Subgroups — C₅₃×#36652 #27244164

Order 13568.

2570 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#525	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2817	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2816	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2815	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2814	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2813	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2812	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2395	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2392	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2388	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#525	—	2	16	no	—
15	C₂×#123×C₅₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂×#121×C₅₃	C₂²	1	2	yes	—
19	C₂×#121×C₅₃	C₂²	1	2	yes	—
20	C₂×#120×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
24	C₂×#117×C₅₃	C₂²	1	2	yes	—
25	C₂×#117×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#400	C₂²	1	4	yes	complemented
27	C₅₃×#399	C₂²	1	4	yes	complemented
28	C₅₃×#398	C₂²	1	4	yes	complemented
29	C₅₃×#398	C₂²	1	4	yes	complemented
30	C₅₃×#397	C₂²	1	8	yes	—
31	C₅₃×#397	C₂²	1	4	yes	—
32	C₅₃×#397	C₂²	1	4	yes	complemented
33	C₅₃×#396	C₂²	1	8	yes	complemented
34	C₅₃×#395	C₂²	1	4	yes	complemented
35	C₅₃×#394	C₂²	1	8	yes	—
36	C₅₃×#394	C₂²	1	8	yes	—
37	C₅₃×#394	C₂²	1	8	yes	—
38	C₅₃×#394	C₂²	1	4	yes	complemented
39	C₅₃×#394	C₂²	1	4	yes	complemented
40	C₅₃×#393	C₂²	1	4	yes	complemented
41	C₅₃×#392	C₂²	1	8	yes	—
42	C₅₃×#392	C₂²	1	4	yes	—
43	C₅₃×#392	C₂²	1	4	yes	complemented
44	C₅₃×#381	C₂²	1	1	yes	characteristic
45	C₅₃×#381	C₂²	1	1	yes	characteristic
46	C₅₃×#380	C₂²	1	4	yes	—
47	C₅₃×#379	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
51	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×#41×C₅₃	—	2	8	no	—
53	C₂×#41×C₅₃	—	4	8	no	—
54	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	—	4	4	no	—
61	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
62	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
67	#123×C₅₃	—	2	16	no	—
68	C₂²⋊Q₈×C₅₃	—	2	8	no	—
69	C₂²⋊Q₈×C₅₃	—	2	8	no	—
70	#121×C₅₃	—	2	8	no	—
71	#121×C₅₃	—	2	8	no	—
72	#120×C₅₃	—	2	16	no	—
73	C₄×D₄×C₅₃	—	2	16	no	—
74	C₄×D₄×C₅₃	—	2	16	no	—
75	C₄×D₄×C₅₃	—	4	8	no	—
76	#117×C₅₃	—	2	8	no	—
77	#117×C₅₃	—	4	4	no	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
97	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
98	#95×C₅₃	C₂³	1	4	yes	—
99	#95×C₅₃	C₂³	1	4	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	8	yes	—
102	#95×C₅₃	C₂³	1	4	yes	—
103	#95×C₅₃	C₂³	1	4	yes	—
104	C₂⁴×C₅₃	—	4	2	no	—
105	C₂⁴×C₅₃	—	4	4	no	—
106	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
108	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
109	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
110	#41×C₅₃	—	4	32	no	—
111	C₂×Q₈×C₅₃	—	2	4	no	—
112	C₂×Q₈×C₅₃	—	4	4	no	—
113	C₂×D₄×C₅₃	—	2	8	no	—
114	C₂×D₄×C₅₃	—	2	8	no	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	2	4	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	4	4	no	—
119	C₂×D₄×C₅₃	—	4	8	no	—
120	C₂×D₄×C₅₃	—	4	8	no	—
121	C₂×D₄×C₅₃	—	4	4	no	—
122	C₂×D₄×C₅₃	—	4	4	no	—
123	C₂×D₄×C₅₃	—	4	8	no	—
124	C₂²×C₄×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	—	2	8	no	—
126	C₂²×C₄×C₅₃	—	2	8	no	—
127	C₂²×C₄×C₅₃	—	2	16	no	—
128	C₂²×C₄×C₅₃	—	4	16	no	—
129	C₂²×C₄×C₅₃	—	4	2	no	—
130	C₂²×C₄×C₅₃	—	4	4	no	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₄⋊C₄×C₅₃	—	2	4	no	—
144	C₄⋊C₄×C₅₃	—	2	4	no	—
145	C₄⋊C₄×C₅₃	—	2	8	no	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	4	8	no	—
148	C₄⋊C₄×C₅₃	—	4	4	no	—
149	C₄⋊C₄×C₅₃	—	4	4	no	—
150	C₂²⋊C₄×C₅₃	—	2	8	no	—
151	C₂²⋊C₄×C₅₃	—	2	8	no	—
152	C₂²⋊C₄×C₅₃	—	2	8	no	—
153	C₂²⋊C₄×C₅₃	—	2	8	no	—
154	C₂²⋊C₄×C₅₃	—	2	8	no	—
155	C₂²⋊C₄×C₅₃	—	4	8	no	—
156	C₂²⋊C₄×C₅₃	—	4	8	no	—
157	C₂²⋊C₄×C₅₃	—	4	8	no	—
158	C₂²⋊C₄×C₅₃	—	4	2	no	—
159	C₂²⋊C₄×C₅₃	—	4	2	no	—
160	C₂²⋊C₄×C₅₃	—	4	8	no	—
161	C₄²×C₅₃	—	2	8	no	—
162	C₄²×C₅₃	—	2	8	no	—
163	C₂³×C₅₃	—	2	4	no	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	2	4	no	—
166	C₂³×C₅₃	—	2	4	no	—
167	C₂³×C₅₃	—	2	4	no	—
168	C₂³×C₅₃	—	2	4	no	—
169	C₂³×C₅₃	—	2	8	no	—
170	C₂³×C₅₃	—	2	4	no	—
171	C₂³×C₅₃	—	2	8	no	—
172	C₂³×C₅₃	—	2	4	no	—
173	C₂³×C₅₃	—	4	4	no	—
174	C₂³×C₅₃	—	4	8	no	—
175	C₂³×C₅₃	—	4	16	no	—
176	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
177	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
178	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
179	Q₈×C₅₃	—	4	8	no	—
180	D₄×C₅₃	—	4	8	no	—
181	D₄×C₅₃	—	4	16	no	—
182	D₄×C₅₃	—	4	16	no	—
183	D₄×C₅₃	—	4	16	no	—
184	D₄×C₅₃	—	4	16	no	—
185	C₂×C₄×C₅₃	—	2	4	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	8	no	—
188	C₂×C₄×C₅₃	—	2	16	no	—
189	C₂×C₄×C₅₃	—	2	4	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	8	no	—
193	C₂×C₄×C₅₃	—	2	4	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	8	no	—
197	C₂×C₄×C₅₃	—	2	8	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	8	no	—
200	C₂×C₄×C₅₃	—	4	8	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	16	no	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
204	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
205	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
206	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
207	#36652	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂²×C₅₃	—	2	4	no	—
209	C₂²×C₅₃	—	2	2	no	—
210	C₂²×C₅₃	—	2	4	no	—
211	C₂²×C₅₃	—	2	4	no	—
212	C₂²×C₅₃	—	2	4	no	—
213	C₂²×C₅₃	—	4	16	no	—
214	C₂²×C₅₃	—	4	8	no	—
215	C₂²×C₅₃	—	4	4	no	—
216	C₂²×C₅₃	—	4	4	no	—
217	C₂²×C₅₃	—	4	4	no	—
218	C₂²×C₅₃	—	4	4	no	—
219	C₂²×C₅₃	—	4	8	no	—
220	C₂²×C₅₃	—	4	8	no	—
221	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
223	C₂²×C₅₃	#585	1	4	yes	—
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	4	no	—
226	C₄×C₅₃	—	4	4	no	—
227	C₄×C₅₃	—	4	4	no	—
228	C₄×C₅₃	—	4	4	no	—
229	C₄×C₅₃	—	4	4	no	—
230	C₄×C₅₃	—	4	4	no	—
231	C₄×C₅₃	#534	1	4	yes	—
232	C₂×#525	C₂×C₅₃	1	2	yes	complemented
233	#2817	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#2816	C₂×C₅₃	1	2	yes	complemented
235	#2815	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#2814	C₂×C₅₃	1	4	yes	complemented
237	#2813	C₂×C₅₃	1	4	yes	complemented
238	#2812	C₂×C₅₃	1	4	yes	complemented
239	#2395	C₂×C₅₃	1	4	yes	complemented
240	#2392	C₂×C₅₃	1	4	yes	complemented
241	#2388	C₂×C₅₃	1	4	yes	complemented
242	#2280	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₂×C₅₃	—	4	4	no	—
244	C₂×C₅₃	—	4	2	no	—
245	C₂×C₅₃	—	8	4	no	—
246	C₂×C₅₃	C₂×#534	1	1	yes	characteristic
247	C₂×C₅₃	#3449	1	2	yes	—
248	C₂×C₅₃	#3529	1	4	yes	—
249	C₂²×#41	C₂²×C₅₃	1	2	yes	—
250	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
251	#525	—	2	16	no	—
252	C₂×#123	C₂²×C₅₃	1	4	yes	—
253	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
254	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
255	C₂×#121	C₂²×C₅₃	1	2	yes	—
256	C₂×#121	C₂²×C₅₃	1	2	yes	—
257	C₂×#120	C₂²×C₅₃	1	4	yes	—
258	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
259	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
260	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
261	C₂×#117	C₂²×C₅₃	1	2	yes	—
262	C₂×#117	C₂²×C₅₃	1	2	yes	—
263	#400	C₂²×C₅₃	1	4	yes	complemented
264	#399	C₂²×C₅₃	1	4	yes	complemented
265	#398	C₂²×C₅₃	1	4	yes	complemented
266	#398	C₂²×C₅₃	1	4	yes	complemented
267	#397	C₂²×C₅₃	1	4	yes	—
268	#397	C₂²×C₅₃	1	8	yes	—
269	#397	C₂²×C₅₃	1	4	yes	complemented
270	#396	C₂²×C₅₃	1	8	yes	complemented
271	#395	C₂²×C₅₃	1	4	yes	complemented
272	#394	C₂²×C₅₃	1	8	yes	—
273	#394	C₂²×C₅₃	1	8	yes	—
274	#394	C₂²×C₅₃	1	8	yes	—
275	#394	C₂²×C₅₃	1	4	yes	complemented
276	#394	C₂²×C₅₃	1	4	yes	complemented
277	#393	C₂²×C₅₃	1	4	yes	complemented
278	#392	C₂²×C₅₃	1	8	yes	—
279	#392	C₂²×C₅₃	1	4	yes	—
280	#392	C₂²×C₅₃	1	4	yes	complemented
281	#381	C₂²×C₅₃	1	1	yes	characteristic
282	#381	C₂²×C₅₃	1	1	yes	characteristic
283	#380	C₂²×C₅₃	1	4	yes	—
284	#379	C₂²×C₅₃	1	2	yes	—
285	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
286	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
288	C₅₃	#36652	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
290	C₂×#41	—	2	8	no	—
291	C₂×#41	—	4	8	no	—
292	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
293	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
294	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
295	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
296	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
297	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
298	C₂³×C₄	—	4	4	no	—
299	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
300	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
301	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
303	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
305	#123	—	2	16	no	—
306	C₂²⋊Q₈	—	2	8	no	—
307	C₂²⋊Q₈	—	2	8	no	—
308	#121	—	2	8	no	—
309	#121	—	2	8	no	—
310	#120	—	2	16	no	—
311	C₄×D₄	—	2	16	no	—
312	C₄×D₄	—	2	16	no	—
313	C₄×D₄	—	4	8	no	—
314	#117	—	2	8	no	—
315	#117	—	4	4	no	—
316	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
319	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
320	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
334	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
335	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
336	#95	C₂³×C₅₃	1	4	yes	—
337	#95	C₂³×C₅₃	1	4	yes	—
338	#95	C₂³×C₅₃	1	8	yes	—
339	#95	C₂³×C₅₃	1	8	yes	—
340	#95	C₂³×C₅₃	1	4	yes	—
341	#95	C₂³×C₅₃	1	4	yes	—
342	C₂⁴	—	4	2	no	—
343	C₂⁴	—	4	4	no	—
344	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
345	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
346	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
347	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
348	#41	—	4	32	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	4	4	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	8	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	4	no	—
357	C₂×D₄	—	4	8	no	—
358	C₂×D₄	—	4	8	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	4	4	no	—
361	C₂×D₄	—	4	8	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	2	8	no	—
365	C₂²×C₄	—	2	16	no	—
366	C₂²×C₄	—	4	16	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
370	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
371	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
372	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
373	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
374	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	—	4	8	no	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	8	no	—
396	C₂²⋊C₄	—	4	8	no	—
397	C₂²⋊C₄	—	4	8	no	—
398	C₂²⋊C₄	—	4	8	no	—
399	C₄²	—	2	8	no	—
400	C₄²	—	2	8	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	4	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	2	4	no	—
405	C₂³	—	2	4	no	—
406	C₂³	—	2	4	no	—
407	C₂³	—	2	8	no	—
408	C₂³	—	2	4	no	—
409	C₂³	—	2	8	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	4	4	no	—
412	C₂³	—	4	8	no	—
413	C₂³	—	4	16	no	—
414	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
415	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
416	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
417	Q₈	—	4	8	no	—
418	D₄	—	4	8	no	—
419	D₄	—	4	16	no	—
420	D₄	—	4	16	no	—
421	D₄	—	4	16	no	—
422	D₄	—	4	16	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	8	no	—
426	C₂×C₄	—	2	16	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	8	no	—
440	C₂×C₄	—	4	16	no	—
441	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
442	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
443	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
444	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	4	16	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	4	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	8	no	—
456	C₂²	—	4	8	no	—
457	C₂²	—	4	4	no	—
458	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
459	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
460	C₂²	C₅₃×#585	1	4	yes	—
461	C₄	—	2	4	no	—
462	C₄	—	2	2	no	—
463	C₄	—	4	4	no	—
464	C₄	—	4	4	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₄	C₅₃×#534	1	4	yes	—
469	C₂	—	4	2	no	—
470	C₂	—	4	4	no	—
471	C₂	—	8	4	no	—
472	C₂	C₂×C₅₃×#534	1	1	yes	characteristic
473	C₂	C₅₃×#3449	1	2	yes	—
474	C₂	C₅₃×#3529	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#525	C₂	semidirect
2. C₅₃×#2817	C₂	semidirect
3. C₅₃×#2816	C₂	semidirect
4. C₅₃×#2815	C₂	semidirect
5. C₅₃×#2814	C₂	semidirect
6. C₅₃×#2813	C₂	semidirect
7. C₅₃×#2812	C₂	semidirect
8. C₅₃×#2395	C₂	semidirect
9. C₅₃×#2392	C₂	semidirect
10. C₅₃×#2388	C₂	semidirect
11. C₅₃×#2280	C₂	semidirect
23. C₂×C₄×D₄×C₅₃	C₂²	semidirect
26. C₅₃×#400	C₂²	semidirect
27. C₅₃×#399	C₂²	semidirect
28. C₅₃×#398	C₂²	semidirect
29. C₅₃×#398	C₂²	semidirect
32. C₅₃×#397	C₂²	semidirect
33. C₅₃×#396	C₂²	semidirect
34. C₅₃×#395	C₂²	semidirect
38. C₅₃×#394	C₂²	semidirect
39. C₅₃×#394	C₂²	semidirect
40. C₅₃×#393	C₂²	semidirect
43. C₅₃×#392	C₂²	semidirect
48. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
61. C₂³×C₄×C₅₃	D₄	semidirect
207. #36652	C₅₃	direct (N × H)
232. C₂×#525	C₂×C₅₃	semidirect
233. #2817	C₂×C₅₃	semidirect
234. #2816	C₂×C₅₃	semidirect
235. #2815	C₂×C₅₃	semidirect
236. #2814	C₂×C₅₃	semidirect
237. #2813	C₂×C₅₃	semidirect
238. #2812	C₂×C₅₃	semidirect
239. #2395	C₂×C₅₃	semidirect
240. #2392	C₂×C₅₃	semidirect
241. #2388	C₂×C₅₃	semidirect
242. #2280	C₂×C₅₃	semidirect
260. C₂×C₄×D₄	C₂²×C₅₃	semidirect
263. #400	C₂²×C₅₃	semidirect
264. #399	C₂²×C₅₃	semidirect
265. #398	C₂²×C₅₃	semidirect
266. #398	C₂²×C₅₃	semidirect
269. #397	C₂²×C₅₃	semidirect
270. #396	C₂²×C₅₃	semidirect
271. #395	C₂²×C₅₃	semidirect
275. #394	C₂²×C₅₃	semidirect
276. #394	C₂²×C₅₃	semidirect
277. #393	C₂²×C₅₃	semidirect
280. #392	C₂²×C₅₃	semidirect
285. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
287. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
288. C₅₃	#36652	direct (N × H)
299. C₂³×C₄	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#525	2
C₅₃×#2817	2
C₅₃×#2816	2
C₅₃×#2815	2
C₅₃×#2814	2
C₅₃×#2813	2
C₅₃×#2812	2
C₅₃×#2395	2
C₅₃×#2392	2
C₅₃×#2388	2
C₅₃×#2280	2
#36652	53

Subgroups — C53×#36652 #27244164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36652 #27244164