Subgroups — C₅₃×#60685 #27268197

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3455	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3179	C₂	1	8	yes	maximal, complemented
3	C₅₃×#3178	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3131	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3054	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3050	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2943	C₂	1	2	yes	maximal, complemented
10	C₄×#135×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	2	yes	—
13	C₂×#135×C₅₃	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#542	C₂²	1	8	yes	—
17	C₅₃×#541	C₂²	1	4	yes	—
18	C₅₃×#540	C₂²	1	4	yes	—
19	C₅₃×#539	—	2	4	no	—
20	C₅₃×#538	—	2	4	no	—
21	C₅₃×#533	C₂²	1	2	yes	—
22	C₅₃×#532	C₂²	1	2	yes	—
23	C₅₃×#529	—	2	8	no	—
24	C₅₃×#525	C₂²	1	4	yes	—
25	C₄×#41×C₅₃	C₂²	1	4	yes	—
26	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#501	C₂²	1	8	yes	complemented
28	C₅₃×#497	C₂²	1	4	yes	complemented
29	C₅₃×#496	C₂²	1	4	yes	complemented
30	C₅₃×#488	C₂²	1	8	yes	complemented
31	C₅₃×#487	C₂²	1	4	yes	complemented
32	C₅₃×#486	C₂²	1	4	yes	complemented
33	C₅₃×#470	C₂²	1	4	yes	—
34	C₅₃×#469	C₂²	1	4	yes	—
35	C₅₃×#468	C₂²	1	8	yes	—
36	C₅₃×#467	C₂²	1	4	yes	—
37	C₅₃×#466	C₂²	1	4	yes	—
38	C₅₃×#465	C₂²	1	8	yes	—
39	C₅₃×#442	C₂²	1	4	yes	complemented
40	C₅₃×#441	C₂²	1	4	yes	complemented
41	C₅₃×#440	C₂²	1	8	yes	complemented
42	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
43	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
44	C₄×D₈×C₅₃	C₂²	1	4	yes	complemented
45	C₅₃×#427	C₂²	1	2	yes	—
46	C₅₃×#427	C₂²	1	2	yes	—
47	C₅₃×#416	C₂²	1	4	yes	—
48	C₅₃×#416	C₂²	1	4	yes	—
49	C₅₃×#405	C₂²	1	8	yes	—
50	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
51	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×#41×C₅₃	C₂³	1	4	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	#135×C₅₃	—	2	8	no	—
57	#135×C₅₃	—	2	8	no	—
58	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
61	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
62	C₂×D₈×C₅₃	C₂³	1	2	yes	—
63	C₂×D₈×C₅₃	C₂³	1	2	yes	—
64	C₂×M₁₆×C₅₃	—	2	4	no	—
65	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
66	C₂²×C₈×C₅₃	—	2	4	no	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
73	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
74	#127×C₅₃	C₂³	1	2	yes	—
75	#126×C₅₃	—	2	8	no	—
76	#126×C₅₃	—	2	8	no	—
77	#125×C₅₃	—	2	8	no	—
78	#125×C₅₃	C₂³	1	4	yes	—
79	#124×C₅₃	—	2	8	no	—
80	#124×C₅₃	C₂³	1	4	yes	—
81	#123×C₅₃	—	2	8	no	—
82	#123×C₅₃	—	2	8	no	—
83	#123×C₅₃	—	2	4	no	—
84	C₂²⋊Q₈×C₅₃	—	2	8	no	—
85	C₂²⋊Q₈×C₅₃	—	2	4	no	—
86	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
87	#121×C₅₃	—	2	8	no	—
88	#121×C₅₃	—	2	4	no	—
89	#121×C₅₃	C₂³	1	8	yes	—
90	#120×C₅₃	—	2	4	no	—
91	C₄×Q₈×C₅₃	—	2	4	no	—
92	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
93	C₄×D₄×C₅₃	—	2	4	no	—
94	C₄×D₄×C₅₃	—	2	8	no	—
95	C₄×D₄×C₅₃	—	2	8	no	—
96	C₄×D₄×C₅₃	C₂³	1	8	yes	—
97	#117×C₅₃	—	2	4	no	—
98	#117×C₅₃	D₄	1	4	yes	—
99	#117×C₅₃	C₂³	1	4	yes	—
100	#117×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
103	C₂×C₄²×C₅₃	—	2	4	no	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	#107×C₅₃	C₂³	1	4	yes	—
107	#107×C₅₃	C₂³	1	4	yes	—
108	#106×C₅₃	C₂³	1	4	yes	—
109	#106×C₅₃	C₂³	1	4	yes	—
110	#103×C₅₃	C₂³	1	8	yes	—
111	#103×C₅₃	C₂³	1	8	yes	—
112	#102×C₅₃	C₂³	1	8	yes	—
113	#102×C₅₃	C₂³	1	8	yes	—
114	#97×C₅₃	C₂³	1	4	yes	—
115	#97×C₅₃	C₂³	1	4	yes	—
116	C₄×C₈×C₅₃	C₂³	1	4	yes	—
117	C₄×C₈×C₅₃	C₂³	1	4	yes	—
118	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
119	#41×C₅₃	—	2	8	no	—
120	#41×C₅₃	—	2	8	no	—
121	C₂×Q₈×C₅₃	—	2	4	no	—
122	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
123	C₂×D₄×C₅₃	—	2	4	no	—
124	C₂×D₄×C₅₃	—	2	8	no	—
125	C₂×D₄×C₅₃	—	2	4	no	—
126	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	8	no	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₅₃	—	2	4	no	—
138	Q₁₆×C₅₃	—	2	4	no	—
139	SD₁₆×C₅₃	—	2	8	no	—
140	SD₁₆×C₅₃	—	2	8	no	—
141	D₈×C₅₃	—	2	4	no	—
142	D₈×C₅₃	—	2	4	no	—
143	M₁₆×C₅₃	—	2	8	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	2	8	no	—
147	C₂×C₈×C₅₃	#41	1	4	yes	—
148	C₂×C₈×C₅₃	#41	1	4	yes	—
149	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
150	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
151	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
152	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
153	C₄⋊C₄×C₅₃	—	2	8	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	2	8	no	—
156	C₄⋊C₄×C₅₃	—	2	8	no	—
157	C₄⋊C₄×C₅₃	—	2	4	no	—
158	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
159	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
160	C₂²⋊C₄×C₅₃	—	2	4	no	—
161	C₂²⋊C₄×C₅₃	—	2	8	no	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	8	no	—
165	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
166	C₄²×C₅₃	—	2	8	no	—
167	C₄²×C₅₃	—	2	4	no	—
168	C₄²×C₅₃	C₂×D₄	1	2	yes	—
169	C₄²×C₅₃	C₂×D₄	1	2	yes	—
170	C₄²×C₅₃	C₂⁴	1	2	yes	—
171	C₄²×C₅₃	C₂⁴	1	2	yes	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	4	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
176	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₅₃	—	2	4	no	—
178	Q₈×C₅₃	—	2	4	no	—
179	Q₈×C₅₃	—	8	4	no	—
180	D₄×C₅₃	—	2	4	no	—
181	D₄×C₅₃	—	2	4	no	—
182	D₄×C₅₃	—	4	8	no	—
183	D₄×C₅₃	—	8	4	no	—
184	D₄×C₅₃	—	8	8	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	4	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	8	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	8	no	—
194	C₂×C₄×C₅₃	—	8	8	no	—
195	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
196	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
200	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
201	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
202	C₈×C₅₃	—	2	2	no	—
203	C₈×C₅₃	—	2	2	no	—
204	C₈×C₅₃	C₂×#41	1	4	yes	—
205	C₈×C₅₃	C₂×#41	1	4	yes	—
206	#60685	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	2	4	no	—
209	C₂²×C₅₃	—	2	2	no	—
210	C₂²×C₅₃	—	4	4	no	—
211	C₂²×C₅₃	—	4	8	no	—
212	C₂²×C₅₃	—	4	4	no	—
213	C₂²×C₅₃	#532	1	2	yes	—
214	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₅₃	—	2	4	no	—
216	C₄×C₅₃	—	2	4	no	—
217	C₄×C₅₃	—	2	2	no	—
218	C₄×C₅₃	—	4	8	no	—
219	C₄×C₅₃	—	4	4	no	—
220	C₄×C₅₃	#532	1	2	yes	—
221	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
222	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
223	#3455	C₂×C₅₃	1	2	yes	complemented
224	#3179	C₂×C₅₃	1	8	yes	complemented
225	#3178	C₂×C₅₃	1	8	yes	complemented
226	#3162	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#3150	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#3131	C₂×C₅₃	1	2	yes	complemented
229	#3054	C₂×C₅₃	1	2	yes	complemented
230	#3050	C₂×C₅₃	1	2	yes	complemented
231	#2943	C₂×C₅₃	1	2	yes	complemented
232	C₄×#135	C₂×C₅₃	1	2	yes	complemented
233	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×C₅₃	—	2	2	no	—
235	C₂×C₅₃	—	4	2	no	—
236	C₂×C₅₃	—	8	4	no	—
237	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
238	C₂×C₅₃	#3584	1	2	yes	—
239	C₂×#135	C₂²×C₅₃	1	2	yes	—
240	C₂×#135	C₂²×C₅₃	1	2	yes	—
241	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
243	#542	C₂²×C₅₃	1	8	yes	—
244	#541	C₂²×C₅₃	1	4	yes	—
245	#540	C₂²×C₅₃	1	4	yes	—
246	#539	—	2	4	no	—
247	#538	—	2	4	no	—
248	#533	C₂²×C₅₃	1	2	yes	—
249	#532	C₂²×C₅₃	1	2	yes	—
250	#529	—	2	8	no	—
251	#525	C₂²×C₅₃	1	4	yes	—
252	C₄×#41	C₂²×C₅₃	1	4	yes	—
253	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
254	#501	C₂²×C₅₃	1	8	yes	complemented
255	#497	C₂²×C₅₃	1	4	yes	complemented
256	#496	C₂²×C₅₃	1	4	yes	complemented
257	#488	C₂²×C₅₃	1	8	yes	complemented
258	#487	C₂²×C₅₃	1	4	yes	complemented
259	#486	C₂²×C₅₃	1	4	yes	complemented
260	#470	C₂²×C₅₃	1	4	yes	—
261	#469	C₂²×C₅₃	1	4	yes	—
262	#468	C₂²×C₅₃	1	8	yes	—
263	#467	C₂²×C₅₃	1	4	yes	—
264	#466	C₂²×C₅₃	1	4	yes	—
265	#465	C₂²×C₅₃	1	8	yes	—
266	#442	C₂²×C₅₃	1	4	yes	complemented
267	#441	C₂²×C₅₃	1	4	yes	complemented
268	#440	C₂²×C₅₃	1	8	yes	complemented
269	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
270	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
271	C₄×D₈	C₂²×C₅₃	1	4	yes	complemented
272	#427	C₂²×C₅₃	1	2	yes	—
273	#427	C₂²×C₅₃	1	2	yes	—
274	#416	C₂²×C₅₃	1	4	yes	—
275	#416	C₂²×C₅₃	1	4	yes	—
276	#405	C₂²×C₅₃	1	8	yes	—
277	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
281	C₅₃	#60685	1	1	yes	characteristic, direct product factor, Sylow
282	C₂×#41	C₂³×C₅₃	1	4	yes	—
283	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
284	#135	—	2	8	no	—
285	#135	—	2	8	no	—
286	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
287	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
288	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
289	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
290	C₂×D₈	C₂³×C₅₃	1	2	yes	—
291	C₂×D₈	C₂³×C₅₃	1	2	yes	—
292	C₂×M₁₆	—	2	4	no	—
293	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
294	C₂²×C₈	—	2	4	no	—
295	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
297	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
300	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
301	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
302	#127	C₂³×C₅₃	1	2	yes	—
303	#126	—	2	8	no	—
304	#126	—	2	8	no	—
305	#125	—	2	8	no	—
306	#125	C₂³×C₅₃	1	4	yes	—
307	#124	—	2	8	no	—
308	#124	C₂³×C₅₃	1	4	yes	—
309	#123	—	2	4	no	—
310	#123	—	2	8	no	—
311	#123	—	2	8	no	—
312	C₂²⋊Q₈	—	2	4	no	—
313	C₂²⋊Q₈	—	2	8	no	—
314	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
315	#121	—	2	4	no	—
316	#121	—	2	8	no	—
317	#121	C₂³×C₅₃	1	8	yes	—
318	#120	—	2	4	no	—
319	C₄×Q₈	—	2	4	no	—
320	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
321	C₄×D₄	—	2	8	no	—
322	C₄×D₄	—	2	8	no	—
323	C₄×D₄	—	2	4	no	—
324	C₄×D₄	C₂³×C₅₃	1	8	yes	—
325	#117	—	2	4	no	—
326	#117	D₄×C₅₃	1	4	yes	—
327	#117	C₂³×C₅₃	1	4	yes	—
328	#117	C₂³×C₅₃	1	4	yes	—
329	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
330	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
331	C₂×C₄²	—	2	4	no	—
332	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
334	#107	C₂³×C₅₃	1	4	yes	—
335	#107	C₂³×C₅₃	1	4	yes	—
336	#106	C₂³×C₅₃	1	4	yes	—
337	#106	C₂³×C₅₃	1	4	yes	—
338	#103	C₂³×C₅₃	1	8	yes	—
339	#103	C₂³×C₅₃	1	8	yes	—
340	#102	C₂³×C₅₃	1	8	yes	—
341	#102	C₂³×C₅₃	1	8	yes	—
342	#97	C₂³×C₅₃	1	4	yes	—
343	#97	C₂³×C₅₃	1	4	yes	—
344	C₄×C₈	C₂³×C₅₃	1	4	yes	—
345	C₄×C₈	C₂³×C₅₃	1	4	yes	—
346	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
347	#41	—	2	8	no	—
348	#41	—	2	8	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
365	Q₁₆	—	2	4	no	—
366	Q₁₆	—	2	4	no	—
367	SD₁₆	—	2	8	no	—
368	SD₁₆	—	2	8	no	—
369	D₈	—	2	4	no	—
370	D₈	—	2	4	no	—
371	M₁₆	—	2	8	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	8	no	—
375	C₂×C₈	#41×C₅₃	1	4	yes	—
376	C₂×C₈	#41×C₅₃	1	4	yes	—
377	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
378	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
379	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
380	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
387	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	8	no	—
396	C₄²	C₂×D₄×C₅₃	1	2	yes	—
397	C₄²	C₂×D₄×C₅₃	1	2	yes	—
398	C₄²	C₂⁴×C₅₃	1	2	yes	—
399	C₄²	C₂⁴×C₅₃	1	2	yes	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	4	2	no	—
403	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
404	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
405	Q₈	—	2	4	no	—
406	Q₈	—	2	4	no	—
407	Q₈	—	8	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	D₄	—	4	8	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	8	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	8	8	no	—
423	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
424	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
428	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
429	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	2	no	—
432	C₈	C₂×#41×C₅₃	1	4	yes	—
433	C₈	C₂×#41×C₅₃	1	4	yes	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	8	no	—
439	C₂²	—	4	4	no	—
440	C₂²	C₅₃×#532	1	2	yes	—
441	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
442	C₄	—	2	4	no	—
443	C₄	—	2	4	no	—
444	C₄	—	2	2	no	—
445	C₄	—	4	8	no	—
446	C₄	—	4	4	no	—
447	C₄	C₅₃×#532	1	2	yes	—
448	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
449	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
450	C₂	—	2	2	no	—
451	C₂	—	4	2	no	—
452	C₂	—	8	4	no	—
453	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
454	C₂	C₅₃×#3584	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 ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3455	C₂	semidirect
2. C₅₃×#3179	C₂	semidirect
3. C₅₃×#3178	C₂	semidirect
4. C₅₃×#3162	C₂	semidirect
5. C₅₃×#3150	C₂	semidirect
6. C₅₃×#3131	C₂	semidirect
7. C₅₃×#3054	C₂	semidirect
8. C₅₃×#3050	C₂	semidirect
9. C₅₃×#2943	C₂	semidirect
10. C₄×#135×C₅₃	C₂	semidirect
11. C₂×C₅₃×#405	C₂	semidirect
27. C₅₃×#501	C₂²	semidirect
28. C₅₃×#497	C₂²	semidirect
29. C₅₃×#496	C₂²	semidirect
30. C₅₃×#488	C₂²	semidirect
31. C₅₃×#487	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
39. C₅₃×#442	C₂²	semidirect
40. C₅₃×#441	C₂²	semidirect
41. C₅₃×#440	C₂²	semidirect
42. C₄×Q₁₆×C₅₃	C₂²	semidirect
43. C₄×SD₁₆×C₅₃	C₂²	semidirect
44. C₄×D₈×C₅₃	C₂²	semidirect
206. #60685	C₅₃	direct (N × H)
223. #3455	C₂×C₅₃	semidirect
224. #3179	C₂×C₅₃	semidirect
225. #3178	C₂×C₅₃	semidirect
226. #3162	C₂×C₅₃	semidirect
227. #3150	C₂×C₅₃	semidirect
228. #3131	C₂×C₅₃	semidirect
229. #3054	C₂×C₅₃	semidirect
230. #3050	C₂×C₅₃	semidirect
231. #2943	C₂×C₅₃	semidirect
232. C₄×#135	C₂×C₅₃	semidirect
233. C₂×#405	C₂×C₅₃	semidirect
254. #501	C₂²×C₅₃	semidirect
255. #497	C₂²×C₅₃	semidirect
256. #496	C₂²×C₅₃	semidirect
257. #488	C₂²×C₅₃	semidirect
258. #487	C₂²×C₅₃	semidirect
259. #486	C₂²×C₅₃	semidirect
266. #442	C₂²×C₅₃	semidirect
267. #441	C₂²×C₅₃	semidirect
268. #440	C₂²×C₅₃	semidirect
269. C₄×Q₁₆	C₂²×C₅₃	semidirect
270. C₄×SD₁₆	C₂²×C₅₃	semidirect
271. C₄×D₈	C₂²×C₅₃	semidirect
281. C₅₃	#60685	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×#3455	2
C₅₃×#3179	2
C₅₃×#3178	2
C₅₃×#3162	2
C₅₃×#3150	2
C₅₃×#3131	2
C₅₃×#3054	2
C₅₃×#3050	2
C₅₃×#2943	2
C₄×#135×C₅₃	2
C₂×C₅₃×#405	2
#60685	53

Subgroups — C53×#60685 #27268197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60685 #27268197