Subgroups — C₅₃×#33651 #27241163

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2913	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2236	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2235	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#536	—	2	1	no	—
15	C₅₃×#520	—	2	1	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#470	—	2	2	no	—
19	C₅₃×#467	C₂²	1	2	yes	—
20	C₅₃×#467	C₂²	1	2	yes	—
21	C₅₃×#428	—	2	1	no	—
22	C₅₃×#428	—	2	2	no	—
23	C₅₃×#427	C₂²	1	2	yes	—
24	C₅₃×#427	C₂²	1	1	yes	characteristic
25	C₅₃×#427	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₃	C₂²	1	2	yes	—
27	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#419	—	2	1	no	—
29	C₅₃×#367	C₂²	1	1	yes	characteristic
30	C₅₃×#367	C₂²	1	2	yes	—
31	C₅₃×#367	C₂²	1	1	yes	characteristic
32	C₅₃×#366	C₂²	1	1	yes	characteristic
33	C₅₃×#366	C₂²	1	2	yes	—
34	C₅₃×#366	C₂²	1	1	yes	characteristic
35	C₅₃×#358	C₂²	1	2	yes	—
36	C₅₃×#358	C₂²	1	2	yes	—
37	C₅₃×#358	C₂²	1	2	yes	—
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂×Q₁₆×C₅₃	—	2	1	no	—
44	C₂×Q₁₆×C₅₃	—	2	2	no	—
45	C₂×Q₁₆×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
47	#131×C₅₃	—	2	2	no	—
48	#131×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	4	1	no	—
54	#124×C₅₃	—	4	1	no	—
55	#123×C₅₃	—	4	2	no	—
56	C₂²⋊Q₈×C₅₃	—	2	2	no	—
57	C₂²⋊Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	—	4	1	no	—
59	C₄×D₄×C₅₃	—	4	2	no	—
60	C₄×D₄×C₅₃	—	4	1	no	—
61	#117×C₅₃	—	2	2	no	—
62	#117×C₅₃	—	2	1	no	—
63	#117×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	M₃₂×C₅₃	—	2	2	no	—
67	C₂×C₁₆×C₅₃	—	2	2	no	—
68	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
71	#107×C₅₃	—	2	2	no	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	C₂³	1	2	yes	—
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#106×C₅₃	—	2	2	no	—
79	#106×C₅₃	—	2	1	no	—
80	#106×C₅₃	—	2	1	no	—
81	#103×C₅₃	—	2	2	no	—
82	#103×C₅₃	—	2	2	no	—
83	#41×C₅₃	—	2	2	no	—
84	#41×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	—	2	2	no	—
86	C₂×Q₈×C₅₃	—	4	1	no	—
87	C₂×Q₈×C₅₃	—	4	2	no	—
88	C₂×Q₈×C₅₃	—	4	2	no	—
89	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
90	C₂×D₄×C₅₃	D₈	1	2	yes	—
91	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	—	4	2	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₅₃	—	2	2	no	—
98	Q₁₆×C₅₃	—	2	2	no	—
99	Q₁₆×C₅₃	—	4	2	no	—
100	M₁₆×C₅₃	—	2	2	no	—
101	M₁₆×C₅₃	—	2	4	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	#41	1	2	yes	—
107	C₂×C₈×C₅₃	#41	1	2	yes	—
108	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₅₃	—	2	1	no	—
110	C₄⋊C₄×C₅₃	—	2	1	no	—
111	C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	4	2	no	—
115	C₄⋊C₄×C₅₃	—	4	1	no	—
116	C₄⋊C₄×C₅₃	—	4	2	no	—
117	C₄⋊C₄×C₅₃	—	4	2	no	—
118	C₂²⋊C₄×C₅₃	—	4	2	no	—
119	C₂²⋊C₄×C₅₃	—	4	1	no	—
120	C₂²⋊C₄×C₅₃	—	4	1	no	—
121	C₂²⋊C₄×C₅₃	—	4	2	no	—
122	C₂²⋊C₄×C₅₃	—	4	2	no	—
123	C₄²×C₅₃	—	4	2	no	—
124	C₄²×C₅₃	—	4	1	no	—
125	C₄²×C₅₃	—	4	1	no	—
126	C₁₆×C₅₃	—	2	1	no	—
127	C₁₆×C₅₃	—	2	1	no	—
128	C₁₆×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	C₂×D₈	1	2	yes	—
130	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
131	Q₈×C₅₃	—	2	2	no	—
132	Q₈×C₅₃	—	4	2	no	—
133	Q₈×C₅₃	—	4	2	no	—
134	Q₈×C₅₃	—	8	2	no	—
135	D₄×C₅₃	—	2	4	no	—
136	D₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	8	2	no	—
146	C₂×C₄×C₅₃	—	16	1	no	—
147	C₂×C₄×C₅₃	—	16	2	no	—
148	C₂×C₄×C₅₃	#123	1	2	yes	—
149	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#137	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	#123	1	2	yes	—
155	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
157	#33651	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	#480	1	2	yes	—
164	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₅₃	—	2	2	no	—
166	C₄×C₅₃	—	8	2	no	—
167	C₄×C₅₃	—	8	1	no	—
168	C₄×C₅₃	—	8	2	no	—
169	C₄×C₅₃	—	8	1	no	—
170	C₄×C₅₃	—	8	1	no	—
171	C₄×C₅₃	—	8	1	no	—
172	C₄×C₅₃	#480	1	2	yes	—
173	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
174	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
175	#3059	C₂×C₅₃	1	1	yes	characteristic
176	#2913	C₂×C₅₃	1	1	yes	characteristic
177	#2237	C₂×C₅₃	1	2	yes	complemented
178	#2237	C₂×C₅₃	1	2	yes	complemented
179	#2236	C₂×C₅₃	1	2	yes	complemented
180	#2235	C₂×C₅₃	1	2	yes	complemented
181	#2159	C₂×C₅₃	1	1	yes	characteristic, complemented
182	#2158	C₂×C₅₃	1	1	yes	characteristic, complemented
183	#2141	C₂×C₅₃	1	1	yes	characteristic, complemented
184	C₂×#358	C₂×C₅₃	1	1	yes	characteristic, complemented
185	#2113	C₂×C₅₃	1	1	yes	characteristic
186	C₂×C₅₃	—	2	2	no	—
187	C₂×C₅₃	—	4	2	no	—
188	C₂×C₅₃	C₂×#480	1	1	yes	characteristic
189	C₂×C₅₃	#3416	1	1	yes	characteristic
190	C₂×C₅₃	#3417	1	1	yes	characteristic
191	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
192	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
193	#536	—	2	1	no	—
194	#520	—	2	1	no	—
195	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
196	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
197	#470	—	2	2	no	—
198	#467	C₂²×C₅₃	1	2	yes	—
199	#467	C₂²×C₅₃	1	2	yes	—
200	#428	—	2	1	no	—
201	#428	—	2	2	no	—
202	#427	C₂²×C₅₃	1	1	yes	characteristic
203	#427	C₂²×C₅₃	1	2	yes	—
204	#427	C₂²×C₅₃	1	1	yes	characteristic
205	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×#107	C₂²×C₅₃	1	2	yes	—
207	#419	—	2	1	no	—
208	#367	C₂²×C₅₃	1	1	yes	characteristic
209	#367	C₂²×C₅₃	1	1	yes	characteristic
210	#367	C₂²×C₅₃	1	2	yes	—
211	#366	C₂²×C₅₃	1	1	yes	characteristic
212	#366	C₂²×C₅₃	1	1	yes	characteristic
213	#366	C₂²×C₅₃	1	2	yes	—
214	#358	C₂²×C₅₃	1	2	yes	—
215	#358	C₂²×C₅₃	1	2	yes	—
216	#358	C₂²×C₅₃	1	2	yes	—
217	#358	C₂²×C₅₃	1	2	yes	—
218	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
219	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
220	C₅₃	#33651	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
222	C₂²×Q₈	—	2	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
227	#131	—	2	2	no	—
228	#131	—	2	2	no	—
229	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
230	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
231	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
232	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	4	1	no	—
234	#124	—	4	1	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	4	1	no	—
239	C₄×D₄	—	4	2	no	—
240	C₄×D₄	—	4	1	no	—
241	#117	—	2	2	no	—
242	#117	—	2	1	no	—
243	#117	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	M₃₂	—	2	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
250	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	C₂³×C₅₃	1	2	yes	—
254	#107	C₂³×C₅₃	1	1	yes	characteristic
255	#107	C₂³×C₅₃	1	1	yes	characteristic
256	#107	C₂³×C₅₃	1	1	yes	characteristic
257	#107	C₂³×C₅₃	1	1	yes	characteristic
258	#106	—	2	2	no	—
259	#106	—	2	1	no	—
260	#106	—	2	1	no	—
261	#103	—	2	2	no	—
262	#103	—	2	2	no	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
270	C₂×D₄	D₈×C₅₃	1	2	yes	—
271	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂×C₈	#41×C₅₃	1	2	yes	—
287	C₂×C₈	#41×C₅₃	1	2	yes	—
288	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₄²	—	4	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	4	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₂³	C₂×D₈×C₅₃	1	2	yes	—
310	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	16	1	no	—
327	C₂×C₄	—	16	2	no	—
328	C₂×C₄	#123×C₅₃	1	2	yes	—
329	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
331	C₂×C₄	#137×C₅₃	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	#123×C₅₃	1	2	yes	—
335	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
336	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	C₅₃×#480	1	2	yes	—
343	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₅₃×#480	1	2	yes	—
352	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
353	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	4	2	no	—
356	C₂	C₂×C₅₃×#480	1	1	yes	characteristic
357	C₂	C₅₃×#3416	1	1	yes	characteristic
358	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	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×#41×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2237	C₂	semidirect
4. C₅₃×#2237	C₂	semidirect
5. C₅₃×#2236	C₂	semidirect
6. C₅₃×#2235	C₂	semidirect
7. C₅₃×#2159	C₂	semidirect
8. C₅₃×#2158	C₂	semidirect
9. C₅₃×#2141	C₂	semidirect
10. C₂×C₅₃×#358	C₂	semidirect
157. #33651	C₅₃	direct (N × H)
177. #2237	C₂×C₅₃	semidirect
178. #2237	C₂×C₅₃	semidirect
179. #2236	C₂×C₅₃	semidirect
180. #2235	C₂×C₅₃	semidirect
181. #2159	C₂×C₅₃	semidirect
182. #2158	C₂×C₅₃	semidirect
183. #2141	C₂×C₅₃	semidirect
184. C₂×#358	C₂×C₅₃	semidirect
220. C₅₃	#33651	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×#3059	2
C₅₃×#2913	2
C₅₃×#2237	2
C₅₃×#2236	2
C₅₃×#2235	2
C₅₃×#2159	2
C₅₃×#2158	2
C₅₃×#2141	2
C₂×C₅₃×#358	2
C₅₃×#2113	2
#33651	53

Subgroups — C53×#33651 #27241163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33651 #27241163