Subgroups — C₅₃×#61321 #27268833

Order 13568.

2126 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3368	C₂	1	6	yes	maximal, complemented
4	C₅₃×#3366	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3362	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3358	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3138	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3077	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3038	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	3	yes	maximal, complemented
12	C₂²×#41×C₅₃	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	3	yes	—
14	Q₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
15	C₅₃×#557	C₂²	1	3	yes	complemented
16	C₅₃×#554	—	2	3	no	—
17	C₅₃×#554	C₂²	1	3	yes	complemented
18	C₅₃×#550	—	2	1	no	—
19	C₅₃×#550	—	2	3	no	—
20	C₅₃×#550	C₂²	1	2	yes	—
21	C₅₃×#550	C₂²	1	6	yes	—
22	D₄×Q₈×C₅₃	—	2	1	no	—
23	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#533	—	2	3	no	—
25	C₅₃×#533	C₂²	1	3	yes	—
26	C₂×#127×C₅₃	C₂²	1	3	yes	—
27	C₂×#121×C₅₃	—	2	3	no	—
28	C₄×#41×C₅₃	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
30	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
32	C₅₃×#490	C₂²	1	12	yes	complemented
33	C₅₃×#486	C₂²	1	6	yes	complemented
34	C₅₃×#486	C₂²	1	6	yes	complemented
35	C₅₃×#478	C₂²	1	12	yes	complemented
36	C₅₃×#476	C₂²	1	6	yes	complemented
37	C₅₃×#474	C₂²	1	6	yes	complemented
38	C₅₃×#460	C₂²	1	6	yes	complemented
39	C₅₃×#460	C₂²	1	6	yes	complemented
40	C₅₃×#459	C₂²	1	6	yes	complemented
41	C₅₃×#459	C₂²	1	6	yes	complemented
42	C₅₃×#446	C₂²	1	6	yes	complemented
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
44	C₅₃×#442	C₂²	1	6	yes	complemented
45	C₅₃×#440	C₂²	1	6	yes	complemented
46	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	complemented
47	C₄×D₈×C₅₃	C₂²	1	6	yes	complemented
48	C₅₃×#428	C₂²	1	3	yes	—
49	C₅₃×#423	C₂²	1	3	yes	—
50	C₅₃×#418	C₂²	1	3	yes	—
51	C₅₃×#418	C₂²	1	6	yes	—
52	C₅₃×#417	C₂²	1	1	yes	characteristic
53	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
54	C₂×#41×C₅₃	—	2	6	no	—
55	C₂×#41×C₅₃	—	2	2	no	—
56	C₂×#41×C₅₃	—	4	8	no	—
57	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₅₃	—	2	1	no	—
59	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	—	2	3	no	—
61	C₂²×D₄×C₅₃	—	2	3	no	—
62	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
63	C₂³×C₄×C₅₃	—	2	3	no	—
64	AGL(1,ℤ/8)×C₅₃	—	2	12	no	—
65	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
66	C₂×D₈×C₅₃	C₂³	1	6	yes	—
67	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
68	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₅₃	—	2	3	no	—
70	C₄⋊Q₈×C₅₃	—	2	3	no	—
71	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
73	#127×C₅₃	—	2	3	no	—
74	#127×C₅₃	—	2	3	no	—
75	#127×C₅₃	C₂³	1	6	yes	—
76	#127×C₅₃	C₂³	1	6	yes	—
77	#125×C₅₃	—	2	6	no	—
78	#125×C₅₃	C₂³	1	6	yes	—
79	C₂²⋊Q₈×C₅₃	—	2	3	no	—
80	C₂²⋊Q₈×C₅₃	—	2	6	no	—
81	C₂²⋊Q₈×C₅₃	—	2	3	no	—
82	C₂²⋊Q₈×C₅₃	—	2	3	no	—
83	C₂²⋊Q₈×C₅₃	—	2	3	no	—
84	#121×C₅₃	—	2	6	no	—
85	#121×C₅₃	—	2	12	no	—
86	#121×C₅₃	—	2	6	no	—
87	C₄×Q₈×C₅₃	—	2	3	no	—
88	C₄×Q₈×C₅₃	—	2	3	no	—
89	C₄×Q₈×C₅₃	—	2	3	no	—
90	C₄×Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₅₃	D₄	1	3	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
95	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
96	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	3	no	—
99	C₄×D₄×C₅₃	—	2	3	no	—
100	C₄×D₄×C₅₃	—	2	3	no	—
101	C₄×D₄×C₅₃	C₂³	1	6	yes	—
102	C₄×D₄×C₅₃	C₂³	1	6	yes	—
103	C₄×D₄×C₅₃	C₂³	1	3	yes	—
104	C₄×D₄×C₅₃	C₂³	1	3	yes	—
105	#117×C₅₃	—	2	3	no	—
106	#117×C₅₃	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	—	2	3	no	—
112	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
113	#107×C₅₃	C₂³	1	6	yes	—
114	#106×C₅₃	C₂³	1	6	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
117	#103×C₅₃	C₂³	1	2	yes	—
118	#102×C₅₃	C₂³	1	6	yes	—
119	#102×C₅₃	C₂³	1	6	yes	—
120	#102×C₅₃	C₂³	1	12	yes	—
121	#102×C₅₃	C₂³	1	12	yes	—
122	#102×C₅₃	C₂³	1	2	yes	—
123	#97×C₅₃	C₂³	1	6	yes	—
124	C₄×C₈×C₅₃	C₂³	1	6	yes	—
125	C₂⁴×C₅₃	—	2	3	no	—
126	#41×C₅₃	—	2	2	no	—
127	#41×C₅₃	—	4	24	no	—
128	#41×C₅₃	—	4	8	no	—
129	#41×C₅₃	#41	1	4	yes	complemented
130	C₂×Q₈×C₅₃	—	2	4	no	—
131	C₂×Q₈×C₅₃	—	2	3	no	—
132	C₂×Q₈×C₅₃	—	2	3	no	—
133	C₂×Q₈×C₅₃	—	4	4	no	—
134	C₂×Q₈×C₅₃	C₂×D₄	1	3	yes	—
135	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₅₃	—	2	1	no	—
138	C₂×D₄×C₅₃	—	2	3	no	—
139	C₂×D₄×C₅₃	—	2	6	no	—
140	C₂×D₄×C₅₃	—	2	6	no	—
141	C₂×D₄×C₅₃	—	2	6	no	—
142	C₂×D₄×C₅₃	—	2	6	no	—
143	C₂×D₄×C₅₃	—	2	6	no	—
144	C₂×D₄×C₅₃	—	2	6	no	—
145	C₂×D₄×C₅₃	—	4	12	no	—
146	C₂×D₄×C₅₃	—	8	6	no	—
147	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
148	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂²×C₄×C₅₃	—	2	3	no	—
152	C₂²×C₄×C₅₃	—	2	6	no	—
153	C₂²×C₄×C₅₃	—	2	6	no	—
154	C₂²×C₄×C₅₃	—	2	3	no	—
155	C₂²×C₄×C₅₃	—	4	12	no	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
159	SD₁₆×C₅₃	—	2	12	no	—
160	D₈×C₅₃	—	2	12	no	—
161	M₁₆×C₅₃	—	2	6	no	—
162	M₁₆×C₅₃	—	2	2	no	—
163	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
164	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
165	C₄⋊C₄×C₅₃	—	2	6	no	—
166	C₄⋊C₄×C₅₃	—	2	3	no	—
167	C₄⋊C₄×C₅₃	—	2	1	no	—
168	C₄⋊C₄×C₅₃	—	2	3	no	—
169	C₄⋊C₄×C₅₃	—	2	3	no	—
170	C₄⋊C₄×C₅₃	—	2	3	no	—
171	C₄⋊C₄×C₅₃	—	2	6	no	—
172	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
174	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
177	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
178	C₂²⋊C₄×C₅₃	—	2	3	no	—
179	C₂²⋊C₄×C₅₃	—	2	6	no	—
180	C₂²⋊C₄×C₅₃	—	2	6	no	—
181	C₂²⋊C₄×C₅₃	—	2	3	no	—
182	C₂²⋊C₄×C₅₃	—	2	3	no	—
183	C₄²×C₅₃	—	2	3	no	—
184	C₄²×C₅₃	—	2	3	no	—
185	C₄²×C₅₃	—	2	3	no	—
186	C₄²×C₅₃	C₂×D₄	1	3	yes	—
187	C₄²×C₅₃	C₂×D₄	1	3	yes	—
188	C₄²×C₅₃	C₂⁴	1	3	yes	—
189	C₄²×C₅₃	C₂⁴	1	3	yes	—
190	C₂³×C₅₃	—	2	1	no	—
191	C₂³×C₅₃	—	2	6	no	—
192	C₂³×C₅₃	—	4	6	no	—
193	C₂³×C₅₃	—	4	6	no	—
194	C₂³×C₅₃	—	4	6	no	—
195	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₅₃	—	2	6	no	—
197	Q₈×C₅₃	—	2	1	no	—
198	Q₈×C₅₃	—	4	6	no	—
199	Q₈×C₅₃	—	8	3	no	—
200	Q₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
201	Q₈×C₅₃	C₂×#41	1	2	yes	complemented
202	D₄×C₅₃	—	2	4	no	—
203	D₄×C₅₃	—	2	6	no	—
204	D₄×C₅₃	—	2	1	no	—
205	D₄×C₅₃	—	2	6	no	—
206	D₄×C₅₃	—	4	12	no	—
207	D₄×C₅₃	—	4	6	no	—
208	D₄×C₅₃	—	4	12	no	—
209	D₄×C₅₃	—	8	6	no	—
210	D₄×C₅₃	—	8	6	no	—
211	D₄×C₅₃	C₂×#41	1	2	yes	complemented
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	6	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	3	no	—
217	C₂×C₄×C₅₃	—	2	3	no	—
218	C₂×C₄×C₅₃	—	4	3	no	—
219	C₂×C₄×C₅₃	—	4	12	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	12	no	—
222	C₂×C₄×C₅₃	—	4	6	no	—
223	C₂×C₄×C₅₃	—	8	3	no	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
230	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
231	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
232	C₈×C₅₃	—	2	6	no	—
233	C₈×C₅₃	—	2	2	no	—
234	#61321	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂²×C₅₃	—	2	1	no	—
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	2	no	—
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	—	4	6	no	—
241	C₂²×C₅₃	—	4	6	no	—
242	C₂²×C₅₃	—	4	6	no	—
243	C₂²×C₅₃	—	8	12	no	—
244	C₂²×C₅₃	#548	1	2	yes	—
245	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₅₃	—	2	3	no	—
247	C₄×C₅₃	—	2	3	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	3	no	—
250	C₄×C₅₃	—	4	3	no	—
251	C₄×C₅₃	—	4	2	no	—
252	C₄×C₅₃	—	4	3	no	—
253	C₄×C₅₃	#548	1	2	yes	—
254	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
255	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
256	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
257	Q₈×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
258	C₂×#550	C₂×C₅₃	1	1	yes	characteristic, complemented
259	#3368	C₂×C₅₃	1	6	yes	complemented
260	#3366	C₂×C₅₃	1	6	yes	complemented
261	#3362	C₂×C₅₃	1	2	yes	complemented
262	#3358	C₂×C₅₃	1	2	yes	complemented
263	#3138	C₂×C₅₃	1	3	yes	complemented
264	#3077	C₂×C₅₃	1	3	yes	complemented
265	#3038	C₂×C₅₃	1	3	yes	complemented
266	Q₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	3	yes	complemented
268	C₂×C₅₃	—	2	2	no	—
269	C₂×C₅₃	—	4	2	no	—
270	C₂×C₅₃	—	8	6	no	—
271	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
272	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂×C₅₃	#3585	1	1	yes	characteristic
274	C₂²×#41	—	2	1	no	—
275	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	3	yes	—
276	Q₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
277	#557	C₂²×C₅₃	1	3	yes	complemented
278	#554	—	2	3	no	—
279	#554	C₂²×C₅₃	1	3	yes	complemented
280	#550	—	2	1	no	—
281	#550	—	2	3	no	—
282	#550	C₂²×C₅₃	1	2	yes	—
283	#550	C₂²×C₅₃	1	6	yes	—
284	D₄×Q₈	—	2	1	no	—
285	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
286	#533	—	2	3	no	—
287	#533	C₂²×C₅₃	1	3	yes	—
288	C₂×#127	C₂²×C₅₃	1	3	yes	—
289	C₂×#121	—	2	3	no	—
290	C₄×#41	C₂²×C₅₃	1	3	yes	—
291	C₂×C₄×Q₈	—	2	1	no	—
292	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
293	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
294	#490	C₂²×C₅₃	1	12	yes	complemented
295	#486	C₂²×C₅₃	1	6	yes	complemented
296	#486	C₂²×C₅₃	1	6	yes	complemented
297	#478	C₂²×C₅₃	1	12	yes	complemented
298	#476	C₂²×C₅₃	1	6	yes	complemented
299	#474	C₂²×C₅₃	1	6	yes	complemented
300	#460	C₂²×C₅₃	1	6	yes	complemented
301	#460	C₂²×C₅₃	1	6	yes	complemented
302	#459	C₂²×C₅₃	1	6	yes	complemented
303	#459	C₂²×C₅₃	1	6	yes	complemented
304	#446	C₂²×C₅₃	1	6	yes	complemented
305	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
306	#442	C₂²×C₅₃	1	6	yes	complemented
307	#440	C₂²×C₅₃	1	6	yes	complemented
308	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	complemented
309	C₄×D₈	C₂²×C₅₃	1	6	yes	complemented
310	#428	C₂²×C₅₃	1	3	yes	—
311	#423	C₂²×C₅₃	1	3	yes	—
312	#418	C₂²×C₅₃	1	6	yes	—
313	#418	C₂²×C₅₃	1	3	yes	—
314	#417	C₂²×C₅₃	1	1	yes	characteristic
315	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
316	C₅₃	#61321	1	1	yes	characteristic, direct product factor, Sylow
317	C₂×#41	—	2	6	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	4	8	no	—
320	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
323	C₂²×D₄	—	2	3	no	—
324	C₂²×D₄	—	2	3	no	—
325	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
326	C₂³×C₄	—	2	3	no	—
327	AGL(1,ℤ/8)	—	2	12	no	—
328	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
329	C₂×D₈	C₂³×C₅₃	1	6	yes	—
330	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
331	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
332	C₄⋊Q₈	—	2	3	no	—
333	C₄⋊Q₈	—	2	3	no	—
334	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
335	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
336	#127	—	2	3	no	—
337	#127	—	2	3	no	—
338	#127	C₂³×C₅₃	1	6	yes	—
339	#127	C₂³×C₅₃	1	6	yes	—
340	#125	—	2	6	no	—
341	#125	C₂³×C₅₃	1	6	yes	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	3	no	—
346	C₂²⋊Q₈	—	2	6	no	—
347	#121	—	2	6	no	—
348	#121	—	2	12	no	—
349	#121	—	2	6	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×Q₈	—	2	3	no	—
352	C₄×Q₈	—	2	3	no	—
353	C₄×Q₈	—	2	3	no	—
354	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
355	C₄×Q₈	D₄×C₅₃	1	3	yes	—
356	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
357	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
358	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	—	2	3	no	—
364	C₄×D₄	C₂³×C₅₃	1	6	yes	—
365	C₄×D₄	C₂³×C₅₃	1	6	yes	—
366	C₄×D₄	C₂³×C₅₃	1	3	yes	—
367	C₄×D₄	C₂³×C₅₃	1	3	yes	—
368	#117	—	2	3	no	—
369	#117	C₂³×C₅₃	1	3	yes	—
370	C₂×C₄⋊C₄	—	2	3	no	—
371	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
372	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	—	2	3	no	—
374	C₂×C₄²	—	2	3	no	—
375	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
376	#107	C₂³×C₅₃	1	6	yes	—
377	#106	C₂³×C₅₃	1	6	yes	—
378	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
379	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
380	#103	C₂³×C₅₃	1	2	yes	—
381	#102	C₂³×C₅₃	1	2	yes	—
382	#102	C₂³×C₅₃	1	6	yes	—
383	#102	C₂³×C₅₃	1	6	yes	—
384	#102	C₂³×C₅₃	1	12	yes	—
385	#102	C₂³×C₅₃	1	12	yes	—
386	#97	C₂³×C₅₃	1	6	yes	—
387	C₄×C₈	C₂³×C₅₃	1	6	yes	—
388	C₂⁴	—	2	3	no	—
389	#41	—	2	2	no	—
390	#41	—	4	24	no	—
391	#41	—	4	8	no	—
392	#41	#41×C₅₃	1	4	yes	complemented
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	2	3	no	—
395	C₂×Q₈	—	2	3	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	C₂×D₄×C₅₃	1	3	yes	—
398	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	6	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	6	no	—
407	C₂×D₄	—	2	6	no	—
408	C₂×D₄	—	4	12	no	—
409	C₂×D₄	—	8	6	no	—
410	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
411	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	6	no	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	—	2	3	no	—
418	C₂²×C₄	—	4	12	no	—
419	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
420	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
421	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
422	SD₁₆	—	2	12	no	—
423	D₈	—	2	12	no	—
424	M₁₆	—	2	6	no	—
425	M₁₆	—	2	2	no	—
426	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
427	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	—	2	3	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	3	no	—
432	C₄⋊C₄	—	2	3	no	—
433	C₄⋊C₄	—	2	3	no	—
434	C₄⋊C₄	—	2	6	no	—
435	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
436	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
437	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
438	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
439	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
440	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
441	C₂²⋊C₄	—	2	3	no	—
442	C₂²⋊C₄	—	2	6	no	—
443	C₂²⋊C₄	—	2	6	no	—
444	C₂²⋊C₄	—	2	3	no	—
445	C₂²⋊C₄	—	2	3	no	—
446	C₄²	—	2	3	no	—
447	C₄²	—	2	3	no	—
448	C₄²	—	2	3	no	—
449	C₄²	C₂×D₄×C₅₃	1	3	yes	—
450	C₄²	C₂×D₄×C₅₃	1	3	yes	—
451	C₄²	C₂⁴×C₅₃	1	3	yes	—
452	C₄²	C₂⁴×C₅₃	1	3	yes	—
453	C₂³	—	2	1	no	—
454	C₂³	—	2	6	no	—
455	C₂³	—	4	6	no	—
456	C₂³	—	4	6	no	—
457	C₂³	—	4	6	no	—
458	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
459	Q₈	—	2	6	no	—
460	Q₈	—	2	1	no	—
461	Q₈	—	4	6	no	—
462	Q₈	—	8	3	no	—
463	Q₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
464	Q₈	C₂×#41×C₅₃	1	2	yes	complemented
465	D₄	—	2	4	no	—
466	D₄	—	2	6	no	—
467	D₄	—	2	1	no	—
468	D₄	—	2	6	no	—
469	D₄	—	4	12	no	—
470	D₄	—	4	6	no	—
471	D₄	—	4	12	no	—
472	D₄	—	8	6	no	—
473	D₄	—	8	6	no	—
474	D₄	C₂×#41×C₅₃	1	2	yes	complemented
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	6	no	—
478	C₂×C₄	—	2	3	no	—
479	C₂×C₄	—	2	3	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	3	no	—
482	C₂×C₄	—	4	12	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	12	no	—
485	C₂×C₄	—	4	6	no	—
486	C₂×C₄	—	8	3	no	—
487	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
488	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
489	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
490	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
492	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
493	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
494	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
495	C₈	—	2	6	no	—
496	C₈	—	2	2	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	6	no	—
503	C₂²	—	4	6	no	—
504	C₂²	—	4	6	no	—
505	C₂²	—	8	12	no	—
506	C₂²	C₅₃×#548	1	2	yes	—
507	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	3	no	—
510	C₄	—	2	3	no	—
511	C₄	—	2	3	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	3	no	—
514	C₄	—	4	3	no	—
515	C₄	C₅₃×#548	1	2	yes	—
516	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	6	yes	—
517	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
518	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
519	C₂	—	2	2	no	—
520	C₂	—	4	2	no	—
521	C₂	—	8	6	no	—
522	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
523	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
524	C₂	C₅₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₅₃	C₂	semidirect
2. C₂×C₅₃×#550	C₂	semidirect
3. C₅₃×#3368	C₂	semidirect
4. C₅₃×#3366	C₂	semidirect
5. C₅₃×#3362	C₂	semidirect
6. C₅₃×#3358	C₂	semidirect
7. C₅₃×#3138	C₂	semidirect
8. C₅₃×#3077	C₂	semidirect
9. C₅₃×#3038	C₂	semidirect
10. Q₈×M₁₆×C₅₃	C₂	semidirect
11. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
14. Q₈²×C₅₃	C₂²	semidirect
15. C₅₃×#557	C₂²	semidirect
17. C₅₃×#554	C₂²	semidirect
23. D₄×Q₈×C₅₃	C₂²	semidirect
32. C₅₃×#490	C₂²	semidirect
33. C₅₃×#486	C₂²	semidirect
34. C₅₃×#486	C₂²	semidirect
35. C₅₃×#478	C₂²	semidirect
36. C₅₃×#476	C₂²	semidirect
37. C₅₃×#474	C₂²	semidirect
38. C₅₃×#460	C₂²	semidirect
39. C₅₃×#460	C₂²	semidirect
40. C₅₃×#459	C₂²	semidirect
41. C₅₃×#459	C₂²	semidirect
42. C₅₃×#446	C₂²	semidirect
43. C₈×Q₈×C₅₃	C₂²	semidirect
44. C₅₃×#442	C₂²	semidirect
45. C₅₃×#440	C₂²	semidirect
46. C₄×SD₁₆×C₅₃	C₂²	semidirect
47. C₄×D₈×C₅₃	C₂²	semidirect
129. #41×C₅₃	#41	semidirect
200. Q₈×C₅₃	AGL(1,ℤ/8)	semidirect
201. Q₈×C₅₃	C₂×#41	semidirect
211. D₄×C₅₃	C₂×#41	semidirect
234. #61321	C₅₃	direct (N × H)
257. Q₈×#41	C₂×C₅₃	semidirect
258. C₂×#550	C₂×C₅₃	semidirect
259. #3368	C₂×C₅₃	semidirect
260. #3366	C₂×C₅₃	semidirect
261. #3362	C₂×C₅₃	semidirect
262. #3358	C₂×C₅₃	semidirect
263. #3138	C₂×C₅₃	semidirect
264. #3077	C₂×C₅₃	semidirect
265. #3038	C₂×C₅₃	semidirect
266. Q₈×M₁₆	C₂×C₅₃	semidirect
267. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
276. Q₈²	C₂²×C₅₃	semidirect
277. #557	C₂²×C₅₃	semidirect
279. #554	C₂²×C₅₃	semidirect
285. D₄×Q₈	C₂²×C₅₃	semidirect
294. #490	C₂²×C₅₃	semidirect
295. #486	C₂²×C₅₃	semidirect
296. #486	C₂²×C₅₃	semidirect
297. #478	C₂²×C₅₃	semidirect
298. #476	C₂²×C₅₃	semidirect
299. #474	C₂²×C₅₃	semidirect
300. #460	C₂²×C₅₃	semidirect
301. #460	C₂²×C₅₃	semidirect
302. #459	C₂²×C₅₃	semidirect
303. #459	C₂²×C₅₃	semidirect
304. #446	C₂²×C₅₃	semidirect
305. C₈×Q₈	C₂²×C₅₃	semidirect
306. #442	C₂²×C₅₃	semidirect
307. #440	C₂²×C₅₃	semidirect
308. C₄×SD₁₆	C₂²×C₅₃	semidirect
309. C₄×D₈	C₂²×C₅₃	semidirect
316. C₅₃	#61321	direct (N × H)
392. #41	#41×C₅₃	semidirect
463. Q₈	AGL(1,ℤ/8)×C₅₃	semidirect
464. Q₈	C₂×#41×C₅₃	semidirect
474. D₄	C₂×#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₅₃	2
C₂×C₅₃×#550	2
C₅₃×#3368	2
C₅₃×#3366	2
C₅₃×#3362	2
C₅₃×#3358	2
C₅₃×#3138	2
C₅₃×#3077	2
C₅₃×#3038	2
Q₈×M₁₆×C₅₃	2
C₄×AGL(1,ℤ/8)×C₅₃	2
#61321	53

Subgroups — C53×#61321 #27268833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61321 #27268833