Subgroups — #27277027

Order 13568.

454 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	C₂×#16810700	C₂	1	1	yes	maximal, characteristic
2	#20274704	C₂	1	1	yes	maximal, characteristic
3	#20274704	C₂	1	1	yes	maximal, characteristic
4	#20273473	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273470	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272590	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	2	yes	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	#16810700	—	2	1	no	—
11	#16810700	—	2	1	no	—
12	#16810700	—	2	1	no	—
13	#16810700	—	2	1	no	—
14	#16810647	C₂²	1	1	yes	characteristic
15	#16810647	C₂²	1	1	yes	characteristic
16	C₂×#13133133	—	2	1	no	—
17	C₂×#13133133	—	2	1	no	—
18	C₂×#13133129	C₄	1	2	yes	—
19	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
20	#16810391	—	2	1	no	—
21	#16810391	—	2	1	no	—
22	#16810375	—	2	1	no	—
23	#16810375	—	2	1	no	—
24	#16810373	—	2	1	no	—
25	#16810373	—	2	1	no	—
26	C₅₃×#423	C₂²	1	1	yes	characteristic
27	C₅₃×#423	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×#11719917	—	2	1	no	—
32	C₂×#11719917	—	2	1	no	—
33	C₂×#11719912	—	2	2	no	—
34	C₂×#11719912	—	2	2	no	—
35	C₂×#11719912	—	4	4	no	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
38	C₄×D₂₁₂	—	2	1	no	—
39	C₄×D₂₁₂	—	2	1	no	—
40	C₄×D₂₁₂	—	2	1	no	—
41	C₄×D₂₁₂	—	2	1	no	—
42	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
43	C₂×#11719921	Q₈	1	2	yes	—
44	C₄⋊#88322	—	2	1	no	—
45	C₄⋊#88322	—	2	1	no	—
46	C₄⋊#88322	—	2	1	no	—
47	C₄⋊#88322	—	2	1	no	—
48	#13133133	—	4	2	no	—
49	#13133133	—	4	2	no	—
50	#13133129	—	2	2	no	—
51	#13133129	—	2	2	no	—
52	C₄×#88322	D₄	1	1	yes	characteristic
53	C₄×#88322	D₄	1	1	yes	characteristic
54	C₄×#88322	D₄	1	1	yes	characteristic
55	C₄×#88322	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	—	2	1	no	—
57	C₂×M₁₆×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₂³×D₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
71	C₂×D₂₁₂	—	2	1	no	—
72	C₂×D₂₁₂	—	2	1	no	—
73	C₂×D₂₁₂	—	2	1	no	—
74	C₂×D₂₁₂	—	2	1	no	—
75	C₂×D₂₁₂	—	2	1	no	—
76	C₂×D₂₁₂	—	2	1	no	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	2	no	—
79	C₂×C₄×D₅₃	—	4	2	no	—
80	C₂×C₄×D₅₃	—	4	2	no	—
81	M₁₆×C₅₃	—	4	1	no	—
82	M₁₆×C₅₃	—	4	1	no	—
83	M₁₆×C₅₃	—	4	1	no	—
84	M₁₆×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	#11719921	—	2	2	no	—
94	#11719921	—	2	2	no	—
95	C₄⋊Dic₅₃	—	2	1	no	—
96	C₄⋊Dic₅₃	—	2	1	no	—
97	#11719917	—	4	1	no	—
98	#11719917	—	4	1	no	—
99	#11719917	—	4	1	no	—
100	#11719917	—	4	1	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	4	1	no	—
103	C₂×#88322	—	4	1	no	—
104	C₂×#88322	—	4	1	no	—
105	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
109	#11719912	—	4	4	no	—
110	#11719912	—	4	4	no	—
111	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×D₅₃	—	2	2	no	—
113	C₂²×D₅₃	—	2	2	no	—
114	C₂²×D₅₃	—	2	2	no	—
115	C₂²×D₅₃	—	4	2	no	—
116	C₂²×D₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	D₂₁₂	—	4	1	no	—
133	D₂₁₂	—	4	1	no	—
134	D₂₁₂	—	4	1	no	—
135	D₂₁₂	—	4	1	no	—
136	C₄×D₅₃	—	4	4	no	—
137	C₄×D₅₃	—	4	2	no	—
138	C₄×D₅₃	—	4	2	no	—
139	C₈×C₅₃	—	4	2	no	—
140	C₈×C₅₃	—	8	1	no	—
141	C₈×C₅₃	—	8	1	no	—
142	#88322	—	2	1	no	—
143	#88322	—	2	1	no	—
144	#88322	—	2	1	no	—
145	#88322	—	8	1	no	—
146	#88322	—	8	1	no	—
147	#88322	#100	1	2	yes	complemented
148	#8618	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	#380	1	1	yes	characteristic
152	C₂²×C₅₃	#428	1	1	yes	characteristic
153	C₂²×C₅₃	#428	1	1	yes	characteristic
154	C₂×D₅₃	—	4	2	no	—
155	C₂×D₅₃	—	4	2	no	—
156	C₂×D₅₃	—	4	2	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₂×D₅₃	—	8	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
163	C₄×C₅₃	#413	1	1	yes	characteristic
164	C₄×C₅₃	#468	1	1	yes	characteristic
165	C₄×C₅₃	#468	1	1	yes	characteristic
166	C₄×C₅₃	#469	1	1	yes	characteristic
167	C₄×C₅₃	#470	1	1	yes	characteristic
168	Dic₅₃	—	4	2	no	—
169	C₂×#436	—	53	1	no	—
170	#1800	—	53	1	no	—
171	#1800	—	53	1	no	—
172	#1567	—	53	1	no	—
173	#1566	—	53	1	no	—
174	#1292	—	53	1	no	—
175	#1292	D₅₃	1	1	yes	characteristic, complemented
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	2	1	no	—
178	C₂×C₅₃	#1800	1	1	yes	characteristic
179	C₂×C₅₃	#1809	1	1	yes	characteristic
180	C₂×C₅₃	#1813	1	1	yes	characteristic
181	D₅₃	—	8	2	no	—
182	C₂²×M₁₆	—	53	2	no	—
183	C₂×C₄×D₄	—	53	1	no	—
184	#436	—	106	1	no	—
185	#436	—	106	1	no	—
186	#436	—	106	1	no	—
187	#436	—	106	1	no	—
188	#423	—	53	1	no	—
189	#423	—	53	1	no	—
190	#423	C₂×D₅₃	1	1	yes	characteristic
191	#423	C₂×D₅₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₂×#100	—	106	1	no	—
194	C₂×#100	—	106	1	no	—
195	C₂×C₂²⋊C₈	—	53	2	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	#343	—	106	1	no	—
198	#343	—	106	1	no	—
199	#335	—	106	1	no	—
200	#335	—	106	1	no	—
201	#334	—	106	1	no	—
202	#334	—	106	1	no	—
203	C₅₃	#8618	1	1	yes	characteristic, complemented, Sylow
204	C₂²×D₄	—	53	1	no	—
205	C₂³×C₄	—	53	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	106	2	no	—
209	C₂×M₁₆	—	106	2	no	—
210	C₂×M₁₆	—	106	1	no	—
211	C₂×M₁₆	—	106	1	no	—
212	C₂×M₁₆	—	212	4	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	C₄×D₅₃	1	2	yes	—
216	C₄×D₄	—	106	1	no	—
217	C₄×D₄	—	106	1	no	—
218	C₄×D₄	—	106	1	no	—
219	C₄×D₄	—	106	1	no	—
220	C₂×C₄⋊C₄	—	53	1	no	—
221	C₂×C₂²⋊C₄	—	53	2	no	—
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	106	1	no	—
230	C₄⋊C₈	—	106	1	no	—
231	C₄⋊C₈	—	106	1	no	—
232	C₄⋊C₈	—	106	1	no	—
233	#100	—	212	2	no	—
234	#100	—	212	2	no	—
235	C₂²⋊C₈	—	106	2	no	—
236	C₂²⋊C₈	—	106	2	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	53	1	no	—
241	C₂⁴	—	53	2	no	—
242	C₂×D₄	—	106	1	no	—
243	C₂×D₄	—	106	1	no	—
244	C₂×D₄	—	106	1	no	—
245	C₂×D₄	—	106	1	no	—
246	C₂×D₄	—	106	1	no	—
247	C₂×D₄	—	106	1	no	—
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	—	212	2	no	—
252	C₂²×C₄	—	212	2	no	—
253	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	212	4	no	—
261	M₁₆	—	212	4	no	—
262	M₁₆	—	212	1	no	—
263	M₁₆	—	212	1	no	—
264	M₁₆	—	212	1	no	—
265	M₁₆	—	212	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	53	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	1	no	—
273	C₂×C₈	—	53	1	no	—
274	C₂×C₈	—	106	2	no	—
275	C₂×C₈	—	212	1	no	—
276	C₂×C₈	—	212	1	no	—
277	C₂×C₈	—	212	1	no	—
278	C₄⋊C₄	—	106	1	no	—
279	C₄⋊C₄	—	106	1	no	—
280	C₂²⋊C₄	—	106	2	no	—
281	C₂²⋊C₄	—	106	2	no	—
282	C₄²	#11719945	1	1	yes	characteristic
283	C₄²	#11719945	1	1	yes	characteristic
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₄²	D₄×D₅₃	1	1	yes	characteristic
286	C₂³	—	106	2	no	—
287	C₂³	—	106	2	no	—
288	C₂³	—	106	2	no	—
289	C₂³	—	212	2	no	—
290	C₂³	—	212	2	no	—
291	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
292	D₄	—	212	1	no	—
293	D₄	—	212	1	no	—
294	D₄	—	212	1	no	—
295	D₄	—	212	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	212	4	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	#13133174	1	1	yes	characteristic
306	C₂×C₄	#13133190	1	1	yes	characteristic
307	C₂×C₄	#13133190	1	1	yes	characteristic
308	C₂×C₄	#13133195	1	1	yes	characteristic
309	C₂×C₄	#13133195	1	1	yes	characteristic
310	C₂×C₄	#13133241	1	1	yes	characteristic
311	C₂×C₄	#13133244	1	1	yes	characteristic
312	C₂×C₄	#13133246	1	1	yes	characteristic
313	C₂×C₄	#13133249	1	1	yes	characteristic
314	C₈	—	4	2	no	—
315	C₈	—	8	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	53	2	no	—
318	C₈	—	106	1	no	—
319	C₈	—	106	1	no	—
320	C₈	—	106	1	no	—
321	C₈	—	424	1	no	—
322	C₈	—	424	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	424	2	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	212	2	no	—
328	C₂²	—	212	2	no	—
329	C₂²	—	212	2	no	—
330	C₂²	#16810482	1	1	yes	characteristic
331	C₂²	#16810662	1	1	yes	characteristic
332	C₂²	#16810664	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	212	2	no	—
337	C₄	#100×D₅₃	1	1	yes	characteristic
338	C₄	#16810606	1	1	yes	characteristic
339	C₄	#16810868	1	1	yes	characteristic
340	C₄	#16810872	1	1	yes	characteristic
341	C₄	#16810875	1	1	yes	characteristic
342	C₄	#16810880	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	424	2	no	—
346	C₂	#20274706	1	1	yes	characteristic
347	C₂	#20274747	1	1	yes	characteristic
348	C₂	#20274764	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810482
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1292	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₂₁₂	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1292	D₅₃
hypocenter	C₅₃	#8618
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
4. #20273473	C₂	semidirect
5. #20273470	C₂	semidirect
6. #20272590	C₂	semidirect
7. C₅₃×#1292	C₂	semidirect
147. #88322	#100	semidirect
175. #1292	D₅₃	semidirect
203. C₅₃	#8618	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810700	2
#20274704	2
#20273473	2
#20273470	2
#20272590	2
C₅₃×#1292	2
#8618	53