C₂	semidirect
3. #20276976	C₂	semidirect
4. #20276845	C₂	semidirect
5. #20276785	C₂	semidirect
6. #20276695	C₂	semidirect
7. #20276674	C₂	semidirect
8. #20276597	C₂	semidirect
9. C₅₃×#2170	C₂	semidirect
12. #16811428	C₂²	semidirect
14. #16811066	C₂²	semidirect
19. C₂×Dic₄₂₄	C₂²	semidirect
21. C₂×#13133162	C₂²	semidirect
23. #16811007	C₂²	semidirect
24. #16810898	C₂²	semidirect
29. #16810741	C₂²	semidirect
30. #16810453	C₂²	semidirect
31. #16810440	C₂²	semidirect
32. #16810437	C₂²	semidirect
33. #16810431	C₂²	semidirect
34. C₅₃×#504	C₂²	semidirect
36. C₅₃×#364	C₂²	semidirect
37. C₅₃×#350	C₂²	semidirect
38. C₄×C₁₆×C₅₃	C₂²	semidirect
185. #2170	D₅₃	semidirect
204. #504	C₂×D₅₃	semidirect
214. #364	C₂×D₅₃	semidirect
218. #350	C₂×D₅₃	semidirect
219. C₄×C₁₆	C₂×D₅₃	semidirect
220. C₅₃	#33857	semidirect

Maximal subgroups

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

subgroup	index
#20287863	2
#20286502	2
#20276976	2
#20276845	2
#20276785	2
#20276695	2
#20276674	2
#20276597	2
C₅₃×#2170	2
#33857	53