Subgroups — C₅₃×#17726 #27225238

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3511	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1554	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1551	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1547	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1543	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1542	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#556	C₄	1	2	yes	—
13	C₅₃×#555	—	2	1	no	—
14	C₅₃×#554	—	2	1	no	—
15	C₅₃×#553	—	2	2	no	—
16	C₅₃×#548	C₄	1	4	yes	—
17	C₅₃×#547	C₄	1	2	yes	—
18	C₅₃×#542	—	2	2	no	—
19	C₅₃×#537	—	2	1	no	—
20	C₅₃×#533	C₂²	1	2	yes	—
21	C₅₃×#532	C₂²	1	2	yes	—
22	C₅₃×#532	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#520	—	2	1	no	—
25	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₅₃	—	2	1	no	—
27	C₈×D₄×C₅₃	—	2	2	no	—
28	C₈×D₄×C₅₃	—	2	1	no	—
29	C₅₃×#433	—	2	2	no	—
30	C₅₃×#432	C₂²	1	2	yes	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	2	yes	—
33	C₅₃×#431	C₂²	1	2	yes	—
34	C₅₃×#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#408	—	2	1	no	—
39	C₅₃×#405	—	2	1	no	—
40	C₅₃×#333	C₂²	1	2	yes	—
41	C₅₃×#333	C₂²	1	2	yes	—
42	C₅₃×#333	C₂²	1	2	yes	—
43	C₅₃×#333	C₂²	1	2	yes	—
44	C₅₃×#332	C₂²	1	1	yes	characteristic
45	C₅₃×#332	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₅₃×#331	C₂²	1	1	yes	characteristic
48	C₅₃×#331	C₂²	1	1	yes	characteristic
49	C₅₃×#331	C₂²	1	2	yes	—
50	C₂×#41×C₅₃	—	2	1	no	—
51	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₅₃	—	2	1	no	—
55	C₂²×C₈×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	#127×C₅₃	C₂³	1	1	yes	characteristic
64	#126×C₅₃	—	2	4	no	—
65	#125×C₅₃	—	2	2	no	—
66	#124×C₅₃	—	2	4	no	—
67	#124×C₅₃	—	2	1	no	—
68	#124×C₅₃	C₂×C₄	1	4	yes	—
69	#123×C₅₃	—	2	4	no	—
70	C₂²⋊Q₈×C₅₃	—	2	4	no	—
71	C₂²⋊Q₈×C₅₃	—	2	2	no	—
72	C₂²⋊Q₈×C₅₃	—	2	2	no	—
73	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
74	#121×C₅₃	—	2	2	no	—
75	#121×C₅₃	—	2	4	no	—
76	#121×C₅₃	—	2	2	no	—
77	#121×C₅₃	—	2	1	no	—
78	#121×C₅₃	—	2	2	no	—
79	#121×C₅₃	C₂×C₄	1	2	yes	—
80	C₄×Q₈×C₅₃	—	2	2	no	—
81	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
84	C₄×D₄×C₅₃	—	2	4	no	—
85	C₄×D₄×C₅₃	—	2	2	no	—
86	C₄×D₄×C₅₃	—	2	2	no	—
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
91	C₄×D₄×C₅₃	D₄	1	2	yes	—
92	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
93	#117×C₅₃	—	2	2	no	—
94	#117×C₅₃	—	2	2	no	—
95	#117×C₅₃	D₄	1	1	yes	characteristic
96	#117×C₅₃	D₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
102	C₄⋊C₈×C₅₃	—	2	1	no	—
103	C₄⋊C₈×C₅₃	—	2	2	no	—
104	C₄⋊C₈×C₅₃	—	2	1	no	—
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
108	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₂²⋊C₈×C₅₃	—	2	2	no	—
113	C₂²⋊C₈×C₅₃	—	2	2	no	—
114	C₂²⋊C₈×C₅₃	—	2	1	no	—
115	C₂²⋊C₈×C₅₃	—	2	2	no	—
116	#97×C₅₃	—	2	2	no	—
117	C₄×C₈×C₅₃	—	2	2	no	—
118	C₄×C₈×C₅₃	—	2	2	no	—
119	C₄×C₈×C₅₃	—	2	1	no	—
120	#41×C₅₃	—	2	2	no	—
121	#41×C₅₃	—	2	2	no	—
122	#41×C₅₃	—	4	2	no	—
123	#41×C₅₃	—	4	2	no	—
124	#41×C₅₃	—	8	4	no	—
125	C₂×Q₈×C₅₃	—	2	2	no	—
126	C₂×Q₈×C₅₃	—	2	1	no	—
127	C₂×Q₈×C₅₃	—	8	2	no	—
128	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₅₃	—	2	2	no	—
131	C₂×D₄×C₅₃	—	2	4	no	—
132	C₂×D₄×C₅₃	—	2	1	no	—
133	C₂×D₄×C₅₃	—	2	1	no	—
134	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
135	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₅₃	—	2	2	no	—
138	C₂²×C₄×C₅₃	—	2	2	no	—
139	C₂²×C₄×C₅₃	—	2	4	no	—
140	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	M₁₆×C₅₃	—	4	2	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	2	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	4	2	no	—
154	C₂×C₈×C₅₃	—	8	1	no	—
155	C₂×C₈×C₅₃	—	8	2	no	—
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	—	2	4	no	—
158	C₄⋊C₄×C₅₃	—	2	2	no	—
159	C₄⋊C₄×C₅₃	—	2	4	no	—
160	C₄⋊C₄×C₅₃	—	2	2	no	—
161	C₄⋊C₄×C₅₃	—	2	2	no	—
162	C₄⋊C₄×C₅₃	—	2	1	no	—
163	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
168	C₂²⋊C₄×C₅₃	—	2	4	no	—
169	C₂²⋊C₄×C₅₃	—	2	4	no	—
170	C₂²⋊C₄×C₅₃	—	2	4	no	—
171	C₂²⋊C₄×C₅₃	—	2	2	no	—
172	C₂²⋊C₄×C₅₃	—	2	2	no	—
173	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
174	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
175	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
176	C₄²×C₅₃	—	2	4	no	—
177	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
179	C₄²×C₅₃	C₂×D₄	1	2	yes	—
180	C₄²×C₅₃	C₂×D₄	1	2	yes	—
181	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂³×C₅₃	—	2	2	no	—
185	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	Q₈×C₅₃	—	2	2	no	—
188	Q₈×C₅₃	—	4	4	no	—
189	Q₈×C₅₃	—	4	1	no	—
190	Q₈×C₅₃	—	4	1	no	—
191	Q₈×C₅₃	—	8	2	no	—
192	D₄×C₅₃	—	2	4	no	—
193	D₄×C₅₃	—	2	2	no	—
194	D₄×C₅₃	—	4	2	no	—
195	D₄×C₅₃	—	4	4	no	—
196	D₄×C₅₃	—	4	1	no	—
197	D₄×C₅₃	—	4	1	no	—
198	D₄×C₅₃	—	4	1	no	—
199	D₄×C₅₃	—	4	1	no	—
200	D₄×C₅₃	—	8	4	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	4	1	no	—
210	C₂×C₄×C₅₃	—	4	2	no	—
211	C₂×C₄×C₅₃	—	4	4	no	—
212	C₂×C₄×C₅₃	—	8	4	no	—
213	C₂×C₄×C₅₃	—	8	2	no	—
214	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	#135	1	2	yes	—
221	C₂×C₄×C₅₃	#135	1	2	yes	—
222	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
225	C₈×C₅₃	—	4	1	no	—
226	C₈×C₅₃	—	4	1	no	—
227	C₈×C₅₃	—	4	2	no	—
228	C₈×C₅₃	—	4	2	no	—
229	C₈×C₅₃	—	4	1	no	—
230	C₈×C₅₃	—	4	1	no	—
231	#17726	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₅₃	—	2	1	no	—
233	C₂²×C₅₃	—	2	1	no	—
234	C₂²×C₅₃	—	2	2	no	—
235	C₂²×C₅₃	—	2	2	no	—
236	C₂²×C₅₃	—	2	2	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	2	no	—
239	C₂²×C₅₃	—	4	2	no	—
240	C₂²×C₅₃	#416	1	2	yes	—
241	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
242	C₄×C₅₃	—	2	1	no	—
243	C₄×C₅₃	—	2	2	no	—
244	C₄×C₅₃	—	2	1	no	—
245	C₄×C₅₃	—	2	2	no	—
246	C₄×C₅₃	—	2	2	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	4	4	no	—
249	C₄×C₅₃	—	4	2	no	—
250	C₄×C₅₃	#416	1	2	yes	—
251	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
252	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
253	C₄×C₅₃	#577	1	1	yes	characteristic
254	C₄×C₅₃	#578	1	1	yes	characteristic
255	#3511	C₂×C₅₃	1	1	yes	characteristic
256	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
257	#2925	C₂×C₅₃	1	1	yes	characteristic, complemented
258	#1554	C₂×C₅₃	1	2	yes	complemented
259	#1551	C₂×C₅₃	1	2	yes	complemented
260	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#1547	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#1543	C₂×C₅₃	1	2	yes	complemented
263	#1542	C₂×C₅₃	1	2	yes	complemented
264	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
266	C₂×C₅₃	—	2	2	no	—
267	C₂×C₅₃	—	4	2	no	—
268	C₂×C₅₃	—	8	2	no	—
269	C₂×C₅₃	#2888	1	1	yes	characteristic
270	C₂×C₅₃	C₂×#416	1	1	yes	characteristic
271	C₂×C₅₃	#2898	1	1	yes	characteristic
272	#556	C₄×C₅₃	1	2	yes	—
273	#555	—	2	1	no	—
274	#554	—	2	1	no	—
275	#553	—	2	2	no	—
276	#548	C₄×C₅₃	1	4	yes	—
277	#547	C₄×C₅₃	1	2	yes	—
278	#542	—	2	2	no	—
279	#537	—	2	1	no	—
280	#533	C₂²×C₅₃	1	2	yes	—
281	#532	C₂²×C₅₃	1	2	yes	—
282	#532	C₂²×C₅₃	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
284	#520	—	2	1	no	—
285	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
286	C₈×Q₈	—	2	1	no	—
287	C₈×D₄	—	2	2	no	—
288	C₈×D₄	—	2	1	no	—
289	#433	—	2	2	no	—
290	#432	C₂²×C₅₃	1	2	yes	—
291	#431	C₂²×C₅₃	1	1	yes	characteristic
292	#431	C₂²×C₅₃	1	2	yes	—
293	#431	C₂²×C₅₃	1	2	yes	—
294	#423	C₂²×C₅₃	1	1	yes	characteristic
295	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
296	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
298	#408	—	2	1	no	—
299	#405	—	2	1	no	—
300	#333	C₂²×C₅₃	1	2	yes	—
301	#333	C₂²×C₅₃	1	2	yes	—
302	#333	C₂²×C₅₃	1	2	yes	—
303	#333	C₂²×C₅₃	1	2	yes	—
304	#332	C₂²×C₅₃	1	1	yes	characteristic
305	#332	C₂²×C₅₃	1	1	yes	characteristic
306	#332	C₂²×C₅₃	1	2	yes	—
307	#331	C₂²×C₅₃	1	1	yes	characteristic
308	#331	C₂²×C₅₃	1	1	yes	characteristic
309	#331	C₂²×C₅₃	1	2	yes	—
310	C₅₃	#17726	1	1	yes	characteristic, direct product factor, Sylow
311	C₂×#41	—	2	1	no	—
312	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
313	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
314	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	1	no	—
320	C₄⋊Q₈	—	2	1	no	—
321	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
322	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
323	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
324	#127	C₂³×C₅₃	1	1	yes	characteristic
325	#126	—	2	4	no	—
326	#125	—	2	2	no	—
327	#124	—	2	4	no	—
328	#124	—	2	1	no	—
329	#124	C₂×C₄×C₅₃	1	4	yes	—
330	#123	—	2	4	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
335	#121	—	2	2	no	—
336	#121	—	2	4	no	—
337	#121	—	2	2	no	—
338	#121	—	2	1	no	—
339	#121	—	2	2	no	—
340	#121	C₂×C₄×C₅₃	1	2	yes	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
343	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
344	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
350	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
351	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
352	C₄×D₄	D₄×C₅₃	1	2	yes	—
353	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
354	#117	—	2	2	no	—
355	#117	—	2	2	no	—
356	#117	D₄×C₅₃	1	1	yes	characteristic
357	#117	D₄×C₅₃	1	2	yes	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
363	C₄⋊C₈	—	2	1	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
368	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
369	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
372	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
373	C₂²⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	—	2	2	no	—
375	C₂²⋊C₈	—	2	1	no	—
376	C₂²⋊C₈	—	2	2	no	—
377	#97	—	2	2	no	—
378	C₄×C₈	—	2	2	no	—
379	C₄×C₈	—	2	2	no	—
380	C₄×C₈	—	2	1	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	4	2	no	—
384	#41	—	4	2	no	—
385	#41	—	8	4	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	8	2	no	—
389	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
390	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
396	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
397	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	4	2	no	—
415	C₂×C₈	—	8	1	no	—
416	C₂×C₈	—	8	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
425	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
426	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
427	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
428	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
435	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
436	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
437	C₄²	—	2	4	no	—
438	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
439	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
440	C₄²	C₂×D₄×C₅₃	1	2	yes	—
441	C₄²	C₂×D₄×C₅₃	1	2	yes	—
442	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
444	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
445	C₂³	—	2	2	no	—
446	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
447	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
448	Q₈	—	2	2	no	—
449	Q₈	—	4	4	no	—
450	Q₈	—	4	1	no	—
451	Q₈	—	4	1	no	—
452	Q₈	—	8	2	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	4	no	—
457	D₄	—	4	1	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	8	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	8	4	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
476	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
480	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
481	C₂×C₄	#135×C₅₃	1	2	yes	—
482	C₂×C₄	#135×C₅₃	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
486	C₈	—	4	1	no	—
487	C₈	—	4	1	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	C₅₃×#416	1	2	yes	—
501	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
502	C₄	—	2	1	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	2	no	—
510	C₄	C₅₃×#416	1	2	yes	—
511	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
512	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
513	C₄	C₅₃×#577	1	1	yes	characteristic
514	C₄	C₅₃×#578	1	1	yes	characteristic
515	C₂	—	2	2	no	—
516	C₂	—	4	2	no	—
517	C₂	—	8	2	no	—
518	C₂	C₅₃×#2888	1	1	yes	characteristic
519	C₂	C₂×C₅₃×#416	1	1	yes	characteristic
520	C₂	C₅₃×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₅₃×#2988	C₂	semidirect
3. C₅₃×#2925	C₂	semidirect
4. C₅₃×#1554	C₂	semidirect
5. C₅₃×#1551	C₂	semidirect
6. C₅₃×#1548	C₂	semidirect
7. C₅₃×#1547	C₂	semidirect
8. C₅₃×#1543	C₂	semidirect
9. C₅₃×#1542	C₂	semidirect
10. C₂×C₅₃×#333	C₂	semidirect
11. C₅₃×#1540	C₂	semidirect
231. #17726	C₅₃	direct (N × H)
256. #2988	C₂×C₅₃	semidirect
257. #2925	C₂×C₅₃	semidirect
258. #1554	C₂×C₅₃	semidirect
259. #1551	C₂×C₅₃	semidirect
260. #1548	C₂×C₅₃	semidirect
261. #1547	C₂×C₅₃	semidirect
262. #1543	C₂×C₅₃	semidirect
263. #1542	C₂×C₅₃	semidirect
264. C₂×#333	C₂×C₅₃	semidirect
265. #1540	C₂×C₅₃	semidirect
310. C₅₃	#17726	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×#3511	2
C₅₃×#2988	2
C₅₃×#2925	2
C₅₃×#1554	2
C₅₃×#1551	2
C₅₃×#1548	2
C₅₃×#1547	2
C₅₃×#1543	2
C₅₃×#1542	2
C₂×C₅₃×#333	2
C₅₃×#1540	2
#17726	53

Subgroups — C53×#17726 #27225238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17726 #27225238