Subgroups — C₅₃×#18773 #27226285

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3077	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1646	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1634	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#555	—	2	2	no	—
11	C₅₃×#555	—	2	2	no	—
12	C₅₃×#555	—	2	2	no	—
13	C₅₃×#555	C₂²	1	4	yes	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#124×C₅₃	—	2	2	no	—
17	C₂×#121×C₅₃	—	2	1	no	—
18	C₂×#121×C₅₃	—	2	1	no	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	—	2	1	no	—
22	C₂×#117×C₅₃	—	2	1	no	—
23	C₅₃×#494	C₂²	1	2	yes	complemented
24	C₅₃×#487	C₂²	1	2	yes	complemented
25	C₅₃×#483	—	2	1	no	—
26	C₅₃×#478	—	2	2	no	—
27	C₅₃×#476	C₂²	1	2	yes	complemented
28	C₅₃×#476	C₂²	1	2	yes	complemented
29	C₅₃×#474	C₂²	1	2	yes	complemented
30	C₅₃×#462	—	2	2	no	—
31	C₅₃×#461	C₂²	1	2	yes	complemented
32	C₅₃×#451	—	2	2	no	—
33	C₅₃×#450	—	2	1	no	—
34	C₄×SD₁₆×C₅₃	—	2	2	no	—
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#428	—	2	1	no	—
38	C₅₃×#423	C₂²	1	1	yes	characteristic
39	C₅₃×#418	—	2	1	no	—
40	C₅₃×#333	C₂²	1	4	yes	complemented
41	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
42	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
44	C₂²×D₄×C₅₃	—	2	2	no	—
45	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	C₂³×C₄×C₅₃	—	2	1	no	—
49	C₂×Q₁₆×C₅₃	—	2	4	no	—
50	C₂×SD₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
55	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
58	#125×C₅₃	—	2	1	no	—
59	#124×C₅₃	—	2	8	no	—
60	#124×C₅₃	—	2	8	no	—
61	C₂²⋊Q₈×C₅₃	—	2	1	no	—
62	C₂²⋊Q₈×C₅₃	—	2	1	no	—
63	#121×C₅₃	—	2	4	no	—
64	#121×C₅₃	—	2	4	no	—
65	#121×C₅₃	—	2	4	no	—
66	#121×C₅₃	—	2	4	no	—
67	C₄×Q₈×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	—	2	1	no	—
69	C₄×D₄×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	C₄×D₄×C₅₃	—	2	1	no	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	D₄	1	2	yes	—
74	C₄×D₄×C₅₃	D₄	1	2	yes	—
75	C₄×D₄×C₅₃	C₂³	1	2	yes	—
76	C₄×D₄×C₅₃	C₂³	1	2	yes	—
77	#117×C₅₃	—	2	4	no	—
78	#117×C₅₃	—	2	4	no	—
79	#117×C₅₃	—	2	4	no	—
80	#117×C₅₃	—	2	4	no	—
81	#117×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
90	C₂×C₄²×C₅₃	—	2	2	no	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	#107×C₅₃	—	2	2	no	—
93	#106×C₅₃	—	2	2	no	—
94	#106×C₅₃	—	2	2	no	—
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	#103×C₅₃	—	2	4	no	—
101	#102×C₅₃	—	2	2	no	—
102	#102×C₅₃	—	2	2	no	—
103	#102×C₅₃	—	2	2	no	—
104	C₂²⋊C₈×C₅₃	—	2	1	no	—
105	C₂²⋊C₈×C₅₃	—	2	1	no	—
106	C₄×C₈×C₅₃	—	2	1	no	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₂⁴×C₅₃	—	2	1	no	—
109	C₂⁴×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	2	no	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	—	8	2	no	—
114	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	2	no	—
119	C₂×D₄×C₅₃	—	2	2	no	—
120	C₂×D₄×C₅₃	—	4	2	no	—
121	C₂×D₄×C₅₃	—	8	4	no	—
122	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	2	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	4	4	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	—	4	2	no	—
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	Q₁₆×C₅₃	—	8	4	no	—
138	SD₁₆×C₅₃	—	4	4	no	—
139	M₁₆×C₅₃	—	4	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	2	1	no	—
152	C₄⋊C₄×C₅₃	—	2	1	no	—
153	C₄⋊C₄×C₅₃	—	4	1	no	—
154	C₄⋊C₄×C₅₃	—	4	1	no	—
155	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₅₃	—	2	8	no	—
158	C₂²⋊C₄×C₅₃	—	2	8	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	2	2	no	—
161	C₂²⋊C₄×C₅₃	—	2	2	no	—
162	C₂²⋊C₄×C₅₃	—	2	2	no	—
163	C₂²⋊C₄×C₅₃	—	4	1	no	—
164	C₄²×C₅₃	—	2	8	no	—
165	C₄²×C₅₃	—	4	1	no	—
166	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	4	2	no	—
173	C₂³×C₅₃	—	4	2	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	—	4	2	no	—
176	C₂³×C₅₃	—	4	2	no	—
177	C₂³×C₅₃	—	4	2	no	—
178	C₂³×C₅₃	#120	1	1	yes	characteristic
179	Q₈×C₅₃	—	4	4	no	—
180	Q₈×C₅₃	—	4	2	no	—
181	Q₈×C₅₃	—	4	2	no	—
182	Q₈×C₅₃	—	8	2	no	—
183	D₄×C₅₃	—	2	2	no	—
184	D₄×C₅₃	—	2	2	no	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	2	no	—
187	D₄×C₅₃	—	4	2	no	—
188	D₄×C₅₃	—	4	2	no	—
189	D₄×C₅₃	—	8	8	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	4	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	4	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	2	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	4	no	—
203	C₂×C₄×C₅₃	—	4	1	no	—
204	C₂×C₄×C₅₃	—	4	1	no	—
205	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
206	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
207	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
208	C₂×C₄×C₅₃	#135	1	2	yes	—
209	C₂×C₄×C₅₃	#137	1	2	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
213	C₈×C₅₃	—	4	2	no	—
214	C₈×C₅₃	—	4	2	no	—
215	C₈×C₅₃	—	4	2	no	—
216	#18773	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	4	2	no	—
220	C₂²×C₅₃	—	4	2	no	—
221	C₂²×C₅₃	—	4	2	no	—
222	C₂²×C₅₃	—	4	2	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	8	4	no	—
226	C₂²×C₅₃	—	8	4	no	—
227	C₂²×C₅₃	#452	1	2	yes	—
228	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	4	2	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	—	4	4	no	—
235	C₄×C₅₃	—	8	1	no	—
236	C₄×C₅₃	—	8	1	no	—
237	C₄×C₅₃	#452	1	2	yes	—
238	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
239	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
240	C₄×C₅₃	#575	1	2	yes	—
241	C₂×#555	C₂×C₅₃	1	1	yes	characteristic
242	#3209	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#3101	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#3077	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#1646	C₂×C₅₃	1	4	yes	complemented
246	#1634	C₂×C₅₃	1	4	yes	complemented
247	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
248	#1486	C₂×C₅₃	1	1	yes	characteristic, complemented
249	#1478	C₂×C₅₃	1	1	yes	characteristic, complemented
250	C₂×C₅₃	—	2	2	no	—
251	C₂×C₅₃	—	8	2	no	—
252	C₂×C₅₃	—	8	2	no	—
253	C₂×C₅₃	C₂×#452	1	1	yes	characteristic
254	C₂×C₅₃	#3005	1	1	yes	characteristic
255	C₂×C₅₃	#3019	1	1	yes	characteristic
256	#555	—	2	2	no	—
257	#555	—	2	2	no	—
258	#555	—	2	2	no	—
259	#555	C₂²×C₅₃	1	4	yes	—
260	#533	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#124	—	2	2	no	—
263	C₂×#121	—	2	1	no	—
264	C₂×#121	—	2	1	no	—
265	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×#117	—	2	1	no	—
268	C₂×#117	—	2	1	no	—
269	#494	C₂²×C₅₃	1	2	yes	complemented
270	#487	C₂²×C₅₃	1	2	yes	complemented
271	#483	—	2	1	no	—
272	#478	—	2	2	no	—
273	#476	C₂²×C₅₃	1	2	yes	complemented
274	#476	C₂²×C₅₃	1	2	yes	complemented
275	#474	C₂²×C₅₃	1	2	yes	complemented
276	#462	—	2	2	no	—
277	#461	C₂²×C₅₃	1	2	yes	complemented
278	#451	—	2	2	no	—
279	#450	—	2	1	no	—
280	C₄×SD₁₆	—	2	2	no	—
281	#431	C₂²×C₅₃	1	1	yes	characteristic
282	#431	C₂²×C₅₃	1	1	yes	characteristic
283	#428	—	2	1	no	—
284	#423	C₂²×C₅₃	1	1	yes	characteristic
285	#418	—	2	1	no	—
286	#333	C₂²×C₅₃	1	4	yes	complemented
287	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
288	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
289	C₅₃	#18773	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
291	C₂²×D₄	—	2	2	no	—
292	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
293	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
294	C₂³×C₄	—	2	1	no	—
295	C₂³×C₄	—	2	1	no	—
296	C₂×Q₁₆	—	2	4	no	—
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₄⋊Q₈	—	2	1	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
302	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
305	#125	—	2	1	no	—
306	#124	—	2	8	no	—
307	#124	—	2	8	no	—
308	C₂²⋊Q₈	—	2	1	no	—
309	C₂²⋊Q₈	—	2	1	no	—
310	#121	—	2	4	no	—
311	#121	—	2	4	no	—
312	#121	—	2	4	no	—
313	#121	—	2	4	no	—
314	C₄×Q₈	—	2	1	no	—
315	C₄×Q₈	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	D₄×C₅₃	1	2	yes	—
321	C₄×D₄	D₄×C₅₃	1	2	yes	—
322	C₄×D₄	C₂³×C₅₃	1	2	yes	—
323	C₄×D₄	C₂³×C₅₃	1	2	yes	—
324	#117	—	2	4	no	—
325	#117	—	2	4	no	—
326	#117	—	2	4	no	—
327	#117	—	2	4	no	—
328	#117	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
339	#107	—	2	2	no	—
340	#106	—	2	2	no	—
341	#106	—	2	2	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	—	2	1	no	—
344	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
345	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
346	#103	—	2	4	no	—
347	#103	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	2	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	C₂³×C₅₃	1	2	yes	—
355	C₂⁴	—	2	1	no	—
356	C₂⁴	—	2	1	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	8	2	no	—
361	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	4	2	no	—
368	C₂×D₄	—	8	4	no	—
369	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
370	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	4	4	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
384	Q₁₆	—	8	4	no	—
385	SD₁₆	—	4	4	no	—
386	M₁₆	—	4	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	4	1	no	—
401	C₄⋊C₄	—	4	1	no	—
402	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
403	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	4	1	no	—
411	C₄²	—	2	8	no	—
412	C₄²	—	4	1	no	—
413	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
415	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
416	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	#120×C₅₃	1	1	yes	characteristic
426	Q₈	—	4	4	no	—
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	8	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	8	8	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
453	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
454	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
455	C₂×C₄	#135×C₅₃	1	2	yes	—
456	C₂×C₄	#137×C₅₃	1	2	yes	—
457	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
460	C₈	—	4	2	no	—
461	C₈	—	4	2	no	—
462	C₈	—	4	2	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	8	4	no	—
472	C₂²	—	8	4	no	—
473	C₂²	C₅₃×#452	1	2	yes	—
474	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	4	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	1	no	—
483	C₄	C₅₃×#452	1	2	yes	—
484	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
485	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
486	C₄	C₅₃×#575	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	8	2	no	—
489	C₂	—	8	2	no	—
490	C₂	C₂×C₅₃×#452	1	1	yes	characteristic
491	C₂	C₅₃×#3005	1	1	yes	characteristic
492	C₂	C₅₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3209	C₂	semidirect
3. C₅₃×#3101	C₂	semidirect
4. C₅₃×#3077	C₂	semidirect
5. C₅₃×#1646	C₂	semidirect
6. C₅₃×#1634	C₂	semidirect
7. C₅₃×#1548	C₂	semidirect
8. C₅₃×#1486	C₂	semidirect
9. C₅₃×#1478	C₂	semidirect
23. C₅₃×#494	C₂²	semidirect
24. C₅₃×#487	C₂²	semidirect
27. C₅₃×#476	C₂²	semidirect
28. C₅₃×#476	C₂²	semidirect
29. C₅₃×#474	C₂²	semidirect
31. C₅₃×#461	C₂²	semidirect
40. C₅₃×#333	C₂²	semidirect
41. C₅₃×D₄⋊C₈	C₂²	semidirect
42. C₅₃×D₄⋊C₈	C₂²	semidirect
216. #18773	C₅₃	direct (N × H)
242. #3209	C₂×C₅₃	semidirect
243. #3101	C₂×C₅₃	semidirect
244. #3077	C₂×C₅₃	semidirect
245. #1646	C₂×C₅₃	semidirect
246. #1634	C₂×C₅₃	semidirect
247. #1548	C₂×C₅₃	semidirect
248. #1486	C₂×C₅₃	semidirect
249. #1478	C₂×C₅₃	semidirect
269. #494	C₂²×C₅₃	semidirect
270. #487	C₂²×C₅₃	semidirect
273. #476	C₂²×C₅₃	semidirect
274. #476	C₂²×C₅₃	semidirect
275. #474	C₂²×C₅₃	semidirect
277. #461	C₂²×C₅₃	semidirect
286. #333	C₂²×C₅₃	semidirect
287. D₄⋊C₈	C₂²×C₅₃	semidirect
288. D₄⋊C₈	C₂²×C₅₃	semidirect
289. C₅₃	#18773	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×#555	2
C₅₃×#3209	2
C₅₃×#3101	2
C₅₃×#3077	2
C₅₃×#1646	2
C₅₃×#1634	2
C₅₃×#1548	2
C₅₃×#1486	2
C₅₃×#1478	2
#18773	53

Subgroups — C53×#18773 #27226285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18773 #27226285