Subgroups — C₅₃×#61413 #27268925

Order 13568.

1886 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3510	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3401	C₂	1	8	yes	maximal, complemented
3	C₅₃×#3396	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3393	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3155	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3043	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3042	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3040	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#137×C₅₃	C₂²	1	2	yes	—
16	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
17	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#559	C₂²	1	8	yes	complemented
22	C₅₃×#556	C₂²	1	8	yes	complemented
23	C₅₃×#551	—	2	8	no	—
24	C₅₃×#542	—	2	4	no	—
25	C₅₃×#536	C₂²	1	2	yes	—
26	C₅₃×#535	C₂²	1	2	yes	—
27	C₅₃×#533	C₂²	1	1	yes	characteristic
28	C₅₃×#529	—	2	4	no	—
29	C₂×#124×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#519	C₂²	1	2	yes	—
31	C₅₃×#518	C₂²	1	2	yes	—
32	C₄×#41×C₅₃	C₂²	1	2	yes	—
33	C₅₃×#497	C₂²	1	4	yes	complemented
34	C₅₃×#496	C₂²	1	4	yes	—
35	C₅₃×#495	C₂²	1	2	yes	—
36	C₅₃×#495	C₂²	1	4	yes	—
37	C₅₃×#495	C₂²	1	4	yes	—
38	C₅₃×#495	C₂²	1	2	yes	complemented
39	C₅₃×#494	C₂²	1	2	yes	complemented
40	C₅₃×#493	C₂²	1	2	yes	—
41	C₅₃×#464	C₂²	1	8	yes	complemented
42	C₅₃×#464	C₂²	1	8	yes	complemented
43	C₅₃×#463	C₂²	1	8	yes	complemented
44	C₅₃×#463	C₂²	1	8	yes	complemented
45	C₅₃×#461	C₂²	1	8	yes	complemented
46	C₅₃×#460	C₂²	1	8	yes	complemented
47	C₅₃×#446	C₂²	1	4	yes	complemented
48	C₈×Q₈×C₅₃	C₂²	1	4	yes	complemented
49	C₅₃×#442	C₂²	1	4	yes	—
50	C₅₃×#441	C₂²	1	4	yes	complemented
51	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
52	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
53	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
54	C₄×D₈×C₅₃	C₂²	1	4	yes	—
55	C₄×D₈×C₅₃	C₂²	1	2	yes	complemented
56	C₅₃×#427	C₂²	1	1	yes	characteristic
57	C₅₃×#424	C₂²	1	1	yes	characteristic
58	C₅₃×#424	C₂²	1	1	yes	characteristic
59	C₅₃×#423	C₂²	1	1	yes	characteristic
60	C₅₃×#419	C₂²	1	2	yes	—
61	C₅₃×#418	C₂²	1	2	yes	—
62	C₅₃×#416	C₂²	1	2	yes	—
63	C₅₃×#405	C₂²	1	1	yes	characteristic
64	C₅₃×#405	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
66	C₂×#41×C₅₃	C₂³	1	2	yes	—
67	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
69	#137×C₅₃	—	4	4	no	—
70	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
71	#135×C₅₃	—	2	4	no	—
72	C₂×Q₁₆×C₅₃	—	2	4	no	—
73	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₅₃	—	4	4	no	—
78	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
82	C₂×D₈×C₅₃	—	2	4	no	—
83	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×D₈×C₅₃	C₂³	1	2	yes	—
85	C₂×D₈×C₅₃	C₂³	1	2	yes	—
86	C₂×D₈×C₅₃	C₂³	1	4	yes	—
87	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
92	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
93	#127×C₅₃	C₂³	1	4	yes	—
94	#126×C₅₃	—	2	8	no	—
95	#126×C₅₃	—	2	8	no	—
96	#125×C₅₃	D₄	1	2	yes	—
97	#124×C₅₃	—	2	2	no	—
98	#124×C₅₃	—	2	2	no	—
99	#124×C₅₃	—	2	8	no	—
100	#124×C₅₃	C₂³	1	8	yes	—
101	#124×C₅₃	C₂³	1	4	yes	—
102	#124×C₅₃	C₂³	1	4	yes	—
103	#123×C₅₃	—	2	8	no	—
104	#123×C₅₃	—	2	8	no	—
105	#123×C₅₃	—	2	4	no	—
106	C₂²⋊Q₈×C₅₃	—	2	4	no	—
107	C₂²⋊Q₈×C₅₃	—	2	8	no	—
108	C₂²⋊Q₈×C₅₃	D₄	1	4	yes	—
109	#121×C₅₃	—	2	4	no	—
110	#120×C₅₃	—	2	4	no	—
111	C₄×Q₈×C₅₃	—	2	4	no	—
112	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
113	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
114	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
115	C₄×D₄×C₅₃	—	2	4	no	—
116	C₄×D₄×C₅₃	—	2	8	no	—
117	C₄×D₄×C₅₃	—	2	4	no	—
118	C₄×D₄×C₅₃	C₂³	1	4	yes	—
119	C₄×D₄×C₅₃	C₂³	1	8	yes	—
120	#117×C₅₃	—	2	8	no	—
121	#117×C₅₃	—	2	4	no	—
122	#117×C₅₃	—	2	2	no	—
123	#117×C₅₃	C₂³	1	2	yes	—
124	#117×C₅₃	C₂³	1	1	yes	characteristic
125	#117×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
127	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
128	C₂×C₄²×C₅₃	—	2	2	no	—
129	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
130	#107×C₅₃	C₂³	1	2	yes	—
131	#106×C₅₃	C₂³	1	2	yes	—
132	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
133	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
134	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
135	#103×C₅₃	C₂³	1	4	yes	—
136	#103×C₅₃	C₂³	1	8	yes	—
137	#102×C₅₃	C₂³	1	4	yes	—
138	#102×C₅₃	C₂³	1	8	yes	—
139	#97×C₅₃	C₂³	1	2	yes	—
140	#97×C₅₃	C₂³	1	2	yes	—
141	C₄×C₈×C₅₃	C₂³	1	2	yes	—
142	C₄×C₈×C₅₃	C₂³	1	2	yes	—
143	C₄×C₈×C₅₃	C₂³	1	2	yes	—
144	C₄×C₈×C₅₃	C₂³	1	2	yes	—
145	C₂⁴×C₅₃	—	2	2	no	—
146	#41×C₅₃	—	2	4	no	—
147	#41×C₅₃	—	2	4	no	—
148	C₂×Q₈×C₅₃	—	2	2	no	—
149	C₂×Q₈×C₅₃	—	2	4	no	—
150	C₂×Q₈×C₅₃	—	2	4	no	—
151	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
152	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
153	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
154	C₂×D₄×C₅₃	—	2	2	no	—
155	C₂×D₄×C₅₃	—	2	2	no	—
156	C₂×D₄×C₅₃	—	2	8	no	—
157	C₂×D₄×C₅₃	—	2	4	no	—
158	C₂×D₄×C₅₃	—	2	4	no	—
159	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
160	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	—	2	2	no	—
162	C₂²×C₄×C₅₃	—	2	2	no	—
163	C₂²×C₄×C₅₃	—	2	4	no	—
164	C₂²×C₄×C₅₃	—	2	8	no	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₅₃	—	2	4	no	—
168	Q₁₆×C₅₃	—	2	4	no	—
169	Q₁₆×C₅₃	—	2	2	no	—
170	Q₁₆×C₅₃	—	4	4	no	—
171	SD₁₆×C₅₃	—	2	4	no	—
172	SD₁₆×C₅₃	—	4	8	no	—
173	SD₁₆×C₅₃	—	4	4	no	—
174	SD₁₆×C₅₃	—	4	4	no	—
175	D₈×C₅₃	—	2	4	no	—
176	D₈×C₅₃	—	2	4	no	—
177	D₈×C₅₃	—	2	2	no	—
178	D₈×C₅₃	—	4	4	no	—
179	M₁₆×C₅₃	—	2	2	no	—
180	M₁₆×C₅₃	—	2	2	no	—
181	C₂×C₈×C₅₃	—	2	2	no	—
182	C₂×C₈×C₅₃	#41	1	4	yes	—
183	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
184	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
185	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
186	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	2	8	no	—
190	C₄⋊C₄×C₅₃	—	2	8	no	—
191	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
193	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₅₃	—	2	2	no	—
198	C₂²⋊C₄×C₅₃	—	2	8	no	—
199	C₂²⋊C₄×C₅₃	—	2	8	no	—
200	C₂²⋊C₄×C₅₃	—	2	8	no	—
201	C₂²⋊C₄×C₅₃	—	2	4	no	—
202	C₂²⋊C₄×C₅₃	—	2	8	no	—
203	C₂²⋊C₄×C₅₃	—	2	8	no	—
204	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
205	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
206	C₄²×C₅₃	—	2	4	no	—
207	C₄²×C₅₃	—	2	2	no	—
208	C₄²×C₅₃	—	2	8	no	—
209	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₅₃	C₂⁴	1	2	yes	—
212	C₄²×C₅₃	C₂⁴	1	2	yes	—
213	C₄²×C₅₃	C₂⁴	1	2	yes	—
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	—	2	4	no	—
216	C₂³×C₅₃	—	4	4	no	—
217	C₂³×C₅₃	—	8	4	no	—
218	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₅₃	—	2	4	no	—
220	Q₈×C₅₃	—	2	2	no	—
221	Q₈×C₅₃	—	2	4	no	—
222	Q₈×C₅₃	—	2	2	no	—
223	Q₈×C₅₃	—	4	4	no	—
224	Q₈×C₅₃	—	4	4	no	—
225	D₄×C₅₃	—	2	4	no	—
226	D₄×C₅₃	—	2	2	no	—
227	D₄×C₅₃	—	2	4	no	—
228	D₄×C₅₃	—	2	2	no	—
229	D₄×C₅₃	—	4	4	no	—
230	D₄×C₅₃	—	4	4	no	—
231	D₄×C₅₃	—	8	8	no	—
232	C₂×C₄×C₅₃	—	2	4	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	4	no	—
235	C₂×C₄×C₅₃	—	4	2	no	—
236	C₂×C₄×C₅₃	—	4	4	no	—
237	C₂×C₄×C₅₃	—	4	4	no	—
238	C₂×C₄×C₅₃	—	8	4	no	—
239	C₂×C₄×C₅₃	—	8	8	no	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
246	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
248	C₈×C₅₃	—	2	2	no	—
249	C₈×C₅₃	—	2	2	no	—
250	C₈×C₅₃	—	2	2	no	—
251	C₈×C₅₃	C₂×#41	1	2	yes	—
252	C₈×C₅₃	C₂×#41	1	2	yes	—
253	#61413	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂²×C₅₃	—	2	2	no	—
255	C₂²×C₅₃	—	4	4	no	—
256	C₂²×C₅₃	—	4	8	no	—
257	C₂²×C₅₃	—	4	2	no	—
258	C₂²×C₅₃	—	4	4	no	—
259	C₂²×C₅₃	—	16	4	no	—
260	C₂²×C₅₃	#550	1	2	yes	—
261	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	2	4	no	—
265	C₄×C₅₃	—	2	4	no	—
266	C₄×C₅₃	—	4	4	no	—
267	C₄×C₅₃	—	4	2	no	—
268	C₄×C₅₃	—	4	4	no	—
269	C₄×C₅₃	#550	1	2	yes	—
270	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
271	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
272	#3510	C₂×C₅₃	1	2	yes	complemented
273	#3401	C₂×C₅₃	1	8	yes	complemented
274	#3396	C₂×C₅₃	1	4	yes	complemented
275	#3393	C₂×C₅₃	1	4	yes	complemented
276	#3156	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#3155	C₂×C₅₃	1	1	yes	characteristic
278	C₂×#495	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#3043	C₂×C₅₃	1	2	yes	complemented
280	#3042	C₂×C₅₃	1	2	yes	complemented
281	#3040	C₂×C₅₃	1	2	yes	complemented
282	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
283	#2950	C₂×C₅₃	1	1	yes	characteristic, complemented
284	#2948	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₄×#135	C₂×C₅₃	1	1	yes	characteristic, complemented
286	C₂×C₅₃	—	2	2	no	—
287	C₂×C₅₃	—	8	2	no	—
288	C₂×C₅₃	—	8	4	no	—
289	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
290	C₂×C₅₃	#3584	1	2	yes	—
291	C₂×#137	C₂²×C₅₃	1	2	yes	—
292	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
293	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
294	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
295	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
296	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
297	#559	C₂²×C₅₃	1	8	yes	complemented
298	#556	C₂²×C₅₃	1	8	yes	complemented
299	#551	—	2	8	no	—
300	#542	—	2	4	no	—
301	#536	C₂²×C₅₃	1	2	yes	—
302	#535	C₂²×C₅₃	1	2	yes	—
303	#533	C₂²×C₅₃	1	1	yes	characteristic
304	#529	—	2	4	no	—
305	C₂×#124	C₂²×C₅₃	1	2	yes	—
306	#519	C₂²×C₅₃	1	2	yes	—
307	#518	C₂²×C₅₃	1	2	yes	—
308	C₄×#41	C₂²×C₅₃	1	2	yes	—
309	#497	C₂²×C₅₃	1	4	yes	complemented
310	#496	C₂²×C₅₃	1	4	yes	—
311	#495	C₂²×C₅₃	1	2	yes	—
312	#495	C₂²×C₅₃	1	4	yes	—
313	#495	C₂²×C₅₃	1	4	yes	—
314	#495	C₂²×C₅₃	1	2	yes	complemented
315	#494	C₂²×C₅₃	1	2	yes	complemented
316	#493	C₂²×C₅₃	1	2	yes	—
317	#464	C₂²×C₅₃	1	8	yes	complemented
318	#464	C₂²×C₅₃	1	8	yes	complemented
319	#463	C₂²×C₅₃	1	8	yes	complemented
320	#463	C₂²×C₅₃	1	8	yes	complemented
321	#461	C₂²×C₅₃	1	8	yes	complemented
322	#460	C₂²×C₅₃	1	8	yes	complemented
323	#446	C₂²×C₅₃	1	4	yes	complemented
324	C₈×Q₈	C₂²×C₅₃	1	4	yes	complemented
325	#442	C₂²×C₅₃	1	4	yes	—
326	#441	C₂²×C₅₃	1	4	yes	complemented
327	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	complemented
328	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
329	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
330	C₄×D₈	C₂²×C₅₃	1	4	yes	—
331	C₄×D₈	C₂²×C₅₃	1	2	yes	complemented
332	#427	C₂²×C₅₃	1	1	yes	characteristic
333	#424	C₂²×C₅₃	1	1	yes	characteristic
334	#424	C₂²×C₅₃	1	1	yes	characteristic
335	#423	C₂²×C₅₃	1	1	yes	characteristic
336	#419	C₂²×C₅₃	1	2	yes	—
337	#418	C₂²×C₅₃	1	2	yes	—
338	#416	C₂²×C₅₃	1	2	yes	—
339	#405	C₂²×C₅₃	1	1	yes	characteristic
340	#405	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
342	C₅₃	#61413	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	C₂³×C₅₃	1	2	yes	—
344	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
345	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
346	#137	—	4	4	no	—
347	AGL(1,ℤ/8)	—	4	4	no	—
348	#135	—	2	4	no	—
349	C₂×Q₁₆	—	2	4	no	—
350	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
351	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
352	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
353	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
354	C₂×SD₁₆	—	4	4	no	—
355	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
356	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
357	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
359	C₂×D₈	—	2	4	no	—
360	C₂×D₈	C₂³×C₅₃	1	2	yes	—
361	C₂×D₈	C₂³×C₅₃	1	2	yes	—
362	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
363	C₂×D₈	C₂³×C₅₃	1	4	yes	—
364	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
365	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
368	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
370	#127	C₂³×C₅₃	1	4	yes	—
371	#126	—	2	8	no	—
372	#126	—	2	8	no	—
373	#125	D₄×C₅₃	1	2	yes	—
374	#124	—	2	8	no	—
375	#124	—	2	2	no	—
376	#124	—	2	2	no	—
377	#124	C₂³×C₅₃	1	8	yes	—
378	#124	C₂³×C₅₃	1	4	yes	—
379	#124	C₂³×C₅₃	1	4	yes	—
380	#123	—	2	8	no	—
381	#123	—	2	8	no	—
382	#123	—	2	4	no	—
383	C₂²⋊Q₈	—	2	8	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₂²⋊Q₈	D₄×C₅₃	1	4	yes	—
386	#121	—	2	4	no	—
387	#120	—	2	4	no	—
388	C₄×Q₈	—	2	4	no	—
389	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
390	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
391	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	C₂³×C₅₃	1	4	yes	—
396	C₄×D₄	C₂³×C₅₃	1	8	yes	—
397	#117	—	2	8	no	—
398	#117	—	2	4	no	—
399	#117	—	2	2	no	—
400	#117	C₂³×C₅₃	1	2	yes	—
401	#117	C₂³×C₅₃	1	1	yes	characteristic
402	#117	C₂³×C₅₃	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	4	no	—
405	C₂×C₄²	—	2	2	no	—
406	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
407	#107	C₂³×C₅₃	1	2	yes	—
408	#106	C₂³×C₅₃	1	2	yes	—
409	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
410	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
411	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
412	#103	C₂³×C₅₃	1	4	yes	—
413	#103	C₂³×C₅₃	1	8	yes	—
414	#102	C₂³×C₅₃	1	4	yes	—
415	#102	C₂³×C₅₃	1	8	yes	—
416	#97	C₂³×C₅₃	1	2	yes	—
417	#97	C₂³×C₅₃	1	2	yes	—
418	C₄×C₈	C₂³×C₅₃	1	2	yes	—
419	C₄×C₈	C₂³×C₅₃	1	2	yes	—
420	C₄×C₈	C₂³×C₅₃	1	2	yes	—
421	C₄×C₈	C₂³×C₅₃	1	2	yes	—
422	C₂⁴	—	2	2	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
429	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
430	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
437	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
444	Q₁₆	—	2	4	no	—
445	Q₁₆	—	2	4	no	—
446	Q₁₆	—	2	2	no	—
447	Q₁₆	—	4	4	no	—
448	SD₁₆	—	2	4	no	—
449	SD₁₆	—	4	8	no	—
450	SD₁₆	—	4	4	no	—
451	SD₁₆	—	4	4	no	—
452	D₈	—	2	4	no	—
453	D₈	—	2	4	no	—
454	D₈	—	2	2	no	—
455	D₈	—	4	4	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	#41×C₅₃	1	4	yes	—
460	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
461	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
462	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
463	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
464	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
469	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
470	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
471	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
472	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
473	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
482	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	2	no	—
485	C₄²	—	2	8	no	—
486	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
488	C₄²	C₂⁴×C₅₃	1	2	yes	—
489	C₄²	C₂⁴×C₅₃	1	2	yes	—
490	C₄²	C₂⁴×C₅₃	1	2	yes	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	8	4	no	—
495	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
496	Q₈	—	2	2	no	—
497	Q₈	—	2	4	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	2	2	no	—
500	Q₈	—	4	4	no	—
501	Q₈	—	4	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	4	4	no	—
507	D₄	—	4	4	no	—
508	D₄	—	8	8	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	8	4	no	—
516	C₂×C₄	—	8	8	no	—
517	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
519	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
521	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
523	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
524	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
525	C₈	—	2	2	no	—
526	C₈	—	2	2	no	—
527	C₈	—	2	2	no	—
528	C₈	C₂×#41×C₅₃	1	2	yes	—
529	C₈	C₂×#41×C₅₃	1	2	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	16	4	no	—
536	C₂²	C₅₃×#550	1	2	yes	—
537	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
538	C₄	—	2	2	no	—
539	C₄	—	2	2	no	—
540	C₄	—	2	4	no	—
541	C₄	—	2	4	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	4	no	—
545	C₄	C₅₃×#550	1	2	yes	—
546	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
547	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
548	C₂	—	2	2	no	—
549	C₂	—	8	4	no	—
550	C₂	—	8	2	no	—
551	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
552	C₂	C₅₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3155	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₅₃×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3510	C₂	semidirect
2. C₅₃×#3401	C₂	semidirect
3. C₅₃×#3396	C₂	semidirect
4. C₅₃×#3393	C₂	semidirect
5. C₅₃×#3156	C₂	semidirect
7. C₂×C₅₃×#495	C₂	semidirect
8. C₅₃×#3043	C₂	semidirect
9. C₅₃×#3042	C₂	semidirect
10. C₅₃×#3040	C₂	semidirect
11. C₅₃×#2960	C₂	semidirect
12. C₅₃×#2950	C₂	semidirect
13. C₅₃×#2948	C₂	semidirect
14. C₄×#135×C₅₃	C₂	semidirect
21. C₅₃×#559	C₂²	semidirect
22. C₅₃×#556	C₂²	semidirect
33. C₅₃×#497	C₂²	semidirect
38. C₅₃×#495	C₂²	semidirect
39. C₅₃×#494	C₂²	semidirect
41. C₅₃×#464	C₂²	semidirect
42. C₅₃×#464	C₂²	semidirect
43. C₅₃×#463	C₂²	semidirect
44. C₅₃×#463	C₂²	semidirect
45. C₅₃×#461	C₂²	semidirect
46. C₅₃×#460	C₂²	semidirect
47. C₅₃×#446	C₂²	semidirect
48. C₈×Q₈×C₅₃	C₂²	semidirect
50. C₅₃×#441	C₂²	semidirect
51. C₄×Q₁₆×C₅₃	C₂²	semidirect
52. C₄×Q₁₆×C₅₃	C₂²	semidirect
53. C₄×SD₁₆×C₅₃	C₂²	semidirect
55. C₄×D₈×C₅₃	C₂²	semidirect
253. #61413	C₅₃	direct (N × H)
272. #3510	C₂×C₅₃	semidirect
273. #3401	C₂×C₅₃	semidirect
274. #3396	C₂×C₅₃	semidirect
275. #3393	C₂×C₅₃	semidirect
276. #3156	C₂×C₅₃	semidirect
278. C₂×#495	C₂×C₅₃	semidirect
279. #3043	C₂×C₅₃	semidirect
280. #3042	C₂×C₅₃	semidirect
281. #3040	C₂×C₅₃	semidirect
282. #2960	C₂×C₅₃	semidirect
283. #2950	C₂×C₅₃	semidirect
284. #2948	C₂×C₅₃	semidirect
285. C₄×#135	C₂×C₅₃	semidirect
297. #559	C₂²×C₅₃	semidirect
298. #556	C₂²×C₅₃	semidirect
309. #497	C₂²×C₅₃	semidirect
314. #495	C₂²×C₅₃	semidirect
315. #494	C₂²×C₅₃	semidirect
317. #464	C₂²×C₅₃	semidirect
318. #464	C₂²×C₅₃	semidirect
319. #463	C₂²×C₅₃	semidirect
320. #463	C₂²×C₅₃	semidirect
321. #461	C₂²×C₅₃	semidirect
322. #460	C₂²×C₅₃	semidirect
323. #446	C₂²×C₅₃	semidirect
324. C₈×Q₈	C₂²×C₅₃	semidirect
326. #441	C₂²×C₅₃	semidirect
327. C₄×Q₁₆	C₂²×C₅₃	semidirect
328. C₄×Q₁₆	C₂²×C₅₃	semidirect
329. C₄×SD₁₆	C₂²×C₅₃	semidirect
331. C₄×D₈	C₂²×C₅₃	semidirect
342. C₅₃	#61413	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×#3510	2
C₅₃×#3401	2
C₅₃×#3396	2
C₅₃×#3393	2
C₅₃×#3156	2
C₅₃×#3155	2
C₂×C₅₃×#495	2
C₅₃×#3043	2
C₅₃×#3042	2
C₅₃×#3040	2
C₅₃×#2960	2
C₅₃×#2950	2
C₅₃×#2948	2
C₄×#135×C₅₃	2
#61413	53

Subgroups — C53×#61413 #27268925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61413 #27268925