Subgroups — C₅₃×#22489 #27230001

Order 13568.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1893	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1875	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1825	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1735	C₂	1	2	yes	maximal, complemented
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#555	—	2	4	no	—
14	C₅₃×#555	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#124×C₅₃	—	2	2	no	—
17	C₂×#121×C₅₃	—	2	1	no	—
18	C₂×#121×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	—	2	1	no	—
22	C₂×#117×C₅₃	C₂²	1	2	yes	—
23	C₂×#117×C₅₃	C₂²	1	2	yes	—
24	C₈×D₄×C₅₃	—	2	4	no	—
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#102×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#390	—	2	1	no	—
32	C₅₃×#388	—	2	2	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#380	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	1	yes	characteristic
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	1	no	—
49	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	—	2	4	no	—
52	C₂²×C₈×C₅₃	—	4	2	no	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
58	#124×C₅₃	—	2	16	no	—
59	#121×C₅₃	—	2	8	no	—
60	#121×C₅₃	—	2	4	no	—
61	#121×C₅₃	C₂×C₄	1	8	yes	—
62	C₄×D₄×C₅₃	—	2	2	no	—
63	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
65	C₄×D₄×C₅₃	D₄	1	4	yes	—
66	#117×C₅₃	—	2	8	no	—
67	#117×C₅₃	—	2	4	no	—
68	#117×C₅₃	—	2	4	no	—
69	#117×C₅₃	—	4	2	no	—
70	#117×C₅₃	C₂×C₄	1	8	yes	—
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
86	C₂×C₄²×C₅₃	—	2	2	no	—
87	C₂×C₄²×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	—	2	1	no	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₅₃	—	2	2	no	—
91	#102×C₅₃	—	2	2	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	4	2	no	—
94	C₂²⋊C₈×C₅₃	—	2	4	no	—
95	C₄×C₈×C₅₃	—	2	2	no	—
96	#95×C₅₃	—	2	2	no	—
97	#95×C₅₃	—	2	1	no	—
98	C₂⁴×C₅₃	—	2	1	no	—
99	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₅₃	—	2	4	no	—
101	C₂×Q₈×C₅₃	—	8	1	no	—
102	C₂×Q₈×C₅₃	—	8	1	no	—
103	C₂×D₄×C₅₃	—	2	8	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	2	1	no	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	8	2	no	—
112	C₂×D₄×C₅₃	—	8	2	no	—
113	C₂×D₄×C₅₃	C₂²⋊C₄	1	4	yes	—
114	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	2	no	—
122	C₂²×C₄×C₅₃	—	2	4	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	4	2	no	—
126	C₂²×C₄×C₅₃	—	4	2	no	—
127	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	4	8	no	—
142	C₂×C₈×C₅₃	#41	1	2	yes	—
143	C₄⋊C₄×C₅₃	—	2	4	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	4	2	no	—
152	C₄⋊C₄×C₅₃	—	8	1	no	—
153	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
154	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
155	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
156	C₄⋊C₄×C₅₃	#41	1	2	yes	—
157	C₂²⋊C₄×C₅₃	—	2	8	no	—
158	C₂²⋊C₄×C₅₃	—	2	8	no	—
159	C₂²⋊C₄×C₅₃	—	2	4	no	—
160	C₂²⋊C₄×C₅₃	—	4	4	no	—
161	C₂²⋊C₄×C₅₃	—	8	2	no	—
162	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
163	C₄²×C₅₃	—	2	8	no	—
164	C₄²×C₅₃	—	4	4	no	—
165	C₄²×C₅₃	—	4	2	no	—
166	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
167	C₄²×C₅₃	C₂×D₄	1	2	yes	—
168	C₂³×C₅₃	—	2	2	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	4	1	no	—
176	C₂³×C₅₃	—	4	1	no	—
177	C₂³×C₅₃	—	4	1	no	—
178	C₂³×C₅₃	—	4	1	no	—
179	C₂³×C₅₃	—	4	1	no	—
180	C₂³×C₅₃	—	4	1	no	—
181	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
182	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₅₃	—	8	4	no	—
184	D₄×C₅₃	—	2	2	no	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	8	no	—
187	D₄×C₅₃	—	4	1	no	—
188	D₄×C₅₃	—	4	1	no	—
189	D₄×C₅₃	—	8	8	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	4	1	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	1	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	—	4	1	no	—
206	C₂×C₄×C₅₃	—	4	2	no	—
207	C₂×C₄×C₅₃	—	4	8	no	—
208	C₂×C₄×C₅₃	—	4	2	no	—
209	C₂×C₄×C₅₃	—	4	1	no	—
210	C₂×C₄×C₅₃	—	4	1	no	—
211	C₂×C₄×C₅₃	—	4	2	no	—
212	C₂×C₄×C₅₃	—	4	1	no	—
213	C₂×C₄×C₅₃	—	4	2	no	—
214	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
215	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₅₃	#135	1	2	yes	—
218	C₂×C₄×C₅₃	#135	1	2	yes	—
219	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	—	4	2	no	—
227	C₈×C₅₃	—	4	1	no	—
228	#22489	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂²×C₅₃	—	2	4	no	—
230	C₂²×C₅₃	—	2	4	no	—
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	4	2	no	—
233	C₂²×C₅₃	—	4	2	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	—	4	2	no	—
236	C₂²×C₅₃	—	4	2	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	2	no	—
239	C₂²×C₅₃	—	8	1	no	—
240	C₂²×C₅₃	—	8	1	no	—
241	C₂²×C₅₃	—	8	1	no	—
242	C₂²×C₅₃	—	8	1	no	—
243	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
244	C₂²×C₅₃	#546	1	1	yes	characteristic
245	C₂²×C₅₃	#547	1	1	yes	characteristic
246	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
247	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
248	C₂²×C₅₃	#576	1	1	yes	characteristic
249	C₂²×C₅₃	#577	1	1	yes	characteristic
250	C₄×C₅₃	—	2	4	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	4	2	no	—
253	C₄×C₅₃	—	4	4	no	—
254	C₄×C₅₃	—	4	2	no	—
255	C₄×C₅₃	—	8	1	no	—
256	C₄×C₅₃	—	8	2	no	—
257	C₄×C₅₃	#416	1	4	yes	—
258	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
259	C₄×C₅₃	#520	1	1	yes	characteristic
260	C₄×C₅₃	#546	1	1	yes	characteristic
261	C₄×C₅₃	#547	1	1	yes	characteristic
262	C₂×#555	C₂×C₅₃	1	1	yes	characteristic
263	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#2354	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#1987	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#1893	C₂×C₅₃	1	2	yes	complemented
268	#1875	C₂×C₅₃	1	2	yes	complemented
269	#1825	C₂×C₅₃	1	2	yes	complemented
270	#1802	C₂×C₅₃	1	1	yes	characteristic, complemented
271	C₄×#102	C₂×C₅₃	1	1	yes	characteristic, complemented
272	#1735	C₂×C₅₃	1	2	yes	complemented
273	C₂×C₅₃	—	4	4	no	—
274	C₂×C₅₃	—	8	2	no	—
275	C₂×C₅₃	#2339	1	1	yes	characteristic
276	C₂×C₅₃	C₂×#416	1	1	yes	characteristic
277	C₂×C₅₃	#2897	1	1	yes	characteristic
278	C₂×C₅₃	#3303	1	1	yes	characteristic
279	C₂×C₅₃	#3305	1	1	yes	characteristic
280	C₂×C₅₃	#3340	1	1	yes	characteristic
281	C₂×C₅₃	#3341	1	1	yes	characteristic
282	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
283	#555	—	2	4	no	—
284	#555	C₄×C₅₃	1	8	yes	complemented
285	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
286	C₂×#124	—	2	2	no	—
287	C₂×#121	—	2	1	no	—
288	C₂×#121	C₂²×C₅₃	1	2	yes	—
289	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
291	C₂×#117	—	2	1	no	—
292	C₂×#117	C₂²×C₅₃	1	2	yes	—
293	C₂×#117	C₂²×C₅₃	1	2	yes	—
294	C₈×D₄	—	2	4	no	—
295	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
296	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
298	C₂×#102	C₂²×C₅₃	1	2	yes	—
299	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
300	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
301	#390	—	2	1	no	—
302	#388	—	2	2	no	—
303	#384	C₂²×C₅₃	1	1	yes	characteristic
304	#380	C₂²×C₅₃	1	2	yes	—
305	C₄×C₄⋊C₄	—	2	1	no	—
306	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
307	#340	C₂²×C₅₃	1	2	yes	—
308	#340	C₂²×C₅₃	1	1	yes	characteristic
309	#340	C₂²×C₅₃	1	2	yes	—
310	#340	C₂²×C₅₃	1	2	yes	—
311	#340	C₂²×C₅₃	1	2	yes	—
312	#340	C₂²×C₅₃	1	2	yes	—
313	#340	C₂²×C₅₃	1	1	yes	characteristic
314	C₅₃	#22489	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
321	C₂²×C₈	—	2	4	no	—
322	C₂²×C₈	—	2	4	no	—
323	C₂²×C₈	—	4	2	no	—
324	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
325	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
329	#124	—	2	16	no	—
330	#121	—	2	8	no	—
331	#121	—	2	4	no	—
332	#121	C₂×C₄×C₅₃	1	8	yes	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
335	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
336	C₄×D₄	D₄×C₅₃	1	4	yes	—
337	#117	—	2	4	no	—
338	#117	—	2	8	no	—
339	#117	—	2	4	no	—
340	#117	—	4	2	no	—
341	#117	C₂×C₄×C₅₃	1	8	yes	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	—	2	1	no	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
357	C₂×C₄²	—	2	2	no	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
361	C₄⋊C₈	—	2	2	no	—
362	#102	—	2	2	no	—
363	#102	—	2	2	no	—
364	#102	—	4	2	no	—
365	C₂²⋊C₈	—	2	4	no	—
366	C₄×C₈	—	2	2	no	—
367	#95	—	2	2	no	—
368	#95	—	2	1	no	—
369	C₂⁴	—	2	1	no	—
370	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	8	1	no	—
373	C₂×Q₈	—	8	1	no	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	8	2	no	—
383	C₂×D₄	—	8	2	no	—
384	C₂×D₄	C₂²⋊C₄×C₅₃	1	4	yes	—
385	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
402	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	4	8	no	—
413	C₂×C₈	#41×C₅₃	1	2	yes	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	—	8	1	no	—
424	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
425	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
426	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
427	C₄⋊C₄	#41×C₅₃	1	2	yes	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	8	2	no	—
433	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
434	C₄²	—	2	8	no	—
435	C₄²	—	4	4	no	—
436	C₄²	—	4	2	no	—
437	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
438	C₄²	C₂×D₄×C₅₃	1	2	yes	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	1	no	—
450	C₂³	—	4	1	no	—
451	C₂³	—	4	1	no	—
452	C₂³	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
453	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
454	Q₈	—	8	4	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	8	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	8	8	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
486	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
487	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
488	C₂×C₄	#135×C₅₃	1	2	yes	—
489	C₂×C₄	#135×C₅₃	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
495	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	4	2	no	—
498	C₈	—	4	1	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	8	1	no	—
510	C₂²	—	8	1	no	—
511	C₂²	—	8	1	no	—
512	C₂²	—	8	1	no	—
513	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
514	C₂²	C₅₃×#546	1	1	yes	characteristic
515	C₂²	C₅₃×#547	1	1	yes	characteristic
516	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
517	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
518	C₂²	C₅₃×#576	1	1	yes	characteristic
519	C₂²	C₅₃×#577	1	1	yes	characteristic
520	C₄	—	2	4	no	—
521	C₄	—	2	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	8	2	no	—
526	C₄	—	8	1	no	—
527	C₄	C₅₃×#416	1	4	yes	—
528	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
529	C₄	C₅₃×#520	1	1	yes	characteristic
530	C₄	C₅₃×#546	1	1	yes	characteristic
531	C₄	C₅₃×#547	1	1	yes	characteristic
532	C₂	—	4	4	no	—
533	C₂	—	8	2	no	—
534	C₂	C₅₃×#2339	1	1	yes	characteristic
535	C₂	C₂×C₅₃×#416	1	1	yes	characteristic
536	C₂	C₅₃×#2897	1	1	yes	characteristic
537	C₂	C₅₃×#3303	1	1	yes	characteristic
538	C₂	C₅₃×#3305	1	1	yes	characteristic
539	C₂	C₅₃×#3340	1	1	yes	characteristic
540	C₂	C₅₃×#3341	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₅₃	C₂	semidirect
3. C₅₃×#2354	C₂	semidirect
4. C₅₃×#1990	C₂	semidirect
5. C₅₃×#1987	C₂	semidirect
6. C₅₃×#1893	C₂	semidirect
7. C₅₃×#1875	C₂	semidirect
8. C₅₃×#1825	C₂	semidirect
9. C₅₃×#1802	C₂	semidirect
10. C₄×#102×C₅₃	C₂	semidirect
11. C₅₃×#1735	C₂	semidirect
14. C₅₃×#555	C₄	semidirect
228. #22489	C₅₃	direct (N × H)
263. C₂×C₈×D₄	C₂×C₅₃	semidirect
264. #2354	C₂×C₅₃	semidirect
265. #1990	C₂×C₅₃	semidirect
266. #1987	C₂×C₅₃	semidirect
267. #1893	C₂×C₅₃	semidirect
268. #1875	C₂×C₅₃	semidirect
269. #1825	C₂×C₅₃	semidirect
270. #1802	C₂×C₅₃	semidirect
271. C₄×#102	C₂×C₅₃	semidirect
272. #1735	C₂×C₅₃	semidirect
284. #555	C₄×C₅₃	semidirect
314. C₅₃	#22489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#555	2
C₂×C₈×D₄×C₅₃	2
C₅₃×#2354	2
C₅₃×#1990	2
C₅₃×#1987	2
C₅₃×#1893	2
C₅₃×#1875	2
C₅₃×#1825	2
C₅₃×#1802	2
C₄×#102×C₅₃	2
C₅₃×#1735	2
#22489	53

Subgroups — C53×#22489 #27230001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22489 #27230001