Subgroups — #27403024

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	2	yes	maximal
2	#20275439	C₂	1	2	yes	maximal
3	#20275439	C₂	1	2	yes	maximal
4	#20275434	C₂	1	2	yes	maximal
5	#20274835	C₂	1	2	yes	maximal
6	#20274832	C₂	1	1	yes	maximal, characteristic
7	#20274572	C₂	1	2	yes	maximal
8	#20274568	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
10	#16810754	—	2	8	no	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	2	yes	—
17	C₂×#13133128	C₂²	1	2	yes	—
18	C₂×#13133128	C₂²	1	2	yes	—
19	C₂×C₄×#88322	C₂²	1	2	yes	—
20	#16810378	C₂²	1	1	yes	characteristic
21	#16810378	C₂²	1	1	yes	characteristic
22	#16810378	C₂²	1	2	yes	—
23	#16810378	C₂²	1	1	yes	characteristic
24	#16810378	C₂²	1	1	yes	characteristic
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#391	C₄	1	1	yes	characteristic
29	C₅₃×#391	C₄	1	1	yes	characteristic
30	C₅₃×#389	C₄	1	1	yes	characteristic
31	C₅₃×#389	C₄	1	1	yes	characteristic
32	C₅₃×#387	C₄	1	2	yes	—
33	C₅₃×#387	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#376	C₂²	1	2	yes	—
37	C₄³×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	2	yes	—
39	C₂²×#88322	C₂³	1	2	yes	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₄⋊#88322	—	2	4	no	—
43	C₄⋊#88322	—	2	4	no	—
44	C₄⋊#88322	—	2	4	no	—
45	C₄⋊#88322	—	2	4	no	—
46	C₄⋊#88322	—	2	4	no	—
47	C₄⋊#88322	—	2	4	no	—
48	#13133128	—	2	4	no	—
49	#13133128	—	2	4	no	—
50	C₄×#88322	—	2	4	no	—
51	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₅₃	D₄	1	4	yes	—
53	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	#95×C₅₃	C₂×C₄	1	2	yes	—
72	#95×C₅₃	C₂×C₄	1	2	yes	—
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	#95×C₅₃	C₂×C₄	1	2	yes	—
75	#95×C₅₃	C₂×C₄	1	2	yes	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	C₂×Q₈×C₅₃	—	4	1	no	—
78	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄⋊C₄×C₅₃	—	4	1	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
95	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₄²×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	—	4	1	no	—
104	C₄²×C₅₃	C₂×D₄	1	4	yes	—
105	C₄²×C₅₃	C₂×D₄	1	2	yes	—
106	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
107	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	#41	1	4	yes	—
120	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	Q₈×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
151	C₂×C₄×C₅₃	#131	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
155	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
156	#88322	—	2	4	no	—
157	#88322	—	4	2	no	—
158	#88322	—	4	2	no	—
159	#88322	—	4	2	no	—
160	#22737	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
163	C₂²×C₅₃	#519	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
165	C₂²×C₅₃	#568	1	1	yes	characteristic
166	C₂²×C₅₃	#568	1	1	yes	characteristic
167	C₂²×C₅₃	#568	1	1	yes	characteristic
168	C₄×C₅₃	—	2	4	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	4	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	4	1	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	2	no	—
176	C₄×C₅₃	—	4	1	no	—
177	C₄×C₅₃	—	4	1	no	—
178	C₄×C₅₃	#548	1	2	yes	—
179	C₄×C₅₃	#557	1	2	yes	—
180	C₂×#446	—	53	2	no	—
181	#2295	D₅₃	1	1	yes	characteristic
182	#1940	—	53	2	no	—
183	#1940	—	53	2	no	—
184	#1939	—	53	2	no	—
185	#1832	—	53	2	no	—
186	#1831	—	53	1	no	—
187	#1768	—	53	2	no	—
188	#1767	—	53	1	no	—
189	C₂×C₅₃	#2351	1	1	yes	characteristic
190	C₂×C₅₃	#2905	1	1	yes	characteristic
191	C₂×C₅₃	#2906	1	1	yes	characteristic
192	C₂×C₅₃	#2967	1	1	yes	characteristic
193	C₂×C₅₃	#2968	1	1	yes	characteristic
194	C₂×C₅₃	#2985	1	1	yes	characteristic
195	C₂×C₅₃	#2987	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
197	#446	—	106	8	no	—
198	C₂×C₄⋊C₈	—	53	2	no	—
199	C₂×C₄⋊C₈	—	53	2	no	—
200	C₂×C₄⋊C₈	—	53	2	no	—
201	C₂×C₄⋊C₈	—	53	2	no	—
202	C₂×C₄⋊C₈	—	53	2	no	—
203	C₂×C₄⋊C₈	—	53	2	no	—
204	C₂×#97	—	53	2	no	—
205	C₂×#97	—	53	2	no	—
206	C₂×C₄×C₈	—	53	2	no	—
207	#391	Dic₅₃	1	1	yes	characteristic
208	#391	Dic₅₃	1	1	yes	characteristic
209	#389	Dic₅₃	1	1	yes	characteristic
210	#389	Dic₅₃	1	1	yes	characteristic
211	#387	Dic₅₃	1	2	yes	—
212	#387	Dic₅₃	1	2	yes	—
213	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
214	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
215	#376	C₂×D₅₃	1	2	yes	—
216	C₄³	C₂×D₅₃	1	1	yes	characteristic
217	#336	—	53	2	no	—
218	#336	—	53	1	no	—
219	#336	—	53	1	no	—
220	#336	—	53	1	no	—
221	#336	—	53	2	no	—
222	#336	—	53	2	no	—
223	#336	—	53	1	no	—
224	C₅₃	#22737	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₈	—	53	2	no	—
227	C₂²×C₈	—	53	2	no	—
228	C₂²×C₈	—	53	2	no	—
229	C₂²×C₈	—	53	2	no	—
230	C₄×Q₈	Dic₁₀₆	1	4	yes	—
231	C₄×Q₈	D₂₁₂	1	4	yes	—
232	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
245	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
247	C₄⋊C₈	—	106	4	no	—
248	C₄⋊C₈	—	106	4	no	—
249	C₄⋊C₈	—	106	4	no	—
250	C₄⋊C₈	—	106	4	no	—
251	C₄⋊C₈	—	106	4	no	—
252	C₄⋊C₈	—	106	4	no	—
253	#97	—	106	4	no	—
254	#97	—	106	4	no	—
255	C₄×C₈	—	106	4	no	—
256	#95	C₂×Dic₅₃	1	2	yes	—
257	#95	C₂×Dic₅₃	1	2	yes	—
258	#95	C₂×Dic₅₃	1	2	yes	—
259	#95	C₂×Dic₅₃	1	2	yes	—
260	#95	C₂×Dic₅₃	1	2	yes	—
261	#95	C₂×Dic₅₃	1	2	yes	—
262	#95	C₂×Dic₅₃	1	2	yes	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
266	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
270	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
272	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
274	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
277	C₂×C₈	—	53	4	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	106	2	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	106	2	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
294	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×Dic₁₀₆	1	2	yes	—
304	C₄²	C₂×Dic₁₀₆	1	4	yes	—
305	C₄²	C₂×D₂₁₂	1	4	yes	—
306	C₄²	C₂×D₂₁₂	1	2	yes	—
307	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
335	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
336	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
337	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
338	C₂×C₄	#13133224	1	2	yes	—
339	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
341	C₂×C₄	#41×D₅₃	1	2	yes	—
342	C₂×C₄	#13133265	1	2	yes	—
343	C₈	—	106	4	no	—
344	C₈	—	212	2	no	—
345	C₈	—	212	2	no	—
346	C₈	—	212	2	no	—
347	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
348	C₂²	#41×Dic₅₃	1	1	yes	characteristic
349	C₂²	#16811101	1	1	yes	characteristic
350	C₂²	C₂×#13133224	1	1	yes	characteristic
351	C₂²	#16811390	1	1	yes	characteristic
352	C₂²	#16811390	1	1	yes	characteristic
353	C₂²	#16811390	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	4	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	4	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	#16811285	1	2	yes	—
365	C₄	#16811349	1	2	yes	—
366	C₂	#20277636	1	1	yes	characteristic
367	C₂	#20283279	1	1	yes	characteristic
368	C₂	#20283283	1	1	yes	characteristic
369	C₂	#20283702	1	1	yes	characteristic
370	C₂	#20283711	1	1	yes	characteristic
371	C₂	#20283810	1	1	yes	characteristic
372	C₂	#20283821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2295	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#22737
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
224. C₅₃	#22737	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
#20275439	2
#20275434	2
#20274835	2
#20274832	2
#20274572	2
#20274568	2
C₅₃×#2295	2
#22737	53