Subgroups — C₅₃×#23113 #27230625

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3035	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1974	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1929	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1876	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1764	C₂	1	2	yes	maximal, complemented
11	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	2	yes	—
13	C₂×#131×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#464	C₄	1	2	yes	complemented
17	C₅₃×#463	C₄	1	2	yes	complemented
18	C₅₃×#462	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#461	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#460	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#459	C₄	1	1	yes	characteristic, complemented
22	C₅₃×#436	—	2	1	no	—
23	C₅₃×#435	—	2	2	no	—
24	C₈×D₄×C₅₃	—	2	1	no	—
25	C₅₃×#433	—	2	2	no	—
26	C₅₃×#432	—	2	1	no	—
27	C₅₃×#431	—	2	1	no	—
28	C₂×#108×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#423	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#104×C₅₃	C₂²	1	2	yes	—
33	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#104×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#416	C₂²	1	2	yes	—
37	C₅₃×#408	C₂²	1	2	yes	—
38	C₅₃×#408	C₂²	1	2	yes	—
39	C₅₃×#405	C₂²	1	2	yes	—
40	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
42	C₄³×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#341	C₂²	1	2	yes	—
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₅₃×#339	C₂²	1	1	yes	characteristic
46	C₅₃×#336	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×#41×C₅₃	C₂³	1	2	yes	—
49	#135×C₅₃	—	4	2	no	—
50	#135×C₅₃	—	4	2	no	—
51	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
55	#131×C₅₃	—	2	4	no	—
56	#131×C₅₃	—	4	2	no	—
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#124×C₅₃	C₂×C₄	1	2	yes	—
67	#121×C₅₃	—	2	1	no	—
68	#121×C₅₃	—	2	1	no	—
69	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₅₃	—	2	1	no	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#117×C₅₃	—	2	1	no	—
78	#117×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	—	2	1	no	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	#108×C₅₃	—	4	2	no	—
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	—	2	1	no	—
88	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
91	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
94	#104×C₅₃	—	2	2	no	—
95	#104×C₅₃	—	2	4	no	—
96	#104×C₅₃	—	2	2	no	—
97	#104×C₅₃	—	4	2	no	—
98	#103×C₅₃	C₂×C₄	1	2	yes	—
99	#103×C₅₃	C₂×C₄	1	2	yes	—
100	#102×C₅₃	C₂×C₄	1	2	yes	—
101	#102×C₅₃	C₂×C₄	1	2	yes	—
102	C₂²⋊C₈×C₅₃	—	2	2	no	—
103	C₂²⋊C₈×C₅₃	—	2	2	no	—
104	C₂²⋊C₈×C₅₃	—	2	2	no	—
105	C₂²⋊C₈×C₅₃	—	2	2	no	—
106	#97×C₅₃	—	2	2	no	—
107	#97×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	—	2	2	no	—
109	C₄×C₈×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	—	4	2	no	—
111	#41×C₅₃	—	2	1	no	—
112	#41×C₅₃	—	2	1	no	—
113	#41×C₅₃	—	2	2	no	—
114	#41×C₅₃	—	2	2	no	—
115	#41×C₅₃	—	2	2	no	—
116	#41×C₅₃	—	2	2	no	—
117	#41×C₅₃	—	2	1	no	—
118	#41×C₅₃	—	2	1	no	—
119	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₄×C₅₃	—	2	1	no	—
122	C₂×D₄×C₅₃	—	2	2	no	—
123	C₂×D₄×C₅₃	—	2	1	no	—
124	C₂×D₄×C₅₃	—	2	1	no	—
125	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	2	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	1	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	Q₁₆×C₅₃	—	4	2	no	—
137	SD₁₆×C₅₃	—	4	2	no	—
138	SD₁₆×C₅₃	—	4	2	no	—
139	D₈×C₅₃	—	4	2	no	—
140	M₁₆×C₅₃	—	2	1	no	—
141	M₁₆×C₅₃	—	2	1	no	—
142	M₁₆×C₅₃	—	2	2	no	—
143	M₁₆×C₅₃	—	2	2	no	—
144	M₁₆×C₅₃	—	2	1	no	—
145	M₁₆×C₅₃	—	4	4	no	—
146	M₁₆×C₅₃	#41	1	2	yes	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	2	no	—
149	C₂×C₈×C₅₃	—	2	2	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	2	2	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	2	no	—
154	C₂×C₈×C₅₃	—	4	4	no	—
155	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₅₃	—	2	1	no	—
160	C₄⋊C₄×C₅₃	—	2	1	no	—
161	C₄⋊C₄×C₅₃	—	2	1	no	—
162	C₄⋊C₄×C₅₃	—	2	1	no	—
163	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₅₃	—	2	1	no	—
166	C₂²⋊C₄×C₅₃	—	2	1	no	—
167	C₂²⋊C₄×C₅₃	—	2	1	no	—
168	C₂²⋊C₄×C₅₃	—	2	2	no	—
169	C₂²⋊C₄×C₅₃	—	2	2	no	—
170	C₄²×C₅₃	—	2	4	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	—	2	1	no	—
173	C₄²×C₅₃	—	2	1	no	—
174	C₄²×C₅₃	—	2	1	no	—
175	C₄²×C₅₃	—	2	1	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	4	2	no	—
178	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₅₃	#41	1	2	yes	—
183	C₂³×C₅₃	—	2	1	no	—
184	C₂³×C₅₃	—	2	2	no	—
185	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
186	Q₈×C₅₃	—	2	2	no	—
187	Q₈×C₅₃	—	2	1	no	—
188	Q₈×C₅₃	—	2	1	no	—
189	Q₈×C₅₃	—	2	2	no	—
190	D₄×C₅₃	—	2	2	no	—
191	D₄×C₅₃	—	2	1	no	—
192	D₄×C₅₃	—	2	1	no	—
193	D₄×C₅₃	—	2	2	no	—
194	D₄×C₅₃	—	4	1	no	—
195	D₄×C₅₃	—	4	2	no	—
196	D₄×C₅₃	—	4	1	no	—
197	D₄×C₅₃	—	4	2	no	—
198	D₄×C₅₃	—	4	2	no	—
199	D₄×C₅₃	—	8	1	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	4	1	no	—
210	C₂×C₄×C₅₃	—	4	1	no	—
211	C₂×C₄×C₅₃	—	4	1	no	—
212	C₂×C₄×C₅₃	—	4	1	no	—
213	C₂×C₄×C₅₃	—	4	2	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	—	4	2	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	8	1	no	—
218	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
228	C₈×C₅₃	—	2	2	no	—
229	C₈×C₅₃	—	2	2	no	—
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	2	2	no	—
232	C₈×C₅₃	—	2	2	no	—
233	C₈×C₅₃	—	4	1	no	—
234	C₈×C₅₃	—	4	2	no	—
235	#23113	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	4	2	no	—
240	C₂²×C₅₃	—	4	1	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	1	no	—
243	C₂²×C₅₃	—	4	2	no	—
244	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₅₃	#553	1	1	yes	characteristic
246	C₂²×C₅₃	#555	1	1	yes	characteristic
247	C₄×C₅₃	—	2	4	no	—
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	2	1	no	—
251	C₄×C₅₃	—	4	2	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	—	4	2	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	—	4	1	no	—
256	C₄×C₅₃	#443	1	2	yes	—
257	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
258	C₄×C₅₃	#518	1	1	yes	characteristic
259	C₄×C₅₃	#546	1	1	yes	characteristic
260	C₄×C₅₃	#550	1	1	yes	characteristic
261	#3035	C₂×C₅₃	1	1	yes	characteristic
262	#2935	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#2925	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#1974	C₂×C₅₃	1	2	yes	complemented
265	#1962	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#1950	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#1929	C₂×C₅₃	1	2	yes	complemented
268	#1876	C₂×C₅₃	1	2	yes	complemented
269	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
270	#1764	C₂×C₅₃	1	2	yes	complemented
271	C₄×#104	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×C₅₃	—	2	1	no	—
273	C₂×C₅₃	—	2	1	no	—
274	C₂×C₅₃	—	8	1	no	—
275	C₂×C₅₃	—	8	2	no	—
276	C₂×C₅₃	#2363	1	1	yes	characteristic
277	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
278	C₂×C₅₃	#2956	1	1	yes	characteristic
279	C₂×#135	C₂²×C₅₃	1	2	yes	—
280	C₂×#131	C₂²×C₅₃	1	2	yes	—
281	#532	C₂²×C₅₃	1	1	yes	characteristic
282	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
283	#464	C₄×C₅₃	1	2	yes	complemented
284	#463	C₄×C₅₃	1	2	yes	complemented
285	#462	C₄×C₅₃	1	1	yes	characteristic, complemented
286	#461	C₄×C₅₃	1	1	yes	characteristic, complemented
287	#460	C₄×C₅₃	1	1	yes	characteristic, complemented
288	#459	C₄×C₅₃	1	1	yes	characteristic, complemented
289	#436	—	2	1	no	—
290	#435	—	2	2	no	—
291	C₈×D₄	—	2	1	no	—
292	#433	—	2	2	no	—
293	#432	—	2	1	no	—
294	#431	—	2	1	no	—
295	C₂×#108	C₂²×C₅₃	1	2	yes	—
296	#423	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
300	C₂×#104	C₂²×C₅₃	1	2	yes	—
301	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×#104	C₂²×C₅₃	1	2	yes	—
303	#416	C₂²×C₅₃	1	2	yes	—
304	#408	C₂²×C₅₃	1	2	yes	—
305	#408	C₂²×C₅₃	1	2	yes	—
306	#405	C₂²×C₅₃	1	2	yes	—
307	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
308	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
309	C₄³	C₂²×C₅₃	1	1	yes	characteristic
310	#341	C₂²×C₅₃	1	2	yes	—
311	#339	C₂²×C₅₃	1	1	yes	characteristic
312	#339	C₂²×C₅₃	1	1	yes	characteristic
313	#336	C₂²×C₅₃	1	2	yes	—
314	C₅₃	#23113	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
316	C₂×#41	C₂³×C₅₃	1	2	yes	—
317	#135	—	4	2	no	—
318	#135	—	4	2	no	—
319	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
320	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
321	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
322	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
323	#131	—	2	4	no	—
324	#131	—	4	2	no	—
325	C₂×M₁₆	—	2	2	no	—
326	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
327	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
328	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
329	C₂²×C₈	—	2	2	no	—
330	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
331	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
332	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
333	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
334	#124	C₂×C₄×C₅₃	1	2	yes	—
335	#121	—	2	1	no	—
336	#121	—	2	1	no	—
337	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
338	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
339	C₄×D₄	—	2	1	no	—
340	C₄×D₄	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
344	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
345	#117	—	2	1	no	—
346	#117	C₂³×C₅₃	1	1	yes	characteristic
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
353	#108	—	4	2	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
357	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
358	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
359	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
360	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
361	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
362	#104	—	2	2	no	—
363	#104	—	2	4	no	—
364	#104	—	2	2	no	—
365	#104	—	4	2	no	—
366	#103	C₂×C₄×C₅₃	1	2	yes	—
367	#103	C₂×C₄×C₅₃	1	2	yes	—
368	#102	C₂×C₄×C₅₃	1	2	yes	—
369	#102	C₂×C₄×C₅₃	1	2	yes	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	#97	—	2	2	no	—
375	#97	—	2	1	no	—
376	C₄×C₈	—	2	2	no	—
377	C₄×C₈	—	2	1	no	—
378	C₄×C₈	—	4	2	no	—
379	#41	—	2	1	no	—
380	#41	—	2	1	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	—	2	1	no	—
386	#41	—	2	1	no	—
387	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
388	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
394	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
402	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
404	Q₁₆	—	4	2	no	—
405	SD₁₆	—	4	2	no	—
406	SD₁₆	—	4	2	no	—
407	D₈	—	4	2	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	4	4	no	—
414	M₁₆	#41×C₅₃	1	2	yes	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	4	4	no	—
423	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
424	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
425	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
426	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
432	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₄²	—	2	4	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	4	2	no	—
446	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
447	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
448	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
449	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
450	C₄²	#41×C₅₃	1	2	yes	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	2	no	—
453	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
454	Q₈	—	2	2	no	—
455	Q₈	—	2	1	no	—
456	Q₈	—	2	1	no	—
457	Q₈	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	1	no	—
461	D₄	—	2	2	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	2	no	—
467	D₄	—	8	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	8	1	no	—
486	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
487	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
488	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
489	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
495	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	2	2	no	—
498	C₈	—	2	2	no	—
499	C₈	—	2	2	no	—
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
512	C₂²	C₅₃×#553	1	1	yes	characteristic
513	C₂²	C₅₃×#555	1	1	yes	characteristic
514	C₄	—	2	4	no	—
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	2	1	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	2	no	—
523	C₄	C₅₃×#443	1	2	yes	—
524	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
525	C₄	C₅₃×#518	1	1	yes	characteristic
526	C₄	C₅₃×#546	1	1	yes	characteristic
527	C₄	C₅₃×#550	1	1	yes	characteristic
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	8	1	no	—
531	C₂	—	8	2	no	—
532	C₂	C₅₃×#2363	1	1	yes	characteristic
533	C₂	C₂×C₅₃×#443	1	1	yes	characteristic
534	C₂	C₅₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2935	C₂	semidirect
3. C₅₃×#2925	C₂	semidirect
4. C₅₃×#1974	C₂	semidirect
5. C₅₃×#1962	C₂	semidirect
6. C₅₃×#1950	C₂	semidirect
7. C₅₃×#1929	C₂	semidirect
8. C₅₃×#1876	C₂	semidirect
9. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
10. C₅₃×#1764	C₂	semidirect
11. C₄×#104×C₅₃	C₂	semidirect
16. C₅₃×#464	C₄	semidirect
17. C₅₃×#463	C₄	semidirect
18. C₅₃×#462	C₄	semidirect
19. C₅₃×#461	C₄	semidirect
20. C₅₃×#460	C₄	semidirect
21. C₅₃×#459	C₄	semidirect
235. #23113	C₅₃	direct (N × H)
262. #2935	C₂×C₅₃	semidirect
263. #2925	C₂×C₅₃	semidirect
264. #1974	C₂×C₅₃	semidirect
265. #1962	C₂×C₅₃	semidirect
266. #1950	C₂×C₅₃	semidirect
267. #1929	C₂×C₅₃	semidirect
268. #1876	C₂×C₅₃	semidirect
269. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
270. #1764	C₂×C₅₃	semidirect
271. C₄×#104	C₂×C₅₃	semidirect
283. #464	C₄×C₅₃	semidirect
284. #463	C₄×C₅₃	semidirect
285. #462	C₄×C₅₃	semidirect
286. #461	C₄×C₅₃	semidirect
287. #460	C₄×C₅₃	semidirect
288. #459	C₄×C₅₃	semidirect
314. C₅₃	#23113	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×#3035	2
C₅₃×#2935	2
C₅₃×#2925	2
C₅₃×#1974	2
C₅₃×#1962	2
C₅₃×#1950	2
C₅₃×#1929	2
C₅₃×#1876	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1764	2
C₄×#104×C₅₃	2
#23113	53

Subgroups — C53×#23113 #27230625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23113 #27230625