Subgroups — #27276904

Order 13568.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810372	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810372	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
15	C₅₃×#391	C₄	1	2	yes	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#389	—	2	1	no	—
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	4	2	no	—
29	C₄⋊#88322	—	4	2	no	—
30	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	—	2	1	no	—
32	C₄⋊Q₈×C₅₃	—	2	1	no	—
33	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
34	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
35	C₄×Q₈×C₅₃	—	4	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#95×C₅₃	—	2	2	no	—
51	#95×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	4	no	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₄⋊C₄×C₅₃	—	4	1	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	—	4	4	no	—
81	C₄²×C₅₃	SD₁₆	1	2	yes	—
82	C₄²×C₅₃	Q₁₆	1	2	yes	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅₃	#41	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂×#88322	—	4	2	no	—
91	C₂×#88322	—	4	2	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₅₃	—	4	4	no	—
95	Q₈×C₅₃	—	4	4	no	—
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	4	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	#100	1	2	yes	—
114	C₂×C₄×C₅₃	#103	1	2	yes	—
115	C₂×C₄×C₅₃	#103	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#135	1	2	yes	—
125	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#137	1	2	yes	—
128	#88322	—	8	2	no	—
129	#88322	—	8	2	no	—
130	#8590	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	#380	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
133	C₂²×C₅₃	#413	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
135	C₂²×C₅₃	#416	1	1	yes	characteristic
136	C₂²×C₅₃	#417	1	1	yes	characteristic
137	C₂²×C₅₃	#419	1	1	yes	characteristic
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	4	4	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	—	8	2	no	—
146	C₄×C₅₃	#460	1	1	yes	characteristic
147	C₄×C₅₃	#462	1	1	yes	characteristic
148	C₄×C₅₃	#464	1	2	yes	—
149	C₄×C₅₃	#475	1	1	yes	characteristic
150	C₄×C₅₃	#477	1	1	yes	characteristic
151	C₄×C₅₃	#479	1	2	yes	—
152	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
153	#1842	—	53	1	no	—
154	#1831	—	53	1	no	—
155	C₂×#333	—	53	2	no	—
156	#1296	—	53	2	no	—
157	C₂×C₅₃	#1542	1	1	yes	characteristic
158	C₂×C₅₃	#1548	1	1	yes	characteristic
159	C₂×C₅₃	#1551	1	1	yes	characteristic
160	C₂×C₅₃	#1559	1	1	yes	characteristic
161	C₂×C₅₃	#1800	1	1	yes	characteristic
162	C₂×C₅₃	#1803	1	1	yes	characteristic
163	C₂×C₅₃	#1805	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
165	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
166	C₂×C₄⋊C₈	—	53	2	no	—
167	C₂×C₄⋊C₈	—	53	2	no	—
168	C₂×C₄⋊C₈	—	106	1	no	—
169	C₂×C₄⋊C₈	—	106	1	no	—
170	#391	Dic₅₃	1	2	yes	—
171	#389	—	2	1	no	—
172	#389	—	2	1	no	—
173	#384	C₂×D₅₃	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄³	C₂×D₅₃	1	1	yes	characteristic
177	#336	—	106	1	no	—
178	#336	—	106	1	no	—
179	#333	—	106	4	no	—
180	C₅₃	#8590	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
187	C₄⋊Q₈	D₂₁₂	1	2	yes	—
188	C₄×Q₈	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	C₄⋊C₈	—	106	2	no	—
204	C₄⋊C₈	—	106	2	no	—
205	C₄⋊C₈	—	106	2	no	—
206	C₄⋊C₈	—	106	2	no	—
207	C₄⋊C₈	—	212	2	no	—
208	C₄⋊C₈	—	212	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	2	no	—
226	C₂×C₈	—	212	2	no	—
227	C₂×C₈	—	212	2	no	—
228	C₂×C₈	—	212	2	no	—
229	C₂×C₈	—	212	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	4	no	—
246	C₄²	#11719924	1	2	yes	—
247	C₄²	#11719925	1	2	yes	—
248	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
249	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
250	C₄²	#11719945	1	1	yes	characteristic
251	C₄²	#11719945	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
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	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	#13133134	1	2	yes	—
273	C₂×C₄	#13133141	1	2	yes	—
274	C₂×C₄	#13133141	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	#13133176	1	1	yes	characteristic
277	C₂×C₄	#13133191	1	1	yes	characteristic
278	C₂×C₄	#13133191	1	1	yes	characteristic
279	C₂×C₄	#13133196	1	1	yes	characteristic
280	C₂×C₄	#13133196	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
283	C₂×C₄	#13133237	1	2	yes	—
284	C₂×C₄	#13133242	1	1	yes	characteristic
285	C₂×C₄	#13133247	1	1	yes	characteristic
286	C₂×C₄	#13133248	1	2	yes	—
287	C₈	—	424	2	no	—
288	C₈	—	424	2	no	—
289	C₂²	#16810483	1	1	yes	characteristic
290	C₂²	C₂×#13133134	1	1	yes	characteristic
291	C₂²	#16810607	1	1	yes	characteristic
292	C₂²	C₂×#13133141	1	1	yes	characteristic
293	C₂²	#16810619	1	1	yes	characteristic
294	C₂²	#16810624	1	1	yes	characteristic
295	C₂²	#16810632	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	#16810823	1	1	yes	characteristic
305	C₄	#16810833	1	1	yes	characteristic
306	C₄	#16810845	1	2	yes	—
307	C₄	#16810908	1	1	yes	characteristic
308	C₄	#16810918	1	1	yes	characteristic
309	C₄	#16810930	1	2	yes	—
310	C₂	#20273372	1	1	yes	characteristic
311	C₂	#20273396	1	1	yes	characteristic
312	C₂	#20273410	1	1	yes	characteristic
313	C₂	#20273445	1	1	yes	characteristic
314	C₂	#20274707	1	1	yes	characteristic
315	C₂	#20274720	1	1	yes	characteristic
316	C₂	#20274730	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#8590
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
180. C₅₃	#8590	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
#20274832	2
C₂×#16810372	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
#8590	53