Subgroups — C₅₃×#17817 #27225329

Order 13568.

684 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	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1574	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1573	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1567	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1566	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₅₃	—	2	1	no	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#432	—	2	1	no	—
11	C₅₃×#432	C₂²	1	2	yes	—
12	C₅₃×#431	—	2	1	no	—
13	C₅₃×#431	C₄	1	2	yes	—
14	C₅₃×#431	C₄	1	2	yes	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#431	C₂²	1	2	yes	—
18	C₅₃×#423	—	2	1	no	—
19	C₅₃×#423	—	2	1	no	—
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	2	yes	—
22	C₅₃×#423	C₂²	1	1	yes	characteristic
23	C₅₃×#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
25	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#407	—	2	1	no	—
28	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
29	C₄×M₁₆×C₅₃	—	2	1	no	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#335	C₂²	1	8	yes	complemented
32	C₅₃×#334	C₂²	1	8	yes	complemented
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₂×M₁₆×C₅₃	—	4	4	no	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	2	no	—
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
61	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
62	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
64	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
65	C₂²⋊C₈×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	—	2	2	no	—
67	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	#97×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
72	C₄×C₈×C₅₃	C₂³	1	4	yes	—
73	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	M₁₆×C₅₃	—	4	4	no	—
97	M₁₆×C₅₃	—	4	2	no	—
98	M₁₆×C₅₃	—	4	4	no	—
99	M₁₆×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	4	no	—
103	C₂×C₈×C₅₃	—	4	4	no	—
104	C₂×C₈×C₅₃	—	4	4	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	#108	1	4	yes	—
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄×C₅₃	#137	1	2	yes	—
158	C₂×C₄×C₅₃	C₂³×C₄	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	C₈×C₅₃	—	2	8	no	—
162	C₈×C₅₃	—	4	4	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	—	4	2	no	—
165	#17817	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	4	4	no	—
174	C₂²×C₅₃	#428	1	2	yes	—
175	C₂²×C₅₃	C₂×#108	1	2	yes	—
176	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	#428	1	2	yes	—
185	C₄×C₅₃	C₂×#108	1	2	yes	—
186	C₄×C₅₃	C₂×#108	1	2	yes	—
187	C₄×C₅₃	#568	1	2	yes	—
188	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
190	#2920	C₂×C₅₃	1	1	yes	characteristic
191	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
192	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
193	#1574	C₂×C₅₃	1	4	yes	complemented
194	#1573	C₂×C₅₃	1	4	yes	complemented
195	#1567	C₂×C₅₃	1	2	yes	complemented
196	#1566	C₂×C₅₃	1	2	yes	complemented
197	C₂×C₅₃	—	2	2	no	—
198	C₂×C₅₃	—	2	2	no	—
199	C₂×C₅₃	—	4	1	no	—
200	C₂×C₅₃	#2905	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
202	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
203	C₂²×M₁₆	—	2	1	no	—
204	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
205	#432	—	2	1	no	—
206	#432	C₂²×C₅₃	1	2	yes	—
207	#431	—	2	1	no	—
208	#431	C₄×C₅₃	1	2	yes	—
209	#431	C₄×C₅₃	1	2	yes	—
210	#431	C₂²×C₅₃	1	2	yes	—
211	#431	C₂²×C₅₃	1	2	yes	—
212	#431	C₂²×C₅₃	1	2	yes	—
213	#423	—	2	1	no	—
214	#423	—	2	1	no	—
215	#423	C₂²×C₅₃	1	1	yes	characteristic
216	#423	C₂²×C₅₃	1	2	yes	—
217	#423	C₂²×C₅₃	1	1	yes	characteristic
218	#423	C₂²×C₅₃	1	2	yes	—
219	C₂×C₄⋊C₈	—	2	1	no	—
220	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
222	#407	—	2	1	no	—
223	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
224	C₄×M₁₆	—	2	1	no	—
225	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
226	#335	C₂²×C₅₃	1	8	yes	complemented
227	#334	C₂²×C₅₃	1	8	yes	complemented
228	C₅₃	#17817	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
231	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	1	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	4	4	no	—
237	C₂²×C₈	—	2	2	no	—
238	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
239	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
246	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
247	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
257	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
258	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
259	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
260	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₂²⋊C₈	—	2	2	no	—
263	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
264	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
265	#97	—	2	2	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
268	C₄×C₈	C₂³×C₅₃	1	4	yes	—
269	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
292	M₁₆	—	4	4	no	—
293	M₁₆	—	4	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	4	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	4	4	no	—
300	C₂×C₈	—	4	4	no	—
301	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
302	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
303	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
304	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
305	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
306	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
307	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
308	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
309	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
310	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
311	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
314	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
315	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
316	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
317	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	4	1	no	—
321	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
322	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
323	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	#108×C₅₃	1	4	yes	—
343	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
353	C₂×C₄	#137×C₅₃	1	2	yes	—
354	C₂×C₄	C₂³×C₄×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	C₈	—	2	8	no	—
358	C₈	—	4	4	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	4	4	no	—
369	C₂²	C₅₃×#428	1	2	yes	—
370	C₂²	C₂×#108×C₅₃	1	2	yes	—
371	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	C₅₃×#428	1	2	yes	—
380	C₄	C₂×#108×C₅₃	1	2	yes	—
381	C₄	C₂×#108×C₅₃	1	2	yes	—
382	C₄	C₅₃×#568	1	2	yes	—
383	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
384	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
385	C₂	—	2	2	no	—
386	C₂	—	2	2	no	—
387	C₂	—	4	1	no	—
388	C₂	C₅₃×#2905	1	1	yes	characteristic
389	C₂	C₂×C₅₃×#428	1	1	yes	characteristic
390	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1574	C₂	semidirect
5. C₅₃×#1573	C₂	semidirect
6. C₅₃×#1567	C₂	semidirect
7. C₅₃×#1566	C₂	semidirect
31. C₅₃×#335	C₂²	semidirect
32. C₅₃×#334	C₂²	semidirect
165. #17817	C₅₃	direct (N × H)
193. #1574	C₂×C₅₃	semidirect
194. #1573	C₂×C₅₃	semidirect
195. #1567	C₂×C₅₃	semidirect
196. #1566	C₂×C₅₃	semidirect
226. #335	C₂²×C₅₃	semidirect
227. #334	C₂²×C₅₃	semidirect
228. C₅₃	#17817	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×#2920	2
C₂×C₅₃×#431	2
C₂×C₅₃×#423	2
C₅₃×#1574	2
C₅₃×#1573	2
C₅₃×#1567	2
C₅₃×#1566	2
#17817	53

Subgroups — C53×#17817 #27225329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17817 #27225329