Subgroups — #27353869

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283676	C₂	1	2	yes	maximal, complemented
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20273628	C₂	1	2	yes	maximal, complemented
4	#20273623	C₂	1	2	yes	maximal, complemented
5	#20273616	C₂	1	4	yes	maximal, complemented
6	#20273095	C₂	1	2	yes	maximal, complemented
7	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₄	1	1	yes	characteristic
10	C₄×#11719945	C₂²	1	2	yes	—
11	#16810750	C₂²	1	4	yes	—
12	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₄×#11719913	C₄	1	2	yes	—
15	C₄×#11719913	C₂²	1	2	yes	—
16	C₄×D₄₂₄	C₄	1	1	yes	characteristic
17	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
18	#16810700	C₂²	1	4	yes	—
19	#16810691	—	2	4	no	—
20	#16810674	—	2	2	no	—
21	#16810661	C₄	1	1	yes	characteristic
22	#16810620	C₄	1	2	yes	—
23	C₄×#11719912	—	2	2	no	—
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	#16810353	C₂²	1	4	yes	—
27	#16810350	C₂²	1	4	yes	—
28	#16810344	C₂²	1	2	yes	—
29	#16810344	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	2	yes	—
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133239	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719913	C₂×C₄	1	2	yes	—
38	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719912	—	2	4	no	—
40	C₂²×#88322	—	2	1	no	—
41	C₂²×#88322	D₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂³	1	2	yes	—
44	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
45	C₄×D₂₁₂	C₂³	1	2	yes	—
46	C₄×C₂²⋊D₅₃	—	2	2	no	—
47	C₄×C₂²⋊D₅₃	—	2	2	no	—
48	#13133176	C₂×C₄	1	2	yes	—
49	#13133174	—	2	2	no	—
50	C₄²×D₅₃	—	2	2	no	—
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	#13133151	C₂×C₄	1	1	yes	characteristic
53	#13133149	C₂×C₄	1	2	yes	—
54	#13133146	—	2	4	no	—
55	#13133142	C₂×C₄	1	2	yes	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133139	C₂×C₄	1	2	yes	—
58	#13133139	C₂×C₄	1	2	yes	—
59	#13133129	—	2	4	no	—
60	#13133127	—	2	4	no	—
61	C₄×#88322	—	2	1	no	—
62	C₄×#88322	D₄	1	2	yes	—
63	C₄×#88322	C₂³	1	2	yes	—
64	C₄×#88322	C₂³	1	2	yes	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	—	2	2	no	—
79	#11719945	—	2	4	no	—
80	#11719945	—	2	4	no	—
81	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₅₃	—	2	2	no	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	—	2	2	no	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	#41	1	1	yes	characteristic
96	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₅₃	—	2	2	no	—
98	#11719921	—	2	2	no	—
99	#11719921	—	2	2	no	—
100	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
102	#11719919	—	2	2	no	—
103	#11719919	—	2	2	no	—
104	C₄×Dic₅₃	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	4	2	no	—
109	C₂×#88322	—	4	2	no	—
110	C₂×#88322	C₂×D₄	1	4	yes	—
111	Dic₂₁₂	—	2	2	no	—
112	D₄₂₄	—	2	2	no	—
113	#11719913	—	2	4	no	—
114	#11719912	—	4	8	no	—
115	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×D₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	#131	1	2	yes	—
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂²⋊D₅₃	—	4	4	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	D₂₁₂	—	2	2	no	—
132	D₂₁₂	—	2	2	no	—
133	C₄×D₅₃	—	4	4	no	—
134	C₄×D₅₃	—	4	4	no	—
135	C₄×D₅₃	—	4	4	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	4	4	no	—
141	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
142	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
143	#88322	—	2	4	no	—
144	#88322	—	4	2	no	—
145	#17890	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	#436	1	2	yes	—
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₅₃	—	4	2	no	—
151	C₂×D₅₃	—	4	2	no	—
152	C₂×D₅₃	—	4	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	#436	1	2	yes	—
156	C₄×C₅₃	#444	1	2	yes	—
157	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
159	C₄×C₅₃	#575	1	2	yes	—
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	4	2	no	—
162	Dic₅₃	—	4	2	no	—
163	#2965	—	53	2	no	—
164	C₄×#135	—	53	1	no	—
165	#1594	—	53	2	no	—
166	#1593	—	53	2	no	—
167	#1592	—	53	4	no	—
168	#1564	D₅₃	1	1	yes	characteristic, complemented
169	#1476	—	53	2	no	—
170	C₂×#322	—	53	1	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
173	C₂×C₅₃	#2943	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
175	D₅₃	—	8	2	no	—
176	C₂×#135	—	53	1	no	—
177	C₄×#41	—	53	2	no	—
178	#446	—	53	4	no	—
179	C₄×Q₁₆	—	53	1	no	—
180	C₄×Q₁₆	—	53	1	no	—
181	C₄×SD₁₆	—	53	2	no	—
182	C₄×SD₁₆	—	53	2	no	—
183	C₄×D₈	—	53	1	no	—
184	C₄×D₈	—	53	1	no	—
185	#436	—	53	4	no	—
186	#435	—	106	4	no	—
187	#431	—	106	2	no	—
188	#431	C₂×D₅₃	1	2	yes	—
189	#427	—	53	1	no	—
190	#416	—	53	2	no	—
191	C₄×M₁₆	—	106	2	no	—
192	C₂×C₄×C₈	—	53	1	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	#335	C₂×D₅₃	1	2	yes	—
196	#334	C₂×D₅₃	1	2	yes	—
197	Q₈⋊C₈	—	53	4	no	—
198	D₄⋊C₈	—	53	4	no	—
199	#322	—	53	2	no	—
200	#322	—	53	2	no	—
201	C₅₃	#17890	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	53	2	no	—
203	#135	—	106	4	no	—
204	C₂×Q₁₆	—	53	1	no	—
205	C₂×SD₁₆	—	53	2	no	—
206	C₂×D₈	—	53	1	no	—
207	C₂×M₁₆	—	106	4	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	106	1	no	—
210	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
212	C₄×Q₈	—	53	2	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×D₄	—	53	2	no	—
215	C₄×D₄	—	53	2	no	—
216	C₄×D₄	—	106	2	no	—
217	C₄×D₄	—	106	2	no	—
218	#117	—	53	2	no	—
219	#117	—	106	2	no	—
220	C₂×C₄²	—	106	2	no	—
221	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
222	#107	—	53	1	no	—
223	#107	—	53	1	no	—
224	#106	—	53	2	no	—
225	C₄⋊C₈	—	106	4	no	—
226	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
227	#103	—	53	2	no	—
228	#103	—	53	2	no	—
229	#102	—	53	2	no	—
230	#102	—	53	2	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₂²⋊C₈	—	106	4	no	—
233	#97	—	106	4	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	106	1	no	—
239	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
240	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	#41	—	106	4	no	—
244	#41	—	106	4	no	—
245	C₂×Q₈	—	53	2	no	—
246	C₂×D₄	—	53	2	no	—
247	C₂×D₄	—	106	2	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	#11719949	1	1	yes	characteristic
254	Q₁₆	—	106	2	no	—
255	SD₁₆	—	106	4	no	—
256	D₈	—	106	2	no	—
257	M₁₆	—	212	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	212	2	no	—
263	C₂×C₈	—	212	2	no	—
264	C₂×C₈	#11719912	1	4	yes	complemented
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₄⋊C₄	—	53	2	no	—
270	C₄⋊C₄	—	53	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₂²⋊C₄	—	106	2	no	—
274	C₂²⋊C₄	—	106	2	no	—
275	C₂²⋊C₄	—	106	2	no	—
276	C₄²	—	106	2	no	—
277	C₄²	—	106	2	no	—
278	C₄²	—	106	2	no	—
279	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₄²	D₄×D₅₃	1	1	yes	characteristic
281	C₄²	#11719949	1	1	yes	characteristic
282	C₄²	C₂³×D₅₃	1	1	yes	characteristic
283	C₂³	—	106	2	no	—
284	C₂³	#13133179	1	1	yes	characteristic
285	Q₈	—	106	2	no	—
286	Q₈	—	106	2	no	—
287	D₄	—	106	2	no	—
288	D₄	—	106	2	no	—
289	D₄	—	212	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	212	4	no	—
296	C₂×C₄	—	212	4	no	—
297	C₂×C₄	—	212	4	no	—
298	C₂×C₄	#13133179	1	1	yes	characteristic
299	C₂×C₄	#13133179	1	1	yes	characteristic
300	C₂×C₄	#13133179	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719912	1	2	yes	—
302	C₂×C₄	#13133225	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	4	no	—
309	C₈	—	106	4	no	—
310	C₈	—	212	2	no	—
311	C₈	C₂×#11719912	1	2	yes	—
312	C₈	C₂×#11719912	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	212	2	no	—
316	C₂²	—	212	2	no	—
317	C₂²	—	212	2	no	—
318	C₂²	#16810700	1	2	yes	—
319	C₂²	C₂×#13133179	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	212	2	no	—
323	C₄	—	212	2	no	—
324	C₄	—	212	2	no	—
325	C₄	#16810700	1	2	yes	—
326	C₄	#16810744	1	2	yes	—
327	C₄	C₂²×#11719912	1	1	yes	characteristic
328	C₄	C₂×#13133225	1	1	yes	characteristic
329	C₄	#16811420	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	424	2	no	—
332	C₂	C₂×#16810700	1	1	yes	characteristic
333	C₂	#20283515	1	1	yes	characteristic
334	C₂	C₂×#16810744	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#17890
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #20283676	C₂	semidirect
3. #20273628	C₂	semidirect
4. #20273623	C₂	semidirect
5. #20273616	C₂	semidirect
6. #20273095	C₂	semidirect
7. C₂×#16810344	C₂	semidirect
8. C₅₃×#1564	C₂	semidirect
88. C₂×C₈×C₅₃	M₁₆	semidirect
168. #1564	D₅₃	semidirect
201. C₅₃	#17890	semidirect
264. C₂×C₈	#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283676	2
C₄×#13133239	2
#20273628	2
#20273623	2
#20273616	2
#20273095	2
C₂×#16810344	2
C₅₃×#1564	2
#17890	53