Subgroups — C₅₃×#60693 #27268205

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3444	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3185	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3184	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3183	C₂	1	8	yes	maximal, complemented
5	C₅₃×#3182	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3131	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3054	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2943	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#533	C₂²	1	4	yes	—
15	C₅₃×#533	C₂²	1	2	yes	—
16	C₅₃×#532	C₂²	1	2	yes	—
17	C₅₃×#532	C₂²	1	4	yes	—
18	C₅₃×#529	—	2	4	no	—
19	C₅₃×#529	—	2	4	no	—
20	C₅₃×#529	C₂²	1	8	yes	—
21	C₅₃×#525	—	2	4	no	—
22	C₅₃×#525	C₂²	1	4	yes	—
23	C₄×#41×C₅₃	—	2	4	no	—
24	C₄×#41×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#500	C₂²	1	2	yes	complemented
27	C₅₃×#499	C₂²	1	4	yes	complemented
28	C₅₃×#498	C₂²	1	2	yes	complemented
29	C₅₃×#495	C₂²	1	4	yes	complemented
30	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
31	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#492	C₂²	1	2	yes	complemented
33	C₅₃×#488	C₂²	1	8	yes	complemented
34	C₅₃×#487	C₂²	1	4	yes	complemented
35	C₅₃×#486	C₂²	1	4	yes	complemented
36	C₅₃×#470	C₂²	1	8	yes	—
37	C₅₃×#469	C₂²	1	8	yes	—
38	C₅₃×#468	C₂²	1	16	yes	—
39	C₅₃×#442	C₂²	1	8	yes	complemented
40	C₅₃×#441	C₂²	1	8	yes	complemented
41	C₅₃×#440	C₂²	1	16	yes	complemented
42	C₅₃×#427	C₂²	1	4	yes	—
43	C₅₃×#416	C₂²	1	8	yes	—
44	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
45	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
46	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
47	C₂×#97×C₅₃	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#41×C₅₃	—	2	4	no	—
51	C₂×#41×C₅₃	C₂³	1	4	yes	—
52	C₂³×C₄×C₅₃	D₄	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	#135×C₅₃	—	2	16	no	—
55	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
56	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
57	C₂×D₈×C₅₃	C₂³	1	4	yes	—
58	C₂×M₁₆×C₅₃	—	2	8	no	—
59	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
60	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
64	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
65	#127×C₅₃	C₂³	1	2	yes	—
66	#126×C₅₃	—	2	8	no	—
67	#125×C₅₃	—	2	4	no	—
68	#125×C₅₃	C₂³	1	4	yes	—
69	#124×C₅₃	—	2	4	no	—
70	#124×C₅₃	C₂³	1	4	yes	—
71	#123×C₅₃	—	2	8	no	—
72	#123×C₅₃	—	2	4	no	—
73	C₂²⋊Q₈×C₅₃	—	2	4	no	—
74	C₂²⋊Q₈×C₅₃	—	2	4	no	—
75	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
76	#121×C₅₃	—	2	4	no	—
77	#121×C₅₃	—	2	4	no	—
78	#121×C₅₃	C₂³	1	8	yes	—
79	#120×C₅₃	—	2	4	no	—
80	C₄×Q₈×C₅₃	—	2	4	no	—
81	C₄×Q₈×C₅₃	—	2	4	no	—
82	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
83	C₄×D₄×C₅₃	—	2	4	no	—
84	C₄×D₄×C₅₃	—	2	4	no	—
85	C₄×D₄×C₅₃	—	2	8	no	—
86	C₄×D₄×C₅₃	—	2	8	no	—
87	C₄×D₄×C₅₃	—	2	8	no	—
88	C₄×D₄×C₅₃	C₂³	1	8	yes	—
89	#117×C₅₃	—	2	8	no	—
90	#117×C₅₃	—	2	4	no	—
91	#117×C₅₃	—	2	4	no	—
92	#117×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄²×C₅₃	—	2	4	no	—
94	C₂×C₄²×C₅₃	D₄	1	4	yes	—
95	C₂×C₄²×C₅₃	D₄	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
100	#107×C₅₃	C₂³	1	8	yes	—
101	#106×C₅₃	C₂³	1	8	yes	—
102	#103×C₅₃	C₂³	1	16	yes	—
103	#102×C₅₃	C₂³	1	16	yes	—
104	#97×C₅₃	C₂³	1	4	yes	—
105	#97×C₅₃	C₂³	1	4	yes	—
106	C₄×C₈×C₅₃	C₂³	1	2	yes	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₄×C₈×C₅₃	C₂³	1	2	yes	—
111	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
112	#41×C₅₃	—	2	16	no	—
113	#41×C₅₃	—	4	16	no	—
114	C₂×Q₈×C₅₃	—	2	4	no	—
115	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
116	C₂×D₄×C₅₃	—	2	4	no	—
117	C₂×D₄×C₅₃	—	2	8	no	—
118	C₂×D₄×C₅₃	—	2	4	no	—
119	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	8	no	—
122	C₂²×C₄×C₅₃	—	2	4	no	—
123	C₂²×C₄×C₅₃	—	4	4	no	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₅₃	—	2	8	no	—
133	SD₁₆×C₅₃	—	2	16	no	—
134	D₈×C₅₃	—	2	8	no	—
135	M₁₆×C₅₃	—	2	8	no	—
136	M₁₆×C₅₃	—	2	8	no	—
137	C₂×C₈×C₅₃	—	2	4	no	—
138	C₂×C₈×C₅₃	#41	1	8	yes	complemented
139	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
140	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
141	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
142	C₄⋊C₄×C₅₃	—	2	4	no	—
143	C₄⋊C₄×C₅₃	—	2	4	no	—
144	C₄⋊C₄×C₅₃	—	2	8	no	—
145	C₄⋊C₄×C₅₃	—	2	8	no	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
148	C₂²⋊C₄×C₅₃	—	2	8	no	—
149	C₂²⋊C₄×C₅₃	—	2	4	no	—
150	C₂²⋊C₄×C₅₃	—	2	8	no	—
151	C₂²⋊C₄×C₅₃	—	2	8	no	—
152	C₂²⋊C₄×C₅₃	—	2	8	no	—
153	C₄²×C₅₃	—	2	8	no	—
154	C₄²×C₅₃	—	2	4	no	—
155	C₄²×C₅₃	C₂×D₄	1	4	yes	—
156	C₄²×C₅₃	C₂×D₄	1	4	yes	—
157	C₄²×C₅₃	C₂×D₄	1	2	yes	—
158	C₄²×C₅₃	C₂×D₄	1	2	yes	—
159	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₄²×C₅₃	C₂⁴	1	2	yes	—
162	C₂³×C₅₃	—	2	2	no	—
163	C₂³×C₅₃	—	2	4	no	—
164	C₂³×C₅₃	—	4	2	no	—
165	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
166	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₅₃	—	2	8	no	—
168	Q₈×C₅₃	—	4	8	no	—
169	D₄×C₅₃	—	2	8	no	—
170	D₄×C₅₃	—	4	16	no	—
171	D₄×C₅₃	—	4	8	no	—
172	D₄×C₅₃	—	4	8	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	8	no	—
177	C₂×C₄×C₅₃	—	2	4	no	—
178	C₂×C₄×C₅₃	—	2	8	no	—
179	C₂×C₄×C₅₃	—	2	4	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	4	no	—
182	C₂×C₄×C₅₃	—	4	16	no	—
183	C₂×C₄×C₅₃	—	4	8	no	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
185	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
190	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
191	C₈×C₅₃	—	2	4	no	—
192	C₈×C₅₃	C₂×#41	1	4	yes	—
193	C₈×C₅₃	C₂×#41	1	4	yes	—
194	#60693	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	2	4	no	—
198	C₂²×C₅₃	—	4	4	no	—
199	C₂²×C₅₃	—	4	4	no	—
200	C₂²×C₅₃	—	4	8	no	—
201	C₂²×C₅₃	#532	1	2	yes	—
202	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
203	C₄×C₅₃	—	2	2	no	—
204	C₄×C₅₃	—	2	4	no	—
205	C₄×C₅₃	—	2	2	no	—
206	C₄×C₅₃	—	4	4	no	—
207	C₄×C₅₃	—	4	8	no	—
208	C₄×C₅₃	#532	1	2	yes	—
209	C₄×C₅₃	#575	1	4	yes	—
210	C₄×C₅₃	C₂²×#41	1	2	yes	—
211	#3444	C₂×C₅₃	1	2	yes	complemented
212	#3185	C₂×C₅₃	1	2	yes	complemented
213	#3184	C₂×C₅₃	1	2	yes	complemented
214	#3183	C₂×C₅₃	1	8	yes	complemented
215	#3182	C₂×C₅₃	1	4	yes	complemented
216	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
217	#3147	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#3131	C₂×C₅₃	1	2	yes	complemented
219	#3054	C₂×C₅₃	1	4	yes	complemented
220	#2943	C₂×C₅₃	1	4	yes	complemented
221	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×C₅₃	—	2	2	no	—
223	C₂×C₅₃	—	4	2	no	—
224	C₂×C₅₃	—	8	4	no	—
225	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
226	C₂×C₅₃	C₂×#575	1	2	yes	—
227	C₂×#135	C₂²×C₅₃	1	4	yes	—
228	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
229	#533	C₂²×C₅₃	1	4	yes	—
230	#533	C₂²×C₅₃	1	2	yes	—
231	#532	C₂²×C₅₃	1	2	yes	—
232	#532	C₂²×C₅₃	1	4	yes	—
233	#529	—	2	4	no	—
234	#529	—	2	4	no	—
235	#529	C₂²×C₅₃	1	8	yes	—
236	#525	—	2	4	no	—
237	#525	C₂²×C₅₃	1	4	yes	—
238	C₄×#41	—	2	4	no	—
239	C₄×#41	C₂²×C₅₃	1	4	yes	—
240	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
241	#500	C₂²×C₅₃	1	2	yes	complemented
242	#499	C₂²×C₅₃	1	4	yes	complemented
243	#498	C₂²×C₅₃	1	2	yes	complemented
244	#495	C₂²×C₅₃	1	4	yes	complemented
245	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
246	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
247	#492	C₂²×C₅₃	1	2	yes	complemented
248	#488	C₂²×C₅₃	1	8	yes	complemented
249	#487	C₂²×C₅₃	1	4	yes	complemented
250	#486	C₂²×C₅₃	1	4	yes	complemented
251	#470	C₂²×C₅₃	1	8	yes	—
252	#469	C₂²×C₅₃	1	8	yes	—
253	#468	C₂²×C₅₃	1	16	yes	—
254	#442	C₂²×C₅₃	1	8	yes	complemented
255	#441	C₂²×C₅₃	1	8	yes	complemented
256	#440	C₂²×C₅₃	1	16	yes	complemented
257	#427	C₂²×C₅₃	1	4	yes	—
258	#416	C₂²×C₅₃	1	8	yes	—
259	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
260	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
261	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
262	C₂×#97	C₂²×C₅₃	1	2	yes	—
263	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
265	C₅₃	#60693	1	1	yes	characteristic, direct product factor, Sylow
266	C₂×#41	—	2	4	no	—
267	C₂×#41	C₂³×C₅₃	1	4	yes	—
268	C₂³×C₄	D₄×C₅₃	1	2	yes	—
269	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
270	#135	—	2	16	no	—
271	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
272	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
273	C₂×D₈	C₂³×C₅₃	1	4	yes	—
274	C₂×M₁₆	—	2	8	no	—
275	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
276	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
277	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
278	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
279	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
280	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
281	#127	C₂³×C₅₃	1	2	yes	—
282	#126	—	2	8	no	—
283	#125	—	2	4	no	—
284	#125	C₂³×C₅₃	1	4	yes	—
285	#124	—	2	4	no	—
286	#124	C₂³×C₅₃	1	4	yes	—
287	#123	—	2	8	no	—
288	#123	—	2	4	no	—
289	C₂²⋊Q₈	—	2	4	no	—
290	C₂²⋊Q₈	—	2	4	no	—
291	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
292	#121	—	2	4	no	—
293	#121	—	2	4	no	—
294	#121	C₂³×C₅₃	1	8	yes	—
295	#120	—	2	4	no	—
296	C₄×Q₈	—	2	4	no	—
297	C₄×Q₈	—	2	4	no	—
298	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
299	C₄×D₄	—	2	8	no	—
300	C₄×D₄	—	2	8	no	—
301	C₄×D₄	—	2	8	no	—
302	C₄×D₄	—	2	4	no	—
303	C₄×D₄	—	2	4	no	—
304	C₄×D₄	C₂³×C₅₃	1	8	yes	—
305	#117	—	2	4	no	—
306	#117	—	2	8	no	—
307	#117	—	2	4	no	—
308	#117	C₂³×C₅₃	1	4	yes	—
309	C₂×C₄²	—	2	4	no	—
310	C₂×C₄²	D₄×C₅₃	1	4	yes	—
311	C₂×C₄²	D₄×C₅₃	1	2	yes	—
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
314	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
316	#107	C₂³×C₅₃	1	8	yes	—
317	#106	C₂³×C₅₃	1	8	yes	—
318	#103	C₂³×C₅₃	1	16	yes	—
319	#102	C₂³×C₅₃	1	16	yes	—
320	#97	C₂³×C₅₃	1	4	yes	—
321	#97	C₂³×C₅₃	1	4	yes	—
322	C₄×C₈	C₂³×C₅₃	1	2	yes	—
323	C₄×C₈	C₂³×C₅₃	1	2	yes	—
324	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₄×C₈	C₂³×C₅₃	1	2	yes	—
327	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
328	#41	—	2	16	no	—
329	#41	—	4	16	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
343	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
347	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
348	Q₁₆	—	2	8	no	—
349	SD₁₆	—	2	16	no	—
350	D₈	—	2	8	no	—
351	M₁₆	—	2	8	no	—
352	M₁₆	—	2	8	no	—
353	C₂×C₈	—	2	4	no	—
354	C₂×C₈	#41×C₅₃	1	8	yes	complemented
355	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
356	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
357	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₄²	—	2	8	no	—
370	C₄²	—	2	4	no	—
371	C₄²	C₂×D₄×C₅₃	1	2	yes	—
372	C₄²	C₂×D₄×C₅₃	1	4	yes	—
373	C₄²	C₂×D₄×C₅₃	1	2	yes	—
374	C₄²	C₂×D₄×C₅₃	1	4	yes	—
375	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
376	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
377	C₄²	C₂⁴×C₅₃	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	4	no	—
380	C₂³	—	4	2	no	—
381	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
382	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
383	Q₈	—	2	8	no	—
384	Q₈	—	4	8	no	—
385	D₄	—	2	8	no	—
386	D₄	—	4	16	no	—
387	D₄	—	4	8	no	—
388	D₄	—	4	8	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	8	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	16	no	—
399	C₂×C₄	—	4	8	no	—
400	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
402	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
403	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
405	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
406	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
407	C₈	—	2	4	no	—
408	C₈	C₂×#41×C₅₃	1	4	yes	—
409	C₈	C₂×#41×C₅₃	1	4	yes	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	4	8	no	—
415	C₂²	—	4	4	no	—
416	C₂²	C₅₃×#532	1	2	yes	—
417	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
418	C₄	—	2	4	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	8	no	—
422	C₄	—	4	4	no	—
423	C₄	C₅₃×#532	1	2	yes	—
424	C₄	C₅₃×#575	1	4	yes	—
425	C₄	C₂²×#41×C₅₃	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	—	8	4	no	—
429	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
430	C₂	C₂×C₅₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂²×C₅₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3444	C₂	semidirect
2. C₅₃×#3185	C₂	semidirect
3. C₅₃×#3184	C₂	semidirect
4. C₅₃×#3183	C₂	semidirect
5. C₅₃×#3182	C₂	semidirect
6. C₅₃×#3160	C₂	semidirect
7. C₅₃×#3147	C₂	semidirect
8. C₅₃×#3131	C₂	semidirect
9. C₅₃×#3054	C₂	semidirect
10. C₅₃×#2943	C₂	semidirect
11. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
26. C₅₃×#500	C₂²	semidirect
27. C₅₃×#499	C₂²	semidirect
28. C₅₃×#498	C₂²	semidirect
29. C₅₃×#495	C₂²	semidirect
30. C₅₃×#494	C₂²	semidirect
31. C₅₃×#493	C₂²	semidirect
32. C₅₃×#492	C₂²	semidirect
33. C₅₃×#488	C₂²	semidirect
34. C₅₃×#487	C₂²	semidirect
35. C₅₃×#486	C₂²	semidirect
39. C₅₃×#442	C₂²	semidirect
40. C₅₃×#441	C₂²	semidirect
41. C₅₃×#440	C₂²	semidirect
138. C₂×C₈×C₅₃	#41	semidirect
194. #60693	C₅₃	direct (N × H)
211. #3444	C₂×C₅₃	semidirect
212. #3185	C₂×C₅₃	semidirect
213. #3184	C₂×C₅₃	semidirect
214. #3183	C₂×C₅₃	semidirect
215. #3182	C₂×C₅₃	semidirect
216. #3160	C₂×C₅₃	semidirect
217. #3147	C₂×C₅₃	semidirect
218. #3131	C₂×C₅₃	semidirect
219. #3054	C₂×C₅₃	semidirect
220. #2943	C₂×C₅₃	semidirect
221. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
241. #500	C₂²×C₅₃	semidirect
242. #499	C₂²×C₅₃	semidirect
243. #498	C₂²×C₅₃	semidirect
244. #495	C₂²×C₅₃	semidirect
245. #494	C₂²×C₅₃	semidirect
246. #493	C₂²×C₅₃	semidirect
247. #492	C₂²×C₅₃	semidirect
248. #488	C₂²×C₅₃	semidirect
249. #487	C₂²×C₅₃	semidirect
250. #486	C₂²×C₅₃	semidirect
254. #442	C₂²×C₅₃	semidirect
255. #441	C₂²×C₅₃	semidirect
256. #440	C₂²×C₅₃	semidirect
265. C₅₃	#60693	direct (N × H)
354. C₂×C₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3444	2
C₅₃×#3185	2
C₅₃×#3184	2
C₅₃×#3183	2
C₅₃×#3182	2
C₅₃×#3160	2
C₅₃×#3147	2
C₅₃×#3131	2
C₅₃×#3054	2
C₅₃×#2943	2
C₂×C₄×M₁₆×C₅₃	2
#60693	53

Subgroups — C53×#60693 #27268205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60693 #27268205