Subgroups — C₅₃×#23669 #27231181

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2316	C₂	1	2	yes	maximal
4	C₅₃×#2088	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2087	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2086	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1736	C₂	1	2	yes	maximal
8	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₅₃	C₂²	1	2	yes	—
12	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#427	—	2	4	no	—
15	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#424	—	2	4	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#389	—	2	2	no	—
22	C₅₃×#389	—	2	2	no	—
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#389	C₂²	1	4	yes	—
25	C₅₃×#380	—	2	4	no	—
26	C₅₃×#340	C₂²	1	2	yes	—
27	C₅₃×#340	C₂²	1	2	yes	—
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₅₃×#340	C₂²	1	4	yes	—
30	C₂³×C₄×C₅₃	—	2	2	no	—
31	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	—	2	4	no	—
34	C₂²×C₈×C₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	D₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	4	no	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	Q₈	1	8	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	#107×C₅₃	—	2	2	no	—
58	#107×C₅₃	—	2	2	no	—
59	#106×C₅₃	—	2	4	no	—
60	C₄⋊C₈×C₅₃	—	2	4	no	—
61	C₄⋊C₈×C₅₃	—	2	4	no	—
62	#95×C₅₃	—	2	4	no	—
63	#95×C₅₃	—	2	4	no	—
64	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	8	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	8	2	no	—
70	C₂²×C₄×C₅₃	—	8	2	no	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	C₂²×C₄×C₅₃	#41	1	4	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₅₃	—	4	4	no	—
80	C₂×C₈×C₅₃	—	2	8	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	8	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	8	4	no	—
96	C₄⋊C₄×C₅₃	—	8	4	no	—
97	C₄⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
98	C₂²⋊C₄×C₅₃	—	2	4	no	—
99	C₂²⋊C₄×C₅₃	—	4	4	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	4	4	no	—
103	C₂³×C₅₃	—	2	1	no	—
104	C₂³×C₅₃	—	2	2	no	—
105	C₂³×C₅₃	—	2	1	no	—
106	C₂³×C₅₃	#121	1	2	yes	—
107	C₂³×C₅₃	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	8	8	no	—
124	C₂×C₄×C₅₃	#121	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
126	C₂×C₄×C₅₃	#123	1	4	yes	—
127	C₂×C₄×C₅₃	#125	1	8	yes	—
128	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
133	C₈×C₅₃	—	2	2	no	—
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	—	4	2	no	—
136	#23669	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₃	—	2	4	no	—
138	C₂²×C₅₃	—	2	4	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	#398	1	4	yes	—
142	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₂²×C₅₃	#576	1	1	yes	characteristic
145	C₂²×C₅₃	#577	1	2	yes	—
146	C₂²×C₅₃	#578	1	1	yes	characteristic
147	C₄×C₅₃	—	4	8	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	#398	1	4	yes	—
153	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#125	1	2	yes	—
155	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
157	C₂×#424	C₂×C₅₃	1	1	yes	characteristic
158	#2316	C₂×C₅₃	1	2	yes	—
159	#2088	C₂×C₅₃	1	4	yes	complemented
160	#2087	C₂×C₅₃	1	2	yes	complemented
161	#2086	C₂×C₅₃	1	2	yes	complemented
162	#1736	C₂×C₅₃	1	2	yes	—
163	C₂×#340	C₂×C₅₃	1	1	yes	characteristic
164	C₂×C₅₃	—	2	4	no	—
165	C₂×C₅₃	C₂×#398	1	1	yes	characteristic
166	C₂×C₅₃	#3060	1	1	yes	characteristic
167	C₂×C₅₃	#3061	1	1	yes	characteristic
168	C₂×C₅₃	#3081	1	2	yes	—
169	C₂×C₅₃	#3082	1	2	yes	—
170	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₂×#117	C₂²×C₅₃	1	2	yes	—
173	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
175	#427	—	2	4	no	—
176	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×#106	C₂²×C₅₃	1	2	yes	—
179	#424	—	2	4	no	—
180	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
182	#389	—	2	2	no	—
183	#389	—	2	2	no	—
184	#389	C₂²×C₅₃	1	4	yes	—
185	#389	C₂²×C₅₃	1	4	yes	—
186	#380	—	2	4	no	—
187	#340	C₂²×C₅₃	1	2	yes	—
188	#340	C₂²×C₅₃	1	4	yes	—
189	#340	C₂²×C₅₃	1	4	yes	—
190	#340	C₂²×C₅₃	1	2	yes	—
191	C₅₃	#23669	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	D₄×C₅₃	1	4	yes	—
198	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
199	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	4	no	—
204	#117	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	Q₈×C₅₃	1	8	yes	—
212	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#106	—	2	4	no	—
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	2	4	no	—
224	#95	—	2	4	no	—
225	#95	—	2	4	no	—
226	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	8	2	no	—
233	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
237	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
238	C₂²×C₄	#41×C₅₃	1	2	yes	—
239	C₂²×C₄	#41×C₅₃	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₄⋊C₄	—	8	4	no	—
259	C₄⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	4	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#121×C₅₃	1	2	yes	—
269	C₂³	C₂²⋊Q₈×C₅₃	1	4	yes	—
270	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	#121×C₅₃	1	2	yes	—
287	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
288	C₂×C₄	#123×C₅₃	1	4	yes	—
289	C₂×C₄	#125×C₅₃	1	8	yes	—
290	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
293	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
294	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₅₃×#398	1	4	yes	—
303	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
304	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
305	C₂²	C₅₃×#576	1	1	yes	characteristic
306	C₂²	C₅₃×#577	1	2	yes	—
307	C₂²	C₅₃×#578	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	8	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₅₃×#398	1	4	yes	—
314	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
315	C₄	C₂×#125×C₅₃	1	2	yes	—
316	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
317	C₂	—	2	4	no	—
318	C₂	C₂×C₅₃×#398	1	1	yes	characteristic
319	C₂	C₅₃×#3060	1	1	yes	characteristic
320	C₂	C₅₃×#3061	1	1	yes	characteristic
321	C₂	C₅₃×#3081	1	2	yes	—
322	C₂	C₅₃×#3082	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#2088	C₂	semidirect
5. C₅₃×#2087	C₂	semidirect
6. C₅₃×#2086	C₂	semidirect
136. #23669	C₅₃	direct (N × H)
159. #2088	C₂×C₅₃	semidirect
160. #2087	C₂×C₅₃	semidirect
161. #2086	C₂×C₅₃	semidirect
191. C₅₃	#23669	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#427	2
C₂×C₅₃×#424	2
C₅₃×#2316	2
C₅₃×#2088	2
C₅₃×#2087	2
C₅₃×#2086	2
C₅₃×#1736	2
C₂×C₅₃×#340	2
#23669	53

Subgroups — C53×#23669 #27231181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23669 #27231181