Subgroups — C₅₃×#24296 #27231808

Order 13568.

1270 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3067	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3020	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2955	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×#1976	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1796	C₂	1	2	yes	maximal, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#578	—	2	2	no	—
14	C₅₃×#578	—	2	2	no	—
15	C₅₃×#578	—	2	2	no	—
16	C₅₃×#578	—	2	2	no	—
17	C₅₃×#575	C₂²	1	1	yes	characteristic
18	C₅₃×#575	C₂²	1	2	yes	—
19	C₂×#137×C₅₃	—	2	2	no	—
20	C₂×#137×C₅₃	C₂²	1	2	yes	—
21	C₂×#135×C₅₃	—	2	1	no	—
22	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×Q₁₆×C₅₃	—	2	1	no	—
24	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#536	C₂²	1	2	yes	—
28	C₅₃×#473	—	2	1	no	—
29	C₅₃×#472	—	2	1	no	—
30	C₅₃×#471	—	2	2	no	—
31	C₅₃×#470	—	2	2	no	—
32	C₅₃×#467	—	2	2	no	—
33	C₅₃×#456	—	2	4	no	—
34	C₅₃×#454	—	2	4	no	—
35	C₅₃×#444	—	2	2	no	—
36	C₅₃×#443	—	2	2	no	—
37	C₅₃×#430	C₂²	1	1	yes	characteristic
38	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#427	C₂²	1	1	yes	characteristic
40	C₅₃×#421	C₂²	1	1	yes	characteristic
41	C₅₃×#421	C₂²	1	1	yes	characteristic
42	C₅₃×#421	C₂²	1	2	yes	—
43	C₅₃×#421	C₂²	1	2	yes	—
44	C₅₃×#419	C₂²	1	2	yes	—
45	C₅₃×#419	C₂²	1	2	yes	—
46	C₅₃×#413	C₂²	1	2	yes	—
47	C₂×#101×C₅₃	C₂²	1	2	yes	—
48	C₅₃×#410	C₂²	1	2	yes	—
49	C₅₃×#405	C₂²	1	1	yes	characteristic
50	C₅₃×#344	C₂²	1	2	yes	—
51	2₋¹⁺⁴×C₅₃	—	2	4	no	—
52	2₋¹⁺⁴×C₅₃	—	2	4	no	—
53	2₋¹⁺⁴×C₅₃	—	2	4	no	—
54	2₋¹⁺⁴×C₅₃	—	4	2	no	—
55	C₂×#41×C₅₃	—	2	2	no	—
56	C₂×#41×C₅₃	—	2	2	no	—
57	C₂×#41×C₅₃	—	4	2	no	—
58	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×#41×C₅₃	C₂³	1	2	yes	—
60	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₅₃	—	2	2	no	—
62	C₂²×Q₈×C₅₃	—	2	2	no	—
63	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
64	#137×C₅₃	—	2	8	no	—
65	#137×C₅₃	—	2	8	no	—
66	#137×C₅₃	—	2	4	no	—
67	#137×C₅₃	—	2	4	no	—
68	#137×C₅₃	—	2	2	no	—
69	#137×C₅₃	—	2	2	no	—
70	#137×C₅₃	—	2	1	no	—
71	#137×C₅₃	D₄	1	2	yes	—
72	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
73	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
74	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
75	AGL(1,ℤ/8)×C₅₃	D₄	1	2	yes	—
76	#135×C₅₃	—	2	4	no	—
77	#135×C₅₃	—	2	4	no	—
78	#135×C₅₃	—	2	1	no	—
79	#135×C₅₃	—	2	1	no	—
80	#135×C₅₃	—	2	1	no	—
81	#135×C₅₃	—	2	1	no	—
82	#135×C₅₃	—	2	2	no	—
83	#135×C₅₃	—	4	2	no	—
84	C₂×Q₁₆×C₅₃	—	2	4	no	—
85	C₂×Q₁₆×C₅₃	—	2	4	no	—
86	C₂×Q₁₆×C₅₃	—	2	1	no	—
87	C₂×Q₁₆×C₅₃	—	2	1	no	—
88	C₂×Q₁₆×C₅₃	—	2	1	no	—
89	C₂×Q₁₆×C₅₃	—	2	1	no	—
90	C₂×Q₁₆×C₅₃	—	2	2	no	—
91	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
92	C₂×Q₁₆×C₅₃	D₄	1	2	yes	—
93	C₂×Q₁₆×C₅₃	D₄	1	2	yes	—
94	C₂×SD₁₆×C₅₃	—	2	2	no	—
95	C₂×SD₁₆×C₅₃	D₄	1	2	yes	—
96	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
97	#131×C₅₃	—	2	1	no	—
98	#131×C₅₃	—	2	1	no	—
99	#131×C₅₃	—	2	1	no	—
100	#131×C₅₃	—	2	1	no	—
101	#131×C₅₃	—	2	1	no	—
102	#131×C₅₃	—	2	1	no	—
103	#131×C₅₃	D₄	1	2	yes	—
104	#131×C₅₃	D₄	1	2	yes	—
105	C₂×M₁₆×C₅₃	—	2	1	no	—
106	C₂×M₁₆×C₅₃	—	2	1	no	—
107	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
109	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₅₃	—	2	1	no	—
112	C₂²×C₈×C₅₃	—	2	1	no	—
113	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₄⋊Q₈×C₅₃	—	2	2	no	—
115	#124×C₅₃	—	2	2	no	—
116	#123×C₅₃	—	4	2	no	—
117	C₂²⋊Q₈×C₅₃	—	2	4	no	—
118	#117×C₅₃	C₂³	1	2	yes	—
119	#117×C₅₃	C₂³	1	1	yes	characteristic
120	#108×C₅₃	—	2	2	no	—
121	#108×C₅₃	—	2	1	no	—
122	#108×C₅₃	—	2	1	no	—
123	#107×C₅₃	—	2	1	no	—
124	#106×C₅₃	—	2	1	no	—
125	#104×C₅₃	—	2	2	no	—
126	#104×C₅₃	—	2	4	no	—
127	#104×C₅₃	—	2	2	no	—
128	#104×C₅₃	—	2	4	no	—
129	#103×C₅₃	—	2	4	no	—
130	#103×C₅₃	—	4	2	no	—
131	#101×C₅₃	—	2	4	no	—
132	#101×C₅₃	—	2	2	no	—
133	#100×C₅₃	—	2	2	no	—
134	#99×C₅₃	—	4	2	no	—
135	#97×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	1	no	—
137	#41×C₅₃	—	2	1	no	—
138	#41×C₅₃	—	2	1	no	—
139	#41×C₅₃	—	2	8	no	—
140	#41×C₅₃	—	2	1	no	—
141	#41×C₅₃	—	2	2	no	—
142	#41×C₅₃	—	2	1	no	—
143	#41×C₅₃	—	2	2	no	—
144	#41×C₅₃	—	2	2	no	—
145	#41×C₅₃	—	2	1	no	—
146	#41×C₅₃	—	2	1	no	—
147	#41×C₅₃	—	2	2	no	—
148	#41×C₅₃	—	4	4	no	—
149	#41×C₅₃	—	4	8	no	—
150	#41×C₅₃	—	4	4	no	—
151	#41×C₅₃	—	4	8	no	—
152	#41×C₅₃	—	4	4	no	—
153	#41×C₅₃	C₂×D₄	1	2	yes	—
154	#41×C₅₃	C₂×D₄	1	2	yes	—
155	C₂×Q₈×C₅₃	—	2	2	no	—
156	C₂×Q₈×C₅₃	—	2	8	no	—
157	C₂×Q₈×C₅₃	—	2	2	no	—
158	C₂×Q₈×C₅₃	—	2	2	no	—
159	C₂×Q₈×C₅₃	—	2	2	no	—
160	C₂×Q₈×C₅₃	—	2	2	no	—
161	C₂×Q₈×C₅₃	—	4	2	no	—
162	C₂×Q₈×C₅₃	—	4	4	no	—
163	C₂×Q₈×C₅₃	—	4	4	no	—
164	C₂×Q₈×C₅₃	—	4	4	no	—
165	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
166	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
167	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
169	C₂×D₄×C₅₃	—	2	1	no	—
170	C₂×D₄×C₅₃	—	2	2	no	—
171	C₂×D₄×C₅₃	—	2	1	no	—
172	C₂×D₄×C₅₃	—	4	2	no	—
173	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
174	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
175	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
176	C₂²×C₄×C₅₃	—	2	1	no	—
177	C₂²×C₄×C₅₃	—	2	2	no	—
178	C₂²×C₄×C₅₃	—	2	1	no	—
179	C₂²×C₄×C₅₃	—	2	2	no	—
180	C₂²×C₄×C₅₃	—	4	2	no	—
181	C₂²×C₄×C₅₃	—	4	2	no	—
182	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	Q₁₆×C₅₃	—	2	8	no	—
184	Q₁₆×C₅₃	—	2	2	no	—
185	Q₁₆×C₅₃	—	2	1	no	—
186	Q₁₆×C₅₃	—	2	2	no	—
187	Q₁₆×C₅₃	—	2	1	no	—
188	Q₁₆×C₅₃	—	2	2	no	—
189	Q₁₆×C₅₃	—	2	2	no	—
190	Q₁₆×C₅₃	—	2	2	no	—
191	Q₁₆×C₅₃	—	4	2	no	—
192	SD₁₆×C₅₃	—	2	8	no	—
193	SD₁₆×C₅₃	—	2	2	no	—
194	SD₁₆×C₅₃	—	2	2	no	—
195	SD₁₆×C₅₃	—	2	2	no	—
196	SD₁₆×C₅₃	—	2	2	no	—
197	SD₁₆×C₅₃	—	4	2	no	—
198	SD₁₆×C₅₃	—	4	2	no	—
199	D₈×C₅₃	—	2	2	no	—
200	D₈×C₅₃	—	2	1	no	—
201	D₈×C₅₃	—	2	1	no	—
202	D₈×C₅₃	—	4	2	no	—
203	M₁₆×C₅₃	—	2	1	no	—
204	M₁₆×C₅₃	—	2	1	no	—
205	M₁₆×C₅₃	—	2	2	no	—
206	M₁₆×C₅₃	—	2	2	no	—
207	M₁₆×C₅₃	—	2	1	no	—
208	M₁₆×C₅₃	—	2	2	no	—
209	M₁₆×C₅₃	—	2	2	no	—
210	M₁₆×C₅₃	—	2	1	no	—
211	M₁₆×C₅₃	—	4	1	no	—
212	M₁₆×C₅₃	—	4	1	no	—
213	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
214	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
215	C₂×C₈×C₅₃	—	2	1	no	—
216	C₂×C₈×C₅₃	—	2	1	no	—
217	C₂×C₈×C₅₃	—	2	2	no	—
218	C₂×C₈×C₅₃	—	2	2	no	—
219	C₂×C₈×C₅₃	—	2	1	no	—
220	C₂×C₈×C₅₃	—	2	2	no	—
221	C₂×C₈×C₅₃	—	4	1	no	—
222	C₂×C₈×C₅₃	—	4	1	no	—
223	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
224	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
229	C₄⋊C₄×C₅₃	—	2	2	no	—
230	C₄⋊C₄×C₅₃	—	2	1	no	—
231	C₄⋊C₄×C₅₃	—	4	4	no	—
232	C₂²⋊C₄×C₅₃	—	2	1	no	—
233	C₂²⋊C₄×C₅₃	—	2	2	no	—
234	C₂²⋊C₄×C₅₃	—	4	4	no	—
235	C₄²×C₅₃	—	2	1	no	—
236	C₄²×C₅₃	—	2	2	no	—
237	C₂³×C₅₃	—	2	1	no	—
238	C₂³×C₅₃	—	2	1	no	—
239	C₂³×C₅₃	—	2	2	no	—
240	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
241	Q₈×C₅₃	—	2	1	no	—
242	Q₈×C₅₃	—	2	2	no	—
243	Q₈×C₅₃	—	2	4	no	—
244	Q₈×C₅₃	—	2	1	no	—
245	Q₈×C₅₃	—	2	2	no	—
246	Q₈×C₅₃	—	2	1	no	—
247	Q₈×C₅₃	—	2	4	no	—
248	Q₈×C₅₃	—	4	8	no	—
249	Q₈×C₅₃	—	4	2	no	—
250	Q₈×C₅₃	—	4	2	no	—
251	Q₈×C₅₃	#121	1	2	yes	—
252	D₄×C₅₃	—	2	2	no	—
253	D₄×C₅₃	—	2	2	no	—
254	D₄×C₅₃	—	2	2	no	—
255	D₄×C₅₃	—	2	1	no	—
256	D₄×C₅₃	—	2	2	no	—
257	D₄×C₅₃	—	2	1	no	—
258	D₄×C₅₃	—	2	1	no	—
259	D₄×C₅₃	—	4	8	no	—
260	D₄×C₅₃	—	4	1	no	—
261	D₄×C₅₃	—	4	2	no	—
262	D₄×C₅₃	—	4	2	no	—
263	D₄×C₅₃	—	4	1	no	—
264	D₄×C₅₃	#121	1	2	yes	—
265	C₂×C₄×C₅₃	—	2	1	no	—
266	C₂×C₄×C₅₃	—	2	2	no	—
267	C₂×C₄×C₅₃	—	2	2	no	—
268	C₂×C₄×C₅₃	—	2	1	no	—
269	C₂×C₄×C₅₃	—	2	4	no	—
270	C₂×C₄×C₅₃	—	2	2	no	—
271	C₂×C₄×C₅₃	—	4	8	no	—
272	C₂×C₄×C₅₃	—	4	8	no	—
273	C₂×C₄×C₅₃	—	4	2	no	—
274	C₂×C₄×C₅₃	—	4	2	no	—
275	C₂×C₄×C₅₃	—	4	1	no	—
276	C₂×C₄×C₅₃	—	4	2	no	—
277	C₂×C₄×C₅₃	—	4	2	no	—
278	C₂×C₄×C₅₃	—	4	1	no	—
279	C₂×C₄×C₅₃	—	4	1	no	—
280	C₂×C₄×C₅₃	—	4	2	no	—
281	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
287	C₈×C₅₃	—	2	2	no	—
288	C₈×C₅₃	—	2	2	no	—
289	C₈×C₅₃	—	4	1	no	—
290	C₈×C₅₃	—	4	2	no	—
291	C₈×C₅₃	#120	1	1	yes	characteristic
292	C₈×C₅₃	#120	1	1	yes	characteristic
293	C₈×C₅₃	#120	1	1	yes	characteristic
294	C₈×C₅₃	#120	1	1	yes	characteristic
295	#24296	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂²×C₅₃	—	2	2	no	—
297	C₂²×C₅₃	—	4	1	no	—
298	C₂²×C₅₃	—	4	2	no	—
299	C₂²×C₅₃	—	4	2	no	—
300	C₂²×C₅₃	—	4	2	no	—
301	C₂²×C₅₃	—	4	1	no	—
302	C₂²×C₅₃	—	8	2	no	—
303	C₂²×C₅₃	#535	1	1	yes	characteristic
304	C₂²×C₅₃	D₄²	1	1	yes	characteristic
305	C₂²×C₅₃	#546	1	1	yes	characteristic
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	4	4	no	—
308	C₄×C₅₃	—	4	2	no	—
309	C₄×C₅₃	—	4	1	no	—
310	C₄×C₅₃	—	8	2	no	—
311	C₄×C₅₃	—	8	1	no	—
312	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
313	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
314	C₄×C₅₃	D₄²	1	1	yes	characteristic
315	C₄×C₅₃	#546	1	1	yes	characteristic
316	C₂×#578	C₂×C₅₃	1	1	yes	characteristic, complemented
317	#3067	C₂×C₅₃	1	1	yes	characteristic, complemented
318	#3059	C₂×C₅₃	1	1	yes	characteristic, complemented
319	#3020	C₂×C₅₃	1	2	yes	complemented
320	#2955	C₂×C₅₃	1	1	yes	characteristic, complemented
321	#2049	C₂×C₅₃	1	2	yes	complemented
322	#2009	C₂×C₅₃	1	2	yes	complemented
323	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
324	#1976	C₂×C₅₃	1	1	yes	characteristic, complemented
325	#1814	C₂×C₅₃	1	1	yes	characteristic, complemented
326	#1796	C₂×C₅₃	1	2	yes	complemented
327	C₂×C₅₃	—	2	1	no	—
328	C₂×C₅₃	—	2	1	no	—
329	C₂×C₅₃	—	2	1	no	—
330	C₂×C₅₃	—	4	2	no	—
331	C₂×C₅₃	—	8	1	no	—
332	C₂×C₅₃	—	8	2	no	—
333	C₂×C₅₃	#2408	1	1	yes	characteristic
334	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
335	#578	—	2	2	no	—
336	#578	—	2	2	no	—
337	#578	—	2	2	no	—
338	#578	—	2	2	no	—
339	#575	C₂²×C₅₃	1	1	yes	characteristic
340	#575	C₂²×C₅₃	1	2	yes	—
341	C₂×#137	—	2	2	no	—
342	C₂×#137	C₂²×C₅₃	1	2	yes	—
343	C₂×#135	—	2	1	no	—
344	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
345	C₂²×Q₁₆	—	2	1	no	—
346	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
347	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
348	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
349	#536	C₂²×C₅₃	1	2	yes	—
350	#473	—	2	1	no	—
351	#472	—	2	1	no	—
352	#471	—	2	2	no	—
353	#470	—	2	2	no	—
354	#467	—	2	2	no	—
355	#456	—	2	4	no	—
356	#454	—	2	4	no	—
357	#444	—	2	2	no	—
358	#443	—	2	2	no	—
359	#430	C₂²×C₅₃	1	1	yes	characteristic
360	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
361	#427	C₂²×C₅₃	1	1	yes	characteristic
362	#421	C₂²×C₅₃	1	2	yes	—
363	#421	C₂²×C₅₃	1	1	yes	characteristic
364	#421	C₂²×C₅₃	1	2	yes	—
365	#421	C₂²×C₅₃	1	1	yes	characteristic
366	#419	C₂²×C₅₃	1	2	yes	—
367	#419	C₂²×C₅₃	1	2	yes	—
368	#413	C₂²×C₅₃	1	2	yes	—
369	C₂×#101	C₂²×C₅₃	1	2	yes	—
370	#410	C₂²×C₅₃	1	2	yes	—
371	#405	C₂²×C₅₃	1	1	yes	characteristic
372	#344	C₂²×C₅₃	1	2	yes	—
373	C₅₃	#24296	1	1	yes	characteristic, direct product factor, Sylow
374	2₋¹⁺⁴	—	2	4	no	—
375	2₋¹⁺⁴	—	2	4	no	—
376	2₋¹⁺⁴	—	2	4	no	—
377	2₋¹⁺⁴	—	4	2	no	—
378	C₂×#41	—	2	2	no	—
379	C₂×#41	—	2	2	no	—
380	C₂×#41	—	4	2	no	—
381	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
382	C₂×#41	C₂³×C₅₃	1	2	yes	—
383	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
384	C₂²×Q₈	—	2	2	no	—
385	C₂²×Q₈	—	2	2	no	—
386	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
387	#137	—	2	8	no	—
388	#137	—	2	8	no	—
389	#137	—	2	4	no	—
390	#137	—	2	1	no	—
391	#137	—	2	2	no	—
392	#137	—	2	2	no	—
393	#137	—	2	4	no	—
394	#137	D₄×C₅₃	1	2	yes	—
395	AGL(1,ℤ/8)	—	2	1	no	—
396	AGL(1,ℤ/8)	—	2	2	no	—
397	AGL(1,ℤ/8)	—	2	2	no	—
398	AGL(1,ℤ/8)	D₄×C₅₃	1	2	yes	—
399	#135	—	2	4	no	—
400	#135	—	2	4	no	—
401	#135	—	2	2	no	—
402	#135	—	2	1	no	—
403	#135	—	2	1	no	—
404	#135	—	2	1	no	—
405	#135	—	2	1	no	—
406	#135	—	4	2	no	—
407	C₂×Q₁₆	—	2	4	no	—
408	C₂×Q₁₆	—	2	4	no	—
409	C₂×Q₁₆	—	2	2	no	—
410	C₂×Q₁₆	—	2	1	no	—
411	C₂×Q₁₆	—	2	1	no	—
412	C₂×Q₁₆	—	2	1	no	—
413	C₂×Q₁₆	—	2	1	no	—
414	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic
415	C₂×Q₁₆	D₄×C₅₃	1	2	yes	—
416	C₂×Q₁₆	D₄×C₅₃	1	2	yes	—
417	C₂×SD₁₆	—	2	2	no	—
418	C₂×SD₁₆	D₄×C₅₃	1	2	yes	—
419	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
420	#131	—	2	1	no	—
421	#131	—	2	1	no	—
422	#131	—	2	1	no	—
423	#131	—	2	1	no	—
424	#131	—	2	1	no	—
425	#131	—	2	1	no	—
426	#131	D₄×C₅₃	1	2	yes	—
427	#131	D₄×C₅₃	1	2	yes	—
428	C₂×M₁₆	—	2	1	no	—
429	C₂×M₁₆	—	2	1	no	—
430	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
431	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
432	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
433	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
434	C₂²×C₈	—	2	1	no	—
435	C₂²×C₈	—	2	1	no	—
436	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
437	C₄⋊Q₈	—	2	2	no	—
438	#124	—	2	2	no	—
439	#123	—	4	2	no	—
440	C₂²⋊Q₈	—	2	4	no	—
441	#117	C₂³×C₅₃	1	2	yes	—
442	#117	C₂³×C₅₃	1	1	yes	characteristic
443	#108	—	2	2	no	—
444	#108	—	2	1	no	—
445	#108	—	2	1	no	—
446	#107	—	2	1	no	—
447	#106	—	2	1	no	—
448	#104	—	2	2	no	—
449	#104	—	2	4	no	—
450	#104	—	2	2	no	—
451	#104	—	2	4	no	—
452	#103	—	2	4	no	—
453	#103	—	4	2	no	—
454	#101	—	2	2	no	—
455	#101	—	2	4	no	—
456	#100	—	2	2	no	—
457	#99	—	4	2	no	—
458	#97	—	2	1	no	—
459	C₄×C₈	—	2	1	no	—
460	#41	—	2	1	no	—
461	#41	—	2	1	no	—
462	#41	—	2	8	no	—
463	#41	—	2	1	no	—
464	#41	—	2	2	no	—
465	#41	—	2	1	no	—
466	#41	—	2	2	no	—
467	#41	—	2	2	no	—
468	#41	—	2	1	no	—
469	#41	—	2	1	no	—
470	#41	—	2	2	no	—
471	#41	—	4	4	no	—
472	#41	—	4	8	no	—
473	#41	—	4	4	no	—
474	#41	—	4	8	no	—
475	#41	—	4	4	no	—
476	#41	C₂×D₄×C₅₃	1	2	yes	—
477	#41	C₂×D₄×C₅₃	1	2	yes	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	8	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	2	no	—
484	C₂×Q₈	—	4	2	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	4	no	—
487	C₂×Q₈	—	4	4	no	—
488	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
489	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
490	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
491	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	4	2	no	—
496	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
497	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
498	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
506	Q₁₆	—	2	8	no	—
507	Q₁₆	—	2	2	no	—
508	Q₁₆	—	2	1	no	—
509	Q₁₆	—	2	2	no	—
510	Q₁₆	—	2	1	no	—
511	Q₁₆	—	2	2	no	—
512	Q₁₆	—	2	2	no	—
513	Q₁₆	—	2	2	no	—
514	Q₁₆	—	4	2	no	—
515	SD₁₆	—	2	8	no	—
516	SD₁₆	—	2	2	no	—
517	SD₁₆	—	2	2	no	—
518	SD₁₆	—	2	2	no	—
519	SD₁₆	—	2	2	no	—
520	SD₁₆	—	4	2	no	—
521	SD₁₆	—	4	2	no	—
522	D₈	—	2	2	no	—
523	D₈	—	2	1	no	—
524	D₈	—	2	1	no	—
525	D₈	—	4	2	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	2	no	—
529	M₁₆	—	2	2	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	2	no	—
532	M₁₆	—	2	2	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	4	1	no	—
535	M₁₆	—	4	1	no	—
536	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
537	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	2	no	—
541	C₂×C₈	—	2	2	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	2	no	—
544	C₂×C₈	—	4	1	no	—
545	C₂×C₈	—	4	1	no	—
546	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
547	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
548	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
549	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
550	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
551	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	4	4	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	—	2	2	no	—
557	C₂²⋊C₄	—	4	4	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	2	no	—
560	C₂³	—	2	1	no	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	2	no	—
563	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
564	Q₈	—	2	1	no	—
565	Q₈	—	2	4	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	1	no	—
568	Q₈	—	2	1	no	—
569	Q₈	—	2	2	no	—
570	Q₈	—	2	4	no	—
571	Q₈	—	4	8	no	—
572	Q₈	—	4	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	#121×C₅₃	1	2	yes	—
575	D₄	—	2	2	no	—
576	D₄	—	2	2	no	—
577	D₄	—	2	2	no	—
578	D₄	—	2	1	no	—
579	D₄	—	2	1	no	—
580	D₄	—	2	2	no	—
581	D₄	—	2	1	no	—
582	D₄	—	4	8	no	—
583	D₄	—	4	1	no	—
584	D₄	—	4	2	no	—
585	D₄	—	4	2	no	—
586	D₄	—	4	1	no	—
587	D₄	#121×C₅₃	1	2	yes	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	4	8	no	—
595	C₂×C₄	—	4	8	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	4	2	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
606	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
609	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
610	C₈	—	2	2	no	—
611	C₈	—	2	2	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	2	no	—
614	C₈	#120×C₅₃	1	1	yes	characteristic
615	C₈	#120×C₅₃	1	1	yes	characteristic
616	C₈	#120×C₅₃	1	1	yes	characteristic
617	C₈	#120×C₅₃	1	1	yes	characteristic
618	C₂²	—	2	2	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	8	2	no	—
625	C₂²	C₅₃×#535	1	1	yes	characteristic
626	C₂²	D₄²×C₅₃	1	1	yes	characteristic
627	C₂²	C₅₃×#546	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	4	4	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	1	no	—
632	C₄	—	8	2	no	—
633	C₄	—	8	1	no	—
634	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
635	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
636	C₄	D₄²×C₅₃	1	1	yes	characteristic
637	C₄	C₅₃×#546	1	1	yes	characteristic
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	4	2	no	—
642	C₂	—	8	1	no	—
643	C₂	—	8	2	no	—
644	C₂	C₅₃×#2408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2408
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
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. C₂×C₅₃×#578	C₂	semidirect
2. C₅₃×#3067	C₂	semidirect
3. C₅₃×#3059	C₂	semidirect
4. C₅₃×#3020	C₂	semidirect
5. C₅₃×#2955	C₂	semidirect
6. C₅₃×#2049	C₂	semidirect
7. C₅₃×#2009	C₂	semidirect
8. C₅₃×#1979	C₂	semidirect
9. C₅₃×#1976	C₂	semidirect
10. C₅₃×#1814	C₂	semidirect
11. C₅₃×#1796	C₂	semidirect
295. #24296	C₅₃	direct (N × H)
316. C₂×#578	C₂×C₅₃	semidirect
317. #3067	C₂×C₅₃	semidirect
318. #3059	C₂×C₅₃	semidirect
319. #3020	C₂×C₅₃	semidirect
320. #2955	C₂×C₅₃	semidirect
321. #2049	C₂×C₅₃	semidirect
322. #2009	C₂×C₅₃	semidirect
323. #1979	C₂×C₅₃	semidirect
324. #1976	C₂×C₅₃	semidirect
325. #1814	C₂×C₅₃	semidirect
326. #1796	C₂×C₅₃	semidirect
373. C₅₃	#24296	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×#578	2
C₅₃×#3067	2
C₅₃×#3059	2
C₅₃×#3020	2
C₅₃×#2955	2
C₅₃×#2049	2
C₅₃×#2009	2
C₅₃×#1979	2
C₅₃×#1976	2
C₅₃×#1814	2
C₅₃×#1796	2
#24296	53

Subgroups — C53×#24296 #27231808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24296 #27231808