Subgroups — C₅₃×#60623 #27268135

Order 13568.

2042 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3141	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3139	C₂	1	3	yes	maximal, complemented
5	C₅₃×#3137	C₂	1	12	yes	maximal, complemented
6	C₅₃×#3131	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₃×#487	C₂	1	3	yes	maximal, complemented
8	C₅₃×#2898	C₂	1	4	yes	maximal, complemented
9	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
11	C₂×#131×C₅₃	C₂²	1	2	yes	—
12	Q₈²×C₅₃	—	2	1	no	—
13	C₅₃×#557	—	2	1	no	—
14	C₅₃×#554	—	2	3	no	—
15	D₄×Q₈×C₅₃	—	2	1	no	—
16	D₄×Q₈×C₅₃	—	2	1	no	—
17	D₄×Q₈×C₅₃	—	2	3	no	—
18	C₅₃×#547	—	2	3	no	—
19	C₅₃×#547	—	2	3	no	—
20	C₅₃×#544	C₂²	1	6	yes	—
21	C₅₃×#536	C₂²	1	6	yes	—
22	C₅₃×#533	C₂²	1	3	yes	—
23	C₅₃×#532	C₂²	1	3	yes	—
24	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
25	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
26	C₅₃×#520	C₂²	1	2	yes	—
27	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#491	C₂²	1	12	yes	—
29	C₅₃×#489	C₂²	1	12	yes	—
30	C₅₃×#488	C₂²	1	12	yes	—
31	C₅₃×#487	C₂²	1	3	yes	—
32	C₅₃×#487	C₂²	1	12	yes	—
33	C₅₃×#487	C₂²	1	6	yes	—
34	C₅₃×#487	C₂²	1	3	yes	—
35	C₅₃×#486	C₂²	1	1	yes	characteristic
36	C₅₃×#486	C₂²	1	3	yes	—
37	C₅₃×#419	C₂²	1	12	yes	—
38	C₅₃×#417	C₂²	1	12	yes	—
39	C₅₃×#416	C₂²	1	12	yes	—
40	C₂×#103×C₅₃	C₂²	1	12	yes	—
41	C₅₃×#405	C₂²	1	6	yes	—
42	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
44	2₋¹⁺⁴×C₅₃	—	2	16	no	—
45	C₂×#41×C₅₃	—	2	6	no	—
46	C₂×#41×C₅₃	C₂³	1	6	yes	—
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₃	—	2	2	no	—
49	C₂²×Q₈×C₅₃	—	2	3	no	—
50	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
51	#131×C₅₃	—	2	8	no	—
52	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
54	C₄⋊Q₈×C₅₃	—	2	6	no	—
55	C₄⋊Q₈×C₅₃	—	2	3	no	—
56	C₄⋊Q₈×C₅₃	—	2	3	no	—
57	C₄⋊Q₈×C₅₃	—	2	3	no	—
58	C₄⋊Q₈×C₅₃	—	2	3	no	—
59	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
64	#127×C₅₃	C₂³	1	1	yes	characteristic
65	#125×C₅₃	—	2	6	no	—
66	#125×C₅₃	C₂³	1	6	yes	—
67	#124×C₅₃	C₂³	1	6	yes	—
68	#123×C₅₃	—	2	12	no	—
69	C₂²⋊Q₈×C₅₃	—	2	6	no	—
70	C₂²⋊Q₈×C₅₃	—	2	6	no	—
71	C₂²⋊Q₈×C₅₃	—	2	12	no	—
72	C₂²⋊Q₈×C₅₃	—	2	3	no	—
73	C₂²⋊Q₈×C₅₃	—	2	3	no	—
74	#121×C₅₃	—	2	3	no	—
75	#121×C₅₃	—	2	3	no	—
76	C₄×Q₈×C₅₃	—	2	3	no	—
77	C₄×Q₈×C₅₃	—	2	3	no	—
78	C₄×Q₈×C₅₃	—	2	2	no	—
79	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
81	C₄×D₄×C₅₃	—	2	3	no	—
82	C₄×D₄×C₅₃	—	2	3	no	—
83	C₄×D₄×C₅₃	—	2	6	no	—
84	C₄×D₄×C₅₃	D₄	1	3	yes	—
85	C₄×D₄×C₅₃	D₄	1	3	yes	—
86	#117×C₅₃	C₂³	1	6	yes	—
87	#117×C₅₃	C₂³	1	3	yes	—
88	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
89	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
90	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
93	#103×C₅₃	C₂³	1	12	yes	—
94	#103×C₅₃	C₂³	1	24	yes	—
95	#103×C₅₃	C₂³	1	12	yes	—
96	#102×C₅₃	C₂³	1	4	yes	—
97	#102×C₅₃	C₂³	1	12	yes	—
98	#97×C₅₃	C₂³	1	6	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	3	yes	—
101	C₄×C₈×C₅₃	C₂³	1	6	yes	—
102	#41×C₅₃	—	2	24	no	—
103	#41×C₅₃	—	4	24	no	—
104	#41×C₅₃	#41	1	8	yes	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	3	no	—
107	C₂×Q₈×C₅₃	—	2	6	no	—
108	C₂×Q₈×C₅₃	—	2	6	no	—
109	C₂×Q₈×C₅₃	—	2	24	no	—
110	C₂×Q₈×C₅₃	—	4	8	no	—
111	C₂×Q₈×C₅₃	—	8	6	no	—
112	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
114	C₂×D₄×C₅₃	—	2	6	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
118	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
119	C₂²×C₄×C₅₃	—	2	6	no	—
120	C₂²×C₄×C₅₃	—	2	6	no	—
121	C₂²×C₄×C₅₃	—	2	6	no	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
125	M₁₆×C₅₃	—	2	12	no	—
126	C₂×C₈×C₅₃	—	2	12	no	—
127	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
128	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
129	C₄⋊C₄×C₅₃	—	2	3	no	—
130	C₄⋊C₄×C₅₃	—	2	3	no	—
131	C₄⋊C₄×C₅₃	—	2	3	no	—
132	C₄⋊C₄×C₅₃	—	2	6	no	—
133	C₄⋊C₄×C₅₃	—	2	6	no	—
134	C₄⋊C₄×C₅₃	—	2	6	no	—
135	C₄⋊C₄×C₅₃	—	2	6	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	6	no	—
138	C₄⋊C₄×C₅₃	—	2	6	no	—
139	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
140	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
141	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
142	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
143	C₂²⋊C₄×C₅₃	—	2	6	no	—
144	C₂²⋊C₄×C₅₃	—	2	6	no	—
145	C₂²⋊C₄×C₅₃	—	2	6	no	—
146	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
147	C₄²×C₅₃	—	2	6	no	—
148	C₄²×C₅₃	C₂×D₄	1	3	yes	—
149	C₄²×C₅₃	C₂×D₄	1	3	yes	—
150	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₄²×C₅₃	C₂⁴	1	3	yes	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
154	Q₈×C₅₃	—	2	12	no	—
155	Q₈×C₅₃	—	4	12	no	—
156	Q₈×C₅₃	—	4	6	no	—
157	Q₈×C₅₃	—	4	2	no	—
158	Q₈×C₅₃	—	8	6	no	—
159	Q₈×C₅₃	#124	1	4	yes	—
160	D₄×C₅₃	—	2	4	no	—
161	D₄×C₅₃	—	4	12	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	2	no	—
164	D₄×C₅₃	#124	1	12	yes	—
165	C₂×C₄×C₅₃	—	2	6	no	—
166	C₂×C₄×C₅₃	—	2	6	no	—
167	C₂×C₄×C₅₃	—	2	6	no	—
168	C₂×C₄×C₅₃	—	2	6	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	4	12	no	—
171	C₂×C₄×C₅₃	—	4	12	no	—
172	C₂×C₄×C₅₃	—	8	6	no	—
173	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
174	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
175	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
179	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	6	no	—
182	#60623	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₅₃	—	2	3	no	—
184	C₂²×C₅₃	—	2	3	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	—	4	2	no	—
187	C₂²×C₅₃	—	4	2	no	—
188	C₂²×C₅₃	C₂×#124	1	6	yes	—
189	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
190	C₄×C₅₃	—	2	3	no	—
191	C₄×C₅₃	—	2	3	no	—
192	C₄×C₅₃	—	4	6	no	—
193	C₄×C₅₃	—	4	6	no	—
194	C₄×C₅₃	—	4	2	no	—
195	C₄×C₅₃	C₂×#124	1	6	yes	—
196	C₄×C₅₃	#577	1	2	yes	—
197	C₄×C₅₃	#578	1	2	yes	—
198	C₄×C₅₃	C₂²×#41	1	2	yes	—
199	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#3499	C₂×C₅₃	1	1	yes	characteristic
201	#3141	C₂×C₅₃	1	3	yes	complemented
202	#3139	C₂×C₅₃	1	3	yes	complemented
203	#3137	C₂×C₅₃	1	12	yes	complemented
204	#3131	C₂×C₅₃	1	3	yes	complemented
205	C₂×#487	C₂×C₅₃	1	3	yes	complemented
206	#2898	C₂×C₅₃	1	4	yes	complemented
207	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	2	6	no	—
209	C₂×C₅₃	—	8	2	no	—
210	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
213	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
214	C₂×#131	C₂²×C₅₃	1	2	yes	—
215	Q₈²	—	2	1	no	—
216	#557	—	2	1	no	—
217	#554	—	2	3	no	—
218	D₄×Q₈	—	2	1	no	—
219	D₄×Q₈	—	2	1	no	—
220	D₄×Q₈	—	2	3	no	—
221	#547	—	2	3	no	—
222	#547	—	2	3	no	—
223	#544	C₂²×C₅₃	1	6	yes	—
224	#536	C₂²×C₅₃	1	6	yes	—
225	#533	C₂²×C₅₃	1	3	yes	—
226	#532	C₂²×C₅₃	1	3	yes	—
227	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
228	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
229	#520	C₂²×C₅₃	1	2	yes	—
230	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
231	#491	C₂²×C₅₃	1	12	yes	—
232	#489	C₂²×C₅₃	1	12	yes	—
233	#488	C₂²×C₅₃	1	12	yes	—
234	#487	C₂²×C₅₃	1	3	yes	—
235	#487	C₂²×C₅₃	1	12	yes	—
236	#487	C₂²×C₅₃	1	6	yes	—
237	#487	C₂²×C₅₃	1	3	yes	—
238	#486	C₂²×C₅₃	1	1	yes	characteristic
239	#486	C₂²×C₅₃	1	3	yes	—
240	#419	C₂²×C₅₃	1	12	yes	—
241	#417	C₂²×C₅₃	1	12	yes	—
242	#416	C₂²×C₅₃	1	12	yes	—
243	C₂×#103	C₂²×C₅₃	1	12	yes	—
244	#405	C₂²×C₅₃	1	6	yes	—
245	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
246	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
247	C₅₃	#60623	1	1	yes	characteristic, direct product factor, Sylow
248	2₋¹⁺⁴	—	2	16	no	—
249	C₂×#41	—	2	6	no	—
250	C₂×#41	C₂³×C₅₃	1	6	yes	—
251	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
252	C₂²×Q₈	—	2	2	no	—
253	C₂²×Q₈	—	2	3	no	—
254	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
255	#131	—	2	8	no	—
256	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
257	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
258	C₄⋊Q₈	—	2	6	no	—
259	C₄⋊Q₈	—	2	3	no	—
260	C₄⋊Q₈	—	2	3	no	—
261	C₄⋊Q₈	—	2	3	no	—
262	C₄⋊Q₈	—	2	3	no	—
263	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
264	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
265	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
266	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
267	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
268	#127	C₂³×C₅₃	1	1	yes	characteristic
269	#125	—	2	6	no	—
270	#125	C₂³×C₅₃	1	6	yes	—
271	#124	C₂³×C₅₃	1	6	yes	—
272	#123	—	2	12	no	—
273	C₂²⋊Q₈	—	2	3	no	—
274	C₂²⋊Q₈	—	2	3	no	—
275	C₂²⋊Q₈	—	2	6	no	—
276	C₂²⋊Q₈	—	2	6	no	—
277	C₂²⋊Q₈	—	2	12	no	—
278	#121	—	2	3	no	—
279	#121	—	2	3	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	—	2	3	no	—
282	C₄×Q₈	—	2	3	no	—
283	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
284	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
285	C₄×D₄	—	2	6	no	—
286	C₄×D₄	—	2	3	no	—
287	C₄×D₄	—	2	3	no	—
288	C₄×D₄	D₄×C₅₃	1	3	yes	—
289	C₄×D₄	D₄×C₅₃	1	3	yes	—
290	#117	C₂³×C₅₃	1	6	yes	—
291	#117	C₂³×C₅₃	1	3	yes	—
292	C₂×C₄⋊C₄	—	2	3	no	—
293	C₂×C₄⋊C₄	—	2	3	no	—
294	C₂×C₄⋊C₄	—	2	3	no	—
295	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
296	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
297	#103	C₂³×C₅₃	1	12	yes	—
298	#103	C₂³×C₅₃	1	24	yes	—
299	#103	C₂³×C₅₃	1	12	yes	—
300	#102	C₂³×C₅₃	1	4	yes	—
301	#102	C₂³×C₅₃	1	12	yes	—
302	#97	C₂³×C₅₃	1	6	yes	—
303	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₅₃	1	3	yes	—
305	C₄×C₈	C₂³×C₅₃	1	6	yes	—
306	#41	—	2	24	no	—
307	#41	—	4	24	no	—
308	#41	#41×C₅₃	1	8	yes	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	3	no	—
311	C₂×Q₈	—	2	6	no	—
312	C₂×Q₈	—	2	6	no	—
313	C₂×Q₈	—	2	24	no	—
314	C₂×Q₈	—	4	8	no	—
315	C₂×Q₈	—	8	6	no	—
316	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
318	C₂×D₄	—	2	6	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
322	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
323	C₂²×C₄	—	2	6	no	—
324	C₂²×C₄	—	2	6	no	—
325	C₂²×C₄	—	2	6	no	—
326	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
327	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
328	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
329	M₁₆	—	2	12	no	—
330	C₂×C₈	—	2	12	no	—
331	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
332	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
333	C₄⋊C₄	—	2	3	no	—
334	C₄⋊C₄	—	2	3	no	—
335	C₄⋊C₄	—	2	3	no	—
336	C₄⋊C₄	—	2	6	no	—
337	C₄⋊C₄	—	2	6	no	—
338	C₄⋊C₄	—	2	6	no	—
339	C₄⋊C₄	—	2	6	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	6	no	—
342	C₄⋊C₄	—	2	6	no	—
343	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
344	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
345	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
346	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
347	C₂²⋊C₄	—	2	6	no	—
348	C₂²⋊C₄	—	2	6	no	—
349	C₂²⋊C₄	—	2	6	no	—
350	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
351	C₄²	—	2	6	no	—
352	C₄²	C₂×D₄×C₅₃	1	3	yes	—
353	C₄²	C₂×D₄×C₅₃	1	3	yes	—
354	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
355	C₄²	C₂⁴×C₅₃	1	3	yes	—
356	C₂³	—	2	2	no	—
357	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
358	Q₈	—	2	12	no	—
359	Q₈	—	4	12	no	—
360	Q₈	—	4	6	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	8	6	no	—
363	Q₈	#124×C₅₃	1	4	yes	—
364	D₄	—	2	4	no	—
365	D₄	—	4	12	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	#124×C₅₃	1	12	yes	—
369	C₂×C₄	—	2	6	no	—
370	C₂×C₄	—	2	6	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	6	no	—
373	C₂×C₄	—	2	6	no	—
374	C₂×C₄	—	4	12	no	—
375	C₂×C₄	—	4	12	no	—
376	C₂×C₄	—	8	6	no	—
377	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
378	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
379	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
380	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
383	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
384	C₈	—	2	2	no	—
385	C₈	—	2	6	no	—
386	C₂²	—	2	3	no	—
387	C₂²	—	2	3	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	C₂×#124×C₅₃	1	6	yes	—
392	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
393	C₄	—	2	3	no	—
394	C₄	—	2	3	no	—
395	C₄	—	4	6	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	6	no	—
398	C₄	C₂×#124×C₅₃	1	6	yes	—
399	C₄	C₅₃×#577	1	2	yes	—
400	C₄	C₅₃×#578	1	2	yes	—
401	C₄	C₂²×#41×C₅₃	1	2	yes	—
402	C₂	—	2	6	no	—
403	C₂	—	8	2	no	—
404	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
405	C₂	C₂×C₅₃×#577	1	1	yes	characteristic
406	C₂	C₂×C₅₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₅₃×#3503	C₂	semidirect
3. C₅₃×#3141	C₂	semidirect
4. C₅₃×#3139	C₂	semidirect
5. C₅₃×#3137	C₂	semidirect
6. C₅₃×#3131	C₂	semidirect
7. C₂×C₅₃×#487	C₂	semidirect
8. C₅₃×#2898	C₂	semidirect
9. C₄×#131×C₅₃	C₂	semidirect
182. #60623	C₅₃	direct (N × H)
199. #3503	C₂×C₅₃	semidirect
201. #3141	C₂×C₅₃	semidirect
202. #3139	C₂×C₅₃	semidirect
203. #3137	C₂×C₅₃	semidirect
204. #3131	C₂×C₅₃	semidirect
205. C₂×#487	C₂×C₅₃	semidirect
206. #2898	C₂×C₅₃	semidirect
207. C₄×#131	C₂×C₅₃	semidirect
247. C₅₃	#60623	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×#3503	2
C₅₃×#3499	2
C₅₃×#3141	2
C₅₃×#3139	2
C₅₃×#3137	2
C₅₃×#3131	2
C₂×C₅₃×#487	2
C₅₃×#2898	2
C₄×#131×C₅₃	2
#60623	53

Subgroups — C53×#60623 #27268135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60623 #27268135