Subgroups — C₅₃×#33920 #27241432

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3334	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2236	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2234	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2220	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2159	C₂	1	2	yes	maximal, complemented
12	C₅₃×#556	—	2	1	no	—
13	C₅₃×#547	—	2	1	no	—
14	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#500	C₂²	1	1	yes	characteristic
17	C₅₃×#500	C₂²	1	1	yes	characteristic
18	C₅₃×#485	—	2	2	no	—
19	C₅₃×#480	—	2	2	no	—
20	C₅₃×#475	—	2	1	no	—
21	C₅₃×#474	—	2	1	no	—
22	C₅₃×#467	C₂²	1	2	yes	—
23	C₅₃×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#427	C₂²	1	2	yes	—
28	C₂×#107×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#367	C₂²	1	1	yes	characteristic
30	C₅₃×#367	C₂²	1	2	yes	—
31	C₅₃×#366	C₂²	1	2	yes	—
32	C₅₃×#366	C₂²	1	2	yes	—
33	C₅₃×#366	C₂²	1	2	yes	—
34	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₅₃×#358	C₂²	1	1	yes	characteristic
36	C₅₃×#358	C₂²	1	2	yes	—
37	C₅₃×#357	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	2	yes	—
39	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#41×C₅₃	—	4	1	no	—
42	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₅₃	—	2	1	no	—
46	C₄⋊Q₈×C₅₃	—	2	1	no	—
47	#126×C₅₃	—	4	2	no	—
48	#124×C₅₃	—	4	1	no	—
49	#123×C₅₃	—	4	2	no	—
50	C₂²⋊Q₈×C₅₃	—	2	2	no	—
51	C₂²⋊Q₈×C₅₃	—	4	1	no	—
52	#121×C₅₃	—	2	2	no	—
53	C₄×Q₈×C₅₃	—	2	1	no	—
54	C₄×Q₈×C₅₃	—	4	1	no	—
55	C₄×D₄×C₅₃	—	2	1	no	—
56	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
57	#117×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	SD₃₂×C₅₃	—	2	2	no	—
60	M₃₂×C₅₃	—	2	4	no	—
61	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	1	no	—
65	#107×C₅₃	—	2	1	no	—
66	#107×C₅₃	—	2	2	no	—
67	#107×C₅₃	C₂³	1	2	yes	—
68	#107×C₅₃	C₂³	1	2	yes	—
69	#107×C₅₃	C₂³	1	1	yes	characteristic
70	#107×C₅₃	C₂³	1	1	yes	characteristic
71	#106×C₅₃	—	2	2	no	—
72	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
73	#103×C₅₃	—	2	1	no	—
74	#103×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	1	no	—
76	#102×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	#41×C₅₃	—	8	4	no	—
80	C₂×Q₈×C₅₃	—	2	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	—	4	1	no	—
83	C₂×D₄×C₅₃	—	2	1	no	—
84	C₂×D₄×C₅₃	—	4	2	no	—
85	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	—	4	1	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	Q₁₆×C₅₃	—	2	1	no	—
91	D₈×C₅₃	—	2	1	no	—
92	D₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
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	D₈	1	2	yes	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	—	4	1	no	—
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	—	2	1	no	—
118	C₁₆×C₅₃	#41	1	2	yes	complemented
119	C₂³×C₅₃	—	4	1	no	—
120	C₂³×C₅₃	C₂×D₈	1	2	yes	—
121	Q₈×C₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	8	2	no	—
124	Q₈×C₅₃	—	8	2	no	—
125	D₄×C₅₃	—	2	2	no	—
126	D₄×C₅₃	—	4	1	no	—
127	D₄×C₅₃	—	4	1	no	—
128	D₄×C₅₃	—	8	4	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	8	4	no	—
137	C₂×C₄×C₅₃	—	8	2	no	—
138	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#33920	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	—	2	2	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	—	8	1	no	—
153	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	8	1	no	—
157	C₄×C₅₃	—	8	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	#509	1	2	yes	—
162	C₄×C₅₃	#547	1	1	yes	characteristic
163	C₄×C₅₃	#578	1	1	yes	characteristic
164	#3342	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#3334	C₂×C₅₃	1	1	yes	characteristic
166	#2252	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#2236	C₂×C₅₃	1	2	yes	complemented
168	#2234	C₂×C₅₃	1	2	yes	complemented
169	#2225	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#2224	C₂×C₅₃	1	1	yes	characteristic, complemented
171	#2220	C₂×C₅₃	1	2	yes	complemented
172	C₄×SD₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
173	#2169	C₂×C₅₃	1	1	yes	characteristic, complemented
174	#2159	C₂×C₅₃	1	2	yes	complemented
175	C₂×C₅₃	—	4	2	no	—
176	C₂×C₅₃	—	16	1	no	—
177	C₂×C₅₃	#3334	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
179	C₂×C₅₃	#3417	1	1	yes	characteristic
180	#556	—	2	1	no	—
181	#547	—	2	1	no	—
182	C₂×SD₃₂	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
184	#500	C₂²×C₅₃	1	1	yes	characteristic
185	#500	C₂²×C₅₃	1	1	yes	characteristic
186	#485	—	2	2	no	—
187	#480	—	2	2	no	—
188	#475	—	2	1	no	—
189	#474	—	2	1	no	—
190	#467	C₂²×C₅₃	1	2	yes	—
191	#466	C₂²×C₅₃	1	2	yes	—
192	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
194	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
195	#427	C₂²×C₅₃	1	2	yes	—
196	C₂×#107	C₂²×C₅₃	1	2	yes	—
197	#367	C₂²×C₅₃	1	1	yes	characteristic
198	#367	C₂²×C₅₃	1	2	yes	—
199	#366	C₂²×C₅₃	1	2	yes	—
200	#366	C₂²×C₅₃	1	2	yes	—
201	#366	C₂²×C₅₃	1	2	yes	—
202	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
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	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
208	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
209	C₅₃	#33920	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×#41	—	4	1	no	—
211	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
212	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	#126	—	4	2	no	—
217	#124	—	4	1	no	—
218	#123	—	4	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
226	#117	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	SD₃₂	—	2	2	no	—
229	M₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
232	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	—	2	1	no	—
236	#107	C₂³×C₅₃	1	2	yes	—
237	#107	C₂³×C₅₃	1	1	yes	characteristic
238	#107	C₂³×C₅₃	1	1	yes	characteristic
239	#107	C₂³×C₅₃	1	2	yes	—
240	#106	—	2	2	no	—
241	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
242	#103	—	2	1	no	—
243	#103	—	2	2	no	—
244	#102	—	2	1	no	—
245	#102	—	2	2	no	—
246	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
247	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
266	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	#41×C₅₃	1	2	yes	complemented
288	C₂³	—	4	1	no	—
289	C₂³	C₂×D₈×C₅₃	1	2	yes	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	8	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
308	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
314	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	1	no	—
329	C₄	C₅₃×#509	1	2	yes	—
330	C₄	C₅₃×#547	1	1	yes	characteristic
331	C₄	C₅₃×#578	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	16	1	no	—
334	C₂	C₅₃×#3334	1	1	yes	characteristic
335	C₂	C₂×C₅₃×#509	1	1	yes	characteristic
336	C₂	C₅₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₅₃×#3334	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×D₄×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3342	C₂	semidirect
3. C₅₃×#2252	C₂	semidirect
4. C₅₃×#2236	C₂	semidirect
5. C₅₃×#2234	C₂	semidirect
6. C₅₃×#2225	C₂	semidirect
7. C₅₃×#2224	C₂	semidirect
8. C₅₃×#2220	C₂	semidirect
9. C₄×SD₃₂×C₅₃	C₂	semidirect
10. C₅₃×#2169	C₂	semidirect
11. C₅₃×#2159	C₂	semidirect
118. C₁₆×C₅₃	#41	semidirect
146. #33920	C₅₃	direct (N × H)
164. #3342	C₂×C₅₃	semidirect
166. #2252	C₂×C₅₃	semidirect
167. #2236	C₂×C₅₃	semidirect
168. #2234	C₂×C₅₃	semidirect
169. #2225	C₂×C₅₃	semidirect
170. #2224	C₂×C₅₃	semidirect
171. #2220	C₂×C₅₃	semidirect
172. C₄×SD₃₂	C₂×C₅₃	semidirect
173. #2169	C₂×C₅₃	semidirect
174. #2159	C₂×C₅₃	semidirect
209. C₅₃	#33920	direct (N × H)
287. C₁₆	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3342	2
C₅₃×#3334	2
C₅₃×#2252	2
C₅₃×#2236	2
C₅₃×#2234	2
C₅₃×#2225	2
C₅₃×#2224	2
C₅₃×#2220	2
C₄×SD₃₂×C₅₃	2
C₅₃×#2169	2
C₅₃×#2159	2
#33920	53

Subgroups — C53×#33920 #27241432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33920 #27241432