Subgroups — C₅₃×#33928 #27241440

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2242	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2210	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
12	C₅₃×#559	—	2	1	no	—
13	C₅₃×#554	—	2	1	no	—
14	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#500	C₂²	1	1	yes	characteristic
17	C₅₃×#499	C₂²	1	2	yes	—
18	C₅₃×#495	C₂²	1	2	yes	—
19	C₅₃×#493	C₂²	1	1	yes	characteristic
20	C₅₃×#478	—	2	2	no	—
21	C₅₃×#476	—	2	1	no	—
22	C₅₃×#463	—	2	2	no	—
23	C₅₃×#460	—	2	1	no	—
24	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₅₃	C₂²	1	2	yes	—
27	C₄×D₈×C₅₃	C₂²	1	2	yes	—
28	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#367	C₂²	1	1	yes	characteristic
30	C₅₃×#366	C₂²	1	2	yes	—
31	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₅₃×#358	C₂²	1	1	yes	characteristic
34	C₅₃×#358	C₂²	1	2	yes	—
35	C₅₃×#357	C₂²	1	1	yes	characteristic
36	C₅₃×#357	C₂²	1	2	yes	—
37	C₅₃×#357	C₂²	1	2	yes	—
38	C₅₃×#357	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×SD₁₆×C₅₃	—	2	2	no	—
43	C₂×D₈×C₅₃	—	2	1	no	—
44	C₂×D₈×C₅₃	C₂³	1	2	yes	—
45	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	—	2	1	no	—
47	#127×C₅₃	—	2	1	no	—
48	#125×C₅₃	—	2	2	no	—
49	#124×C₅₃	—	2	2	no	—
50	#124×C₅₃	—	4	1	no	—
51	#123×C₅₃	—	4	2	no	—
52	C₂²⋊Q₈×C₅₃	—	4	2	no	—
53	C₂²⋊Q₈×C₅₃	—	4	1	no	—
54	C₄×Q₈×C₅₃	—	2	1	no	—
55	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
56	C₄×D₄×C₅₃	—	2	1	no	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	—	4	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
61	SD₃₂×C₅₃	—	2	2	no	—
62	D₁₆×C₅₃	—	2	4	no	—
63	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
66	#107×C₅₃	—	2	1	no	—
67	#107×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	C₂³	1	2	yes	—
69	#107×C₅₃	C₂³	1	1	yes	characteristic
70	#106×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
73	#103×C₅₃	—	2	1	no	—
74	#102×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂²×C₄×C₅₃	—	4	1	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	Q₁₆×C₅₃	—	2	1	no	—
91	SD₁₆×C₅₃	—	8	2	no	—
92	D₈×C₅₃	—	2	2	no	—
93	D₈×C₅₃	—	2	1	no	—
94	D₈×C₅₃	—	2	2	no	—
95	D₈×C₅₃	—	8	1	no	—
96	D₈×C₅₃	#41	1	2	yes	—
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	2	no	—
107	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
109	C₂²⋊C₄×C₅₃	—	4	2	no	—
110	C₂²⋊C₄×C₅₃	—	4	1	no	—
111	C₂²⋊C₄×C₅₃	—	4	2	no	—
112	C₂²⋊C₄×C₅₃	—	4	2	no	—
113	C₂²⋊C₄×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	—	4	1	no	—
115	C₄²×C₅₃	C₂×D₄	1	2	yes	—
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₅₃	—	2	2	no	—
118	C₁₆×C₅₃	—	2	1	no	—
119	C₁₆×C₅₃	—	2	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂³×C₅₃	—	4	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	4	1	no	—
125	Q₈×C₅₃	—	16	1	no	—
126	D₄×C₅₃	—	2	2	no	—
127	D₄×C₅₃	—	4	2	no	—
128	D₄×C₅₃	—	4	1	no	—
129	D₄×C₅₃	—	4	2	no	—
130	D₄×C₅₃	—	8	2	no	—
131	D₄×C₅₃	—	16	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	8	2	no	—
137	C₂×C₄×C₅₃	—	16	2	no	—
138	C₂×C₄×C₅₃	—	16	2	no	—
139	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
148	#33928	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₃	—	4	2	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	—	8	2	no	—
155	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	8	1	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	—	8	2	no	—
163	C₄×C₅₃	#508	1	2	yes	—
164	C₄×C₅₃	#548	1	1	yes	characteristic
165	C₄×C₅₃	#577	1	1	yes	characteristic
166	#3396	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#3380	C₂×C₅₃	1	1	yes	characteristic, complemented
168	#2242	C₂×C₅₃	1	2	yes	complemented
169	#2240	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#2228	C₂×C₅₃	1	2	yes	complemented
171	#2227	C₂×C₅₃	1	1	yes	characteristic, complemented
172	#2210	C₂×C₅₃	1	2	yes	complemented
173	#2206	C₂×C₅₃	1	1	yes	characteristic
174	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
175	C₄×SD₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
176	C₄×D₁₆	C₂×C₅₃	1	2	yes	complemented
177	C₂×C₅₃	—	8	2	no	—
178	C₂×C₅₃	—	16	2	no	—
179	C₂×C₅₃	#3358	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
181	C₂×C₅₃	#3415	1	1	yes	characteristic
182	#559	—	2	1	no	—
183	#554	—	2	1	no	—
184	C₂×SD₃₂	C₂²×C₅₃	1	1	yes	characteristic
185	C₂×D₁₆	C₂²×C₅₃	1	2	yes	—
186	#500	C₂²×C₅₃	1	1	yes	characteristic
187	#499	C₂²×C₅₃	1	2	yes	—
188	#495	C₂²×C₅₃	1	2	yes	—
189	#493	C₂²×C₅₃	1	1	yes	characteristic
190	#478	—	2	2	no	—
191	#476	—	2	1	no	—
192	#463	—	2	2	no	—
193	#460	—	2	1	no	—
194	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
196	C₄×D₈	C₂²×C₅₃	1	2	yes	—
197	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
198	C₄×D₈	C₂²×C₅₃	1	2	yes	—
199	#367	C₂²×C₅₃	1	1	yes	characteristic
200	#366	C₂²×C₅₃	1	2	yes	—
201	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
203	#358	C₂²×C₅₃	1	1	yes	characteristic
204	#358	C₂²×C₅₃	1	2	yes	—
205	#357	C₂²×C₅₃	1	1	yes	characteristic
206	#357	C₂²×C₅₃	1	2	yes	—
207	#357	C₂²×C₅₃	1	2	yes	—
208	#357	C₂²×C₅₃	1	2	yes	—
209	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
210	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
211	C₅₃	#33928	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	C₂³×C₅₃	1	2	yes	—
216	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	#127	—	2	1	no	—
219	#125	—	2	2	no	—
220	#124	—	2	2	no	—
221	#124	—	4	1	no	—
222	#123	—	4	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	4	1	no	—
231	C₂×C₄⋊C₄	—	4	1	no	—
232	SD₃₂	—	2	2	no	—
233	D₁₆	—	2	4	no	—
234	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
237	#107	—	2	1	no	—
238	#107	C₂³×C₅₃	1	2	yes	—
239	#107	C₂³×C₅₃	1	1	yes	characteristic
240	#107	C₂³×C₅₃	1	1	yes	characteristic
241	#106	—	2	2	no	—
242	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
243	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
244	#103	—	2	1	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	1	no	—
248	C₄×C₈	C₂³×C₅₃	1	2	yes	—
249	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	2	no	—
266	D₈	—	8	1	no	—
267	D₈	#41×C₅₃	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
279	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×D₄×C₅₃	1	2	yes	—
287	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	Q₈	—	16	1	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	16	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
312	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
318	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	2	no	—
333	C₄	C₅₃×#508	1	2	yes	—
334	C₄	C₅₃×#548	1	1	yes	characteristic
335	C₄	C₅₃×#577	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₅₃×#3358	1	1	yes	characteristic
339	C₂	C₂×C₅₃×#508	1	1	yes	characteristic
340	C₂	C₅₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₅₃×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×Q₈×C₅₃ ◁ C₈×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3396	C₂	semidirect
2. C₅₃×#3380	C₂	semidirect
3. C₅₃×#2242	C₂	semidirect
4. C₅₃×#2240	C₂	semidirect
5. C₅₃×#2228	C₂	semidirect
6. C₅₃×#2227	C₂	semidirect
7. C₅₃×#2210	C₂	semidirect
9. Q₈×C₁₆×C₅₃	C₂	semidirect
10. C₄×SD₃₂×C₅₃	C₂	semidirect
11. C₄×D₁₆×C₅₃	C₂	semidirect
148. #33928	C₅₃	direct (N × H)
166. #3396	C₂×C₅₃	semidirect
167. #3380	C₂×C₅₃	semidirect
168. #2242	C₂×C₅₃	semidirect
169. #2240	C₂×C₅₃	semidirect
170. #2228	C₂×C₅₃	semidirect
171. #2227	C₂×C₅₃	semidirect
172. #2210	C₂×C₅₃	semidirect
174. Q₈×C₁₆	C₂×C₅₃	semidirect
175. C₄×SD₃₂	C₂×C₅₃	semidirect
176. C₄×D₁₆	C₂×C₅₃	semidirect
211. C₅₃	#33928	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3396	2
C₅₃×#3380	2
C₅₃×#2242	2
C₅₃×#2240	2
C₅₃×#2228	2
C₅₃×#2227	2
C₅₃×#2210	2
C₅₃×#2206	2
Q₈×C₁₆×C₅₃	2
C₄×SD₃₂×C₅₃	2
C₄×D₁₆×C₅₃	2
#33928	53

Subgroups — C53×#33928 #27241440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33928 #27241440