Subgroups — C₅₃×#17654 #27225166

Order 13568.

960 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×#3518	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1531	C₂	1	3	yes	maximal, complemented
5	C₅₃×#1529	C₂	1	3	yes	maximal, complemented
6	C₅₃×#1526	C₂	1	3	yes	maximal, complemented
7	C₅₃×#1524	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₅₃	—	2	1	no	—
9	Q₈²×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#556	—	2	3	no	—
11	C₅₃×#556	C₄	1	3	yes	—
12	C₅₃×#555	C₄	1	3	yes	—
13	D₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#548	—	2	1	no	—
15	C₅₃×#542	—	2	3	no	—
16	C₅₃×#533	C₂²	1	3	yes	—
17	C₅₃×#529	C₂²	1	3	yes	—
18	C₅₃×#519	—	2	1	no	—
19	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₅₃	—	2	1	no	—
21	C₅₃×#436	—	2	3	no	—
22	C₅₃×#432	C₂²	1	3	yes	—
23	C₅₃×#405	—	2	3	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
25	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
26	C₂×#97×C₅₃	C₂²	1	3	yes	—
27	C₅₃×#330	C₂²	1	3	yes	—
28	C₅₃×#330	C₂²	1	3	yes	—
29	C₅₃×#330	C₂²	1	6	yes	—
30	C₅₃×#329	C₂²	1	1	yes	characteristic
31	C₅₃×#329	C₂²	1	3	yes	—
32	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₅₃	—	2	1	no	—
34	#131×C₅₃	—	2	8	no	—
35	C₂×M₁₆×C₅₃	—	2	3	no	—
36	C₂×M₁₆×C₅₃	—	2	3	no	—
37	C₂²×C₈×C₅₃	—	2	3	no	—
38	C₄⋊Q₈×C₅₃	—	2	3	no	—
39	C₄⋊Q₈×C₅₃	C₂×C₄	1	3	yes	—
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	3	yes	—
41	#126×C₅₃	—	2	6	no	—
42	#126×C₅₃	—	2	3	no	—
43	#125×C₅₃	C₂³	1	3	yes	—
44	#124×C₅₃	—	2	6	no	—
45	#124×C₅₃	—	2	3	no	—
46	#124×C₅₃	C₂³	1	1	yes	characteristic
47	#123×C₅₃	—	2	3	no	—
48	C₂²⋊Q₈×C₅₃	—	2	3	no	—
49	C₂²⋊Q₈×C₅₃	—	2	3	no	—
50	C₂²⋊Q₈×C₅₃	—	2	3	no	—
51	C₂²⋊Q₈×C₅₃	C₂×C₄	1	3	yes	—
52	#121×C₅₃	—	2	3	no	—
53	#121×C₅₃	C₂×C₄	1	3	yes	—
54	C₄×Q₈×C₅₃	—	2	3	no	—
55	C₄×Q₈×C₅₃	—	2	3	no	—
56	C₄×Q₈×C₅₃	—	2	3	no	—
57	C₄×Q₈×C₅₃	—	2	1	no	—
58	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
60	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×D₄×C₅₃	—	2	3	no	—
62	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
63	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
64	C₄×D₄×C₅₃	D₄	1	3	yes	—
65	#117×C₅₃	—	2	3	no	—
66	#117×C₅₃	—	2	3	no	—
67	#117×C₅₃	C₂×C₄	1	6	yes	—
68	#117×C₅₃	D₄	1	3	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
70	C₄⋊C₈×C₅₃	—	2	3	no	—
71	C₂²⋊C₈×C₅₃	—	2	3	no	—
72	#97×C₅₃	—	2	3	no	—
73	#97×C₅₃	C₂³	1	1	yes	characteristic
74	#97×C₅₃	C₂³	1	3	yes	—
75	#97×C₅₃	C₂³	1	1	yes	characteristic
76	#97×C₅₃	C₂³	1	3	yes	—
77	C₄×C₈×C₅₃	—	2	3	no	—
78	C₄×C₈×C₅₃	—	2	3	no	—
79	#41×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	—	2	3	no	—
81	C₂×Q₈×C₅₃	—	2	3	no	—
82	C₂×Q₈×C₅₃	—	4	4	no	—
83	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×D₄×C₅₃	—	2	3	no	—
86	C₂×D₄×C₅₃	C₂²⋊C₄	1	3	yes	—
87	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	2	3	no	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
92	M₁₆×C₅₃	—	2	12	no	—
93	M₁₆×C₅₃	—	4	6	no	—
94	C₂×C₈×C₅₃	—	2	12	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	—	2	3	no	—
97	C₂×C₈×C₅₃	—	2	1	no	—
98	C₂×C₈×C₅₃	—	2	3	no	—
99	C₄⋊C₄×C₅₃	—	2	3	no	—
100	C₄⋊C₄×C₅₃	—	2	3	no	—
101	C₄⋊C₄×C₅₃	—	2	3	no	—
102	C₄⋊C₄×C₅₃	—	2	3	no	—
103	C₄⋊C₄×C₅₃	—	2	3	no	—
104	C₄⋊C₄×C₅₃	—	2	3	no	—
105	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
106	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
107	C₄⋊C₄×C₅₃	C₂²×C₄	1	3	yes	—
108	C₄⋊C₄×C₅₃	C₂²×C₄	1	3	yes	—
109	C₂²⋊C₄×C₅₃	—	2	6	no	—
110	C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂²⋊C₄×C₅₃	—	2	3	no	—
112	C₂²⋊C₄×C₅₃	—	2	3	no	—
113	C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₂²⋊C₄×C₅₃	—	2	1	no	—
115	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
116	C₄²×C₅₃	—	2	6	no	—
117	C₄²×C₅₃	—	2	3	no	—
118	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
119	C₄²×C₅₃	C₂×D₄	1	3	yes	—
120	C₄²×C₅₃	C₂×D₄	1	3	yes	—
121	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂³×C₅₃	—	2	1	no	—
123	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
124	Q₈×C₅₃	—	4	1	no	—
125	Q₈×C₅₃	—	4	6	no	—
126	Q₈×C₅₃	—	8	3	no	—
127	Q₈×C₅₃	#104	1	4	yes	—
128	D₄×C₅₃	—	2	6	no	—
129	D₄×C₅₃	—	4	1	no	—
130	D₄×C₅₃	—	8	3	no	—
131	C₂×C₄×C₅₃	—	2	6	no	—
132	C₂×C₄×C₅₃	—	2	3	no	—
133	C₂×C₄×C₅₃	—	2	3	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	4	3	no	—
136	C₂×C₄×C₅₃	—	4	3	no	—
137	C₂×C₄×C₅₃	—	4	6	no	—
138	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
139	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
140	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	—	2	6	no	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	—	4	3	no	—
148	#17654	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₃	—	2	3	no	—
150	C₂²×C₅₃	—	2	3	no	—
151	C₂²×C₅₃	—	2	3	no	—
152	C₂²×C₅₃	—	4	1	no	—
153	C₂²×C₅₃	—	4	1	no	—
154	C₂²×C₅₃	—	4	1	no	—
155	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	3	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	3	no	—
161	C₄×C₅₃	—	4	1	no	—
162	C₄×C₅₃	—	4	3	no	—
163	C₄×C₅₃	—	4	3	no	—
164	C₄×C₅₃	C₂×#104	1	6	yes	—
165	#3518	C₂×C₅₃	1	1	yes	characteristic
166	#2971	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
168	#1531	C₂×C₅₃	1	3	yes	complemented
169	#1529	C₂×C₅₃	1	3	yes	complemented
170	#1526	C₂×C₅₃	1	3	yes	complemented
171	#1524	C₂×C₅₃	1	3	yes	complemented
172	C₂×C₅₃	—	4	3	no	—
173	C₂×C₅₃	—	8	1	no	—
174	C₂×C₅₃	#2889	1	1	yes	characteristic
175	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
176	C₂×C₅₃	#2901	1	1	yes	characteristic
177	C₂×#131	—	2	1	no	—
178	Q₈²	C₄×C₅₃	1	1	yes	characteristic
179	#556	—	2	3	no	—
180	#556	C₄×C₅₃	1	3	yes	—
181	#555	C₄×C₅₃	1	3	yes	—
182	D₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
183	#548	—	2	1	no	—
184	#542	—	2	3	no	—
185	#533	C₂²×C₅₃	1	3	yes	—
186	#529	C₂²×C₅₃	1	3	yes	—
187	#519	—	2	1	no	—
188	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
189	C₈×Q₈	—	2	1	no	—
190	#436	—	2	3	no	—
191	#432	C₂²×C₅₃	1	3	yes	—
192	#405	—	2	3	no	—
193	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
194	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
195	C₂×#97	C₂²×C₅₃	1	3	yes	—
196	#330	C₂²×C₅₃	1	3	yes	—
197	#330	C₂²×C₅₃	1	3	yes	—
198	#330	C₂²×C₅₃	1	6	yes	—
199	#329	C₂²×C₅₃	1	1	yes	characteristic
200	#329	C₂²×C₅₃	1	3	yes	—
201	C₅₃	#17654	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
203	C₂²×Q₈	—	2	1	no	—
204	#131	—	2	8	no	—
205	C₂×M₁₆	—	2	3	no	—
206	C₂×M₁₆	—	2	3	no	—
207	C₂²×C₈	—	2	3	no	—
208	C₄⋊Q₈	—	2	3	no	—
209	C₄⋊Q₈	C₂×C₄×C₅₃	1	3	yes	—
210	C₄⋊Q₈	C₂×C₄×C₅₃	1	3	yes	—
211	#126	—	2	6	no	—
212	#126	—	2	3	no	—
213	#125	C₂³×C₅₃	1	3	yes	—
214	#124	—	2	3	no	—
215	#124	—	2	6	no	—
216	#124	C₂³×C₅₃	1	1	yes	characteristic
217	#123	—	2	3	no	—
218	C₂²⋊Q₈	—	2	3	no	—
219	C₂²⋊Q₈	—	2	3	no	—
220	C₂²⋊Q₈	—	2	3	no	—
221	C₂²⋊Q₈	C₂×C₄×C₅₃	1	3	yes	—
222	#121	—	2	3	no	—
223	#121	C₂×C₄×C₅₃	1	3	yes	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	3	no	—
226	C₄×Q₈	—	2	3	no	—
227	C₄×Q₈	—	2	3	no	—
228	C₄×Q₈	C₂×C₄×C₅₃	1	3	yes	—
229	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
231	C₄×D₄	—	2	3	no	—
232	C₄×D₄	C₂×C₄×C₅₃	1	3	yes	—
233	C₄×D₄	C₂×C₄×C₅₃	1	3	yes	—
234	C₄×D₄	D₄×C₅₃	1	3	yes	—
235	#117	—	2	3	no	—
236	#117	—	2	3	no	—
237	#117	C₂×C₄×C₅₃	1	6	yes	—
238	#117	D₄×C₅₃	1	3	yes	—
239	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
240	C₄⋊C₈	—	2	3	no	—
241	C₂²⋊C₈	—	2	3	no	—
242	#97	—	2	3	no	—
243	#97	C₂³×C₅₃	1	1	yes	characteristic
244	#97	C₂³×C₅₃	1	3	yes	—
245	#97	C₂³×C₅₃	1	1	yes	characteristic
246	#97	C₂³×C₅₃	1	3	yes	—
247	C₄×C₈	—	2	3	no	—
248	C₄×C₈	—	2	3	no	—
249	#41	—	2	4	no	—
250	C₂×Q₈	—	2	3	no	—
251	C₂×Q₈	—	2	3	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
254	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
255	C₂×D₄	—	2	3	no	—
256	C₂×D₄	C₂²⋊C₄×C₅₃	1	3	yes	—
257	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	—	2	3	no	—
259	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
260	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
261	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
262	M₁₆	—	2	12	no	—
263	M₁₆	—	4	6	no	—
264	C₂×C₈	—	2	12	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	3	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	3	no	—
269	C₄⋊C₄	—	2	3	no	—
270	C₄⋊C₄	—	2	3	no	—
271	C₄⋊C₄	—	2	3	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₄⋊C₄	—	2	3	no	—
274	C₄⋊C₄	—	2	3	no	—
275	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
276	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
277	C₄⋊C₄	C₂²×C₄×C₅₃	1	3	yes	—
278	C₄⋊C₄	C₂²×C₄×C₅₃	1	3	yes	—
279	C₂²⋊C₄	—	2	6	no	—
280	C₂²⋊C₄	—	2	3	no	—
281	C₂²⋊C₄	—	2	3	no	—
282	C₂²⋊C₄	—	2	3	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
286	C₄²	—	2	6	no	—
287	C₄²	—	2	3	no	—
288	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
289	C₄²	C₂×D₄×C₅₃	1	3	yes	—
290	C₄²	C₂×D₄×C₅₃	1	3	yes	—
291	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	6	no	—
296	Q₈	—	8	3	no	—
297	Q₈	#104×C₅₃	1	4	yes	—
298	D₄	—	2	6	no	—
299	D₄	—	4	1	no	—
300	D₄	—	8	3	no	—
301	C₂×C₄	—	2	6	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	3	no	—
304	C₂×C₄	—	2	3	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	3	no	—
307	C₂×C₄	—	4	6	no	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
309	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
311	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	6	no	—
316	C₈	—	2	2	no	—
317	C₈	—	4	3	no	—
318	C₂²	—	2	3	no	—
319	C₂²	—	2	3	no	—
320	C₂²	—	2	3	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	3	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	3	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	3	no	—
332	C₄	—	4	3	no	—
333	C₄	C₂×#104×C₅₃	1	6	yes	—
334	C₂	—	4	3	no	—
335	C₂	—	8	1	no	—
336	C₂	C₅₃×#2889	1	1	yes	characteristic
337	C₂	C₂²×#104×C₅₃	1	1	yes	characteristic
338	C₂	C₅₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3518	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2971	C₂	semidirect
3. C₅₃×#2875	C₂	semidirect
4. C₅₃×#1531	C₂	semidirect
5. C₅₃×#1529	C₂	semidirect
6. C₅₃×#1526	C₂	semidirect
7. C₅₃×#1524	C₂	semidirect
148. #17654	C₅₃	direct (N × H)
166. #2971	C₂×C₅₃	semidirect
167. #2875	C₂×C₅₃	semidirect
168. #1531	C₂×C₅₃	semidirect
169. #1529	C₂×C₅₃	semidirect
170. #1526	C₂×C₅₃	semidirect
171. #1524	C₂×C₅₃	semidirect
201. C₅₃	#17654	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×#3518	2
C₅₃×#2971	2
C₅₃×#2875	2
C₅₃×#1531	2
C₅₃×#1529	2
C₅₃×#1526	2
C₅₃×#1524	2
#17654	53

Subgroups — C53×#17654 #27225166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17654 #27225166