Subgroups — C₅₃×#22686 #27230198

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#444	C₂	1	2	yes	maximal
2	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2906	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1946	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1866	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1806	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#131×C₅₃	C₂²	1	4	yes	—
11	C₄×#41×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#444	—	2	8	no	—
13	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
14	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₄×SD₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₄×SD₁₆×C₅₃	C₄	1	2	yes	complemented
17	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×#108×C₅₃	C₂²	1	2	yes	—
20	C₂×#108×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#427	C₂²	1	1	yes	characteristic
22	C₅₃×#424	—	2	4	no	—
23	C₅₃×#424	C₂²	1	4	yes	—
24	C₅₃×#423	—	2	2	no	—
25	C₅₃×#423	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#104×C₅₃	C₂²	1	4	yes	—
29	C₅₃×#416	C₂²	1	2	yes	—
30	C₂×#97×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#339	C₂²	1	4	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₂×#41×C₅₃	C₂³	1	2	yes	—
36	#135×C₅₃	D₄	1	4	yes	—
37	#135×C₅₃	Q₈	1	4	yes	—
38	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#131×C₅₃	—	2	8	no	—
42	#131×C₅₃	—	4	4	no	—
43	C₂×M₁₆×C₅₃	—	2	4	no	—
44	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
45	C₂²×C₈×C₅₃	—	2	4	no	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₅₃	—	2	2	no	—
52	C₄×D₄×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
55	#117×C₅₃	—	2	2	no	—
56	#117×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	—	2	2	no	—
59	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#108×C₅₃	—	2	4	no	—
62	#108×C₅₃	—	4	2	no	—
63	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#106×C₅₃	C₂×C₄	1	2	yes	—
66	C₄⋊C₈×C₅₃	—	2	4	no	—
67	C₄⋊C₈×C₅₃	—	2	4	no	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
71	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
72	#104×C₅₃	—	2	8	no	—
73	#103×C₅₃	C₂×C₄	1	2	yes	—
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	#102×C₅₃	C₂×C₄	1	2	yes	—
76	#102×C₅₃	C₂×C₄	1	2	yes	—
77	#97×C₅₃	—	2	4	no	—
78	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	#95×C₅₃	—	2	1	no	—
83	#41×C₅₃	—	2	2	no	—
84	#41×C₅₃	—	2	2	no	—
85	#41×C₅₃	C₂×D₄	1	4	yes	—
86	#41×C₅₃	C₂×Q₈	1	4	yes	—
87	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
98	SD₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
99	D₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
100	M₁₆×C₅₃	—	2	4	no	—
101	M₁₆×C₅₃	—	2	4	no	—
102	M₁₆×C₅₃	—	4	8	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	4	no	—
108	C₂×C₈×C₅₃	—	4	8	no	—
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
114	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	4	2	no	—
119	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂²⋊C₄×C₅₃	—	2	2	no	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
137	D₄×C₅₃	—	2	4	no	—
138	D₄×C₅₃	—	2	4	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	8	2	no	—
153	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	4	no	—
166	C₈×C₅₃	—	4	4	no	—
167	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
168	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
169	#22686	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	2	4	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₅₃	#547	1	1	yes	characteristic
177	C₂²×C₅₃	#554	1	1	yes	characteristic
178	C₄×C₅₃	—	2	4	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	4	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	—	8	2	no	—
185	C₄×C₅₃	#444	1	2	yes	complemented
186	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅₃	#520	1	1	yes	characteristic
188	C₄×C₅₃	D₄²	1	1	yes	characteristic
189	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
190	C₂×#444	C₂×C₅₃	1	2	yes	—
191	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
192	#2906	C₂×C₅₃	1	2	yes	complemented
193	#1946	C₂×C₅₃	1	2	yes	complemented
194	#1866	C₂×C₅₃	1	4	yes	complemented
195	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#1806	C₂×C₅₃	1	2	yes	complemented
197	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₂×C₅₃	—	2	1	no	—
199	C₂×C₅₃	—	2	1	no	—
200	C₂×C₅₃	—	4	4	no	—
201	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
203	C₂×C₅₃	#2957	1	1	yes	characteristic
204	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
205	C₂×#131	C₂²×C₅₃	1	4	yes	—
206	C₄×#41	C₂²×C₅₃	1	2	yes	—
207	#444	—	2	8	no	—
208	C₄×Q₁₆	C₄×C₅₃	1	1	yes	characteristic, complemented
209	C₄×Q₁₆	C₄×C₅₃	1	1	yes	characteristic, complemented
210	C₄×SD₁₆	C₄×C₅₃	1	2	yes	complemented
211	C₄×SD₁₆	C₄×C₅₃	1	2	yes	complemented
212	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic, complemented
213	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic, complemented
214	C₂×#108	C₂²×C₅₃	1	2	yes	—
215	C₂×#108	C₂²×C₅₃	1	2	yes	—
216	#427	C₂²×C₅₃	1	1	yes	characteristic
217	#424	—	2	4	no	—
218	#424	C₂²×C₅₃	1	4	yes	—
219	#423	—	2	2	no	—
220	#423	C₂²×C₅₃	1	2	yes	—
221	C₂×C₄⋊C₈	—	2	2	no	—
222	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×#104	C₂²×C₅₃	1	4	yes	—
224	#416	C₂²×C₅₃	1	2	yes	—
225	C₂×#97	C₂²×C₅₃	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
228	#339	C₂²×C₅₃	1	4	yes	—
229	#336	C₂²×C₅₃	1	2	yes	—
230	C₅₃	#22686	1	1	yes	characteristic, direct product factor, Sylow
231	C₂×#41	C₂³×C₅₃	1	2	yes	—
232	#135	D₄×C₅₃	1	4	yes	—
233	#135	Q₈×C₅₃	1	4	yes	—
234	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
236	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
237	#131	—	2	8	no	—
238	#131	—	4	4	no	—
239	C₂×M₁₆	—	2	4	no	—
240	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
241	C₂²×C₈	—	2	4	no	—
242	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
244	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
245	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
246	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
247	C₄×D₄	—	2	2	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
250	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
251	#117	—	2	2	no	—
252	#117	C₂³×C₅₃	1	2	yes	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
256	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
257	#108	—	2	4	no	—
258	#108	—	4	2	no	—
259	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
260	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
261	#106	C₂×C₄×C₅₃	1	2	yes	—
262	C₄⋊C₈	—	2	4	no	—
263	C₄⋊C₈	—	2	4	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
267	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
268	#104	—	2	8	no	—
269	#103	C₂×C₄×C₅₃	1	2	yes	—
270	#103	C₂×C₄×C₅₃	1	2	yes	—
271	#102	C₂×C₄×C₅₃	1	2	yes	—
272	#102	C₂×C₄×C₅₃	1	2	yes	—
273	#97	—	2	4	no	—
274	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
278	#95	—	2	1	no	—
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	C₂×D₄×C₅₃	1	4	yes	—
282	#41	C₂×Q₈×C₅₃	1	4	yes	—
283	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
293	Q₁₆	C₄⋊C₄×C₅₃	1	4	yes	complemented
294	SD₁₆	C₄⋊C₄×C₅₃	1	8	yes	complemented
295	D₈	C₄⋊C₄×C₅₃	1	4	yes	complemented
296	M₁₆	—	2	4	no	—
297	M₁₆	—	2	4	no	—
298	M₁₆	—	4	8	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	4	8	no	—
305	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
308	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
310	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
311	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
316	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
326	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
331	Q₈	—	2	2	no	—
332	Q₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
333	D₄	—	2	4	no	—
334	D₄	—	2	4	no	—
335	D₄	—	2	2	no	—
336	D₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
357	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
358	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	4	no	—
362	C₈	—	4	4	no	—
363	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
364	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	4	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
371	C₂²	C₅₃×#547	1	1	yes	characteristic
372	C₂²	C₅₃×#554	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	8	2	no	—
380	C₄	C₅₃×#444	1	2	yes	complemented
381	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
382	C₄	C₅₃×#520	1	1	yes	characteristic
383	C₄	D₄²×C₅₃	1	1	yes	characteristic
384	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	4	4	no	—
388	C₂	D₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
389	C₂	C₂×C₅₃×#444	1	1	yes	characteristic
390	C₂	C₅₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2906	C₂	semidirect
4. C₅₃×#1946	C₂	semidirect
5. C₅₃×#1866	C₂	semidirect
6. C₅₃×#1848	C₂	semidirect
7. C₅₃×#1806	C₂	semidirect
8. C₅₃×#1770	C₂	semidirect
13. C₄×Q₁₆×C₅₃	C₄	semidirect
14. C₄×Q₁₆×C₅₃	C₄	semidirect
15. C₄×SD₁₆×C₅₃	C₄	semidirect
16. C₄×SD₁₆×C₅₃	C₄	semidirect
17. C₄×D₈×C₅₃	C₄	semidirect
18. C₄×D₈×C₅₃	C₄	semidirect
97. Q₁₆×C₅₃	C₄⋊C₄	semidirect
98. SD₁₆×C₅₃	C₄⋊C₄	semidirect
99. D₈×C₅₃	C₄⋊C₄	semidirect
169. #22686	C₅₃	direct (N × H)
185. C₄×C₅₃	#444	semidirect
192. #2906	C₂×C₅₃	semidirect
193. #1946	C₂×C₅₃	semidirect
194. #1866	C₂×C₅₃	semidirect
195. #1848	C₂×C₅₃	semidirect
196. #1806	C₂×C₅₃	semidirect
197. #1770	C₂×C₅₃	semidirect
208. C₄×Q₁₆	C₄×C₅₃	semidirect
209. C₄×Q₁₆	C₄×C₅₃	semidirect
210. C₄×SD₁₆	C₄×C₅₃	semidirect
211. C₄×SD₁₆	C₄×C₅₃	semidirect
212. C₄×D₈	C₄×C₅₃	semidirect
213. C₄×D₈	C₄×C₅₃	semidirect
230. C₅₃	#22686	direct (N × H)
293. Q₁₆	C₄⋊C₄×C₅₃	semidirect
294. SD₁₆	C₄⋊C₄×C₅₃	semidirect
295. D₈	C₄⋊C₄×C₅₃	semidirect
380. C₄	C₅₃×#444	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#444	2
C₄×#135×C₅₃	2
C₅₃×#2906	2
C₅₃×#1946	2
C₅₃×#1866	2
C₅₃×#1848	2
C₅₃×#1806	2
C₅₃×#1770	2
#22686	53

Subgroups — C53×#22686 #27230198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22686 #27230198