Subgroups — C₅₃×#22455 #27229967

Order 13568.

1378 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#576	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2948	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2883	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1976	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1904	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1795	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#576	—	2	4	no	—
11	C₅₃×#576	—	2	4	no	—
12	C₂×AGL(1,ℤ/8)×C₅₃	C₄	1	4	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
14	C₂×#135×C₅₃	C₄	1	2	yes	complemented
15	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₅₃	C₄	1	2	yes	complemented
17	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#518	C₂²	1	4	yes	complemented
20	C₅₃×#442	—	2	4	no	—
21	C₄×D₈×C₅₃	—	2	4	no	—
22	C₅₃×#427	C₂²	1	2	yes	complemented
23	C₅₃×#421	C₂²	1	4	yes	complemented
24	C₅₃×#418	C₂²	1	4	yes	complemented
25	C₅₃×#413	C₂²	1	1	yes	characteristic
26	C₅₃×#410	C₂²	1	2	yes	—
27	C₅₃×#410	C₂²	1	1	yes	characteristic
28	C₅₃×#410	C₂²	1	4	yes	complemented
29	C₂×#99×C₅₃	—	2	8	no	—
30	C₅₃×#405	C₂²	1	2	yes	complemented
31	C₅₃×#344	C₂²	1	2	yes	complemented
32	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	complemented
33	H(5,𝔽₂)×C₅₃	—	2	8	no	—
34	H(5,𝔽₂)×C₅₃	—	4	4	no	—
35	C₂×#41×C₅₃	—	2	2	no	—
36	C₂×#41×C₅₃	C₂×C₄	1	4	yes	complemented
37	C₂×#41×C₅₃	C₂³	1	2	yes	—
38	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₅₃	—	2	4	no	—
40	C₂²×D₄×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
42	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
43	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
44	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
45	#135×C₅₃	—	2	4	no	—
46	#135×C₅₃	—	2	4	no	—
47	#135×C₅₃	—	2	2	no	—
48	#135×C₅₃	—	2	2	no	—
49	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
50	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
52	C₂×D₈×C₅₃	—	2	4	no	—
53	C₂×D₈×C₅₃	—	2	4	no	—
54	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
57	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₂×D₈×C₅₃	D₄	1	4	yes	—
59	C₂×D₈×C₅₃	D₄	1	4	yes	—
60	#131×C₅₃	—	2	2	no	—
61	#131×C₅₃	—	2	2	no	—
62	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
63	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₅₃	—	2	8	no	—
68	C₄×D₄×C₅₃	—	4	4	no	—
69	#117×C₅₃	C₂³	1	4	yes	—
70	#117×C₅₃	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
72	#107×C₅₃	—	2	2	no	—
73	#106×C₅₃	—	2	2	no	—
74	#104×C₅₃	—	4	4	no	—
75	#102×C₅₃	—	2	8	no	—
76	#101×C₅₃	—	2	1	no	—
77	#100×C₅₃	—	2	1	no	—
78	#99×C₅₃	—	2	8	no	—
79	#99×C₅₃	—	2	4	no	—
80	#99×C₅₃	D₄	1	4	yes	—
81	#97×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₂⁴×C₅₃	—	2	2	no	—
84	C₂⁴×C₅₃	—	2	4	no	—
85	#41×C₅₃	—	2	2	no	—
86	#41×C₅₃	—	2	2	no	—
87	#41×C₅₃	—	2	8	no	—
88	#41×C₅₃	—	2	4	no	—
89	#41×C₅₃	—	2	4	no	—
90	#41×C₅₃	—	4	4	no	—
91	#41×C₅₃	—	4	8	no	—
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
94	C₂×D₄×C₅₃	—	2	8	no	—
95	C₂×D₄×C₅₃	—	2	4	no	—
96	C₂×D₄×C₅₃	—	2	2	no	—
97	C₂×D₄×C₅₃	—	2	4	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	4	no	—
100	C₂×D₄×C₅₃	—	4	4	no	—
101	C₂×D₄×C₅₃	—	4	8	no	—
102	C₂×D₄×C₅₃	—	4	16	no	—
103	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
105	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₅₃	C₂×D₄	1	8	yes	—
107	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	4	8	no	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₅₃	—	2	1	no	—
114	Q₁₆×C₅₃	—	2	1	no	—
115	SD₁₆×C₅₃	—	2	8	no	—
116	SD₁₆×C₅₃	—	2	4	no	—
117	D₈×C₅₃	—	2	8	no	—
118	D₈×C₅₃	—	2	1	no	—
119	D₈×C₅₃	—	2	1	no	—
120	D₈×C₅₃	C₂²⋊C₄	1	4	yes	—
121	D₈×C₅₃	C₂²⋊C₄	1	4	yes	—
122	D₈×C₅₃	C₂²⋊C₄	1	8	yes	—
123	M₁₆×C₅₃	—	2	2	no	—
124	M₁₆×C₅₃	—	2	4	no	—
125	M₁₆×C₅₃	—	2	2	no	—
126	M₁₆×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	4	no	—
137	C₂²⋊C₄×C₅₃	—	2	4	no	—
138	C₂²⋊C₄×C₅₃	—	4	8	no	—
139	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
140	C₄²×C₅₃	—	2	2	no	—
141	C₄²×C₅₃	—	2	4	no	—
142	C₂³×C₅₃	—	2	4	no	—
143	C₂³×C₅₃	—	2	8	no	—
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	4	8	no	—
146	C₂³×C₅₃	—	16	2	no	—
147	C₂³×C₅₃	—	16	4	no	—
148	C₂³×C₅₃	C₄×D₄	1	2	yes	—
149	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	Q₈×C₅₃	—	2	2	no	—
152	Q₈×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	2	4	no	—
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	4	no	—
156	D₄×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	2	8	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	4	8	no	—
160	D₄×C₅₃	—	4	8	no	—
161	D₄×C₅₃	—	4	4	no	—
162	D₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	4	no	—
167	C₂×C₄×C₅₃	—	4	8	no	—
168	C₂×C₄×C₅₃	—	4	4	no	—
169	C₂×C₄×C₅₃	—	4	4	no	—
170	C₂×C₄×C₅₃	—	16	8	no	—
171	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
176	C₈×C₅₃	—	2	2	no	—
177	C₈×C₅₃	—	4	2	no	—
178	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
181	#22455	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	2	8	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	—	4	2	no	—
187	C₂²×C₅₃	—	8	8	no	—
188	C₂²×C₅₃	—	8	4	no	—
189	C₂²×C₅₃	—	8	2	no	—
190	C₂²×C₅₃	—	16	8	no	—
191	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₅₃	D₄²	1	2	yes	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	4	4	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	8	4	no	—
197	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₄×C₅₃	#518	1	1	yes	characteristic
199	C₄×C₅₃	#546	1	2	yes	—
200	C₂×#576	C₂×C₅₃	1	1	yes	characteristic
201	#2948	C₂×C₅₃	1	2	yes	complemented
202	#2883	C₂×C₅₃	1	2	yes	complemented
203	#1976	C₂×C₅₃	1	2	yes	complemented
204	#1904	C₂×C₅₃	1	4	yes	complemented
205	#1812	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#1795	C₂×C₅₃	1	2	yes	complemented
207	#1757	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	4	2	no	—
211	C₂×C₅₃	—	4	8	no	—
212	C₂×C₅₃	—	8	2	no	—
213	C₂×C₅₃	D₄×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
215	#576	—	2	4	no	—
216	#576	—	2	4	no	—
217	C₂×AGL(1,ℤ/8)	C₄×C₅₃	1	4	yes	complemented
218	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
219	C₂×#135	C₄×C₅₃	1	2	yes	complemented
220	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
221	C₂²×D₈	C₄×C₅₃	1	2	yes	complemented
222	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
224	#518	C₂²×C₅₃	1	4	yes	complemented
225	#442	—	2	4	no	—
226	C₄×D₈	—	2	4	no	—
227	#427	C₂²×C₅₃	1	2	yes	complemented
228	#421	C₂²×C₅₃	1	4	yes	complemented
229	#418	C₂²×C₅₃	1	4	yes	complemented
230	#413	C₂²×C₅₃	1	1	yes	characteristic
231	#410	C₂²×C₅₃	1	2	yes	—
232	#410	C₂²×C₅₃	1	1	yes	characteristic
233	#410	C₂²×C₅₃	1	4	yes	complemented
234	C₂×#99	—	2	8	no	—
235	#405	C₂²×C₅₃	1	2	yes	complemented
236	#344	C₂²×C₅₃	1	2	yes	complemented
237	H(3,ℤ/4)	C₂²×C₅₃	1	2	yes	complemented
238	C₅₃	#22455	1	1	yes	characteristic, direct product factor, Sylow
239	H(5,𝔽₂)	—	2	8	no	—
240	H(5,𝔽₂)	—	4	4	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	C₂×C₄×C₅₃	1	4	yes	complemented
243	C₂×#41	C₂³×C₅₃	1	2	yes	—
244	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
245	C₂²×D₄	—	2	4	no	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
248	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
249	AGL(1,ℤ/8)	—	2	8	no	—
250	AGL(1,ℤ/8)	—	2	16	no	—
251	#135	—	2	4	no	—
252	#135	—	2	4	no	—
253	#135	—	2	2	no	—
254	#135	—	2	2	no	—
255	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
256	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
257	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
258	C₂×D₈	—	2	4	no	—
259	C₂×D₈	—	2	4	no	—
260	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
261	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
262	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
263	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
264	C₂×D₈	D₄×C₅₃	1	4	yes	—
265	C₂×D₈	D₄×C₅₃	1	4	yes	—
266	#131	—	2	2	no	—
267	#131	—	2	2	no	—
268	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
269	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
270	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
271	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	complemented
272	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
273	C₄×D₄	—	2	8	no	—
274	C₄×D₄	—	4	4	no	—
275	#117	C₂³×C₅₃	1	4	yes	—
276	#117	C₂³×C₅₃	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	8	no	—
278	#107	—	2	2	no	—
279	#106	—	2	2	no	—
280	#104	—	4	4	no	—
281	#102	—	2	8	no	—
282	#101	—	2	1	no	—
283	#100	—	2	1	no	—
284	#99	—	2	4	no	—
285	#99	—	2	8	no	—
286	#99	D₄×C₅₃	1	4	yes	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	C₂⁴	—	2	2	no	—
290	C₂⁴	—	2	4	no	—
291	#41	—	2	2	no	—
292	#41	—	2	2	no	—
293	#41	—	2	8	no	—
294	#41	—	2	4	no	—
295	#41	—	2	4	no	—
296	#41	—	4	4	no	—
297	#41	—	4	8	no	—
298	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
299	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂×D₄	—	4	8	no	—
308	C₂×D₄	—	4	16	no	—
309	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
310	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
311	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
312	C₂×D₄	C₂×D₄×C₅₃	1	8	yes	—
313	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	8	no	—
317	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
319	Q₁₆	—	2	1	no	—
320	Q₁₆	—	2	1	no	—
321	SD₁₆	—	2	8	no	—
322	SD₁₆	—	2	4	no	—
323	D₈	—	2	8	no	—
324	D₈	—	2	1	no	—
325	D₈	—	2	1	no	—
326	D₈	C₂²⋊C₄×C₅₃	1	4	yes	—
327	D₈	C₂²⋊C₄×C₅₃	1	4	yes	—
328	D₈	C₂²⋊C₄×C₅₃	1	8	yes	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	2	4	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	4	8	no	—
345	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	4	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	16	2	no	—
353	C₂³	—	16	4	no	—
354	C₂³	C₄×D₄×C₅₃	1	2	yes	—
355	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	Q₈	—	2	2	no	—
357	Q₈	—	2	2	no	—
358	Q₈	—	2	2	no	—
359	D₄	—	2	4	no	—
360	D₄	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	D₄	—	2	8	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	8	no	—
367	D₄	—	4	4	no	—
368	D₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	16	8	no	—
377	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
378	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
380	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
381	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
385	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	8	8	no	—
393	C₂²	—	8	4	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	16	8	no	—
396	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
397	C₂²	D₄²×C₅₃	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	2	no	—
401	C₄	—	8	4	no	—
402	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
403	C₄	C₅₃×#518	1	1	yes	characteristic
404	C₄	C₅₃×#546	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	1	no	—
407	C₂	—	4	2	no	—
408	C₂	—	4	8	no	—
409	C₂	—	8	2	no	—
410	C₂	D₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#576	C₂
LCS	G ▷ C₂×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
2. C₅₃×#2948	C₂	semidirect
3. C₅₃×#2883	C₂	semidirect
4. C₅₃×#1976	C₂	semidirect
5. C₅₃×#1904	C₂	semidirect
6. C₅₃×#1812	C₂	semidirect
7. C₅₃×#1795	C₂	semidirect
8. C₅₃×#1757	C₂	semidirect
12. C₂×AGL(1,ℤ/8)×C₅₃	C₄	semidirect
14. C₂×#135×C₅₃	C₄	semidirect
16. C₂²×D₈×C₅₃	C₄	semidirect
19. C₅₃×#518	C₂²	semidirect
22. C₅₃×#427	C₂²	semidirect
23. C₅₃×#421	C₂²	semidirect
24. C₅₃×#418	C₂²	semidirect
28. C₅₃×#410	C₂²	semidirect
30. C₅₃×#405	C₂²	semidirect
31. C₅₃×#344	C₂²	semidirect
32. C₅₃×H(3,ℤ/4)	C₂²	semidirect
36. C₂×#41×C₅₃	C₂×C₄	semidirect
41. C₂²×D₄×C₅₃	C₂×C₄	semidirect
49. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
50. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
51. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
56. C₂×D₈×C₅₃	C₂×C₄	semidirect
57. C₂×D₈×C₅₃	C₂×C₄	semidirect
62. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
65. C₂²×C₈×C₅₃	C₂×C₄	semidirect
181. #22455	C₅₃	direct (N × H)
201. #2948	C₂×C₅₃	semidirect
202. #2883	C₂×C₅₃	semidirect
203. #1976	C₂×C₅₃	semidirect
204. #1904	C₂×C₅₃	semidirect
205. #1812	C₂×C₅₃	semidirect
206. #1795	C₂×C₅₃	semidirect
207. #1757	C₂×C₅₃	semidirect
217. C₂×AGL(1,ℤ/8)	C₄×C₅₃	semidirect
219. C₂×#135	C₄×C₅₃	semidirect
221. C₂²×D₈	C₄×C₅₃	semidirect
224. #518	C₂²×C₅₃	semidirect
227. #427	C₂²×C₅₃	semidirect
228. #421	C₂²×C₅₃	semidirect
229. #418	C₂²×C₅₃	semidirect
233. #410	C₂²×C₅₃	semidirect
235. #405	C₂²×C₅₃	semidirect
236. #344	C₂²×C₅₃	semidirect
237. H(3,ℤ/4)	C₂²×C₅₃	semidirect
238. C₅₃	#22455	direct (N × H)
242. C₂×#41	C₂×C₄×C₅₃	semidirect
247. C₂²×D₄	C₂×C₄×C₅₃	semidirect
255. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
256. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
257. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
262. C₂×D₈	C₂×C₄×C₅₃	semidirect
263. C₂×D₈	C₂×C₄×C₅₃	semidirect
268. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
271. C₂²×C₈	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
C₂×C₅₃×#576	2
C₅₃×#2948	2
C₅₃×#2883	2
C₅₃×#1976	2
C₅₃×#1904	2
C₅₃×#1812	2
C₅₃×#1795	2
C₅₃×#1757	2
#22455	53

Subgroups — C53×#22455 #27229967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22455 #27229967