Subgroups — C₅₃×#26321 #27233833

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2548	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2092	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2073	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1982	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1926	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
11	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#124×C₅₃	—	2	1	no	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#394	—	2	2	no	—
25	C₅₃×#391	—	2	1	no	—
26	C₅₃×#390	—	2	1	no	—
27	C₅₃×#390	C₂²	1	2	yes	—
28	C₅₃×#390	C₂²	1	2	yes	—
29	C₅₃×#387	—	2	2	no	—
30	C₅₃×#384	C₂²	1	1	yes	characteristic
31	C₅₃×#382	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	—	2	1	no	—
44	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#127×C₅₃	—	2	2	no	—
49	#127×C₅₃	—	2	2	no	—
50	#124×C₅₃	—	4	4	no	—
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	—	2	1	no	—
67	C₂×C₄²×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	—	2	2	no	—
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	—	4	2	no	—
72	#106×C₅₃	—	2	2	no	—
73	#106×C₅₃	—	2	2	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	4	no	—
76	#102×C₅₃	—	2	4	no	—
77	#102×C₅₃	—	4	2	no	—
78	#102×C₅₃	—	4	2	no	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	1	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	—	2	2	no	—
85	C₂⁴×C₅₃	—	2	2	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	4	2	no	—
96	C₂×D₄×C₅₃	—	4	2	no	—
97	C₂×D₄×C₅₃	—	4	2	no	—
98	C₂×D₄×C₅₃	—	4	2	no	—
99	C₂×D₄×C₅₃	—	8	1	no	—
100	C₂×D₄×C₅₃	—	8	1	no	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	#41	1	2	yes	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	4	2	no	—
132	C₄⋊C₄×C₅₃	—	8	1	no	—
133	C₄⋊C₄×C₅₃	#41	1	2	yes	—
134	C₂²⋊C₄×C₅₃	—	4	4	no	—
135	C₂²⋊C₄×C₅₃	—	4	4	no	—
136	C₂²⋊C₄×C₅₃	—	8	2	no	—
137	C₂²⋊C₄×C₅₃	—	8	2	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	—	4	2	no	—
141	C₄²×C₅₃	—	4	2	no	—
142	C₄²×C₅₃	—	8	2	no	—
143	C₄²×C₅₃	#41	1	2	yes	—
144	C₄²×C₅₃	#41	1	2	yes	—
145	C₂³×C₅₃	—	4	2	no	—
146	C₂³×C₅₃	—	4	2	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	2	no	—
149	C₂³×C₅₃	—	4	2	no	—
150	C₂³×C₅₃	—	4	2	no	—
151	C₂³×C₅₃	—	4	2	no	—
152	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₅₃	—	4	4	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	4	2	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	4	2	no	—
158	D₄×C₅₃	—	8	4	no	—
159	D₄×C₅₃	—	8	4	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
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	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	4	1	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	#123	1	2	yes	—
184	C₂×C₄×C₅₃	#123	1	2	yes	—
185	C₂×C₄×C₅₃	#135	1	2	yes	—
186	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	4	1	no	—
195	C₈×C₅₃	—	4	2	no	—
196	#26321	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂²×C₅₃	—	8	2	no	—
198	C₂²×C₅₃	—	8	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	8	2	no	—
201	C₂²×C₅₃	—	8	2	no	—
202	C₂²×C₅₃	—	8	2	no	—
203	C₂²×C₅₃	—	8	2	no	—
204	C₂²×C₅₃	C₂×#123	1	1	yes	characteristic
205	C₂²×C₅₃	#548	1	1	yes	characteristic
206	C₂²×C₅₃	#550	1	1	yes	characteristic
207	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
208	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂²×C₅₃	#576	1	1	yes	characteristic
210	C₂²×C₅₃	#577	1	1	yes	characteristic
211	C₄×C₅₃	—	2	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	2	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	8	1	no	—
216	C₄×C₅₃	—	8	1	no	—
217	C₄×C₅₃	—	8	1	no	—
218	C₄×C₅₃	—	8	2	no	—
219	C₄×C₅₃	#482	1	4	yes	—
220	C₄×C₅₃	#529	1	1	yes	characteristic
221	C₄×C₅₃	#542	1	1	yes	characteristic
222	C₄×C₅₃	#556	1	1	yes	characteristic
223	C₄×C₅₃	#559	1	1	yes	characteristic
224	#2548	C₂×C₅₃	1	1	yes	characteristic
225	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#2092	C₂×C₅₃	1	2	yes	complemented
227	#2073	C₂×C₅₃	1	2	yes	complemented
228	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1982	C₂×C₅₃	1	1	yes	characteristic
230	#1957	C₂×C₅₃	1	1	yes	characteristic
231	#1947	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1926	C₂×C₅₃	1	2	yes	complemented
233	#1925	C₂×C₅₃	1	2	yes	complemented
234	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₂×C₅₃	—	16	2	no	—
236	C₂×C₅₃	#2448	1	1	yes	characteristic
237	C₂×C₅₃	C₂×#482	1	1	yes	characteristic
238	C₂×C₅₃	#3096	1	1	yes	characteristic
239	C₂×C₅₃	#3361	1	1	yes	characteristic
240	C₂×C₅₃	#3368	1	1	yes	characteristic
241	C₂×C₅₃	#3394	1	1	yes	characteristic
242	C₂×C₅₃	#3399	1	1	yes	characteristic
243	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#124	—	2	1	no	—
245	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#107	C₂²×C₅₃	1	2	yes	—
247	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×#102	C₂²×C₅₃	1	2	yes	—
251	C₂×#102	C₂²×C₅₃	1	2	yes	—
252	C₂×#102	C₂²×C₅₃	1	2	yes	—
253	C₂×#102	C₂²×C₅₃	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
255	#394	—	2	2	no	—
256	#391	—	2	1	no	—
257	#390	—	2	1	no	—
258	#390	C₂²×C₅₃	1	2	yes	—
259	#390	C₂²×C₅₃	1	2	yes	—
260	#387	—	2	2	no	—
261	#384	C₂²×C₅₃	1	1	yes	characteristic
262	#382	C₂²×C₅₃	1	2	yes	—
263	C₄×C₄⋊C₄	—	2	1	no	—
264	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
265	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
267	#340	C₂²×C₅₃	1	1	yes	characteristic
268	#340	C₂²×C₅₃	1	2	yes	—
269	#340	C₂²×C₅₃	1	1	yes	characteristic
270	#336	C₂²×C₅₃	1	2	yes	—
271	#336	C₂²×C₅₃	1	2	yes	—
272	C₅₃	#26321	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
277	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
279	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
280	#127	—	2	2	no	—
281	#127	—	2	2	no	—
282	#124	—	4	4	no	—
283	C₄×Q₈	—	2	2	no	—
284	C₄×Q₈	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₄²	—	2	1	no	—
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
302	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
303	#107	—	4	2	no	—
304	#106	—	2	2	no	—
305	#106	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#102	—	4	2	no	—
310	#102	—	4	2	no	—
311	C₄×C₈	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	2	no	—
315	#95	—	2	1	no	—
316	#95	—	2	2	no	—
317	C₂⁴	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	4	2	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	2	no	—
331	C₂×D₄	—	8	1	no	—
332	C₂×D₄	—	8	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	#41×C₅₃	1	2	yes	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	—	8	1	no	—
365	C₄⋊C₄	#41×C₅₃	1	2	yes	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	—	4	4	no	—
368	C₂²⋊C₄	—	8	2	no	—
369	C₂²⋊C₄	—	8	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	2	no	—
373	C₄²	—	4	2	no	—
374	C₄²	—	8	2	no	—
375	C₄²	#41×C₅₃	1	2	yes	—
376	C₄²	#41×C₅₃	1	2	yes	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
385	Q₈	—	4	4	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	8	4	no	—
391	D₄	—	8	4	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	#123×C₅₃	1	2	yes	—
416	C₂×C₄	#123×C₅₃	1	2	yes	—
417	C₂×C₄	#135×C₅₃	1	2	yes	—
418	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
419	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
423	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
424	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
425	C₈	—	2	2	no	—
426	C₈	—	4	1	no	—
427	C₈	—	4	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	C₂×#123×C₅₃	1	1	yes	characteristic
436	C₂²	C₅₃×#548	1	1	yes	characteristic
437	C₂²	C₅₃×#550	1	1	yes	characteristic
438	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
439	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
440	C₂²	C₅₃×#576	1	1	yes	characteristic
441	C₂²	C₅₃×#577	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	8	1	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	1	no	—
449	C₄	—	8	2	no	—
450	C₄	C₅₃×#482	1	4	yes	—
451	C₄	C₅₃×#529	1	1	yes	characteristic
452	C₄	C₅₃×#542	1	1	yes	characteristic
453	C₄	C₅₃×#556	1	1	yes	characteristic
454	C₄	C₅₃×#559	1	1	yes	characteristic
455	C₂	—	16	2	no	—
456	C₂	C₅₃×#2448	1	1	yes	characteristic
457	C₂	C₂×C₅₃×#482	1	1	yes	characteristic
458	C₂	C₅₃×#3096	1	1	yes	characteristic
459	C₂	C₅₃×#3361	1	1	yes	characteristic
460	C₂	C₅₃×#3368	1	1	yes	characteristic
461	C₂	C₅₃×#3394	1	1	yes	characteristic
462	C₂	C₅₃×#3399	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2355	C₂	semidirect
3. C₅₃×#2092	C₂	semidirect
4. C₅₃×#2073	C₂	semidirect
5. C₅₃×#1984	C₂	semidirect
8. C₅₃×#1947	C₂	semidirect
9. C₅₃×#1926	C₂	semidirect
10. C₅₃×#1925	C₂	semidirect
11. C₄×#106×C₅₃	C₂	semidirect
196. #26321	C₅₃	direct (N × H)
225. #2355	C₂×C₅₃	semidirect
226. #2092	C₂×C₅₃	semidirect
227. #2073	C₂×C₅₃	semidirect
228. #1984	C₂×C₅₃	semidirect
231. #1947	C₂×C₅₃	semidirect
232. #1926	C₂×C₅₃	semidirect
233. #1925	C₂×C₅₃	semidirect
234. C₄×#106	C₂×C₅₃	semidirect
272. C₅₃	#26321	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×#2548	2
C₅₃×#2355	2
C₅₃×#2092	2
C₅₃×#2073	2
C₅₃×#1984	2
C₅₃×#1982	2
C₅₃×#1957	2
C₅₃×#1947	2
C₅₃×#1926	2
C₅₃×#1925	2
C₄×#106×C₅₃	2
#26321	53

Subgroups — C53×#26321 #27233833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26321 #27233833