Subgroups — C₅₃×#62002 #27269514

Order 13568.

1872 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3483	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3392	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3363	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3329	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3278	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3254	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3206	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3107	C₂	1	3	yes	maximal, complemented
10	C₅₃×#3039	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₅₃	C₂²	1	6	yes	—
13	Q₈²×C₅₃	—	2	1	no	—
14	Q₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#557	C₂²	1	2	yes	—
16	C₅₃×#555	—	2	3	no	—
17	C₅₃×#555	C₂²	1	3	yes	—
18	C₅₃×#554	—	2	3	no	—
19	C₅₃×#550	—	2	1	no	—
20	C₅₃×#550	C₂²	1	3	yes	—
21	C₅₃×#550	C₂²	1	1	yes	characteristic
22	D₄×Q₈×C₅₃	C₂²	1	6	yes	—
23	C₅₃×#544	—	2	3	no	—
24	C₅₃×#533	C₂²	1	3	yes	—
25	C₅₃×#532	—	2	3	no	—
26	C₅₃×#532	C₂²	1	3	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
28	C₅₃×#520	—	2	1	no	—
29	C₄×#41×C₅₃	—	2	1	no	—
30	C₄×#41×C₅₃	C₂²	1	3	yes	—
31	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#492	C₂²	1	6	yes	—
33	C₅₃×#487	C₂²	1	6	yes	—
34	C₅₃×#485	C₂²	1	6	yes	—
35	C₅₃×#477	C₂²	1	6	yes	—
36	C₅₃×#475	C₂²	1	1	yes	characteristic
37	C₅₃×#474	C₂²	1	3	yes	—
38	C₅₃×#468	C₂²	1	12	yes	—
39	C₅₃×#462	C₂²	1	6	yes	—
40	C₅₃×#462	C₂²	1	3	yes	—
41	C₅₃×#461	C₂²	1	3	yes	—
42	C₅₃×#461	C₂²	1	6	yes	—
43	C₅₃×#461	C₂²	1	6	yes	—
44	C₅₃×#460	C₂²	1	6	yes	—
45	C₅₃×#452	C₂²	1	6	yes	—
46	C₅₃×#448	C₂²	1	12	yes	—
47	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
48	C₄×Q₁₆×C₅₃	C₂²	1	6	yes	—
49	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
50	C₅₃×#436	C₂²	1	6	yes	—
51	C₅₃×#431	C₂²	1	3	yes	—
52	C₅₃×#423	C₂²	1	3	yes	—
53	C₅₃×#419	C₂²	1	6	yes	—
54	C₂×#41×C₅₃	—	2	1	no	—
55	C₂×#41×C₅₃	—	2	3	no	—
56	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
57	C₂×#41×C₅₃	C₂³	1	3	yes	—
58	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
59	#137×C₅₃	—	4	12	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
61	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
62	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
63	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
64	C₄⋊Q₈×C₅₃	—	2	3	no	—
65	C₄⋊Q₈×C₅₃	—	2	6	no	—
66	C₄⋊Q₈×C₅₃	—	2	3	no	—
67	C₄⋊Q₈×C₅₃	—	2	3	no	—
68	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
72	#127×C₅₃	—	2	3	no	—
73	#127×C₅₃	C₂³	1	3	yes	—
74	#125×C₅₃	—	2	6	no	—
75	#125×C₅₃	—	2	3	no	—
76	#124×C₅₃	—	2	6	no	—
77	#124×C₅₃	—	2	3	no	—
78	C₂²⋊Q₈×C₅₃	—	2	12	no	—
79	C₂²⋊Q₈×C₅₃	—	2	3	no	—
80	C₂²⋊Q₈×C₅₃	C₂³	1	6	yes	—
81	#121×C₅₃	—	2	3	no	—
82	#121×C₅₃	—	2	6	no	—
83	#121×C₅₃	—	2	3	no	—
84	#121×C₅₃	—	2	3	no	—
85	#121×C₅₃	C₂³	1	6	yes	—
86	C₄×Q₈×C₅₃	—	2	3	no	—
87	C₄×Q₈×C₅₃	—	2	1	no	—
88	C₄×Q₈×C₅₃	—	2	2	no	—
89	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	6	yes	—
94	C₄×D₄×C₅₃	—	2	3	no	—
95	C₄×D₄×C₅₃	—	2	3	no	—
96	C₄×D₄×C₅₃	—	2	6	no	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	3	no	—
99	C₄×D₄×C₅₃	D₄	1	3	yes	—
100	C₄×D₄×C₅₃	C₂³	1	3	yes	—
101	C₄×D₄×C₅₃	C₂³	1	3	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	#117×C₅₃	—	2	3	no	—
104	#117×C₅₃	—	2	3	no	—
105	#117×C₅₃	—	2	3	no	—
106	#117×C₅₃	D₄	1	3	yes	—
107	#117×C₅₃	C₂³	1	3	yes	—
108	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
109	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
110	C₂×C₄²×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
112	#107×C₅₃	C₂³	1	6	yes	—
113	#106×C₅₃	C₂³	1	2	yes	—
114	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
117	#103×C₅₃	C₂³	1	6	yes	—
118	#103×C₅₃	C₂³	1	6	yes	—
119	#103×C₅₃	C₂³	1	12	yes	—
120	#103×C₅₃	C₂³	1	2	yes	—
121	#102×C₅₃	C₂³	1	6	yes	—
122	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
123	C₄×C₈×C₅₃	C₂³	1	6	yes	—
124	#41×C₅₃	—	2	4	no	—
125	#41×C₅₃	—	2	6	no	—
126	#41×C₅₃	—	4	3	no	—
127	#41×C₅₃	—	4	4	no	—
128	#41×C₅₃	—	8	6	no	—
129	C₂×Q₈×C₅₃	—	2	1	no	—
130	C₂×Q₈×C₅₃	—	2	6	no	—
131	C₂×Q₈×C₅₃	—	2	3	no	—
132	C₂×Q₈×C₅₃	—	4	6	no	—
133	C₂×Q₈×C₅₃	—	4	6	no	—
134	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
136	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₅₃	—	2	3	no	—
138	C₂×D₄×C₅₃	—	2	6	no	—
139	C₂×D₄×C₅₃	—	2	3	no	—
140	C₂×D₄×C₅₃	—	2	3	no	—
141	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
142	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
143	C₂²×C₄×C₅₃	—	2	6	no	—
144	C₂²×C₄×C₅₃	—	2	3	no	—
145	C₂²×C₄×C₅₃	—	2	3	no	—
146	C₂²×C₄×C₅₃	—	2	3	no	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
148	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
150	Q₁₆×C₅₃	—	2	12	no	—
151	SD₁₆×C₅₃	—	4	6	no	—
152	SD₁₆×C₅₃	—	4	12	no	—
153	M₁₆×C₅₃	—	4	6	no	—
154	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
155	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
156	C₄⋊C₄×C₅₃	—	2	3	no	—
157	C₄⋊C₄×C₅₃	—	2	3	no	—
158	C₄⋊C₄×C₅₃	—	2	6	no	—
159	C₄⋊C₄×C₅₃	—	2	6	no	—
160	C₄⋊C₄×C₅₃	—	2	1	no	—
161	C₄⋊C₄×C₅₃	—	2	1	no	—
162	C₄⋊C₄×C₅₃	—	2	3	no	—
163	C₄⋊C₄×C₅₃	—	2	3	no	—
164	C₄⋊C₄×C₅₃	—	2	6	no	—
165	C₄⋊C₄×C₅₃	—	2	6	no	—
166	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
169	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
171	C₂²⋊C₄×C₅₃	—	2	6	no	—
172	C₂²⋊C₄×C₅₃	—	2	3	no	—
173	C₂²⋊C₄×C₅₃	—	2	6	no	—
174	C₂²⋊C₄×C₅₃	—	2	3	no	—
175	C₂²⋊C₄×C₅₃	—	2	3	no	—
176	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₄²×C₅₃	—	2	3	no	—
179	C₄²×C₅₃	—	2	3	no	—
180	C₄²×C₅₃	—	2	6	no	—
181	C₄²×C₅₃	C₂×D₄	1	3	yes	—
182	C₄²×C₅₃	C₂×D₄	1	3	yes	—
183	C₄²×C₅₃	C₂⁴	1	3	yes	—
184	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂³×C₅₃	—	2	3	no	—
186	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
187	Q₈×C₅₃	—	2	1	no	—
188	Q₈×C₅₃	—	2	6	no	—
189	Q₈×C₅₃	—	2	6	no	—
190	Q₈×C₅₃	—	4	2	no	—
191	Q₈×C₅₃	—	8	6	no	—
192	Q₈×C₅₃	—	8	3	no	—
193	Q₈×C₅₃	#137	1	4	yes	complemented
194	Q₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
195	D₄×C₅₃	—	2	6	no	—
196	D₄×C₅₃	—	2	3	no	—
197	D₄×C₅₃	—	2	3	no	—
198	D₄×C₅₃	—	4	6	no	—
199	D₄×C₅₃	—	4	6	no	—
200	D₄×C₅₃	—	8	3	no	—
201	D₄×C₅₃	—	8	6	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	6	no	—
204	C₂×C₄×C₅₃	—	2	6	no	—
205	C₂×C₄×C₅₃	—	2	3	no	—
206	C₂×C₄×C₅₃	—	2	3	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	6	no	—
209	C₂×C₄×C₅₃	—	2	3	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	4	6	no	—
212	C₂×C₄×C₅₃	—	4	6	no	—
213	C₂×C₄×C₅₃	—	4	6	no	—
214	C₂×C₄×C₅₃	—	8	6	no	—
215	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
221	C₈×C₅₃	—	2	6	no	—
222	C₈×C₅₃	—	2	2	no	—
223	#62002	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂²×C₅₃	—	2	3	no	—
225	C₂²×C₅₃	—	2	3	no	—
226	C₂²×C₅₃	—	2	3	no	—
227	C₂²×C₅₃	—	4	3	no	—
228	C₂²×C₅₃	—	4	3	no	—
229	C₂²×C₅₃	—	4	3	no	—
230	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₅₃	—	2	3	no	—
232	C₄×C₅₃	—	2	2	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	3	no	—
235	C₄×C₅₃	—	4	6	no	—
236	C₄×C₅₃	—	4	1	no	—
237	C₄×C₅₃	—	4	3	no	—
238	C₄×C₅₃	—	4	2	no	—
239	C₄×C₅₃	C₂×#137	1	6	yes	—
240	C₄×C₅₃	#577	1	2	yes	—
241	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
242	C₄×C₅₃	#585	1	1	yes	characteristic
243	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#3483	C₂×C₅₃	1	1	yes	characteristic
245	#3392	C₂×C₅₃	1	2	yes	complemented
246	#3363	C₂×C₅₃	1	2	yes	complemented
247	#3329	C₂×C₅₃	1	6	yes	complemented
248	#3278	C₂×C₅₃	1	6	yes	complemented
249	#3254	C₂×C₅₃	1	3	yes	complemented
250	#3206	C₂×C₅₃	1	3	yes	complemented
251	#3107	C₂×C₅₃	1	3	yes	complemented
252	#3039	C₂×C₅₃	1	3	yes	complemented
253	#2991	C₂×C₅₃	1	1	yes	characteristic, complemented
254	C₂×C₅₃	—	4	3	no	—
255	C₂×C₅₃	—	8	3	no	—
256	C₂×C₅₃	#3488	1	1	yes	characteristic
257	C₂×C₅₃	C₂²×#137	1	1	yes	characteristic
258	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
259	C₂×#137	C₂²×C₅₃	1	6	yes	—
260	Q₈²	—	2	1	no	—
261	Q₈²	C₂²×C₅₃	1	1	yes	characteristic
262	#557	C₂²×C₅₃	1	2	yes	—
263	#555	—	2	3	no	—
264	#555	C₂²×C₅₃	1	3	yes	—
265	#554	—	2	3	no	—
266	#550	—	2	1	no	—
267	#550	C₂²×C₅₃	1	3	yes	—
268	#550	C₂²×C₅₃	1	1	yes	characteristic
269	D₄×Q₈	C₂²×C₅₃	1	6	yes	—
270	#544	—	2	3	no	—
271	#533	C₂²×C₅₃	1	3	yes	—
272	#532	—	2	3	no	—
273	#532	C₂²×C₅₃	1	3	yes	—
274	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
275	#520	—	2	1	no	—
276	C₄×#41	—	2	1	no	—
277	C₄×#41	C₂²×C₅₃	1	3	yes	—
278	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
279	#492	C₂²×C₅₃	1	6	yes	—
280	#487	C₂²×C₅₃	1	6	yes	—
281	#485	C₂²×C₅₃	1	6	yes	—
282	#477	C₂²×C₅₃	1	6	yes	—
283	#475	C₂²×C₅₃	1	1	yes	characteristic
284	#474	C₂²×C₅₃	1	3	yes	—
285	#468	C₂²×C₅₃	1	12	yes	—
286	#462	C₂²×C₅₃	1	3	yes	—
287	#462	C₂²×C₅₃	1	6	yes	—
288	#461	C₂²×C₅₃	1	6	yes	—
289	#461	C₂²×C₅₃	1	3	yes	—
290	#461	C₂²×C₅₃	1	6	yes	—
291	#460	C₂²×C₅₃	1	6	yes	—
292	#452	C₂²×C₅₃	1	6	yes	—
293	#448	C₂²×C₅₃	1	12	yes	—
294	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
295	C₄×Q₁₆	C₂²×C₅₃	1	6	yes	—
296	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
297	#436	C₂²×C₅₃	1	6	yes	—
298	#431	C₂²×C₅₃	1	3	yes	—
299	#423	C₂²×C₅₃	1	3	yes	—
300	#419	C₂²×C₅₃	1	6	yes	—
301	C₅₃	#62002	1	1	yes	characteristic, direct product factor, Sylow
302	C₂×#41	—	2	1	no	—
303	C₂×#41	—	2	3	no	—
304	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
305	C₂×#41	C₂³×C₅₃	1	3	yes	—
306	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
307	#137	—	4	12	no	—
308	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
309	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
310	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
311	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
312	C₄⋊Q₈	—	2	3	no	—
313	C₄⋊Q₈	—	2	6	no	—
314	C₄⋊Q₈	—	2	3	no	—
315	C₄⋊Q₈	—	2	3	no	—
316	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
317	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
318	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
319	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
320	#127	—	2	3	no	—
321	#127	C₂³×C₅₃	1	3	yes	—
322	#125	—	2	6	no	—
323	#125	—	2	3	no	—
324	#124	—	2	6	no	—
325	#124	—	2	3	no	—
326	C₂²⋊Q₈	—	2	3	no	—
327	C₂²⋊Q₈	—	2	12	no	—
328	C₂²⋊Q₈	C₂³×C₅₃	1	6	yes	—
329	#121	—	2	3	no	—
330	#121	—	2	3	no	—
331	#121	—	2	3	no	—
332	#121	—	2	6	no	—
333	#121	C₂³×C₅₃	1	6	yes	—
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	3	no	—
337	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
338	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
339	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
340	C₄×Q₈	C₂³×C₅₃	1	6	yes	—
341	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
342	C₄×D₄	—	2	6	no	—
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	D₄×C₅₃	1	3	yes	—
348	C₄×D₄	C₂³×C₅₃	1	3	yes	—
349	C₄×D₄	C₂³×C₅₃	1	3	yes	—
350	C₄×D₄	C₂³×C₅₃	1	3	yes	—
351	#117	—	2	3	no	—
352	#117	—	2	3	no	—
353	#117	—	2	3	no	—
354	#117	D₄×C₅₃	1	3	yes	—
355	#117	C₂³×C₅₃	1	3	yes	—
356	C₂×C₄⋊C₄	—	2	3	no	—
357	C₂×C₄⋊C₄	—	2	3	no	—
358	C₂×C₄²	—	2	3	no	—
359	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
360	#107	C₂³×C₅₃	1	6	yes	—
361	#106	C₂³×C₅₃	1	2	yes	—
362	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
363	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
364	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
365	#103	C₂³×C₅₃	1	6	yes	—
366	#103	C₂³×C₅₃	1	12	yes	—
367	#103	C₂³×C₅₃	1	2	yes	—
368	#103	C₂³×C₅₃	1	6	yes	—
369	#102	C₂³×C₅₃	1	6	yes	—
370	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
371	C₄×C₈	C₂³×C₅₃	1	6	yes	—
372	#41	—	2	4	no	—
373	#41	—	2	6	no	—
374	#41	—	4	3	no	—
375	#41	—	4	4	no	—
376	#41	—	8	6	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	6	no	—
379	C₂×Q₈	—	2	3	no	—
380	C₂×Q₈	—	4	6	no	—
381	C₂×Q₈	—	4	6	no	—
382	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
383	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
384	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
385	C₂×D₄	—	2	3	no	—
386	C₂×D₄	—	2	6	no	—
387	C₂×D₄	—	2	3	no	—
388	C₂×D₄	—	2	3	no	—
389	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
390	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
391	C₂²×C₄	—	2	6	no	—
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	—	2	3	no	—
394	C₂²×C₄	—	2	3	no	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
396	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
397	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
398	Q₁₆	—	2	12	no	—
399	SD₁₆	—	4	12	no	—
400	SD₁₆	—	4	6	no	—
401	M₁₆	—	4	6	no	—
402	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
403	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
404	C₄⋊C₄	—	2	3	no	—
405	C₄⋊C₄	—	2	3	no	—
406	C₄⋊C₄	—	2	6	no	—
407	C₄⋊C₄	—	2	6	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	6	no	—
413	C₄⋊C₄	—	2	6	no	—
414	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
415	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
417	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
418	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
419	C₂²⋊C₄	—	2	6	no	—
420	C₂²⋊C₄	—	2	3	no	—
421	C₂²⋊C₄	—	2	6	no	—
422	C₂²⋊C₄	—	2	3	no	—
423	C₂²⋊C₄	—	2	3	no	—
424	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	3	no	—
427	C₄²	—	2	3	no	—
428	C₄²	—	2	6	no	—
429	C₄²	C₂×D₄×C₅₃	1	3	yes	—
430	C₄²	C₂×D₄×C₅₃	1	3	yes	—
431	C₄²	C₂⁴×C₅₃	1	3	yes	—
432	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
433	C₂³	—	2	3	no	—
434	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
435	Q₈	—	2	1	no	—
436	Q₈	—	2	6	no	—
437	Q₈	—	2	6	no	—
438	Q₈	—	4	2	no	—
439	Q₈	—	8	6	no	—
440	Q₈	—	8	3	no	—
441	Q₈	#137×C₅₃	1	4	yes	complemented
442	Q₈	H(5,𝔽₂)×C₅₃	1	2	yes	—
443	D₄	—	2	6	no	—
444	D₄	—	2	3	no	—
445	D₄	—	2	3	no	—
446	D₄	—	4	6	no	—
447	D₄	—	4	6	no	—
448	D₄	—	8	3	no	—
449	D₄	—	8	6	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	6	no	—
452	C₂×C₄	—	2	6	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	3	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	3	no	—
458	C₂×C₄	—	2	6	no	—
459	C₂×C₄	—	4	6	no	—
460	C₂×C₄	—	4	6	no	—
461	C₂×C₄	—	4	6	no	—
462	C₂×C₄	—	8	6	no	—
463	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
464	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
466	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
467	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
468	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
469	C₈	—	2	6	no	—
470	C₈	—	2	2	no	—
471	C₂²	—	2	3	no	—
472	C₂²	—	2	3	no	—
473	C₂²	—	2	3	no	—
474	C₂²	—	4	3	no	—
475	C₂²	—	4	3	no	—
476	C₂²	—	4	3	no	—
477	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
478	C₄	—	2	3	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	3	no	—
482	C₄	—	4	6	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	3	no	—
485	C₄	—	4	2	no	—
486	C₄	C₂×#137×C₅₃	1	6	yes	—
487	C₄	C₅₃×#577	1	2	yes	—
488	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
489	C₄	C₅₃×#585	1	1	yes	characteristic
490	C₂	—	4	3	no	—
491	C₂	—	8	3	no	—
492	C₂	C₅₃×#3488	1	1	yes	characteristic
493	C₂	C₂²×#137×C₅₃	1	1	yes	characteristic
494	C₂	C₂×C₅₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3483	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3503	C₂	semidirect
3. C₅₃×#3392	C₂	semidirect
4. C₅₃×#3363	C₂	semidirect
5. C₅₃×#3329	C₂	semidirect
6. C₅₃×#3278	C₂	semidirect
7. C₅₃×#3254	C₂	semidirect
8. C₅₃×#3206	C₂	semidirect
9. C₅₃×#3107	C₂	semidirect
10. C₅₃×#3039	C₂	semidirect
11. C₅₃×#2991	C₂	semidirect
193. Q₈×C₅₃	#137	semidirect
223. #62002	C₅₃	direct (N × H)
243. #3503	C₂×C₅₃	semidirect
245. #3392	C₂×C₅₃	semidirect
246. #3363	C₂×C₅₃	semidirect
247. #3329	C₂×C₅₃	semidirect
248. #3278	C₂×C₅₃	semidirect
249. #3254	C₂×C₅₃	semidirect
250. #3206	C₂×C₅₃	semidirect
251. #3107	C₂×C₅₃	semidirect
252. #3039	C₂×C₅₃	semidirect
253. #2991	C₂×C₅₃	semidirect
301. C₅₃	#62002	direct (N × H)
441. Q₈	#137×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3503	2
C₅₃×#3483	2
C₅₃×#3392	2
C₅₃×#3363	2
C₅₃×#3329	2
C₅₃×#3278	2
C₅₃×#3254	2
C₅₃×#3206	2
C₅₃×#3107	2
C₅₃×#3039	2
C₅₃×#2991	2
#62002	53

Subgroups — C53×#62002 #27269514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62002 #27269514