Subgroups — #27276896

Order 13568.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810368	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	#16810529	—	2	1	no	—
12	#16810521	—	2	1	no	—
13	#16810521	C₄	1	2	yes	—
14	#16810493	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
17	#16810368	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#336	—	2	1	no	—
21	C₅₃×#336	—	2	1	no	—
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	1	no	—
25	#13133217	—	2	1	no	—
26	#13133217	D₄	1	2	yes	—
27	#13133217	Q₈	1	2	yes	—
28	C₄×Dic₁₀₆	—	4	2	no	—
29	C₄×Dic₁₀₆	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×#11719919	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	4	2	no	—
40	C₄⋊#88322	—	4	2	no	—
41	#13133124	—	2	2	no	—
42	#13133124	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	2	2	no	—
61	C₂²×Dic₅₃	—	2	2	no	—
62	C₂×Dic₁₀₆	—	2	2	no	—
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	—	4	1	no	—
65	C₂×Dic₁₀₆	—	4	1	no	—
66	C₂×Dic₁₀₆	—	4	1	no	—
67	C₂×Dic₁₀₆	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	D₈	1	2	yes	—
78	C₄²×C₅₃	SD₁₆	1	2	yes	—
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	4	1	no	—
87	C₄⋊Dic₅₃	—	4	1	no	—
88	C₄⋊Dic₅₃	—	4	2	no	—
89	C₄⋊Dic₅₃	—	4	2	no	—
90	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
91	#11719919	—	4	4	no	—
92	C₄×Dic₅₃	—	4	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
108	C₂×C₄×C₅₃	#101	1	2	yes	—
109	C₂×C₄×C₅₃	#102	1	2	yes	—
110	C₂×C₄×C₅₃	#102	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#135	1	2	yes	—
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	2	yes	—
122	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	4	no	—
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	—	4	2	no	—
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	Dic₁₀₆	—	4	2	no	—
130	Dic₁₀₆	—	4	2	no	—
131	Dic₁₀₆	—	4	2	no	—
132	Dic₁₀₆	—	4	2	no	—
133	C₈×C₅₃	—	8	2	no	—
134	#88322	—	8	2	no	—
135	#8589	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	#380	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
138	C₂²×C₅₃	#413	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
140	C₂²×C₅₃	#416	1	1	yes	characteristic
141	C₂²×C₅₃	#417	1	1	yes	characteristic
142	C₂²×C₅₃	#419	1	1	yes	characteristic
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	4	1	no	—
149	C₄×C₅₃	#459	1	1	yes	characteristic
150	C₄×C₅₃	#461	1	1	yes	characteristic
151	C₄×C₅₃	#464	1	2	yes	—
152	C₄×C₅₃	#474	1	1	yes	characteristic
153	C₄×C₅₃	#476	1	1	yes	characteristic
154	C₄×C₅₃	#479	1	2	yes	—
155	Dic₅₃	—	4	4	no	—
156	Dic₅₃	—	8	2	no	—
157	C₄×C₄⋊Q₈	—	53	1	no	—
158	#1842	—	53	1	no	—
159	#1831	D₅₃	1	1	yes	characteristic
160	C₂×#332	—	53	2	no	—
161	#1296	—	53	2	no	—
162	C₂×C₅₃	#1540	1	1	yes	characteristic
163	C₂×C₅₃	#1546	1	1	yes	characteristic
164	C₂×C₅₃	#1552	1	1	yes	characteristic
165	C₂×C₅₃	#1559	1	1	yes	characteristic
166	C₂×C₅₃	#1801	1	1	yes	characteristic
167	C₂×C₅₃	#1802	1	1	yes	characteristic
168	C₂×C₅₃	#1805	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	—	53	1	no	—
170	C₂×C₄×Q₈	—	106	1	no	—
171	C₂×C₄⋊C₈	—	53	2	no	—
172	C₂×C₄⋊C₈	—	106	1	no	—
173	C₂×C₄⋊C₈	—	106	1	no	—
174	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
175	#391	—	106	1	no	—
176	#389	—	53	2	no	—
177	#389	—	106	1	no	—
178	#384	—	53	1	no	—
179	C₄×C₄⋊C₄	—	53	2	no	—
180	C₄×C₄⋊C₄	—	106	1	no	—
181	C₄³	C₂×D₅₃	1	1	yes	characteristic
182	#336	—	2	1	no	—
183	#336	—	2	1	no	—
184	#332	—	106	4	no	—
185	C₅₃	#8589	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	—	106	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₄⋊Q₈	—	53	2	no	—
190	C₄⋊Q₈	—	53	2	no	—
191	C₄⋊Q₈	—	106	1	no	—
192	C₄⋊Q₈	—	106	1	no	—
193	C₄×Q₈	—	212	2	no	—
194	C₄×Q₈	—	212	2	no	—
195	C₂×C₄⋊C₄	—	53	2	no	—
196	C₂×C₄⋊C₄	—	53	2	no	—
197	C₂×C₄⋊C₄	—	106	1	no	—
198	C₂×C₄⋊C₄	—	106	1	no	—
199	C₂×C₄⋊C₄	—	106	1	no	—
200	C₂×C₄⋊C₄	—	106	2	no	—
201	C₂×C₄²	—	106	2	no	—
202	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
205	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	106	2	no	—
212	C₄⋊C₈	—	106	2	no	—
213	C₄⋊C₈	—	212	2	no	—
214	C₄⋊C₈	—	212	2	no	—
215	#95	—	106	2	no	—
216	#95	—	106	2	no	—
217	C₂×Q₈	—	106	2	no	—
218	C₂×Q₈	—	106	2	no	—
219	C₂×Q₈	—	212	1	no	—
220	C₂×Q₈	—	212	1	no	—
221	C₂×Q₈	—	212	1	no	—
222	C₂×Q₈	—	212	1	no	—
223	C₂²×C₄	—	106	2	no	—
224	C₂²×C₄	—	106	2	no	—
225	C₂²×C₄	#11719921	1	1	yes	characteristic
226	C₂²×C₄	#11719921	1	1	yes	characteristic
227	C₂²×C₄	#11719921	1	1	yes	characteristic
228	C₂²×C₄	#11719921	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	212	2	no	—
236	C₂×C₈	—	212	2	no	—
237	C₂×C₈	—	212	2	no	—
238	C₄⋊C₄	—	53	4	no	—
239	C₄⋊C₄	—	106	2	no	—
240	C₄⋊C₄	—	106	2	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₄⋊C₄	—	212	2	no	—
243	C₄⋊C₄	—	212	4	no	—
244	C₄⋊C₄	—	212	1	no	—
245	C₄⋊C₄	—	212	1	no	—
246	C₄⋊C₄	—	212	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	212	4	no	—
254	C₄²	#11719913	1	2	yes	—
255	C₄²	D₄₂₄	1	2	yes	—
256	C₄²	D₄×D₅₃	1	1	yes	characteristic
257	C₄²	#11719947	1	1	yes	characteristic
258	C₄²	Q₈×D₅₃	1	1	yes	characteristic
259	C₄²	#11719949	1	1	yes	characteristic
260	C₂³	C₂×#11719921	1	1	yes	characteristic
261	Q₈	—	212	2	no	—
262	Q₈	—	212	2	no	—
263	Q₈	—	212	2	no	—
264	Q₈	—	212	2	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	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	106	4	no	—
277	C₂×C₄	—	212	2	no	—
278	C₂×C₄	—	212	2	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	#13133135	1	2	yes	—
283	C₂×C₄	#13133139	1	2	yes	—
284	C₂×C₄	#13133139	1	2	yes	—
285	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
286	C₂×C₄	#13133175	1	1	yes	characteristic
287	C₂×C₄	#13133188	1	1	yes	characteristic
288	C₂×C₄	#13133192	1	1	yes	characteristic
289	C₂×C₄	#13133193	1	1	yes	characteristic
290	C₂×C₄	#13133197	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
293	C₂×C₄	#13133237	1	2	yes	—
294	C₂×C₄	#13133245	1	1	yes	characteristic
295	C₂×C₄	#13133247	1	2	yes	—
296	C₂×C₄	#13133250	1	1	yes	characteristic
297	C₈	—	8	2	no	—
298	C₈	—	424	2	no	—
299	C₂²	#16810480	1	1	yes	characteristic
300	C₂²	C₂×#13133135	1	1	yes	characteristic
301	C₂²	#16810605	1	1	yes	characteristic
302	C₂²	C₂×#13133139	1	1	yes	characteristic
303	C₂²	#16810617	1	1	yes	characteristic
304	C₂²	#16810625	1	1	yes	characteristic
305	C₂²	#16810631	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	424	2	no	—
313	C₄	—	212	4	no	—
314	C₄	#16810819	1	1	yes	characteristic
315	C₄	#16810829	1	1	yes	characteristic
316	C₄	#16810842	1	2	yes	—
317	C₄	#16810904	1	1	yes	characteristic
318	C₄	#16810914	1	1	yes	characteristic
319	C₄	#16810927	1	2	yes	—
320	C₂	#20273362	1	1	yes	characteristic
321	C₂	#20273386	1	1	yes	characteristic
322	C₂	#20273416	1	1	yes	characteristic
323	C₂	#20273446	1	1	yes	characteristic
324	C₂	#20274708	1	1	yes	characteristic
325	C₂	#20274716	1	1	yes	characteristic
326	C₂	#20274727	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8589
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
185. C₅₃	#8589	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20274867	2
C₂×#16810368	2
#20272599	2
C₅₃×#1831	2
#8589	53