Subgroups — C₅₃×#3515 #20272452

Order 6784.

1014 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#553	C₂	1	2	yes	maximal, complemented
5	C₅₃×#552	C₂	1	2	yes	maximal, complemented
6	C₅₃×#552	C₂	1	4	yes	maximal, complemented
7	C₅₃×#552	C₂	1	2	yes	maximal, complemented
8	C₅₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#547	C₂	1	2	yes	maximal, complemented
10	C₅₃×#541	C₂	1	2	yes	maximal, complemented
11	C₅₃×#539	C₂	1	4	yes	maximal, complemented
12	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#529	C₂	1	2	yes	maximal, complemented
14	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#125×C₅₃	C₂	1	2	yes	maximal, complemented
16	C₅₃×#520	C₂	1	2	yes	maximal, complemented
17	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#41×C₅₃	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
23	#126×C₅₃	C₂²	1	4	yes	—
24	#126×C₅₃	C₂²	1	2	yes	—
25	#126×C₅₃	C₂²	1	4	yes	—
26	#126×C₅₃	C₂²	1	4	yes	—
27	#125×C₅₃	C₂²	1	1	yes	characteristic
28	#125×C₅₃	C₂²	1	2	yes	—
29	#125×C₅₃	C₂²	1	4	yes	—
30	#125×C₅₃	C₂²	1	4	yes	—
31	#125×C₅₃	C₂²	1	2	yes	—
32	#125×C₅₃	C₂²	1	2	yes	—
33	#125×C₅₃	C₂²	1	4	yes	—
34	#125×C₅₃	C₂²	1	4	yes	—
35	#125×C₅₃	C₂²	1	2	yes	—
36	#124×C₅₃	C₂²	1	1	yes	characteristic
37	#123×C₅₃	C₂²	1	2	yes	—
38	#123×C₅₃	C₂²	1	4	yes	—
39	#123×C₅₃	C₂²	1	4	yes	—
40	#123×C₅₃	C₂²	1	4	yes	—
41	#123×C₅₃	C₂²	1	4	yes	—
42	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
44	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
45	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
46	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
47	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
48	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
49	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
50	#121×C₅₃	C₂²	1	2	yes	—
51	#121×C₅₃	C₂²	1	1	yes	characteristic
52	#121×C₅₃	C₂²	1	4	yes	—
53	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
56	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
57	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
58	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
60	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₅₃	C₂²	1	2	yes	—
62	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
63	C₄×D₄×C₅₃	C₂²	1	2	yes	—
64	C₄×D₄×C₅₃	C₂²	1	4	yes	—
65	C₄×D₄×C₅₃	C₂²	1	2	yes	—
66	C₄×D₄×C₅₃	C₂²	1	4	yes	—
67	C₄×D₄×C₅₃	C₂²	1	2	yes	—
68	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
69	C₄×D₄×C₅₃	C₂²	1	2	yes	—
70	#117×C₅₃	C₂²	1	2	yes	—
71	#117×C₅₃	C₂²	1	1	yes	characteristic
72	#117×C₅₃	C₂²	1	2	yes	—
73	#117×C₅₃	C₂²	1	4	yes	—
74	#117×C₅₃	C₂²	1	4	yes	—
75	#117×C₅₃	C₂²	1	2	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
81	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
83	#41×C₅₃	—	2	4	no	—
84	#41×C₅₃	—	2	8	no	—
85	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
88	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
89	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₅₃	C₂³	1	2	yes	—
91	C₂×D₄×C₅₃	C₂³	1	4	yes	—
92	C₂×D₄×C₅₃	C₂³	1	2	yes	—
93	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
99	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
103	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
104	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
106	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
117	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
122	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
124	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
137	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
139	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
140	C₄²×C₅₃	C₂³	1	1	yes	characteristic
141	C₄²×C₅₃	C₂³	1	2	yes	—
142	C₄²×C₅₃	C₂³	1	1	yes	characteristic
143	C₄²×C₅₃	C₂³	1	1	yes	characteristic
144	C₄²×C₅₃	C₂³	1	2	yes	—
145	C₄²×C₅₃	C₂³	1	1	yes	characteristic
146	C₄²×C₅₃	C₂³	1	2	yes	—
147	C₄²×C₅₃	C₂³	1	4	yes	—
148	C₄²×C₅₃	C₂³	1	4	yes	—
149	C₄²×C₅₃	C₂³	1	2	yes	—
150	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂³×C₅₃	C₂⁴	1	2	yes	—
152	C₂³×C₅₃	C₂⁴	1	2	yes	—
153	Q₈×C₅₃	—	2	2	no	—
154	Q₈×C₅₃	—	2	4	no	—
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	2	4	no	—
158	D₄×C₅₃	—	2	8	no	—
159	D₄×C₅₃	—	2	2	no	—
160	D₄×C₅₃	—	4	2	no	—
161	D₄×C₅₃	#41	1	4	yes	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	8	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	4	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	4	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	4	no	—
172	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
176	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
177	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
178	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
179	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
180	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
184	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂²×C₅₃	—	2	2	no	—
187	C₂²×C₅₃	—	2	2	no	—
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	2	2	no	—
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	2	no	—
192	C₂²×C₅₃	—	2	2	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	C₂×#41	1	4	yes	—
195	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	2	no	—
198	C₄×C₅₃	—	2	2	no	—
199	C₄×C₅₃	—	2	4	no	—
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	2	2	no	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	2	no	—
205	C₄×C₅₃	—	2	2	no	—
206	C₄×C₅₃	—	2	4	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	C₂×#41	1	2	yes	—
209	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
210	#3515	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₅₃	—	2	4	no	—
212	C₂×C₅₃	—	4	1	no	—
213	C₂×C₅₃	—	4	2	no	—
214	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
215	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
216	C₂×C₅₃	#585	1	1	yes	characteristic
217	#557	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#554	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#554	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#553	C₂×C₅₃	1	2	yes	complemented
221	#552	C₂×C₅₃	1	2	yes	complemented
222	#552	C₂×C₅₃	1	4	yes	complemented
223	#552	C₂×C₅₃	1	2	yes	complemented
224	#547	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#547	C₂×C₅₃	1	2	yes	complemented
226	#541	C₂×C₅₃	1	2	yes	complemented
227	#539	C₂×C₅₃	1	4	yes	complemented
228	#533	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#529	C₂×C₅₃	1	2	yes	complemented
230	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
231	C₂×#125	C₂×C₅₃	1	2	yes	complemented
232	#520	C₂×C₅₃	1	2	yes	complemented
233	C₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₅₃	#3515	1	1	yes	characteristic, direct product factor, Sylow
235	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
236	C₂×#41	C₂²×C₅₃	1	2	yes	—
237	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
238	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
240	#126	C₂²×C₅₃	1	4	yes	—
241	#126	C₂²×C₅₃	1	2	yes	—
242	#126	C₂²×C₅₃	1	4	yes	—
243	#126	C₂²×C₅₃	1	4	yes	—
244	#125	C₂²×C₅₃	1	1	yes	characteristic
245	#125	C₂²×C₅₃	1	2	yes	—
246	#125	C₂²×C₅₃	1	4	yes	—
247	#125	C₂²×C₅₃	1	4	yes	—
248	#125	C₂²×C₅₃	1	2	yes	—
249	#125	C₂²×C₅₃	1	2	yes	—
250	#125	C₂²×C₅₃	1	4	yes	—
251	#125	C₂²×C₅₃	1	4	yes	—
252	#125	C₂²×C₅₃	1	2	yes	—
253	#124	C₂²×C₅₃	1	1	yes	characteristic
254	#123	C₂²×C₅₃	1	2	yes	—
255	#123	C₂²×C₅₃	1	4	yes	—
256	#123	C₂²×C₅₃	1	4	yes	—
257	#123	C₂²×C₅₃	1	4	yes	—
258	#123	C₂²×C₅₃	1	4	yes	—
259	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
260	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
261	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
262	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
263	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
264	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
265	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
266	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
267	#121	C₂²×C₅₃	1	2	yes	—
268	#121	C₂²×C₅₃	1	1	yes	characteristic
269	#121	C₂²×C₅₃	1	4	yes	—
270	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
271	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
272	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
273	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
274	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
275	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
276	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
277	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
278	C₄×D₄	C₂²×C₅₃	1	2	yes	—
279	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
280	C₄×D₄	C₂²×C₅₃	1	2	yes	—
281	C₄×D₄	C₂²×C₅₃	1	4	yes	—
282	C₄×D₄	C₂²×C₅₃	1	2	yes	—
283	C₄×D₄	C₂²×C₅₃	1	4	yes	—
284	C₄×D₄	C₂²×C₅₃	1	2	yes	—
285	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
286	C₄×D₄	C₂²×C₅₃	1	2	yes	—
287	#117	C₂²×C₅₃	1	1	yes	characteristic
288	#117	C₂²×C₅₃	1	2	yes	—
289	#117	C₂²×C₅₃	1	2	yes	—
290	#117	C₂²×C₅₃	1	4	yes	—
291	#117	C₂²×C₅₃	1	4	yes	—
292	#117	C₂²×C₅₃	1	2	yes	—
293	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
294	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
295	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
296	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
297	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
298	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
299	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
300	#41	—	2	4	no	—
301	#41	—	2	8	no	—
302	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
303	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
305	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
306	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
307	C₂×D₄	C₂³×C₅₃	1	2	yes	—
308	C₂×D₄	C₂³×C₅₃	1	4	yes	—
309	C₂×D₄	C₂³×C₅₃	1	2	yes	—
310	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
313	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
315	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
316	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
317	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
318	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
320	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
321	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
323	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
324	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
325	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
326	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
327	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
328	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
329	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
330	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
331	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
332	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
333	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
334	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
335	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
336	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
337	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
338	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
339	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
340	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
341	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
342	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
343	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
344	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
346	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
348	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
349	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
350	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
351	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
352	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
353	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
356	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
357	C₄²	C₂³×C₅₃	1	1	yes	characteristic
358	C₄²	C₂³×C₅₃	1	2	yes	—
359	C₄²	C₂³×C₅₃	1	1	yes	characteristic
360	C₄²	C₂³×C₅₃	1	1	yes	characteristic
361	C₄²	C₂³×C₅₃	1	2	yes	—
362	C₄²	C₂³×C₅₃	1	1	yes	characteristic
363	C₄²	C₂³×C₅₃	1	2	yes	—
364	C₄²	C₂³×C₅₃	1	4	yes	—
365	C₄²	C₂³×C₅₃	1	4	yes	—
366	C₄²	C₂³×C₅₃	1	2	yes	—
367	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
368	C₂³	C₂⁴×C₅₃	1	2	yes	—
369	C₂³	C₂⁴×C₅₃	1	2	yes	—
370	Q₈	—	2	2	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	2	2	no	—
373	Q₈	—	4	2	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	8	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	2	no	—
378	D₄	#41×C₅₃	1	4	yes	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
393	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
394	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
395	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
397	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
398	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
399	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
400	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
401	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
402	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	1	no	—
411	C₂²	C₂×#41×C₅₃	1	4	yes	—
412	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	2	4	no	—
417	C₄	—	2	1	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	1	no	—
421	C₄	—	2	2	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	1	no	—
425	C₄	C₂×#41×C₅₃	1	2	yes	—
426	C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
427	C₂	—	2	4	no	—
428	C₂	—	4	1	no	—
429	C₂	—	4	2	no	—
430	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
431	C₂	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
432	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#557	C₂	semidirect
2. C₅₃×#554	C₂	semidirect
3. C₅₃×#554	C₂	semidirect
4. C₅₃×#553	C₂	semidirect
5. C₅₃×#552	C₂	semidirect
6. C₅₃×#552	C₂	semidirect
7. C₅₃×#552	C₂	semidirect
8. C₅₃×#547	C₂	semidirect
9. C₅₃×#547	C₂	semidirect
10. C₅₃×#541	C₂	semidirect
11. C₅₃×#539	C₂	semidirect
12. C₅₃×#533	C₂	semidirect
13. C₅₃×#529	C₂	semidirect
14. C₅₃×#529	C₂	semidirect
15. C₂×#125×C₅₃	C₂	semidirect
16. C₅₃×#520	C₂	semidirect
17. C₄×#41×C₅₃	C₂	semidirect
210. #3515	C₅₃	direct (N × H)
217. #557	C₂×C₅₃	semidirect
218. #554	C₂×C₅₃	semidirect
219. #554	C₂×C₅₃	semidirect
220. #553	C₂×C₅₃	semidirect
221. #552	C₂×C₅₃	semidirect
222. #552	C₂×C₅₃	semidirect
223. #552	C₂×C₅₃	semidirect
224. #547	C₂×C₅₃	semidirect
225. #547	C₂×C₅₃	semidirect
226. #541	C₂×C₅₃	semidirect
227. #539	C₂×C₅₃	semidirect
228. #533	C₂×C₅₃	semidirect
229. #529	C₂×C₅₃	semidirect
230. #529	C₂×C₅₃	semidirect
231. C₂×#125	C₂×C₅₃	semidirect
232. #520	C₂×C₅₃	semidirect
233. C₄×#41	C₂×C₅₃	semidirect
234. C₅₃	#3515	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×#557	2
C₅₃×#554	2
C₅₃×#553	2
C₅₃×#552	2
C₅₃×#547	2
C₅₃×#541	2
C₅₃×#539	2
C₅₃×#533	2
C₅₃×#529	2
C₂×#125×C₅₃	2
C₅₃×#520	2
C₄×#41×C₅₃	2
#3515	53

Subgroups — C53×#3515 #20272452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3515 #20272452