Subgroups — C₅₃×#23042 #27230554

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 468 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×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2956	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1974	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×#1881	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1858	C₂	1	2	yes	maximal, complemented
10	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#575	C₂²	1	2	yes	—
13	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#568	C₂²	1	2	yes	—
15	C₅₃×#568	C₂²	1	2	yes	—
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#473	C₄	1	2	yes	complemented
18	C₅₃×#472	C₄	1	2	yes	complemented
19	C₅₃×#471	C₄	1	4	yes	complemented
20	C₅₃×#444	—	2	2	no	—
21	C₅₃×#444	—	2	4	no	—
22	C₅₃×#443	—	2	2	no	—
23	C₅₃×#443	—	2	4	no	—
24	C₅₃×#430	—	2	2	no	—
25	C₅₃×#430	C₂²	1	2	yes	—
26	C₅₃×#430	C₂²	1	2	yes	—
27	C₂×#108×C₅₃	—	2	2	no	—
28	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#421	C₂²	1	2	yes	—
32	C₂×#104×C₅₃	C₂²	1	2	yes	—
33	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#416	C₂²	1	1	yes	characteristic
35	C₅₃×#413	C₂²	1	2	yes	—
36	C₅₃×#408	C₂²	1	2	yes	—
37	C₅₃×#405	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#344	C₂²	1	2	yes	—
41	C₅₃×#341	C₂²	1	1	yes	characteristic
42	C₅₃×#341	C₂²	1	2	yes	—
43	C₅₃×#339	C₂²	1	1	yes	characteristic
44	C₂×#41×C₅₃	C₂³	1	2	yes	—
45	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
46	#137×C₅₃	—	2	2	no	—
47	#137×C₅₃	C₂×C₄	1	4	yes	—
48	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
49	AGL(1,ℤ/8)×C₅₃	C₂×C₄	1	4	yes	—
50	#135×C₅₃	—	2	1	no	—
51	#135×C₅₃	—	2	1	no	—
52	#135×C₅₃	—	2	1	no	—
53	#135×C₅₃	—	2	1	no	—
54	#135×C₅₃	—	2	4	no	—
55	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#131×C₅₃	—	2	1	no	—
59	#131×C₅₃	—	2	1	no	—
60	#131×C₅₃	—	2	4	no	—
61	#131×C₅₃	—	2	4	no	—
62	#131×C₅₃	—	2	4	no	—
63	#131×C₅₃	—	4	2	no	—
64	#131×C₅₃	C₂×C₄	1	2	yes	—
65	#131×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₅₃	—	2	1	no	—
67	C₂×M₁₆×C₅₃	—	2	4	no	—
68	C₂×M₁₆×C₅₃	—	4	2	no	—
69	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
70	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
72	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
73	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	—	2	1	no	—
75	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	#117×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#108×C₅₃	—	2	4	no	—
80	#108×C₅₃	—	2	4	no	—
81	#108×C₅₃	—	2	1	no	—
82	#108×C₅₃	—	2	1	no	—
83	#108×C₅₃	C₂×C₄	1	2	yes	—
84	#108×C₅₃	C₂×C₄	1	2	yes	—
85	#108×C₅₃	D₄	1	2	yes	—
86	#108×C₅₃	D₄	1	2	yes	—
87	#104×C₅₃	—	2	4	no	—
88	#104×C₅₃	—	2	2	no	—
89	#104×C₅₃	—	2	2	no	—
90	#104×C₅₃	D₄	1	2	yes	—
91	#104×C₅₃	D₄	1	2	yes	—
92	#103×C₅₃	—	2	1	no	—
93	#102×C₅₃	—	2	1	no	—
94	#101×C₅₃	C₂×C₄	1	4	yes	—
95	#100×C₅₃	C₂×C₄	1	4	yes	—
96	C₂²⋊C₈×C₅₃	—	4	2	no	—
97	#97×C₅₃	—	2	1	no	—
98	#97×C₅₃	—	2	2	no	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	—	2	2	no	—
103	#41×C₅₃	—	2	1	no	—
104	#41×C₅₃	—	2	1	no	—
105	#41×C₅₃	—	2	2	no	—
106	#41×C₅₃	—	2	2	no	—
107	#41×C₅₃	—	2	2	no	—
108	#41×C₅₃	—	2	2	no	—
109	#41×C₅₃	C₂²×C₄	1	2	yes	—
110	#41×C₅₃	C₂×D₄	1	2	yes	—
111	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₅₃	—	2	4	no	—
122	Q₁₆×C₅₃	—	2	1	no	—
123	Q₁₆×C₅₃	—	2	1	no	—
124	SD₁₆×C₅₃	—	2	4	no	—
125	SD₁₆×C₅₃	—	2	4	no	—
126	SD₁₆×C₅₃	—	2	2	no	—
127	SD₁₆×C₅₃	—	2	2	no	—
128	D₈×C₅₃	—	2	4	no	—
129	D₈×C₅₃	—	2	1	no	—
130	D₈×C₅₃	—	2	1	no	—
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	—	2	2	no	—
136	M₁₆×C₅₃	—	2	2	no	—
137	M₁₆×C₅₃	—	2	2	no	—
138	M₁₆×C₅₃	—	2	2	no	—
139	M₁₆×C₅₃	—	4	4	no	—
140	M₁₆×C₅₃	—	4	4	no	—
141	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
142	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
143	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
144	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	2	2	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	2	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	1	no	—
154	C₂×C₈×C₅₃	—	2	2	no	—
155	C₂×C₈×C₅₃	—	4	4	no	—
156	C₂×C₈×C₅₃	—	4	4	no	—
157	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
162	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	—	2	1	no	—
164	C₂²⋊C₄×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₂³×C₅₃	—	2	1	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
172	Q₈×C₅₃	—	2	1	no	—
173	Q₈×C₅₃	—	2	2	no	—
174	Q₈×C₅₃	—	2	2	no	—
175	Q₈×C₅₃	C₄×D₄	1	2	yes	—
176	D₄×C₅₃	—	2	2	no	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	1	no	—
181	D₄×C₅₃	—	4	2	no	—
182	D₄×C₅₃	—	4	2	no	—
183	D₄×C₅₃	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	1	no	—
194	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
200	C₈×C₅₃	—	2	2	no	—
201	C₈×C₅₃	—	2	4	no	—
202	C₈×C₅₃	—	2	2	no	—
203	C₈×C₅₃	—	2	4	no	—
204	C₈×C₅₃	—	4	2	no	—
205	C₈×C₅₃	—	4	1	no	—
206	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
207	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
208	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
209	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
210	#23042	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	2	2	no	—
212	C₂²×C₅₃	—	4	1	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	8	1	no	—
215	C₂²×C₅₃	—	8	2	no	—
216	C₂²×C₅₃	#520	1	1	yes	characteristic
217	C₂²×C₅₃	#547	1	1	yes	characteristic
218	C₂²×C₅₃	#548	1	1	yes	characteristic
219	C₄×C₅₃	—	2	2	no	—
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	2	no	—
223	C₄×C₅₃	—	8	1	no	—
224	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₅₃	D₄²	1	1	yes	characteristic
227	C₄×C₅₃	#553	1	1	yes	characteristic
228	#3067	C₂×C₅₃	1	1	yes	characteristic
229	#2957	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#2956	C₂×C₅₃	1	2	yes	complemented
231	#2916	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1974	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1932	C₂×C₅₃	1	2	yes	complemented
235	#1881	C₂×C₅₃	1	2	yes	complemented
236	#1858	C₂×C₅₃	1	2	yes	complemented
237	C₄×#108	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#1764	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×C₅₃	—	2	1	no	—
240	C₂×C₅₃	—	2	1	no	—
241	C₂×C₅₃	—	2	1	no	—
242	C₂×C₅₃	—	4	2	no	—
243	C₂×C₅₃	—	8	2	no	—
244	C₂×C₅₃	#2361	1	1	yes	characteristic
245	#575	C₂²×C₅₃	1	2	yes	—
246	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
247	#568	C₂²×C₅₃	1	2	yes	—
248	#568	C₂²×C₅₃	1	2	yes	—
249	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
250	#473	C₄×C₅₃	1	2	yes	complemented
251	#472	C₄×C₅₃	1	2	yes	complemented
252	#471	C₄×C₅₃	1	4	yes	complemented
253	#444	—	2	2	no	—
254	#444	—	2	4	no	—
255	#443	—	2	2	no	—
256	#443	—	2	4	no	—
257	#430	—	2	2	no	—
258	#430	C₂²×C₅₃	1	2	yes	—
259	#430	C₂²×C₅₃	1	2	yes	—
260	C₂×#108	—	2	2	no	—
261	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
264	#421	C₂²×C₅₃	1	2	yes	—
265	C₂×#104	C₂²×C₅₃	1	2	yes	—
266	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
267	#416	C₂²×C₅₃	1	1	yes	characteristic
268	#413	C₂²×C₅₃	1	2	yes	—
269	#408	C₂²×C₅₃	1	2	yes	—
270	#405	C₂²×C₅₃	1	1	yes	characteristic
271	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
272	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
273	#344	C₂²×C₅₃	1	2	yes	—
274	#341	C₂²×C₅₃	1	1	yes	characteristic
275	#341	C₂²×C₅₃	1	2	yes	—
276	#339	C₂²×C₅₃	1	1	yes	characteristic
277	C₅₃	#23042	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×#41	C₂³×C₅₃	1	2	yes	—
279	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
280	#137	—	2	2	no	—
281	#137	C₂×C₄×C₅₃	1	4	yes	—
282	AGL(1,ℤ/8)	—	2	2	no	—
283	AGL(1,ℤ/8)	C₂×C₄×C₅₃	1	4	yes	—
284	#135	—	2	4	no	—
285	#135	—	2	1	no	—
286	#135	—	2	1	no	—
287	#135	—	2	1	no	—
288	#135	—	2	1	no	—
289	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
290	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
291	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
292	#131	—	2	1	no	—
293	#131	—	2	1	no	—
294	#131	—	2	4	no	—
295	#131	—	2	4	no	—
296	#131	—	2	4	no	—
297	#131	—	4	2	no	—
298	#131	C₂×C₄×C₅₃	1	2	yes	—
299	#131	C₂×C₄×C₅₃	1	2	yes	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	—	2	4	no	—
302	C₂×M₁₆	—	4	2	no	—
303	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
304	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
305	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
306	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
307	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
311	#117	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	#108	—	2	4	no	—
314	#108	—	2	4	no	—
315	#108	—	2	1	no	—
316	#108	—	2	1	no	—
317	#108	C₂×C₄×C₅₃	1	2	yes	—
318	#108	C₂×C₄×C₅₃	1	2	yes	—
319	#108	D₄×C₅₃	1	2	yes	—
320	#108	D₄×C₅₃	1	2	yes	—
321	#104	—	2	4	no	—
322	#104	—	2	2	no	—
323	#104	—	2	2	no	—
324	#104	D₄×C₅₃	1	2	yes	—
325	#104	D₄×C₅₃	1	2	yes	—
326	#103	—	2	1	no	—
327	#102	—	2	1	no	—
328	#101	C₂×C₄×C₅₃	1	4	yes	—
329	#100	C₂×C₄×C₅₃	1	4	yes	—
330	C₂²⋊C₈	—	4	2	no	—
331	#97	—	2	1	no	—
332	#97	—	2	2	no	—
333	C₄×C₈	—	2	1	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	—	2	2	no	—
337	#41	—	2	1	no	—
338	#41	—	2	1	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	2	2	no	—
343	#41	C₂²×C₄×C₅₃	1	2	yes	—
344	#41	C₂×D₄×C₅₃	1	2	yes	—
345	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
346	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
350	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
355	Q₁₆	—	2	4	no	—
356	Q₁₆	—	2	1	no	—
357	Q₁₆	—	2	1	no	—
358	SD₁₆	—	2	4	no	—
359	SD₁₆	—	2	4	no	—
360	SD₁₆	—	2	2	no	—
361	SD₁₆	—	2	2	no	—
362	D₈	—	2	4	no	—
363	D₈	—	2	1	no	—
364	D₈	—	2	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	2	2	no	—
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	2	no	—
373	M₁₆	—	4	4	no	—
374	M₁₆	—	4	4	no	—
375	M₁₆	C₂²×C₄×C₅₃	1	2	yes	—
376	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
377	M₁₆	C₂×D₄×C₅₃	1	4	yes	—
378	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	4	4	no	—
390	C₂×C₈	—	4	4	no	—
391	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
392	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
393	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
394	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
396	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
397	C₄⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
403	C₂³	—	2	1	no	—
404	C₂³	—	2	2	no	—
405	C₂³	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
406	Q₈	—	2	1	no	—
407	Q₈	—	2	2	no	—
408	Q₈	—	2	2	no	—
409	Q₈	C₄×D₄×C₅₃	1	2	yes	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	2	2	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	1	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	C₄×D₄×C₅₃	1	2	yes	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
433	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	2	4	no	—
436	C₈	—	2	2	no	—
437	C₈	—	2	4	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	1	no	—
440	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
441	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
442	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
443	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
444	C₂²	—	2	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₅₃×#520	1	1	yes	characteristic
450	C₂²	C₅₃×#547	1	1	yes	characteristic
451	C₂²	C₅₃×#548	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
458	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
459	C₄	D₄²×C₅₃	1	1	yes	characteristic
460	C₄	C₅₃×#553	1	1	yes	characteristic
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	2	1	no	—
464	C₂	—	4	2	no	—
465	C₂	—	8	2	no	—
466	C₂	C₅₃×#2361	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	#2361
prime radical	C₅₃×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2957	C₂	semidirect
3. C₅₃×#2956	C₂	semidirect
4. C₅₃×#2916	C₂	semidirect
5. C₅₃×#1974	C₂	semidirect
6. C₅₃×#1953	C₂	semidirect
7. C₅₃×#1932	C₂	semidirect
8. C₅₃×#1881	C₂	semidirect
9. C₅₃×#1858	C₂	semidirect
10. C₄×#108×C₅₃	C₂	semidirect
11. C₅₃×#1764	C₂	semidirect
17. C₅₃×#473	C₄	semidirect
18. C₅₃×#472	C₄	semidirect
19. C₅₃×#471	C₄	semidirect
210. #23042	C₅₃	direct (N × H)
229. #2957	C₂×C₅₃	semidirect
230. #2956	C₂×C₅₃	semidirect
231. #2916	C₂×C₅₃	semidirect
232. #1974	C₂×C₅₃	semidirect
233. #1953	C₂×C₅₃	semidirect
234. #1932	C₂×C₅₃	semidirect
235. #1881	C₂×C₅₃	semidirect
236. #1858	C₂×C₅₃	semidirect
237. C₄×#108	C₂×C₅₃	semidirect
238. #1764	C₂×C₅₃	semidirect
250. #473	C₄×C₅₃	semidirect
251. #472	C₄×C₅₃	semidirect
252. #471	C₄×C₅₃	semidirect
277. C₅₃	#23042	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×#3067	2
C₅₃×#2957	2
C₅₃×#2956	2
C₅₃×#2916	2
C₅₃×#1974	2
C₅₃×#1953	2
C₅₃×#1932	2
C₅₃×#1881	2
C₅₃×#1858	2
C₄×#108×C₅₃	2
C₅₃×#1764	2
#23042	53

Subgroups — C53×#23042 #27230554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23042 #27230554