Subgroups — C₅₃×#22810 #27230322

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2357	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1970	C₂	1	2	yes	maximal
4	C₅₃×#1926	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1804	C₂	1	2	yes	maximal, complemented
7	C₄×#107×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
11	C₄×D₈×C₅₃	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#102×C₅₃	C₂²	1	4	yes	—
17	C₂×#102×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#390	C₂²	1	4	yes	—
21	C₅₃×#388	—	2	4	no	—
22	C₅₃×#382	C₂²	1	2	yes	—
23	C₅₃×#381	—	2	2	no	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
27	C₅₃×#340	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	2	no	—
32	C₂×D₈×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₅₃	—	2	4	no	—
38	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	—	2	4	no	—
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	2	4	no	—
49	#107×C₅₃	C₂×C₄	1	4	yes	—
50	#106×C₅₃	—	4	2	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	#102×C₅₃	—	4	4	no	—
53	#102×C₅₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₃	—	2	4	no	—
55	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	4	no	—
58	#95×C₅₃	—	2	2	no	—
59	#95×C₅₃	D₄	1	1	yes	characteristic
60	#95×C₅₃	D₄	1	1	yes	characteristic
61	C₂⁴×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	2	2	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	8	no	—
70	C₂²×C₄×C₅₃	—	2	4	no	—
71	C₂²×C₄×C₅₃	—	8	4	no	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	D₈×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	#41	1	2	yes	—
86	C₂×C₈×C₅₃	#41	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₅₃	—	2	8	no	—
93	C₂²⋊C₄×C₅₃	—	4	8	no	—
94	C₂²⋊C₄×C₅₃	—	8	4	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	8	4	no	—
98	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	#41	1	2	yes	—
100	C₂³×C₅₃	—	2	4	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	—	4	4	no	—
104	C₂³×C₅₃	—	4	4	no	—
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	—	4	2	no	—
109	D₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	—	4	4	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	—	8	8	no	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	4	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	4	2	no	—
134	C₈×C₅₃	#117	1	4	yes	—
135	#22810	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	—	4	4	no	—
139	C₂²×C₅₃	—	8	4	no	—
140	C₂²×C₅₃	—	8	4	no	—
141	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₅₃	#547	1	1	yes	characteristic
143	C₂²×C₅₃	#550	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#135	1	2	yes	—
145	C₂²×C₅₃	#576	1	2	yes	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	4	4	no	—
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₅₃	#520	1	1	yes	characteristic
154	C₄×C₅₃	#555	1	1	yes	characteristic
155	C₄×C₅₃	#556	1	1	yes	characteristic
156	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
157	#2357	C₂×C₅₃	1	2	yes	complemented
158	#1970	C₂×C₅₃	1	2	yes	—
159	#1926	C₂×C₅₃	1	4	yes	complemented
160	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
161	#1804	C₂×C₅₃	1	2	yes	complemented
162	C₄×#107	C₂×C₅₃	1	2	yes	complemented
163	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
164	C₂×C₅₃	—	8	4	no	—
165	C₂×C₅₃	#2354	1	1	yes	characteristic
166	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
167	C₂×C₅₃	#2948	1	1	yes	characteristic
168	C₂×C₅₃	#3339	1	2	yes	—
169	C₂×C₅₃	#3396	1	2	yes	—
170	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
171	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
172	C₄×D₈	C₄×C₅₃	1	8	yes	complemented
173	C₂×#107	C₂²×C₅₃	1	2	yes	—
174	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×#106	C₂²×C₅₃	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×#102	C₂²×C₅₃	1	4	yes	—
178	C₂×#102	C₂²×C₅₃	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
181	#390	C₂²×C₅₃	1	4	yes	—
182	#388	—	2	4	no	—
183	#382	C₂²×C₅₃	1	2	yes	—
184	#381	—	2	2	no	—
185	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
186	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
187	C₄×C₂²⋊C₄	—	2	2	no	—
188	#340	C₂²×C₅₃	1	4	yes	—
189	#336	C₂²×C₅₃	1	2	yes	—
190	C₅₃	#22810	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
196	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
201	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	—	2	4	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
209	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#107	C₂×C₄×C₅₃	1	4	yes	—
212	#106	—	4	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#102	—	4	4	no	—
215	#102	C₂×C₄×C₅₃	1	8	yes	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	#95	—	2	2	no	—
221	#95	D₄×C₅₃	1	1	yes	characteristic
222	#95	D₄×C₅₃	1	1	yes	characteristic
223	C₂⁴	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	8	4	no	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
239	D₈	—	2	8	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
247	C₂×C₈	#41×C₅₃	1	2	yes	—
248	C₂×C₈	#41×C₅₃	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	—	4	8	no	—
256	C₂²⋊C₄	—	8	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	8	4	no	—
260	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
261	C₄²	#41×C₅₃	1	2	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
268	D₄	—	2	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	8	no	—
285	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
287	C₂×C₄	#135×C₅₃	1	4	yes	—
288	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
292	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
293	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	4	2	no	—
296	C₈	#117×C₅₃	1	4	yes	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
303	C₂²	C₅₃×#547	1	1	yes	characteristic
304	C₂²	C₅₃×#550	1	1	yes	characteristic
305	C₂²	C₂×#135×C₅₃	1	2	yes	—
306	C₂²	C₅₃×#576	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
314	C₄	C₅₃×#520	1	1	yes	characteristic
315	C₄	C₅₃×#555	1	1	yes	characteristic
316	C₄	C₅₃×#556	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	C₅₃×#2354	1	1	yes	characteristic
319	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
320	C₂	C₅₃×#2948	1	1	yes	characteristic
321	C₂	C₅₃×#3339	1	2	yes	—
322	C₂	C₅₃×#3396	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2357	C₂	semidirect
4. C₅₃×#1926	C₂	semidirect
5. C₅₃×#1846	C₂	semidirect
6. C₅₃×#1804	C₂	semidirect
7. C₄×#107×C₅₃	C₂	semidirect
8. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
11. C₄×D₈×C₅₃	C₄	semidirect
135. #22810	C₅₃	direct (N × H)
157. #2357	C₂×C₅₃	semidirect
159. #1926	C₂×C₅₃	semidirect
160. #1846	C₂×C₅₃	semidirect
161. #1804	C₂×C₅₃	semidirect
162. C₄×#107	C₂×C₅₃	semidirect
163. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
172. C₄×D₈	C₄×C₅₃	semidirect
190. C₅₃	#22810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
C₅₃×#2357	2
C₅₃×#1970	2
C₅₃×#1926	2
C₅₃×#1846	2
C₅₃×#1804	2
C₄×#107×C₅₃	2
C₈×C₄⋊C₄×C₅₃	2
#22810	53

Subgroups — C53×#22810 #27230322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22810 #27230322