Subgroups — C₅₃×#21558 #27229070

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3067	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2955	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1932	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1906	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1757	C₂	1	2	yes	maximal, complemented
12	C₅₃×#575	C₂²	1	2	yes	—
13	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₂²	1	2	yes	—
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#473	C₄	1	2	yes	complemented
19	C₅₃×#472	C₄	1	2	yes	complemented
20	C₅₃×#471	C₄	1	4	yes	complemented
21	C₅₃×#444	—	2	4	no	—
22	C₅₃×#443	—	2	4	no	—
23	C₅₃×#443	—	2	1	no	—
24	C₅₃×#443	—	2	1	no	—
25	C₅₃×#443	—	2	1	no	—
26	C₅₃×#443	—	2	1	no	—
27	C₅₃×#430	C₂²	1	2	yes	—
28	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#427	C₂²	1	1	yes	characteristic
31	C₅₃×#421	C₂²	1	2	yes	—
32	C₅₃×#421	C₂²	1	2	yes	—
33	C₂×#104×C₅₃	C₂²	1	2	yes	—
34	C₅₃×#413	C₂²	1	2	yes	—
35	C₅₃×#410	C₂²	1	2	yes	—
36	C₅₃×#405	—	2	1	no	—
37	C₅₃×#405	—	2	1	no	—
38	C₅₃×#405	C₂²	1	1	yes	characteristic
39	C₅₃×#405	C₂²	1	2	yes	—
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₅₃	—	2	1	no	—
42	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₅₃	—	2	1	no	—
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#344	C₂²	1	2	yes	—
46	C₅₃×#344	C₂²	1	2	yes	—
47	C₅₃×#339	C₂²	1	2	yes	—
48	C₂×#41×C₅₃	C₂³	1	2	yes	—
49	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
50	#137×C₅₃	—	2	2	no	—
51	#137×C₅₃	C₂×C₄	1	4	yes	—
52	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
53	AGL(1,ℤ/8)×C₅₃	C₂×C₄	1	4	yes	—
54	#135×C₅₃	—	2	1	no	—
55	#135×C₅₃	—	2	1	no	—
56	#135×C₅₃	—	2	1	no	—
57	#135×C₅₃	—	2	1	no	—
58	#135×C₅₃	—	2	4	no	—
59	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
61	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
62	#131×C₅₃	—	2	1	no	—
63	#131×C₅₃	—	2	1	no	—
64	#131×C₅₃	—	2	1	no	—
65	#131×C₅₃	—	2	1	no	—
66	#131×C₅₃	—	2	2	no	—
67	#131×C₅₃	—	2	2	no	—
68	#131×C₅₃	—	2	2	no	—
69	#131×C₅₃	—	2	2	no	—
70	#131×C₅₃	C₂×C₄	1	2	yes	—
71	#131×C₅₃	C₂×C₄	1	2	yes	—
72	#131×C₅₃	D₄	1	2	yes	—
73	#131×C₅₃	D₄	1	2	yes	—
74	C₂×M₁₆×C₅₃	—	2	1	no	—
75	C₂×M₁₆×C₅₃	—	2	1	no	—
76	C₂×M₁₆×C₅₃	—	2	2	no	—
77	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
80	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	—	2	1	no	—
82	C₂²×C₈×C₅₃	—	2	1	no	—
83	C₂²×C₈×C₅₃	—	2	2	no	—
84	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	—	2	2	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	#117×C₅₃	—	2	1	no	—
90	#117×C₅₃	C₂³	1	2	yes	—
91	#117×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	—	2	1	no	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	#108×C₅₃	—	2	4	no	—
95	#108×C₅₃	—	2	1	no	—
96	#108×C₅₃	—	2	1	no	—
97	#108×C₅₃	C₂×C₄	1	2	yes	complemented
98	#108×C₅₃	C₂×C₄	1	2	yes	complemented
99	#107×C₅₃	—	2	1	no	—
100	#106×C₅₃	—	2	1	no	—
101	#104×C₅₃	—	2	4	no	—
102	#104×C₅₃	—	2	2	no	—
103	#104×C₅₃	—	2	2	no	—
104	#104×C₅₃	—	2	4	no	—
105	#101×C₅₃	C₂×C₄	1	4	yes	complemented
106	#100×C₅₃	C₂×C₄	1	4	yes	complemented
107	#99×C₅₃	—	2	4	no	—
108	#97×C₅₃	—	2	1	no	—
109	#97×C₅₃	—	2	1	no	—
110	#97×C₅₃	—	2	2	no	—
111	#97×C₅₃	—	2	1	no	—
112	C₄×C₈×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	—	2	1	no	—
114	C₄×C₈×C₅₃	—	2	2	no	—
115	C₄×C₈×C₅₃	—	2	1	no	—
116	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
117	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
118	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
119	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
120	#41×C₅₃	—	2	1	no	—
121	#41×C₅₃	—	2	1	no	—
122	#41×C₅₃	—	2	2	no	—
123	#41×C₅₃	—	2	2	no	—
124	#41×C₅₃	—	2	2	no	—
125	#41×C₅₃	—	2	2	no	—
126	#41×C₅₃	C₂²×C₄	1	2	yes	—
127	#41×C₅₃	C₂×D₄	1	2	yes	—
128	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
130	C₂×D₄×C₅₃	—	2	1	no	—
131	C₂×D₄×C₅₃	—	2	2	no	—
132	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
133	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	—	2	1	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	1	no	—
137	C₂²×C₄×C₅₃	—	2	2	no	—
138	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₅₃	—	2	4	no	—
140	Q₁₆×C₅₃	—	2	1	no	—
141	Q₁₆×C₅₃	—	2	1	no	—
142	SD₁₆×C₅₃	—	2	4	no	—
143	SD₁₆×C₅₃	—	2	4	no	—
144	SD₁₆×C₅₃	—	2	2	no	—
145	SD₁₆×C₅₃	—	2	2	no	—
146	D₈×C₅₃	—	2	4	no	—
147	D₈×C₅₃	—	2	1	no	—
148	D₈×C₅₃	—	2	1	no	—
149	M₁₆×C₅₃	—	2	1	no	—
150	M₁₆×C₅₃	—	2	1	no	—
151	M₁₆×C₅₃	—	2	2	no	—
152	M₁₆×C₅₃	—	2	2	no	—
153	M₁₆×C₅₃	—	2	2	no	—
154	M₁₆×C₅₃	—	2	2	no	—
155	M₁₆×C₅₃	—	2	2	no	—
156	M₁₆×C₅₃	—	2	2	no	—
157	M₁₆×C₅₃	—	2	2	no	—
158	M₁₆×C₅₃	—	4	2	no	—
159	M₁₆×C₅₃	—	4	2	no	—
160	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
161	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
162	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	2	no	—
166	C₂×C₈×C₅₃	—	2	2	no	—
167	C₂×C₈×C₅₃	—	2	2	no	—
168	C₂×C₈×C₅₃	—	2	2	no	—
169	C₂×C₈×C₅₃	—	2	2	no	—
170	C₂×C₈×C₅₃	—	2	2	no	—
171	C₂×C₈×C₅₃	—	4	2	no	—
172	C₂×C₈×C₅₃	—	4	2	no	—
173	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
182	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
183	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
184	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
185	C₄⋊C₄×C₅₃	—	2	2	no	—
186	C₄⋊C₄×C₅₃	—	2	1	no	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	#41	1	2	yes	—
191	C₄²×C₅₃	—	2	2	no	—
192	C₄²×C₅₃	—	2	1	no	—
193	C₄²×C₅₃	—	2	1	no	—
194	C₄²×C₅₃	—	2	2	no	—
195	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
197	C₂³×C₅₃	—	2	1	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
200	Q₈×C₅₃	—	2	1	no	—
201	Q₈×C₅₃	—	2	2	no	—
202	Q₈×C₅₃	—	2	2	no	—
203	Q₈×C₅₃	C₄×D₄	1	2	yes	—
204	D₄×C₅₃	—	2	2	no	—
205	D₄×C₅₃	—	2	2	no	—
206	D₄×C₅₃	—	2	2	no	—
207	D₄×C₅₃	—	2	2	no	—
208	D₄×C₅₃	—	2	1	no	—
209	D₄×C₅₃	—	4	2	no	—
210	D₄×C₅₃	—	4	2	no	—
211	D₄×C₅₃	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	4	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	—	4	2	no	—
223	C₂×C₄×C₅₃	—	4	2	no	—
224	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	2	4	no	—
232	C₈×C₅₃	—	2	2	no	—
233	C₈×C₅₃	—	4	2	no	—
234	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
235	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
236	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
237	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
238	C₈×C₅₃	#121	1	1	yes	characteristic
239	C₈×C₅₃	#121	1	1	yes	characteristic
240	C₈×C₅₃	#121	1	1	yes	characteristic
241	C₈×C₅₃	#121	1	1	yes	characteristic
242	#21558	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	2	no	—
246	C₂²×C₅₃	—	4	2	no	—
247	C₂²×C₅₃	—	8	2	no	—
248	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
249	C₂²×C₅₃	#525	1	1	yes	characteristic
250	C₂²×C₅₃	#532	1	1	yes	characteristic
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	4	2	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	—	4	2	no	—
256	C₄×C₅₃	—	8	2	no	—
257	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
258	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
259	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
260	C₄×C₅₃	#529	1	1	yes	characteristic
261	#3067	C₂×C₅₃	1	1	yes	characteristic
262	#2955	C₂×C₅₃	1	2	yes	complemented
263	C₂×#443	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#1976	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#1932	C₂×C₅₃	1	2	yes	complemented
268	#1906	C₂×C₅₃	1	2	yes	complemented
269	#1814	C₂×C₅₃	1	1	yes	characteristic, complemented
270	#1772	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#1757	C₂×C₅₃	1	2	yes	complemented
272	C₂×C₅₃	—	2	1	no	—
273	C₂×C₅₃	—	2	1	no	—
274	C₂×C₅₃	—	2	1	no	—
275	C₂×C₅₃	—	4	2	no	—
276	C₂×C₅₃	—	8	2	no	—
277	C₂×C₅₃	C₄×#121	1	1	yes	characteristic
278	#575	C₂²×C₅₃	1	2	yes	—
279	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×#131	C₂²×C₅₃	1	2	yes	—
281	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
282	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
283	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
284	#473	C₄×C₅₃	1	2	yes	complemented
285	#472	C₄×C₅₃	1	2	yes	complemented
286	#471	C₄×C₅₃	1	4	yes	complemented
287	#444	—	2	4	no	—
288	#443	—	2	4	no	—
289	#443	—	2	1	no	—
290	#443	—	2	1	no	—
291	#443	—	2	1	no	—
292	#443	—	2	1	no	—
293	#430	C₂²×C₅₃	1	2	yes	—
294	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
295	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
296	#427	C₂²×C₅₃	1	1	yes	characteristic
297	#421	C₂²×C₅₃	1	2	yes	—
298	#421	C₂²×C₅₃	1	2	yes	—
299	C₂×#104	C₂²×C₅₃	1	2	yes	—
300	#413	C₂²×C₅₃	1	2	yes	—
301	#410	C₂²×C₅₃	1	2	yes	—
302	#405	—	2	1	no	—
303	#405	—	2	1	no	—
304	#405	C₂²×C₅₃	1	1	yes	characteristic
305	#405	C₂²×C₅₃	1	1	yes	characteristic
306	#405	C₂²×C₅₃	1	2	yes	—
307	C₄×M₁₆	—	2	1	no	—
308	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
309	C₂×C₄×C₈	—	2	1	no	—
310	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
311	#344	C₂²×C₅₃	1	2	yes	—
312	#344	C₂²×C₅₃	1	2	yes	—
313	#339	C₂²×C₅₃	1	2	yes	—
314	C₅₃	#21558	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#41	C₂³×C₅₃	1	2	yes	—
316	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
317	#137	—	2	2	no	—
318	#137	C₂×C₄×C₅₃	1	4	yes	—
319	AGL(1,ℤ/8)	—	2	2	no	—
320	AGL(1,ℤ/8)	C₂×C₄×C₅₃	1	4	yes	—
321	#135	—	2	4	no	—
322	#135	—	2	1	no	—
323	#135	—	2	1	no	—
324	#135	—	2	1	no	—
325	#135	—	2	1	no	—
326	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
327	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
328	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
329	#131	—	2	1	no	—
330	#131	—	2	1	no	—
331	#131	—	2	1	no	—
332	#131	—	2	1	no	—
333	#131	—	2	2	no	—
334	#131	—	2	2	no	—
335	#131	—	2	2	no	—
336	#131	—	2	2	no	—
337	#131	C₂×C₄×C₅₃	1	2	yes	—
338	#131	C₂×C₄×C₅₃	1	2	yes	—
339	#131	D₄×C₅₃	1	2	yes	—
340	#131	D₄×C₅₃	1	2	yes	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
345	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
346	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
347	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
348	C₂²×C₈	—	2	1	no	—
349	C₂²×C₈	—	2	1	no	—
350	C₂²×C₈	—	2	2	no	—
351	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
353	C₄×Q₈	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	—	2	2	no	—
356	#117	—	2	1	no	—
357	#117	C₂³×C₅₃	1	2	yes	—
358	#117	C₂³×C₅₃	1	1	yes	characteristic
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
361	#108	—	2	4	no	—
362	#108	—	2	1	no	—
363	#108	—	2	1	no	—
364	#108	C₂×C₄×C₅₃	1	2	yes	complemented
365	#108	C₂×C₄×C₅₃	1	2	yes	complemented
366	#107	—	2	1	no	—
367	#106	—	2	1	no	—
368	#104	—	2	4	no	—
369	#104	—	2	4	no	—
370	#104	—	2	2	no	—
371	#104	—	2	2	no	—
372	#101	C₂×C₄×C₅₃	1	4	yes	complemented
373	#100	C₂×C₄×C₅₃	1	4	yes	complemented
374	#99	—	2	4	no	—
375	#97	—	2	1	no	—
376	#97	—	2	1	no	—
377	#97	—	2	2	no	—
378	#97	—	2	1	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	—	2	1	no	—
383	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
384	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
385	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
386	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	C₂²×C₄×C₅₃	1	2	yes	—
394	#41	C₂×D₄×C₅₃	1	2	yes	—
395	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
396	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
400	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
406	Q₁₆	—	2	4	no	—
407	Q₁₆	—	2	1	no	—
408	Q₁₆	—	2	1	no	—
409	SD₁₆	—	2	4	no	—
410	SD₁₆	—	2	4	no	—
411	SD₁₆	—	2	2	no	—
412	SD₁₆	—	2	2	no	—
413	D₈	—	2	4	no	—
414	D₈	—	2	1	no	—
415	D₈	—	2	1	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	2	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	2	no	—
421	M₁₆	—	2	2	no	—
422	M₁₆	—	2	2	no	—
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	4	2	no	—
426	M₁₆	—	4	2	no	—
427	M₁₆	C₂²×C₄×C₅₃	1	2	yes	—
428	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
429	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	4	2	no	—
439	C₂×C₈	—	4	2	no	—
440	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
441	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
443	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
444	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
445	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
446	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
448	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
449	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
450	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
451	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	#41×C₅₃	1	2	yes	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	2	no	—
462	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
464	C₂³	—	2	1	no	—
465	C₂³	—	2	2	no	—
466	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
467	Q₈	—	2	1	no	—
468	Q₈	—	2	2	no	—
469	Q₈	—	2	2	no	—
470	Q₈	C₄×D₄×C₅₃	1	2	yes	—
471	D₄	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	2	no	—
474	D₄	—	2	2	no	—
475	D₄	—	2	1	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	C₄×D₄×C₅₃	1	2	yes	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
495	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	2	4	no	—
499	C₈	—	2	2	no	—
500	C₈	—	4	2	no	—
501	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
502	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
503	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
504	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
505	C₈	#121×C₅₃	1	1	yes	characteristic
506	C₈	#121×C₅₃	1	1	yes	characteristic
507	C₈	#121×C₅₃	1	1	yes	characteristic
508	C₈	#121×C₅₃	1	1	yes	characteristic
509	C₂²	—	2	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	8	2	no	—
514	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
515	C₂²	C₅₃×#525	1	1	yes	characteristic
516	C₂²	C₅₃×#532	1	1	yes	characteristic
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	2	no	—
522	C₄	—	8	2	no	—
523	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
524	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
525	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
526	C₄	C₅₃×#529	1	1	yes	characteristic
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	4	2	no	—
531	C₂	—	8	2	no	—
532	C₂	C₄×#121×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#121
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	C₄×#121
prime radical	C₅₃×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2955	C₂	semidirect
3. C₂×C₅₃×#443	C₂	semidirect
4. C₄×#131×C₅₃	C₂	semidirect
5. C₅₃×#1976	C₂	semidirect
6. C₅₃×#1953	C₂	semidirect
7. C₅₃×#1932	C₂	semidirect
8. C₅₃×#1906	C₂	semidirect
9. C₅₃×#1814	C₂	semidirect
10. C₅₃×#1772	C₂	semidirect
11. C₅₃×#1757	C₂	semidirect
18. C₅₃×#473	C₄	semidirect
19. C₅₃×#472	C₄	semidirect
20. C₅₃×#471	C₄	semidirect
59. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
60. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
61. C₂×D₈×C₅₃	C₂×C₄	semidirect
97. #108×C₅₃	C₂×C₄	semidirect
98. #108×C₅₃	C₂×C₄	semidirect
105. #101×C₅₃	C₂×C₄	semidirect
106. #100×C₅₃	C₂×C₄	semidirect
242. #21558	C₅₃	direct (N × H)
262. #2955	C₂×C₅₃	semidirect
263. C₂×#443	C₂×C₅₃	semidirect
264. C₄×#131	C₂×C₅₃	semidirect
265. #1976	C₂×C₅₃	semidirect
266. #1953	C₂×C₅₃	semidirect
267. #1932	C₂×C₅₃	semidirect
268. #1906	C₂×C₅₃	semidirect
269. #1814	C₂×C₅₃	semidirect
270. #1772	C₂×C₅₃	semidirect
271. #1757	C₂×C₅₃	semidirect
284. #473	C₄×C₅₃	semidirect
285. #472	C₄×C₅₃	semidirect
286. #471	C₄×C₅₃	semidirect
314. C₅₃	#21558	direct (N × H)
326. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
327. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
328. C₂×D₈	C₂×C₄×C₅₃	semidirect
364. #108	C₂×C₄×C₅₃	semidirect
365. #108	C₂×C₄×C₅₃	semidirect
372. #101	C₂×C₄×C₅₃	semidirect
373. #100	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3067	2
C₅₃×#2955	2
C₂×C₅₃×#443	2
C₄×#131×C₅₃	2
C₅₃×#1976	2
C₅₃×#1953	2
C₅₃×#1932	2
C₅₃×#1906	2
C₅₃×#1814	2
C₅₃×#1772	2
C₅₃×#1757	2
#21558	53

Subgroups — C53×#21558 #27229070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21558 #27229070