Subgroups — #27284080

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272555	C₂	1	2	yes	maximal
4	#20272555	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	#16810521	C₄	1	1	yes	characteristic
10	#16810521	C₄	1	1	yes	characteristic
11	#16810507	—	2	1	no	—
12	#16810507	—	2	1	no	—
13	#16810489	C₄	1	1	yes	characteristic
14	C₄×#11719919	—	2	1	no	—
15	C₄×#11719919	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810378	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133209	—	2	2	no	—
25	#13133209	C₈	1	4	yes	complemented
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×#11719919	—	2	1	no	—
32	C₂×#11719919	—	2	1	no	—
33	C₂×#11719919	—	2	1	no	—
34	C₂×C₄×Dic₅₃	—	2	1	no	—
35	C₂×C₄×Dic₅₃	—	2	1	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	Q₈	1	4	yes	—
38	C₄×#88322	—	4	2	no	—
39	#13133124	—	2	1	no	—
40	#13133124	—	2	1	no	—
41	#13133124	—	2	1	no	—
42	#13133124	—	2	1	no	—
43	#13133124	—	2	1	no	—
44	#13133124	—	2	1	no	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	1	no	—
63	C₂²×Dic₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	2	1	no	—
65	C₂²×Dic₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	C₂×C₈	1	2	yes	—
76	C₄²×C₅₃	M₁₆	1	2	yes	—
77	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₄⋊Dic₅₃	—	2	1	no	—
80	C₄⋊Dic₅₃	—	2	1	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	—	2	2	no	—
83	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
85	#11719919	—	2	4	no	—
86	#11719919	—	2	4	no	—
87	#11719919	—	4	2	no	—
88	#11719919	—	4	2	no	—
89	C₄×Dic₅₃	—	4	2	no	—
90	C₄×Dic₅₃	—	4	1	no	—
91	C₄×Dic₅₃	—	4	1	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₅₃	#104	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#131	1	2	yes	—
117	C₂×C₄×C₅₃	#135	1	2	yes	—
118	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
120	C₂×Dic₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	—	4	1	no	—
123	C₂×Dic₅₃	—	4	1	no	—
124	C₂×Dic₅₃	—	4	1	no	—
125	C₂×Dic₅₃	—	4	1	no	—
126	C₂×Dic₅₃	—	4	1	no	—
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₈×C₅₃	—	4	4	no	—
133	#88322	—	4	4	no	—
134	#9866	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	#391	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₅₃	#408	1	1	yes	characteristic
138	C₂²×C₅₃	#416	1	1	yes	characteristic
139	C₂²×C₅₃	#417	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
141	C₂²×C₅₃	#421	1	1	yes	characteristic
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	4	1	no	—
148	C₄×C₅₃	C₈×Q₈	1	2	yes	—
149	C₄×C₅₃	#446	1	2	yes	—
150	C₄×C₅₃	#478	1	2	yes	—
151	C₄×C₅₃	#479	1	2	yes	—
152	Dic₅₃	—	4	2	no	—
153	Dic₅₃	—	4	2	no	—
154	Dic₅₃	—	8	1	no	—
155	Dic₅₃	—	8	1	no	—
156	C₄×#125	—	53	1	no	—
157	C₄×C₄⋊C₈	—	53	1	no	—
158	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
159	#1279	—	53	2	no	—
160	#1279	—	53	2	no	—
161	C₂×C₅₃	#1476	1	1	yes	characteristic
162	C₂×C₅₃	#1477	1	1	yes	characteristic
163	C₂×C₅₃	#1482	1	1	yes	characteristic
164	C₂×C₅₃	#1483	1	1	yes	characteristic
165	C₂×C₅₃	#1984	1	1	yes	characteristic
166	C₂×C₅₃	#1986	1	1	yes	characteristic
167	C₂×C₅₃	#1989	1	1	yes	characteristic
168	C₂×#125	—	53	1	no	—
169	C₂×C₄⋊C₈	—	2	1	no	—
170	C₂×C₄⋊C₈	—	53	2	no	—
171	C₂×C₄×C₈	—	106	1	no	—
172	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
173	#389	—	53	1	no	—
174	#389	—	53	1	no	—
175	#387	—	106	1	no	—
176	#387	—	106	1	no	—
177	#382	—	53	1	no	—
178	C₄×C₄⋊C₄	—	53	1	no	—
179	C₄×C₄⋊C₄	—	53	1	no	—
180	C₄×C₄⋊C₄	—	106	1	no	—
181	C₄×C₄⋊C₄	—	106	1	no	—
182	C₄³	C₂×D₅₃	1	1	yes	characteristic
183	#336	—	2	1	no	—
184	#336	—	106	1	no	—
185	C₅₃	#9866	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	106	2	no	—
188	#125	—	53	4	no	—
189	#125	—	106	2	no	—
190	C₂×C₄⋊C₄	—	53	1	no	—
191	C₂×C₄⋊C₄	—	53	1	no	—
192	C₂×C₄⋊C₄	—	53	1	no	—
193	C₂×C₄⋊C₄	—	53	1	no	—
194	C₂×C₄⋊C₄	—	106	1	no	—
195	C₂×C₄⋊C₄	—	106	1	no	—
196	C₂×C₄⋊C₄	—	106	1	no	—
197	C₂×C₄⋊C₄	—	106	1	no	—
198	C₂×C₄²	—	106	1	no	—
199	C₂×C₄²	—	106	1	no	—
200	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
203	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	53	4	no	—
209	C₄⋊C₈	—	106	2	no	—
210	C₄×C₈	—	2	4	no	—
211	C₄×C₈	—	212	2	no	—
212	#95	—	106	1	no	—
213	#95	—	106	1	no	—
214	#95	—	106	1	no	—
215	#95	—	106	1	no	—
216	#95	—	106	1	no	—
217	#95	—	106	1	no	—
218	C₂²×C₄	—	106	1	no	—
219	C₂²×C₄	—	106	1	no	—
220	C₂²×C₄	—	106	1	no	—
221	C₂²×C₄	—	106	1	no	—
222	C₂²×C₄	#11719921	1	1	yes	characteristic
223	C₂²×C₄	#11719921	1	1	yes	characteristic
224	C₂²×C₄	#11719921	1	1	yes	characteristic
225	C₂²×C₄	#11719921	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	212	2	no	—
234	C₂×C₈	—	212	2	no	—
235	C₄⋊C₄	—	53	2	no	—
236	C₄⋊C₄	—	53	2	no	—
237	C₄⋊C₄	—	106	4	no	—
238	C₄⋊C₄	—	106	4	no	—
239	C₄⋊C₄	—	106	1	no	—
240	C₄⋊C₄	—	106	1	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₄⋊C₄	—	106	2	no	—
243	C₄⋊C₄	—	212	2	no	—
244	C₄⋊C₄	—	212	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	212	2	no	—
252	C₄²	—	212	1	no	—
253	C₄²	—	212	1	no	—
254	C₄²	C₈×D₅₃	1	2	yes	—
255	C₄²	#11719912	1	2	yes	—
256	C₄²	#11719947	1	2	yes	—
257	C₄²	Q₈×D₅₃	1	2	yes	—
258	C₂³	C₂×#11719921	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	106	2	no	—
271	C₂×C₄	—	106	2	no	—
272	C₂×C₄	—	212	1	no	—
273	C₂×C₄	—	212	1	no	—
274	C₂×C₄	—	212	1	no	—
275	C₂×C₄	—	212	1	no	—
276	C₂×C₄	—	212	1	no	—
277	C₂×C₄	—	212	1	no	—
278	C₂×C₄	—	212	1	no	—
279	C₂×C₄	—	212	1	no	—
280	C₂×C₄	—	212	1	no	—
281	C₂×C₄	—	212	1	no	—
282	C₂×C₄	#13133129	1	2	yes	—
283	C₂×C₄	#13133129	1	2	yes	—
284	C₂×C₄	#13133145	1	2	yes	—
285	C₂×C₄	#13133182	1	2	yes	—
286	C₂×C₄	#13133197	1	2	yes	—
287	C₂×C₄	#13133204	1	1	yes	characteristic
288	C₂×C₄	#13133215	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
291	C₂×C₄	#13133225	1	2	yes	—
292	C₂×C₄	#13133239	1	2	yes	—
293	C₂×C₄	#13133245	1	1	yes	characteristic
294	C₂×C₄	#13133250	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	212	4	no	—
297	C₂²	#16810528	1	1	yes	characteristic
298	C₂²	C₂×#13133129	1	1	yes	characteristic
299	C₂²	#16810591	1	1	yes	characteristic
300	C₂²	#16810620	1	1	yes	characteristic
301	C₂²	#16810625	1	1	yes	characteristic
302	C₂²	C₂×#13133145	1	1	yes	characteristic
303	C₂²	#16810641	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	424	1	no	—
311	C₄	—	424	1	no	—
312	C₄	—	212	2	no	—
313	C₄	—	212	2	no	—
314	C₄	#16810747	1	2	yes	—
315	C₄	#16810750	1	2	yes	—
316	C₄	#16810924	1	2	yes	—
317	C₄	#16810931	1	2	yes	—
318	C₂	#20273098	1	1	yes	characteristic
319	C₂	#20273103	1	1	yes	characteristic
320	C₂	#20273124	1	1	yes	characteristic
321	C₂	#20273129	1	1	yes	characteristic
322	C₂	#20275650	1	1	yes	characteristic
323	C₂	#20275658	1	1	yes	characteristic
324	C₂	#20275670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#9866
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. #13133209	C₈	semidirect
185. C₅₃	#9866	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
C₄×C₄⋊#88322	2
#20272555	2
C₄×C₄⋊C₈×C₅₃	2
#9866	53