Subgroups — C₅₃×#60704 #27268216

Order 13568.

1974 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3179	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3176	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3174	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#3055	C₂	1	4	yes	maximal, complemented
11	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#542	C₂²	1	4	yes	complemented
16	C₅₃×#541	C₂²	1	2	yes	complemented
17	C₅₃×#540	—	2	4	no	—
18	C₅₃×#540	C₂²	1	4	yes	—
19	C₅₃×#540	C₂²	1	4	yes	—
20	C₅₃×#540	C₂²	1	2	yes	complemented
21	C₅₃×#539	—	2	2	no	—
22	C₅₃×#538	—	2	2	no	—
23	C₅₃×#533	C₂²	1	1	yes	characteristic
24	C₅₃×#532	C₂²	1	1	yes	characteristic
25	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#529	—	2	4	no	—
27	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#525	C₂²	1	2	yes	—
29	C₂×#123×C₅₃	—	2	2	no	—
30	C₂×#121×C₅₃	—	2	2	no	—
31	C₂×#121×C₅₃	C₂²	1	2	yes	—
32	C₄×#41×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
34	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#501	C₂²	1	4	yes	complemented
36	C₅₃×#499	C₂²	1	4	yes	complemented
37	C₅₃×#496	C₂²	1	4	yes	complemented
38	C₅₃×#493	C₂²	1	2	yes	complemented
39	C₅₃×#492	C₂²	1	2	yes	complemented
40	C₅₃×#491	C₂²	1	2	yes	complemented
41	C₅₃×#490	C₂²	1	4	yes	complemented
42	C₅₃×#490	C₂²	1	2	yes	complemented
43	C₅₃×#488	C₂²	1	4	yes	complemented
44	C₅₃×#486	C₂²	1	4	yes	complemented
45	C₅₃×#469	C₂²	1	8	yes	complemented
46	C₅₃×#468	C₂²	1	8	yes	complemented
47	C₅₃×#466	C₂²	1	8	yes	complemented
48	C₅₃×#465	C₂²	1	8	yes	complemented
49	C₅₃×#442	C₂²	1	8	yes	complemented
50	C₅₃×#440	C₂²	1	8	yes	complemented
51	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
52	C₄×D₈×C₅₃	C₂²	1	8	yes	complemented
53	C₅₃×#428	C₂²	1	4	yes	—
54	C₅₃×#418	C₂²	1	4	yes	—
55	C₅₃×#417	C₂²	1	4	yes	—
56	C₅₃×#405	C₂²	1	4	yes	complemented
57	C₅₃×#405	C₂²	1	4	yes	complemented
58	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
59	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
60	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×#41×C₅₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₅₃	—	2	2	no	—
63	C₂²×D₄×C₅₃	—	2	2	no	—
64	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	—	2	2	no	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
68	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
69	C₂×D₈×C₅₃	C₂³	1	8	yes	—
70	C₂×M₁₆×C₅₃	—	2	8	no	—
71	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
72	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
73	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
75	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
77	#127×C₅₃	—	2	2	no	—
78	#127×C₅₃	C₂³	1	2	yes	—
79	#127×C₅₃	C₂³	1	2	yes	—
80	#127×C₅₃	C₂³	1	1	yes	characteristic
81	#126×C₅₃	—	2	4	no	—
82	#126×C₅₃	—	2	4	no	—
83	#125×C₅₃	—	2	4	no	—
84	#125×C₅₃	—	2	2	no	—
85	#125×C₅₃	C₂³	1	2	yes	—
86	#125×C₅₃	C₂³	1	2	yes	—
87	#125×C₅₃	C₂³	1	2	yes	—
88	#124×C₅₃	—	2	4	no	—
89	#124×C₅₃	C₂³	1	2	yes	—
90	#123×C₅₃	—	2	8	no	—
91	#123×C₅₃	—	2	8	no	—
92	#123×C₅₃	—	2	4	no	—
93	#123×C₅₃	—	2	4	no	—
94	#123×C₅₃	—	2	2	no	—
95	C₂²⋊Q₈×C₅₃	—	2	4	no	—
96	C₂²⋊Q₈×C₅₃	—	2	2	no	—
97	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
98	#121×C₅₃	—	2	8	no	—
99	#121×C₅₃	—	2	8	no	—
100	#121×C₅₃	—	2	4	no	—
101	#121×C₅₃	—	2	4	no	—
102	#121×C₅₃	—	2	2	no	—
103	#121×C₅₃	C₂³	1	8	yes	—
104	#121×C₅₃	C₂³	1	4	yes	—
105	#120×C₅₃	—	2	2	no	—
106	C₄×Q₈×C₅₃	—	2	2	no	—
107	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
108	C₄×D₄×C₅₃	—	2	4	no	—
109	C₄×D₄×C₅₃	—	2	2	no	—
110	C₄×D₄×C₅₃	—	2	4	no	—
111	C₄×D₄×C₅₃	—	2	4	no	—
112	C₄×D₄×C₅₃	C₂³	1	8	yes	—
113	C₄×D₄×C₅₃	C₂³	1	4	yes	—
114	#117×C₅₃	—	2	2	no	—
115	#117×C₅₃	D₄	1	4	yes	—
116	#117×C₅₃	C₂³	1	2	yes	—
117	#117×C₅₃	C₂³	1	2	yes	—
118	#117×C₅₃	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
125	C₂×C₄²×C₅₃	—	2	2	no	—
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
128	#107×C₅₃	C₂³	1	8	yes	—
129	#106×C₅₃	C₂³	1	8	yes	—
130	#103×C₅₃	C₂³	1	8	yes	—
131	#102×C₅₃	C₂³	1	8	yes	—
132	#102×C₅₃	C₂³	1	8	yes	—
133	#102×C₅₃	C₂³	1	8	yes	—
134	#97×C₅₃	C₂³	1	2	yes	—
135	#97×C₅₃	C₂³	1	2	yes	—
136	#97×C₅₃	C₂³	1	4	yes	—
137	C₄×C₈×C₅₃	C₂³	1	2	yes	—
138	C₄×C₈×C₅₃	C₂³	1	2	yes	—
139	C₄×C₈×C₅₃	C₂³	1	4	yes	—
140	C₂⁴×C₅₃	—	2	2	no	—
141	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
142	#41×C₅₃	—	2	8	no	—
143	C₂×Q₈×C₅₃	—	2	2	no	—
144	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
145	C₂×D₄×C₅₃	—	2	2	no	—
146	C₂×D₄×C₅₃	—	2	4	no	—
147	C₂×D₄×C₅₃	—	2	8	no	—
148	C₂×D₄×C₅₃	—	2	2	no	—
149	C₂×D₄×C₅₃	—	2	2	no	—
150	C₂×D₄×C₅₃	—	2	4	no	—
151	C₂×D₄×C₅₃	—	2	4	no	—
152	C₂×D₄×C₅₃	—	2	8	no	—
153	C₂×D₄×C₅₃	—	8	4	no	—
154	C₂×D₄×C₅₃	—	8	4	no	—
155	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	2	no	—
159	C₂²×C₄×C₅₃	—	2	2	no	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	8	no	—
162	C₂²×C₄×C₅₃	—	8	4	no	—
163	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	SD₁₆×C₅₃	—	2	16	no	—
170	D₈×C₅₃	—	2	16	no	—
171	M₁₆×C₅₃	—	2	4	no	—
172	M₁₆×C₅₃	—	2	8	no	—
173	M₁₆×C₅₃	#41	1	8	yes	—
174	C₂×C₈×C₅₃	—	2	8	no	—
175	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
176	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₅₃	—	2	4	no	—
178	C₄⋊C₄×C₅₃	—	2	4	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	2	4	no	—
181	C₄⋊C₄×C₅₃	—	2	2	no	—
182	C₄⋊C₄×C₅₃	—	2	4	no	—
183	C₄⋊C₄×C₅₃	—	2	4	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
187	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	4	no	—
191	C₂²⋊C₄×C₅₃	—	2	8	no	—
192	C₂²⋊C₄×C₅₃	—	2	8	no	—
193	C₂²⋊C₄×C₅₃	—	2	4	no	—
194	C₂²⋊C₄×C₅₃	—	2	4	no	—
195	C₂²⋊C₄×C₅₃	—	2	4	no	—
196	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
197	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
198	C₄²×C₅₃	—	2	2	no	—
199	C₄²×C₅₃	—	2	4	no	—
200	C₄²×C₅₃	C₂×D₄	1	2	yes	—
201	C₄²×C₅₃	C₂×D₄	1	2	yes	—
202	C₄²×C₅₃	C₂⁴	1	2	yes	—
203	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
204	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
205	C₂³×C₅₃	—	2	2	no	—
206	C₂³×C₅₃	—	2	2	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	4	2	no	—
209	C₂³×C₅₃	—	4	4	no	—
210	C₂³×C₅₃	—	4	4	no	—
211	C₂³×C₅₃	—	4	4	no	—
212	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
213	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₅₃	—	2	4	no	—
215	Q₈×C₅₃	—	8	2	no	—
216	D₄×C₅₃	—	2	8	no	—
217	D₄×C₅₃	—	2	4	no	—
218	D₄×C₅₃	—	4	4	no	—
219	D₄×C₅₃	—	4	4	no	—
220	D₄×C₅₃	—	8	2	no	—
221	D₄×C₅₃	—	8	4	no	—
222	D₄×C₅₃	—	8	4	no	—
223	D₄×C₅₃	—	8	4	no	—
224	D₄×C₅₃	—	8	8	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	4	no	—
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	4	2	no	—
233	C₂×C₄×C₅₃	—	4	2	no	—
234	C₂×C₄×C₅₃	—	4	4	no	—
235	C₂×C₄×C₅₃	—	4	4	no	—
236	C₂×C₄×C₅₃	—	8	4	no	—
237	C₂×C₄×C₅₃	—	8	8	no	—
238	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
244	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
245	C₈×C₅₃	—	2	4	no	—
246	C₈×C₅₃	C₂×#41	1	8	yes	—
247	#60704	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂²×C₅₃	—	2	2	no	—
249	C₂²×C₅₃	—	2	4	no	—
250	C₂²×C₅₃	—	2	2	no	—
251	C₂²×C₅₃	—	4	4	no	—
252	C₂²×C₅₃	—	4	4	no	—
253	C₂²×C₅₃	—	4	4	no	—
254	C₂²×C₅₃	—	4	2	no	—
255	C₂²×C₅₃	—	4	4	no	—
256	C₂²×C₅₃	—	8	2	no	—
257	C₂²×C₅₃	—	8	8	no	—
258	C₂²×C₅₃	#532	1	2	yes	—
259	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
260	C₄×C₅₃	—	2	4	no	—
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	4	4	no	—
265	C₄×C₅₃	—	4	4	no	—
266	C₄×C₅₃	—	4	2	no	—
267	C₄×C₅₃	#532	1	2	yes	—
268	C₄×C₅₃	C₂²×#41	1	2	yes	—
269	#3455	C₂×C₅₃	1	1	yes	characteristic, complemented
270	C₂×#540	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#3179	C₂×C₅₃	1	8	yes	complemented
272	#3176	C₂×C₅₃	1	4	yes	complemented
273	#3174	C₂×C₅₃	1	4	yes	complemented
274	#3166	C₂×C₅₃	1	1	yes	characteristic, complemented
275	#3151	C₂×C₅₃	1	1	yes	characteristic, complemented
276	#3142	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#3138	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#3055	C₂×C₅₃	1	4	yes	complemented
279	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	4	yes	complemented
280	#2873	C₂×C₅₃	1	1	yes	characteristic, complemented
281	C₂×C₅₃	—	2	2	no	—
282	C₂×C₅₃	—	4	2	no	—
283	C₂×C₅₃	—	8	2	no	—
284	C₂×C₅₃	—	8	4	no	—
285	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
286	C₂×C₅₃	#3585	1	2	yes	—
287	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	—
288	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
289	#542	C₂²×C₅₃	1	4	yes	complemented
290	#541	C₂²×C₅₃	1	2	yes	complemented
291	#540	—	2	4	no	—
292	#540	C₂²×C₅₃	1	4	yes	—
293	#540	C₂²×C₅₃	1	4	yes	—
294	#540	C₂²×C₅₃	1	2	yes	complemented
295	#539	—	2	2	no	—
296	#538	—	2	2	no	—
297	#533	C₂²×C₅₃	1	1	yes	characteristic
298	#532	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
300	#529	—	2	4	no	—
301	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
302	#525	C₂²×C₅₃	1	2	yes	—
303	C₂×#123	—	2	2	no	—
304	C₂×#121	—	2	2	no	—
305	C₂×#121	C₂²×C₅₃	1	2	yes	—
306	C₄×#41	C₂²×C₅₃	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
308	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
309	#501	C₂²×C₅₃	1	4	yes	complemented
310	#499	C₂²×C₅₃	1	4	yes	complemented
311	#496	C₂²×C₅₃	1	4	yes	complemented
312	#493	C₂²×C₅₃	1	2	yes	complemented
313	#492	C₂²×C₅₃	1	2	yes	complemented
314	#491	C₂²×C₅₃	1	2	yes	complemented
315	#490	C₂²×C₅₃	1	4	yes	complemented
316	#490	C₂²×C₅₃	1	2	yes	complemented
317	#488	C₂²×C₅₃	1	4	yes	complemented
318	#486	C₂²×C₅₃	1	4	yes	complemented
319	#469	C₂²×C₅₃	1	8	yes	complemented
320	#468	C₂²×C₅₃	1	8	yes	complemented
321	#466	C₂²×C₅₃	1	8	yes	complemented
322	#465	C₂²×C₅₃	1	8	yes	complemented
323	#442	C₂²×C₅₃	1	8	yes	complemented
324	#440	C₂²×C₅₃	1	8	yes	complemented
325	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
326	C₄×D₈	C₂²×C₅₃	1	8	yes	complemented
327	#428	C₂²×C₅₃	1	4	yes	—
328	#418	C₂²×C₅₃	1	4	yes	—
329	#417	C₂²×C₅₃	1	4	yes	—
330	#405	C₂²×C₅₃	1	4	yes	complemented
331	#405	C₂²×C₅₃	1	4	yes	complemented
332	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
333	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
334	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
335	C₅₃	#60704	1	1	yes	characteristic, direct product factor, Sylow
336	C₂×#41	C₂³×C₅₃	1	2	yes	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
340	C₂³×C₄	—	2	2	no	—
341	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	AGL(1,ℤ/8)	—	2	16	no	—
343	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
344	C₂×D₈	C₂³×C₅₃	1	8	yes	—
345	C₂×M₁₆	—	2	8	no	—
346	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
347	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
348	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
349	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
350	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
352	#127	—	2	2	no	—
353	#127	C₂³×C₅₃	1	2	yes	—
354	#127	C₂³×C₅₃	1	2	yes	—
355	#127	C₂³×C₅₃	1	1	yes	characteristic
356	#126	—	2	4	no	—
357	#126	—	2	4	no	—
358	#125	—	2	4	no	—
359	#125	—	2	2	no	—
360	#125	C₂³×C₅₃	1	2	yes	—
361	#125	C₂³×C₅₃	1	2	yes	—
362	#125	C₂³×C₅₃	1	2	yes	—
363	#124	—	2	4	no	—
364	#124	C₂³×C₅₃	1	2	yes	—
365	#123	—	2	2	no	—
366	#123	—	2	8	no	—
367	#123	—	2	8	no	—
368	#123	—	2	4	no	—
369	#123	—	2	4	no	—
370	C₂²⋊Q₈	—	2	2	no	—
371	C₂²⋊Q₈	—	2	4	no	—
372	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
373	#121	—	2	4	no	—
374	#121	—	2	2	no	—
375	#121	—	2	8	no	—
376	#121	—	2	8	no	—
377	#121	—	2	4	no	—
378	#121	C₂³×C₅₃	1	8	yes	—
379	#121	C₂³×C₅₃	1	4	yes	—
380	#120	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	C₂³×C₅₃	1	8	yes	—
388	C₄×D₄	C₂³×C₅₃	1	4	yes	—
389	#117	—	2	2	no	—
390	#117	D₄×C₅₃	1	4	yes	—
391	#117	C₂³×C₅₃	1	2	yes	—
392	#117	C₂³×C₅₃	1	2	yes	—
393	#117	C₂³×C₅₃	1	2	yes	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
400	C₂×C₄²	—	2	2	no	—
401	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
403	#107	C₂³×C₅₃	1	8	yes	—
404	#106	C₂³×C₅₃	1	8	yes	—
405	#103	C₂³×C₅₃	1	8	yes	—
406	#102	C₂³×C₅₃	1	8	yes	—
407	#102	C₂³×C₅₃	1	8	yes	—
408	#102	C₂³×C₅₃	1	8	yes	—
409	#97	C₂³×C₅₃	1	4	yes	—
410	#97	C₂³×C₅₃	1	2	yes	—
411	#97	C₂³×C₅₃	1	2	yes	—
412	C₄×C₈	C₂³×C₅₃	1	4	yes	—
413	C₄×C₈	C₂³×C₅₃	1	2	yes	—
414	C₄×C₈	C₂³×C₅₃	1	2	yes	—
415	C₂⁴	—	2	2	no	—
416	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
417	#41	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	8	no	—
428	C₂×D₄	—	8	4	no	—
429	C₂×D₄	—	8	4	no	—
430	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
431	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	8	4	no	—
438	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
444	SD₁₆	—	2	16	no	—
445	D₈	—	2	16	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	8	no	—
448	M₁₆	#41×C₅₃	1	8	yes	—
449	C₂×C₈	—	2	8	no	—
450	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
451	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
461	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
472	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	4	no	—
475	C₄²	C₂×D₄×C₅₃	1	2	yes	—
476	C₄²	C₂×D₄×C₅₃	1	2	yes	—
477	C₄²	C₂⁴×C₅₃	1	2	yes	—
478	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
479	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	4	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
488	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
489	Q₈	—	2	4	no	—
490	Q₈	—	8	2	no	—
491	D₄	—	2	4	no	—
492	D₄	—	2	8	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	4	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	4	no	—
497	D₄	—	8	4	no	—
498	D₄	—	8	4	no	—
499	D₄	—	8	8	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	8	4	no	—
512	C₂×C₄	—	8	8	no	—
513	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
514	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
515	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
517	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
519	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
520	C₈	—	2	4	no	—
521	C₈	C₂×#41×C₅₃	1	8	yes	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	8	no	—
532	C₂²	C₅₃×#532	1	2	yes	—
533	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
534	C₄	—	2	4	no	—
535	C₄	—	2	2	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	2	no	—
538	C₄	—	4	4	no	—
539	C₄	—	4	4	no	—
540	C₄	—	4	2	no	—
541	C₄	C₅₃×#532	1	2	yes	—
542	C₄	C₂²×#41×C₅₃	1	2	yes	—
543	C₂	—	2	2	no	—
544	C₂	—	4	2	no	—
545	C₂	—	8	4	no	—
546	C₂	—	8	2	no	—
547	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
548	C₂	C₅₃×#3585	1	2	yes	—

	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×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3455	C₂	semidirect
2. C₂×C₅₃×#540	C₂	semidirect
3. C₅₃×#3179	C₂	semidirect
4. C₅₃×#3176	C₂	semidirect
5. C₅₃×#3174	C₂	semidirect
6. C₅₃×#3166	C₂	semidirect
7. C₅₃×#3151	C₂	semidirect
8. C₅₃×#3142	C₂	semidirect
9. C₅₃×#3138	C₂	semidirect
10. C₅₃×#3055	C₂	semidirect
11. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
12. C₅₃×#2873	C₂	semidirect
15. C₅₃×#542	C₂²	semidirect
16. C₅₃×#541	C₂²	semidirect
20. C₅₃×#540	C₂²	semidirect
35. C₅₃×#501	C₂²	semidirect
36. C₅₃×#499	C₂²	semidirect
37. C₅₃×#496	C₂²	semidirect
38. C₅₃×#493	C₂²	semidirect
39. C₅₃×#492	C₂²	semidirect
40. C₅₃×#491	C₂²	semidirect
41. C₅₃×#490	C₂²	semidirect
42. C₅₃×#490	C₂²	semidirect
43. C₅₃×#488	C₂²	semidirect
44. C₅₃×#486	C₂²	semidirect
45. C₅₃×#469	C₂²	semidirect
46. C₅₃×#468	C₂²	semidirect
47. C₅₃×#466	C₂²	semidirect
48. C₅₃×#465	C₂²	semidirect
49. C₅₃×#442	C₂²	semidirect
50. C₅₃×#440	C₂²	semidirect
51. C₄×SD₁₆×C₅₃	C₂²	semidirect
52. C₄×D₈×C₅₃	C₂²	semidirect
56. C₅₃×#405	C₂²	semidirect
57. C₅₃×#405	C₂²	semidirect
247. #60704	C₅₃	direct (N × H)
269. #3455	C₂×C₅₃	semidirect
270. C₂×#540	C₂×C₅₃	semidirect
271. #3179	C₂×C₅₃	semidirect
272. #3176	C₂×C₅₃	semidirect
273. #3174	C₂×C₅₃	semidirect
274. #3166	C₂×C₅₃	semidirect
275. #3151	C₂×C₅₃	semidirect
276. #3142	C₂×C₅₃	semidirect
277. #3138	C₂×C₅₃	semidirect
278. #3055	C₂×C₅₃	semidirect
279. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
280. #2873	C₂×C₅₃	semidirect
289. #542	C₂²×C₅₃	semidirect
290. #541	C₂²×C₅₃	semidirect
294. #540	C₂²×C₅₃	semidirect
309. #501	C₂²×C₅₃	semidirect
310. #499	C₂²×C₅₃	semidirect
311. #496	C₂²×C₅₃	semidirect
312. #493	C₂²×C₅₃	semidirect
313. #492	C₂²×C₅₃	semidirect
314. #491	C₂²×C₅₃	semidirect
315. #490	C₂²×C₅₃	semidirect
316. #490	C₂²×C₅₃	semidirect
317. #488	C₂²×C₅₃	semidirect
318. #486	C₂²×C₅₃	semidirect
319. #469	C₂²×C₅₃	semidirect
320. #468	C₂²×C₅₃	semidirect
321. #466	C₂²×C₅₃	semidirect
322. #465	C₂²×C₅₃	semidirect
323. #442	C₂²×C₅₃	semidirect
324. #440	C₂²×C₅₃	semidirect
325. C₄×SD₁₆	C₂²×C₅₃	semidirect
326. C₄×D₈	C₂²×C₅₃	semidirect
330. #405	C₂²×C₅₃	semidirect
331. #405	C₂²×C₅₃	semidirect
335. C₅₃	#60704	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×#3455	2
C₂×C₅₃×#540	2
C₅₃×#3179	2
C₅₃×#3176	2
C₅₃×#3174	2
C₅₃×#3166	2
C₅₃×#3151	2
C₅₃×#3142	2
C₅₃×#3138	2
C₅₃×#3055	2
C₄×AGL(1,ℤ/8)×C₅₃	2
C₅₃×#2873	2
#60704	53

Subgroups — C53×#60704 #27268216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60704 #27268216