Subgroups — C₅₃×#10154 #27217666

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₅₃	—	2	2	no	—
9	C₅₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#494	—	2	2	no	—
13	C₅₃×#494	—	2	2	no	—
14	C₅₃×#492	—	2	1	no	—
15	C₅₃×#489	—	2	2	no	—
16	C₅₃×#487	—	2	1	no	—
17	C₅₃×#487	—	2	1	no	—
18	C₅₃×#486	—	2	1	no	—
19	C₅₃×#461	—	2	2	no	—
20	C₅₃×#448	—	2	2	no	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#423	C₂²	1	1	yes	characteristic
23	C₅₃×#417	—	2	2	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#343	—	2	2	no	—
27	C₅₃×#332	—	2	2	no	—
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
32	C₂×Q₁₆×C₅₃	—	2	8	no	—
33	C₂×Q₁₆×C₅₃	—	4	8	no	—
34	C₂×SD₁₆×C₅₃	—	2	4	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₄⋊Q₈×C₅₃	—	2	1	no	—
39	C₄⋊Q₈×C₅₃	—	2	1	no	—
40	C₄⋊Q₈×C₅₃	—	2	1	no	—
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	1	no	—
43	C₄⋊Q₈×C₅₃	—	2	1	no	—
44	C₄⋊Q₈×C₅₃	—	2	1	no	—
45	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
47	#127×C₅₃	D₄	1	1	yes	characteristic
48	#124×C₅₃	—	2	1	no	—
49	#121×C₅₃	—	2	1	no	—
50	#121×C₅₃	—	2	1	no	—
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
61	#103×C₅₃	—	2	4	no	—
62	#103×C₅₃	—	2	4	no	—
63	#102×C₅₃	—	2	4	no	—
64	C₂²⋊C₈×C₅₃	—	2	1	no	—
65	#97×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	D₄	1	2	yes	—
70	#41×C₅₃	—	4	2	no	—
71	#41×C₅₃	—	4	2	no	—
72	C₂×Q₈×C₅₃	—	2	1	no	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×Q₈×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	4	4	no	—
77	C₂×Q₈×C₅₃	—	4	4	no	—
78	C₂×Q₈×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	2	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	4	1	no	—
85	C₂²×C₄×C₅₃	—	4	1	no	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
89	Q₁₆×C₅₃	—	4	16	no	—
90	SD₁₆×C₅₃	—	8	4	no	—
91	M₁₆×C₅₃	—	2	2	no	—
92	M₁₆×C₅₃	—	2	2	no	—
93	M₁₆×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	—	2	4	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	4	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	—	2	1	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	—	4	1	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₂²⋊C₄×C₅₃	—	4	1	no	—
110	C₂²⋊C₄×C₅₃	—	4	1	no	—
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₅₃	—	4	1	no	—
116	C₂³×C₅₃	#121	1	1	yes	characteristic
117	Q₈×C₅₃	—	4	4	no	—
118	Q₈×C₅₃	—	4	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	4	4	no	—
121	Q₈×C₅₃	—	8	2	no	—
122	D₄×C₅₃	—	4	2	no	—
123	D₄×C₅₃	—	4	2	no	—
124	D₄×C₅₃	—	4	2	no	—
125	D₄×C₅₃	—	8	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	8	2	no	—
132	C₂×C₄×C₅₃	—	8	1	no	—
133	C₂×C₄×C₅₃	—	8	1	no	—
134	C₂×C₄×C₅₃	—	8	2	no	—
135	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#135	1	2	yes	—
142	C₂×C₄×C₅₃	#137	1	2	yes	—
143	C₈×C₅₃	—	2	4	no	—
144	C₈×C₅₃	—	4	2	no	—
145	C₈×C₅₃	—	4	2	no	—
146	C₈×C₅₃	—	8	1	no	—
147	#10154	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	—	8	1	no	—
153	C₂²×C₅₃	#392	1	1	yes	characteristic
154	C₂²×C₅₃	#464	1	2	yes	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	1	no	—
160	C₄×C₅₃	#452	1	2	yes	—
161	C₄×C₅₃	#453	1	2	yes	—
162	C₄×C₅₃	#473	1	2	yes	—
163	C₄×C₅₃	#495	1	1	yes	characteristic
164	C₄×C₅₃	#497	1	1	yes	characteristic
165	#3206	C₂×C₅₃	1	1	yes	characteristic
166	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#3141	C₂×C₅₃	1	1	yes	characteristic
168	#1997	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1492	C₂×C₅₃	1	1	yes	characteristic
171	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	16	1	no	—
174	C₂×C₅₃	#2002	1	1	yes	characteristic
175	C₂×C₅₃	#2015	1	1	yes	characteristic
176	C₂×C₅₃	#2020	1	1	yes	characteristic
177	C₂²×Q₁₆	—	2	2	no	—
178	#532	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
180	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
181	#494	—	2	2	no	—
182	#494	—	2	2	no	—
183	#492	—	2	1	no	—
184	#489	—	2	2	no	—
185	#487	—	2	1	no	—
186	#487	—	2	1	no	—
187	#486	—	2	1	no	—
188	#461	—	2	2	no	—
189	#448	—	2	2	no	—
190	#431	C₂²×C₅₃	1	1	yes	characteristic
191	#423	C₂²×C₅₃	1	1	yes	characteristic
192	#417	—	2	2	no	—
193	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
195	#343	—	2	2	no	—
196	#332	—	2	2	no	—
197	D₄⋊C₈	—	2	2	no	—
198	C₅₃	#10154	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×#41	—	2	1	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
202	C₂×Q₁₆	—	2	8	no	—
203	C₂×Q₁₆	—	4	8	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
207	C₂²×C₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
216	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
217	#127	D₄×C₅₃	1	1	yes	characteristic
218	#124	—	2	1	no	—
219	#121	—	2	1	no	—
220	#121	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	D₄×C₅₃	1	2	yes	—
240	#41	—	4	2	no	—
241	#41	—	4	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
259	Q₁₆	—	4	16	no	—
260	SD₁₆	—	8	4	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	4	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
285	C₂³	—	4	1	no	—
286	C₂³	#121×C₅₃	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
306	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
307	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
308	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
309	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
310	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
311	C₂×C₄	#135×C₅₃	1	2	yes	—
312	C₂×C₄	#137×C₅₃	1	2	yes	—
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	—	8	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₅₃×#392	1	1	yes	characteristic
323	C₂²	C₅₃×#464	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	1	no	—
329	C₄	C₅₃×#452	1	2	yes	—
330	C₄	C₅₃×#453	1	2	yes	—
331	C₄	C₅₃×#473	1	2	yes	—
332	C₄	C₅₃×#495	1	1	yes	characteristic
333	C₄	C₅₃×#497	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	16	1	no	—
336	C₂	C₅₃×#2002	1	1	yes	characteristic
337	C₂	C₅₃×#2015	1	1	yes	characteristic
338	C₂	C₅₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#494	C₂	semidirect
4. C₅₃×#1997	C₂	semidirect
5. C₅₃×#1549	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
147. #10154	C₅₃	direct (N × H)
166. C₂×#494	C₂×C₅₃	semidirect
168. #1997	C₂×C₅₃	semidirect
169. #1549	C₂×C₅₃	semidirect
171. #1278	C₂×C₅₃	semidirect
198. C₅₃	#10154	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×#3206	2
C₂×C₅₃×#494	2
C₅₃×#3141	2
C₅₃×#1997	2
C₅₃×#1549	2
C₅₃×#1492	2
C₅₃×#1278	2
#10154	53

Subgroups — C53×#10154 #27217666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10154 #27217666