Subgroups — C₅₃×#25448 #27232960

Order 13568.

860 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3069	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3067	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2049	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2009	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1915	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#575	C₂²	1	2	yes	—
13	C₂×#137×C₅₃	C₂²	1	2	yes	—
14	C₂×#137×C₅₃	C₂²	1	2	yes	—
15	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#568	C₂²	1	2	yes	—
18	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#536	C₂²	1	2	yes	—
20	C₅₃×#497	—	2	2	no	—
21	C₅₃×#495	—	2	1	no	—
22	C₅₃×#494	—	2	1	no	—
23	C₅₃×#473	—	2	1	no	—
24	C₅₃×#473	—	2	2	no	—
25	C₅₃×#473	—	2	2	no	—
26	C₅₃×#473	—	2	4	no	—
27	C₅₃×#472	—	2	1	no	—
28	C₅₃×#471	—	2	2	no	—
29	C₅₃×#471	—	2	4	no	—
30	C₅₃×#444	—	2	2	no	—
31	C₅₃×#443	—	2	2	no	—
32	C₅₃×#430	C₂²	1	2	yes	—
33	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#421	C₂²	1	2	yes	—
36	C₅₃×#419	C₂²	1	2	yes	—
37	C₅₃×#413	C₂²	1	2	yes	—
38	C₅₃×#413	C₂²	1	1	yes	characteristic
39	C₂×#101×C₅₃	C₂²	1	2	yes	—
40	C₂×#101×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#410	C₂²	1	1	yes	characteristic
42	C₅₃×#405	C₂²	1	1	yes	characteristic
43	C₅₃×#405	C₂²	1	1	yes	characteristic
44	C₅₃×#344	C₂²	1	1	yes	characteristic
45	C₅₃×#344	C₂²	1	2	yes	—
46	C₅₃×#344	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	C₂³	1	2	yes	—
48	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
50	#137×C₅₃	—	2	4	no	—
51	#137×C₅₃	—	2	4	no	—
52	#137×C₅₃	—	2	2	no	—
53	#137×C₅₃	—	2	2	no	—
54	#137×C₅₃	—	4	2	no	—
55	#137×C₅₃	—	4	2	no	—
56	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
57	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
58	#135×C₅₃	—	2	1	no	—
59	#135×C₅₃	—	2	1	no	—
60	#135×C₅₃	—	2	1	no	—
61	#135×C₅₃	—	2	1	no	—
62	#135×C₅₃	—	4	2	no	—
63	C₂×Q₁₆×C₅₃	—	2	1	no	—
64	C₂×Q₁₆×C₅₃	—	2	1	no	—
65	C₂×Q₁₆×C₅₃	—	2	1	no	—
66	C₂×Q₁₆×C₅₃	—	2	1	no	—
67	C₂×Q₁₆×C₅₃	—	2	1	no	—
68	C₂×Q₁₆×C₅₃	—	2	1	no	—
69	C₂×Q₁₆×C₅₃	—	2	2	no	—
70	C₂×Q₁₆×C₅₃	—	2	2	no	—
71	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₅₃	—	2	2	no	—
73	C₂×SD₁₆×C₅₃	—	2	2	no	—
74	C₂×SD₁₆×C₅₃	D₄	1	2	yes	—
75	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
76	#131×C₅₃	—	2	4	no	—
77	#131×C₅₃	—	2	4	no	—
78	#131×C₅₃	—	2	1	no	—
79	#131×C₅₃	—	2	1	no	—
80	#131×C₅₃	—	2	1	no	—
81	#131×C₅₃	—	2	1	no	—
82	C₂×M₁₆×C₅₃	—	2	1	no	—
83	C₂×M₁₆×C₅₃	—	4	2	no	—
84	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
86	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
88	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	—	2	1	no	—
90	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₅₃	—	2	2	no	—
92	#124×C₅₃	—	2	2	no	—
93	#123×C₅₃	—	2	4	no	—
94	C₂²⋊Q₈×C₅₃	—	4	2	no	—
95	#117×C₅₃	C₂³	1	1	yes	characteristic
96	#117×C₅₃	C₂³	1	1	yes	characteristic
97	#108×C₅₃	—	2	4	no	—
98	#108×C₅₃	—	2	1	no	—
99	#108×C₅₃	—	2	1	no	—
100	#108×C₅₃	D₄	1	2	yes	—
101	#108×C₅₃	D₄	1	2	yes	—
102	#104×C₅₃	—	2	4	no	—
103	#103×C₅₃	—	4	2	no	—
104	#101×C₅₃	—	2	4	no	—
105	#101×C₅₃	—	2	4	no	—
106	#101×C₅₃	—	2	2	no	—
107	#101×C₅₃	D₄	1	2	yes	—
108	#100×C₅₃	—	2	2	no	—
109	#100×C₅₃	D₄	1	2	yes	—
110	#99×C₅₃	—	2	2	no	—
111	#97×C₅₃	—	2	1	no	—
112	#97×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	—	2	1	no	—
114	C₄×C₈×C₅₃	—	2	1	no	—
115	#41×C₅₃	—	2	2	no	—
116	#41×C₅₃	—	2	1	no	—
117	#41×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	1	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	—	2	1	no	—
121	#41×C₅₃	—	2	2	no	—
122	#41×C₅₃	—	2	1	no	—
123	C₂×Q₈×C₅₃	—	2	2	no	—
124	C₂×Q₈×C₅₃	—	2	2	no	—
125	C₂×Q₈×C₅₃	—	2	2	no	—
126	C₂×Q₈×C₅₃	—	2	2	no	—
127	C₂×Q₈×C₅₃	—	2	2	no	—
128	C₂×Q₈×C₅₃	—	2	2	no	—
129	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₅₃	—	2	1	no	—
132	C₂×D₄×C₅₃	—	2	2	no	—
133	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	—	2	1	no	—
136	C₂²×C₄×C₅₃	—	2	2	no	—
137	C₂²×C₄×C₅₃	—	2	2	no	—
138	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₅₃	—	2	1	no	—
140	Q₁₆×C₅₃	—	2	1	no	—
141	Q₁₆×C₅₃	—	4	4	no	—
142	Q₁₆×C₅₃	—	4	4	no	—
143	Q₁₆×C₅₃	—	4	2	no	—
144	Q₁₆×C₅₃	—	4	1	no	—
145	Q₁₆×C₅₃	—	4	1	no	—
146	Q₁₆×C₅₃	—	4	1	no	—
147	Q₁₆×C₅₃	—	4	1	no	—
148	SD₁₆×C₅₃	—	2	2	no	—
149	SD₁₆×C₅₃	—	2	2	no	—
150	SD₁₆×C₅₃	—	4	4	no	—
151	SD₁₆×C₅₃	—	4	4	no	—
152	SD₁₆×C₅₃	—	4	2	no	—
153	SD₁₆×C₅₃	—	4	2	no	—
154	D₈×C₅₃	—	2	1	no	—
155	D₈×C₅₃	—	2	1	no	—
156	D₈×C₅₃	—	4	2	no	—
157	M₁₆×C₅₃	—	2	1	no	—
158	M₁₆×C₅₃	—	2	2	no	—
159	M₁₆×C₅₃	—	2	1	no	—
160	M₁₆×C₅₃	—	2	1	no	—
161	M₁₆×C₅₃	—	2	2	no	—
162	M₁₆×C₅₃	—	2	1	no	—
163	M₁₆×C₅₃	—	2	2	no	—
164	M₁₆×C₅₃	—	4	4	no	—
165	M₁₆×C₅₃	—	4	1	no	—
166	M₁₆×C₅₃	—	4	1	no	—
167	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
168	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
169	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
170	C₂×C₈×C₅₃	—	2	1	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	2	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	2	no	—
175	C₂×C₈×C₅₃	—	4	4	no	—
176	C₂×C₈×C₅₃	—	4	1	no	—
177	C₂×C₈×C₅₃	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
183	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	4	4	no	—
187	C₂²⋊C₄×C₅₃	—	2	1	no	—
188	C₂²⋊C₄×C₅₃	—	4	4	no	—
189	C₂²⋊C₄×C₅₃	#41	1	2	yes	—
190	C₄²×C₅₃	—	2	1	no	—
191	C₄²×C₅₃	—	2	1	no	—
192	C₂³×C₅₃	—	2	1	no	—
193	C₂³×C₅₃	—	2	2	no	—
194	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
195	Q₈×C₅₃	—	2	2	no	—
196	Q₈×C₅₃	—	2	1	no	—
197	Q₈×C₅₃	—	2	2	no	—
198	Q₈×C₅₃	—	2	1	no	—
199	Q₈×C₅₃	—	4	2	no	—
200	Q₈×C₅₃	—	4	2	no	—
201	Q₈×C₅₃	—	4	2	no	—
202	Q₈×C₅₃	—	4	2	no	—
203	D₄×C₅₃	—	2	2	no	—
204	D₄×C₅₃	—	2	1	no	—
205	D₄×C₅₃	—	2	2	no	—
206	D₄×C₅₃	—	2	1	no	—
207	D₄×C₅₃	—	4	2	no	—
208	D₄×C₅₃	—	4	1	no	—
209	D₄×C₅₃	—	4	1	no	—
210	D₄×C₅₃	—	4	2	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	—	4	1	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	4	2	no	—
218	C₂×C₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂²×D₄	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	4	no	—
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	4	1	no	—
232	C₈×C₅₃	—	4	2	no	—
233	C₈×C₅₃	#121	1	1	yes	characteristic
234	C₈×C₅₃	#121	1	1	yes	characteristic
235	C₈×C₅₃	#121	1	1	yes	characteristic
236	C₈×C₅₃	#121	1	1	yes	characteristic
237	#25448	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	—	4	2	no	—
241	C₂²×C₅₃	—	8	2	no	—
242	C₂²×C₅₃	—	8	1	no	—
243	C₂²×C₅₃	#532	1	1	yes	characteristic
244	C₂²×C₅₃	#546	1	1	yes	characteristic
245	C₂²×C₅₃	#550	1	1	yes	characteristic
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	1	no	—
248	C₄×C₅₃	—	4	2	no	—
249	C₄×C₅₃	—	8	1	no	—
250	C₄×C₅₃	—	8	2	no	—
251	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
252	C₄×C₅₃	#540	1	1	yes	characteristic
253	C₄×C₅₃	D₄²	1	1	yes	characteristic
254	C₄×C₅₃	D₄²	1	1	yes	characteristic
255	#3156	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#3069	C₂×C₅₃	1	2	yes	complemented
257	#3067	C₂×C₅₃	1	1	yes	characteristic
258	C₂×#473	C₂×C₅₃	1	1	yes	characteristic, complemented
259	#2957	C₂×C₅₃	1	1	yes	characteristic, complemented
260	#2049	C₂×C₅₃	1	2	yes	complemented
261	#2009	C₂×C₅₃	1	2	yes	complemented
262	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#1915	C₂×C₅₃	1	2	yes	complemented
265	#1757	C₂×C₅₃	1	1	yes	characteristic, complemented
266	C₂×C₅₃	—	2	1	no	—
267	C₂×C₅₃	—	2	1	no	—
268	C₂×C₅₃	—	2	1	no	—
269	C₂×C₅₃	—	8	1	no	—
270	C₂×C₅₃	—	8	2	no	—
271	C₂×C₅₃	#2428	1	1	yes	characteristic
272	#575	C₂²×C₅₃	1	2	yes	—
273	C₂×#137	C₂²×C₅₃	1	2	yes	—
274	C₂×#137	C₂²×C₅₃	1	2	yes	—
275	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
276	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
277	#568	C₂²×C₅₃	1	2	yes	—
278	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
279	#536	C₂²×C₅₃	1	2	yes	—
280	#497	—	2	2	no	—
281	#495	—	2	1	no	—
282	#494	—	2	1	no	—
283	#473	—	2	1	no	—
284	#473	—	2	2	no	—
285	#473	—	2	2	no	—
286	#473	—	2	4	no	—
287	#472	—	2	1	no	—
288	#471	—	2	2	no	—
289	#471	—	2	4	no	—
290	#444	—	2	2	no	—
291	#443	—	2	2	no	—
292	#430	C₂²×C₅₃	1	2	yes	—
293	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
295	#421	C₂²×C₅₃	1	2	yes	—
296	#419	C₂²×C₅₃	1	2	yes	—
297	#413	C₂²×C₅₃	1	2	yes	—
298	#413	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×#101	C₂²×C₅₃	1	2	yes	—
300	C₂×#101	C₂²×C₅₃	1	2	yes	—
301	#410	C₂²×C₅₃	1	1	yes	characteristic
302	#405	C₂²×C₅₃	1	1	yes	characteristic
303	#405	C₂²×C₅₃	1	1	yes	characteristic
304	#344	C₂²×C₅₃	1	1	yes	characteristic
305	#344	C₂²×C₅₃	1	1	yes	characteristic
306	#344	C₂²×C₅₃	1	2	yes	—
307	C₅₃	#25448	1	1	yes	characteristic, direct product factor, Sylow
308	C₂×#41	C₂³×C₅₃	1	2	yes	—
309	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
310	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
311	#137	—	2	2	no	—
312	#137	—	2	2	no	—
313	#137	—	2	4	no	—
314	#137	—	2	4	no	—
315	#137	—	4	2	no	—
316	#137	—	4	2	no	—
317	AGL(1,ℤ/8)	—	2	2	no	—
318	AGL(1,ℤ/8)	—	2	2	no	—
319	#135	—	2	1	no	—
320	#135	—	2	1	no	—
321	#135	—	2	1	no	—
322	#135	—	2	1	no	—
323	#135	—	4	2	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	—	2	2	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	—	2	1	no	—
332	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	D₄×C₅₃	1	2	yes	—
336	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
337	#131	—	2	4	no	—
338	#131	—	2	4	no	—
339	#131	—	2	1	no	—
340	#131	—	2	1	no	—
341	#131	—	2	1	no	—
342	#131	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	4	2	no	—
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	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
349	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
350	C₂²×C₈	—	2	1	no	—
351	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
352	C₄⋊Q₈	—	2	2	no	—
353	#124	—	2	2	no	—
354	#123	—	2	4	no	—
355	C₂²⋊Q₈	—	4	2	no	—
356	#117	C₂³×C₅₃	1	1	yes	characteristic
357	#117	C₂³×C₅₃	1	1	yes	characteristic
358	#108	—	2	4	no	—
359	#108	—	2	1	no	—
360	#108	—	2	1	no	—
361	#108	D₄×C₅₃	1	2	yes	—
362	#108	D₄×C₅₃	1	2	yes	—
363	#104	—	2	4	no	—
364	#103	—	4	2	no	—
365	#101	—	2	2	no	—
366	#101	—	2	4	no	—
367	#101	—	2	4	no	—
368	#101	D₄×C₅₃	1	2	yes	—
369	#100	—	2	2	no	—
370	#100	D₄×C₅₃	1	2	yes	—
371	#99	—	2	2	no	—
372	#97	—	2	1	no	—
373	#97	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	2	1	no	—
376	#41	—	2	2	no	—
377	#41	—	2	1	no	—
378	#41	—	2	2	no	—
379	#41	—	2	1	no	—
380	#41	—	2	2	no	—
381	#41	—	2	1	no	—
382	#41	—	2	2	no	—
383	#41	—	2	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
391	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
395	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
400	Q₁₆	—	2	1	no	—
401	Q₁₆	—	2	1	no	—
402	Q₁₆	—	4	4	no	—
403	Q₁₆	—	4	4	no	—
404	Q₁₆	—	4	2	no	—
405	Q₁₆	—	4	1	no	—
406	Q₁₆	—	4	1	no	—
407	Q₁₆	—	4	1	no	—
408	Q₁₆	—	4	1	no	—
409	SD₁₆	—	2	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	4	4	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	2	no	—
414	SD₁₆	—	4	2	no	—
415	D₈	—	2	1	no	—
416	D₈	—	2	1	no	—
417	D₈	—	4	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	2	1	no	—
422	M₁₆	—	2	2	no	—
423	M₁₆	—	2	1	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	4	4	no	—
426	M₁₆	—	4	1	no	—
427	M₁₆	—	4	1	no	—
428	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
429	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
430	M₁₆	C₂×D₄×C₅₃	1	4	yes	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	4	4	no	—
437	C₂×C₈	—	4	1	no	—
438	C₂×C₈	—	4	1	no	—
439	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
443	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
444	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	#41×C₅₃	1	2	yes	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₂³	—	2	1	no	—
454	C₂³	—	2	2	no	—
455	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
456	Q₈	—	2	2	no	—
457	Q₈	—	2	1	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	1	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	1	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
487	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₈	#121×C₅₃	1	1	yes	characteristic
495	C₈	#121×C₅₃	1	1	yes	characteristic
496	C₈	#121×C₅₃	1	1	yes	characteristic
497	C₈	#121×C₅₃	1	1	yes	characteristic
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	1	no	—
503	C₂²	C₅₃×#532	1	1	yes	characteristic
504	C₂²	C₅₃×#546	1	1	yes	characteristic
505	C₂²	C₅₃×#550	1	1	yes	characteristic
506	C₄	—	4	2	no	—
507	C₄	—	4	1	no	—
508	C₄	—	4	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
512	C₄	C₅₃×#540	1	1	yes	characteristic
513	C₄	D₄²×C₅₃	1	1	yes	characteristic
514	C₄	D₄²×C₅₃	1	1	yes	characteristic
515	C₂	—	2	1	no	—
516	C₂	—	2	1	no	—
517	C₂	—	2	1	no	—
518	C₂	—	8	1	no	—
519	C₂	—	8	2	no	—
520	C₂	C₅₃×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3156	C₂	semidirect
2. C₅₃×#3069	C₂	semidirect
4. C₂×C₅₃×#473	C₂	semidirect
5. C₅₃×#2957	C₂	semidirect
6. C₅₃×#2049	C₂	semidirect
7. C₅₃×#2009	C₂	semidirect
8. C₅₃×#1979	C₂	semidirect
9. C₅₃×#1953	C₂	semidirect
10. C₅₃×#1915	C₂	semidirect
11. C₅₃×#1757	C₂	semidirect
237. #25448	C₅₃	direct (N × H)
255. #3156	C₂×C₅₃	semidirect
256. #3069	C₂×C₅₃	semidirect
258. C₂×#473	C₂×C₅₃	semidirect
259. #2957	C₂×C₅₃	semidirect
260. #2049	C₂×C₅₃	semidirect
261. #2009	C₂×C₅₃	semidirect
262. #1979	C₂×C₅₃	semidirect
263. #1953	C₂×C₅₃	semidirect
264. #1915	C₂×C₅₃	semidirect
265. #1757	C₂×C₅₃	semidirect
307. C₅₃	#25448	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×#3156	2
C₅₃×#3069	2
C₅₃×#3067	2
C₂×C₅₃×#473	2
C₅₃×#2957	2
C₅₃×#2049	2
C₅₃×#2009	2
C₅₃×#1979	2
C₅₃×#1953	2
C₅₃×#1915	2
C₅₃×#1757	2
#25448	53

Subgroups — C53×#25448 #27232960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25448 #27232960