Subgroups — C₅₃×#60827 #27268339

Order 13568.

2154 conjugacy classes of subgroups.

The subgroup lattice has 670 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	1	yes	maximal, characteristic, complemented
2	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3224	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3219	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3217	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#3043	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3041	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3009	C₂	1	2	yes	maximal, complemented
13	C₅₃×#3008	C₂	1	2	yes	maximal, complemented
14	C₅₃×#2923	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
16	C₂×#137×C₅₃	C₂²	1	2	yes	complemented
17	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	complemented
18	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
19	C₂²×D₈×C₅₃	C₂²	1	2	yes	complemented
20	C₅₃×#559	—	2	2	no	—
21	C₅₃×#556	—	2	2	no	—
22	C₅₃×#551	—	2	4	no	—
23	C₅₃×#550	—	2	2	no	—
24	D₄²×C₅₃	—	2	2	no	—
25	D₄²×C₅₃	—	2	4	no	—
26	C₅₃×#542	C₂²	1	4	yes	complemented
27	C₅₃×#536	C₂²	1	1	yes	characteristic
28	C₅₃×#535	C₂²	1	2	yes	—
29	C₅₃×#535	C₂²	1	1	yes	characteristic
30	C₅₃×#535	C₂²	1	4	yes	complemented
31	C₅₃×#533	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#532	C₂²	1	1	yes	characteristic
33	C₅₃×#532	C₂²	1	2	yes	complemented
34	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
36	C₅₃×#529	C₂²	1	4	yes	complemented
37	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#519	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#518	C₂²	1	1	yes	characteristic, complemented
40	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#497	C₂²	1	2	yes	complemented
42	C₅₃×#496	C₂²	1	2	yes	complemented
43	C₅₃×#496	C₂²	1	4	yes	complemented
44	C₅₃×#496	C₂²	1	4	yes	complemented
45	C₅₃×#495	C₂²	1	4	yes	complemented
46	C₅₃×#491	C₂²	1	2	yes	complemented
47	C₅₃×#490	C₂²	1	2	yes	complemented
48	C₅₃×#490	C₂²	1	4	yes	complemented
49	C₅₃×#489	C₂²	1	4	yes	complemented
50	C₅₃×#486	C₂²	1	4	yes	complemented
51	C₅₃×#464	C₂²	1	8	yes	complemented
52	C₅₃×#463	C₂²	1	8	yes	complemented
53	C₅₃×#460	C₂²	1	8	yes	complemented
54	C₅₃×#459	C₂²	1	8	yes	complemented
55	C₅₃×#452	C₂²	1	8	yes	complemented
56	C₅₃×#450	C₂²	1	8	yes	complemented
57	C₅₃×#449	C₂²	1	8	yes	complemented
58	C₅₃×#447	C₂²	1	8	yes	complemented
59	C₅₃×#433	C₂²	1	4	yes	complemented
60	C₅₃×#432	C₂²	1	4	yes	complemented
61	C₅₃×#424	C₂²	1	2	yes	complemented
62	C₅₃×#418	C₂²	1	2	yes	complemented
63	C₅₃×#418	C₂²	1	2	yes	complemented
64	C₅₃×#416	C₂²	1	2	yes	complemented
65	C₅₃×#416	C₂²	1	2	yes	complemented
66	C₅₃×#408	C₂²	1	2	yes	—
67	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
68	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
69	H(5,𝔽₂)×C₅₃	—	4	4	no	—
70	H(5,𝔽₂)×C₅₃	—	4	4	no	—
71	C₂×#41×C₅₃	—	2	2	no	—
72	C₂×#41×C₅₃	C₂³	1	2	yes	—
73	C₂×#41×C₅₃	C₂³	1	4	yes	—
74	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₅₃	—	2	4	no	—
77	C₂²×D₄×C₅₃	—	2	2	no	—
78	C₂²×D₄×C₅₃	—	2	2	no	—
79	C₂²×D₄×C₅₃	—	2	4	no	—
80	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
82	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
83	#137×C₅₃	—	4	4	no	—
84	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
85	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₅₃	—	4	4	no	—
87	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
90	C₂×D₈×C₅₃	—	4	4	no	—
91	C₂×D₈×C₅₃	C₂³	1	4	yes	—
92	C₂×D₈×C₅₃	C₂³	1	8	yes	—
93	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
95	C₄⋊Q₈×C₅₃	—	2	1	no	—
96	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
97	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
98	#127×C₅₃	—	2	2	no	—
99	#127×C₅₃	—	2	4	no	—
100	#127×C₅₃	—	2	1	no	—
101	#127×C₅₃	C₂³	1	2	yes	—
102	#127×C₅₃	C₂³	1	2	yes	—
103	#127×C₅₃	C₂³	1	1	yes	characteristic
104	#127×C₅₃	C₂³	1	4	yes	—
105	#126×C₅₃	—	2	4	no	—
106	#126×C₅₃	—	2	4	no	—
107	#125×C₅₃	C₂³	1	2	yes	—
108	#124×C₅₃	—	2	4	no	—
109	#124×C₅₃	—	2	2	no	—
110	#124×C₅₃	—	2	1	no	—
111	#124×C₅₃	—	2	1	no	—
112	#124×C₅₃	C₂³	1	2	yes	—
113	#124×C₅₃	C₂³	1	2	yes	—
114	#124×C₅₃	C₂³	1	2	yes	—
115	#124×C₅₃	C₂³	1	4	yes	—
116	#123×C₅₃	—	2	4	no	—
117	#123×C₅₃	—	2	2	no	—
118	#123×C₅₃	—	2	4	no	—
119	C₂²⋊Q₈×C₅₃	—	2	4	no	—
120	C₂²⋊Q₈×C₅₃	—	2	2	no	—
121	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
122	#121×C₅₃	—	2	4	no	—
123	#121×C₅₃	—	2	8	no	—
124	#121×C₅₃	—	2	2	no	—
125	#121×C₅₃	—	2	4	no	—
126	#121×C₅₃	C₂³	1	4	yes	—
127	#120×C₅₃	—	2	4	no	—
128	#120×C₅₃	—	2	8	no	—
129	#120×C₅₃	—	2	2	no	—
130	C₄×Q₈×C₅₃	—	2	2	no	—
131	C₄×Q₈×C₅₃	—	2	2	no	—
132	C₄×Q₈×C₅₃	D₄	1	2	yes	—
133	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
134	C₄×D₄×C₅₃	—	2	2	no	—
135	C₄×D₄×C₅₃	—	2	2	no	—
136	C₄×D₄×C₅₃	—	2	4	no	—
137	C₄×D₄×C₅₃	—	2	4	no	—
138	C₄×D₄×C₅₃	D₄	1	4	yes	—
139	C₄×D₄×C₅₃	D₄	1	2	yes	—
140	C₄×D₄×C₅₃	C₂³	1	4	yes	—
141	#117×C₅₃	—	2	2	no	—
142	#117×C₅₃	—	2	4	no	—
143	#117×C₅₃	C₂³	1	1	yes	characteristic
144	#117×C₅₃	C₂³	1	1	yes	characteristic
145	#117×C₅₃	C₂³	1	1	yes	characteristic
146	#117×C₅₃	C₂³	1	2	yes	—
147	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
149	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
151	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
152	#103×C₅₃	C₂³	1	4	yes	—
153	#103×C₅₃	C₂³	1	4	yes	—
154	#102×C₅₃	C₂³	1	8	yes	—
155	#102×C₅₃	C₂³	1	8	yes	—
156	#102×C₅₃	C₂³	1	4	yes	—
157	#102×C₅₃	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
159	#97×C₅₃	C₂³	1	2	yes	—
160	#97×C₅₃	C₂³	1	2	yes	—
161	#97×C₅₃	C₂³	1	2	yes	—
162	C₄×C₈×C₅₃	C₂³	1	2	yes	—
163	C₂⁴×C₅₃	—	2	1	no	—
164	C₂⁴×C₅₃	—	2	4	no	—
165	C₂⁴×C₅₃	—	2	2	no	—
166	#41×C₅₃	—	2	4	no	—
167	#41×C₅₃	—	2	4	no	—
168	#41×C₅₃	—	2	4	no	—
169	#41×C₅₃	—	4	4	no	—
170	#41×C₅₃	—	4	4	no	—
171	#41×C₅₃	—	4	8	no	—
172	C₂×Q₈×C₅₃	—	2	1	no	—
173	C₂×Q₈×C₅₃	—	2	2	no	—
174	C₂×Q₈×C₅₃	—	2	4	no	—
175	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
176	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
177	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
178	C₂×D₄×C₅₃	—	2	4	no	—
179	C₂×D₄×C₅₃	—	2	2	no	—
180	C₂×D₄×C₅₃	—	2	4	no	—
181	C₂×D₄×C₅₃	—	2	1	no	—
182	C₂×D₄×C₅₃	—	2	2	no	—
183	C₂×D₄×C₅₃	—	2	4	no	—
184	C₂×D₄×C₅₃	—	2	4	no	—
185	C₂×D₄×C₅₃	—	2	2	no	—
186	C₂×D₄×C₅₃	—	2	8	no	—
187	C₂×D₄×C₅₃	—	2	4	no	—
188	C₂×D₄×C₅₃	—	2	4	no	—
189	C₂×D₄×C₅₃	—	2	4	no	—
190	C₂×D₄×C₅₃	—	2	4	no	—
191	C₂×D₄×C₅₃	—	4	4	no	—
192	C₂×D₄×C₅₃	—	4	4	no	—
193	C₂×D₄×C₅₃	—	4	16	no	—
194	C₂×D₄×C₅₃	—	4	8	no	—
195	C₂×D₄×C₅₃	—	8	4	no	—
196	C₂×D₄×C₅₃	—	8	8	no	—
197	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
198	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
199	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
200	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
201	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅₃	—	2	1	no	—
203	C₂²×C₄×C₅₃	—	2	4	no	—
204	C₂²×C₄×C₅₃	—	2	2	no	—
205	C₂²×C₄×C₅₃	—	2	4	no	—
206	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
207	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
208	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
210	Q₁₆×C₅₃	—	4	4	no	—
211	SD₁₆×C₅₃	—	4	8	no	—
212	SD₁₆×C₅₃	—	4	4	no	—
213	SD₁₆×C₅₃	—	4	4	no	—
214	D₈×C₅₃	—	4	8	no	—
215	D₈×C₅₃	—	4	4	no	—
216	M₁₆×C₅₃	—	2	4	no	—
217	C₂×C₈×C₅₃	—	2	4	no	—
218	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
219	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₅₃	—	2	2	no	—
221	C₄⋊C₄×C₅₃	—	2	4	no	—
222	C₄⋊C₄×C₅₃	—	2	4	no	—
223	C₄⋊C₄×C₅₃	—	2	2	no	—
224	C₄⋊C₄×C₅₃	—	2	2	no	—
225	C₄⋊C₄×C₅₃	—	2	2	no	—
226	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
227	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
230	C₂²⋊C₄×C₅₃	—	2	1	no	—
231	C₂²⋊C₄×C₅₃	—	2	1	no	—
232	C₂²⋊C₄×C₅₃	—	2	4	no	—
233	C₂²⋊C₄×C₅₃	—	2	4	no	—
234	C₂²⋊C₄×C₅₃	—	2	4	no	—
235	C₂²⋊C₄×C₅₃	—	2	4	no	—
236	C₂²⋊C₄×C₅₃	—	2	4	no	—
237	C₂²⋊C₄×C₅₃	—	2	8	no	—
238	C₂²⋊C₄×C₅₃	—	2	4	no	—
239	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
240	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
241	C₄²×C₅₃	—	2	4	no	—
242	C₄²×C₅₃	—	2	1	no	—
243	C₄²×C₅₃	—	2	1	no	—
244	C₄²×C₅₃	C₂×D₄	1	4	yes	—
245	C₄²×C₅₃	C₂×D₄	1	2	yes	—
246	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₄²×C₅₃	C₂⁴	1	2	yes	—
249	C₂³×C₅₃	—	2	2	no	—
250	C₂³×C₅₃	—	2	4	no	—
251	C₂³×C₅₃	—	2	2	no	—
252	C₂³×C₅₃	—	4	8	no	—
253	C₂³×C₅₃	—	4	4	no	—
254	C₂³×C₅₃	—	4	2	no	—
255	C₂³×C₅₃	—	8	4	no	—
256	C₂³×C₅₃	—	8	8	no	—
257	C₂³×C₅₃	—	8	2	no	—
258	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
259	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
260	Q₈×C₅₃	—	2	2	no	—
261	Q₈×C₅₃	—	2	4	no	—
262	Q₈×C₅₃	—	2	2	no	—
263	Q₈×C₅₃	—	2	2	no	—
264	Q₈×C₅₃	—	4	2	no	—
265	Q₈×C₅₃	—	8	2	no	—
266	D₄×C₅₃	—	2	4	no	—
267	D₄×C₅₃	—	2	2	no	—
268	D₄×C₅₃	—	2	4	no	—
269	D₄×C₅₃	—	2	4	no	—
270	D₄×C₅₃	—	2	4	no	—
271	D₄×C₅₃	—	2	2	no	—
272	D₄×C₅₃	—	2	2	no	—
273	D₄×C₅₃	—	4	8	no	—
274	D₄×C₅₃	—	4	4	no	—
275	D₄×C₅₃	—	4	8	no	—
276	D₄×C₅₃	—	4	8	no	—
277	D₄×C₅₃	—	4	4	no	—
278	D₄×C₅₃	—	4	2	no	—
279	D₄×C₅₃	—	4	4	no	—
280	D₄×C₅₃	—	8	8	no	—
281	D₄×C₅₃	—	8	4	no	—
282	D₄×C₅₃	—	8	4	no	—
283	C₂×C₄×C₅₃	—	2	4	no	—
284	C₂×C₄×C₅₃	—	2	2	no	—
285	C₂×C₄×C₅₃	—	2	2	no	—
286	C₂×C₄×C₅₃	—	2	2	no	—
287	C₂×C₄×C₅₃	—	2	2	no	—
288	C₂×C₄×C₅₃	—	4	4	no	—
289	C₂×C₄×C₅₃	—	4	2	no	—
290	C₂×C₄×C₅₃	—	4	8	no	—
291	C₂×C₄×C₅₃	—	4	2	no	—
292	C₂×C₄×C₅₃	—	4	4	no	—
293	C₂×C₄×C₅₃	—	8	4	no	—
294	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
296	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
297	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
298	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
300	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
301	C₈×C₅₃	—	2	4	no	—
302	C₈×C₅₃	—	2	4	no	—
303	#60827	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂²×C₅₃	—	2	2	no	—
305	C₂²×C₅₃	—	2	4	no	—
306	C₂²×C₅₃	—	2	2	no	—
307	C₂²×C₅₃	—	4	4	no	—
308	C₂²×C₅₃	—	4	8	no	—
309	C₂²×C₅₃	—	4	4	no	—
310	C₂²×C₅₃	—	4	4	no	—
311	C₂²×C₅₃	—	4	2	no	—
312	C₂²×C₅₃	—	4	2	no	—
313	C₂²×C₅₃	—	16	2	no	—
314	C₂²×C₅₃	—	16	4	no	—
315	C₂²×C₅₃	—	16	8	no	—
316	C₂²×C₅₃	#535	1	2	yes	—
317	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
318	C₄×C₅₃	—	2	2	no	—
319	C₄×C₅₃	—	2	4	no	—
320	C₄×C₅₃	—	2	2	no	—
321	C₄×C₅₃	—	2	2	no	—
322	C₄×C₅₃	—	4	2	no	—
323	C₄×C₅₃	—	4	2	no	—
324	C₄×C₅₃	—	4	2	no	—
325	C₄×C₅₃	—	4	2	no	—
326	C₄×C₅₃	#535	1	2	yes	—
327	C₄×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
328	#3510	C₂×C₅₃	1	1	yes	characteristic, complemented
329	#3498	C₂×C₅₃	1	1	yes	characteristic, complemented
330	#3224	C₂×C₅₃	1	8	yes	complemented
331	#3219	C₂×C₅₃	1	4	yes	complemented
332	#3217	C₂×C₅₃	1	4	yes	complemented
333	#3153	C₂×C₅₃	1	1	yes	characteristic, complemented
334	C₂×#496	C₂×C₅₃	1	1	yes	characteristic, complemented
335	#3138	C₂×C₅₃	1	1	yes	characteristic, complemented
336	#3135	C₂×C₅₃	1	1	yes	characteristic, complemented
337	#3043	C₂×C₅₃	1	2	yes	complemented
338	#3041	C₂×C₅₃	1	2	yes	complemented
339	#3009	C₂×C₅₃	1	2	yes	complemented
340	#3008	C₂×C₅₃	1	2	yes	complemented
341	#2923	C₂×C₅₃	1	1	yes	characteristic, complemented
342	C₂×C₅₃	—	2	2	no	—
343	C₂×C₅₃	—	4	2	no	—
344	C₂×C₅₃	—	8	2	no	—
345	C₂×C₅₃	—	8	4	no	—
346	C₂×C₅₃	—	8	2	no	—
347	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
348	C₂×C₅₃	#3585	1	2	yes	—
349	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
350	C₂×#137	C₂²×C₅₃	1	2	yes	complemented
351	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	complemented
352	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
353	C₂²×D₈	C₂²×C₅₃	1	2	yes	complemented
354	#559	—	2	2	no	—
355	#556	—	2	2	no	—
356	#551	—	2	4	no	—
357	#550	—	2	2	no	—
358	D₄²	—	2	2	no	—
359	D₄²	—	2	4	no	—
360	#542	C₂²×C₅₃	1	4	yes	complemented
361	#536	C₂²×C₅₃	1	1	yes	characteristic
362	#535	C₂²×C₅₃	1	2	yes	—
363	#535	C₂²×C₅₃	1	1	yes	characteristic
364	#535	C₂²×C₅₃	1	4	yes	complemented
365	#533	C₂²×C₅₃	1	1	yes	characteristic, complemented
366	#532	C₂²×C₅₃	1	1	yes	characteristic
367	#532	C₂²×C₅₃	1	2	yes	complemented
368	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
369	C₂×#127	C₂²×C₅₃	1	2	yes	complemented
370	#529	C₂²×C₅₃	1	4	yes	complemented
371	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic, complemented
372	#519	C₂²×C₅₃	1	1	yes	characteristic, complemented
373	#518	C₂²×C₅₃	1	1	yes	characteristic, complemented
374	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
375	#497	C₂²×C₅₃	1	2	yes	complemented
376	#496	C₂²×C₅₃	1	4	yes	complemented
377	#496	C₂²×C₅₃	1	4	yes	complemented
378	#496	C₂²×C₅₃	1	2	yes	complemented
379	#495	C₂²×C₅₃	1	4	yes	complemented
380	#491	C₂²×C₅₃	1	2	yes	complemented
381	#490	C₂²×C₅₃	1	2	yes	complemented
382	#490	C₂²×C₅₃	1	4	yes	complemented
383	#489	C₂²×C₅₃	1	4	yes	complemented
384	#486	C₂²×C₅₃	1	4	yes	complemented
385	#464	C₂²×C₅₃	1	8	yes	complemented
386	#463	C₂²×C₅₃	1	8	yes	complemented
387	#460	C₂²×C₅₃	1	8	yes	complemented
388	#459	C₂²×C₅₃	1	8	yes	complemented
389	#452	C₂²×C₅₃	1	8	yes	complemented
390	#450	C₂²×C₅₃	1	8	yes	complemented
391	#449	C₂²×C₅₃	1	8	yes	complemented
392	#447	C₂²×C₅₃	1	8	yes	complemented
393	#433	C₂²×C₅₃	1	4	yes	complemented
394	#432	C₂²×C₅₃	1	4	yes	complemented
395	#424	C₂²×C₅₃	1	2	yes	complemented
396	#418	C₂²×C₅₃	1	2	yes	complemented
397	#418	C₂²×C₅₃	1	2	yes	complemented
398	#416	C₂²×C₅₃	1	2	yes	complemented
399	#416	C₂²×C₅₃	1	2	yes	complemented
400	#408	C₂²×C₅₃	1	2	yes	—
401	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
402	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic, complemented
403	C₅₃	#60827	1	1	yes	characteristic, direct product factor, Sylow
404	H(5,𝔽₂)	—	4	4	no	—
405	H(5,𝔽₂)	—	4	4	no	—
406	C₂×#41	—	2	2	no	—
407	C₂×#41	C₂³×C₅₃	1	2	yes	—
408	C₂×#41	C₂³×C₅₃	1	4	yes	—
409	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
410	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
411	C₂²×D₄	—	2	4	no	—
412	C₂²×D₄	—	2	2	no	—
413	C₂²×D₄	—	2	2	no	—
414	C₂²×D₄	—	2	4	no	—
415	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
417	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
418	#137	—	4	4	no	—
419	AGL(1,ℤ/8)	—	4	4	no	—
420	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
421	C₂×SD₁₆	—	4	4	no	—
422	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
423	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
424	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
425	C₂×D₈	—	4	4	no	—
426	C₂×D₈	C₂³×C₅₃	1	8	yes	—
427	C₂×D₈	C₂³×C₅₃	1	4	yes	—
428	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
429	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
430	C₄⋊Q₈	—	2	1	no	—
431	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
432	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
433	#127	—	2	1	no	—
434	#127	—	2	2	no	—
435	#127	—	2	4	no	—
436	#127	C₂³×C₅₃	1	2	yes	—
437	#127	C₂³×C₅₃	1	2	yes	—
438	#127	C₂³×C₅₃	1	1	yes	characteristic
439	#127	C₂³×C₅₃	1	4	yes	—
440	#126	—	2	4	no	—
441	#126	—	2	4	no	—
442	#125	C₂³×C₅₃	1	2	yes	—
443	#124	—	2	1	no	—
444	#124	—	2	1	no	—
445	#124	—	2	4	no	—
446	#124	—	2	2	no	—
447	#124	C₂³×C₅₃	1	2	yes	—
448	#124	C₂³×C₅₃	1	2	yes	—
449	#124	C₂³×C₅₃	1	2	yes	—
450	#124	C₂³×C₅₃	1	4	yes	—
451	#123	—	2	2	no	—
452	#123	—	2	4	no	—
453	#123	—	2	4	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	4	no	—
456	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
457	#121	—	2	2	no	—
458	#121	—	2	4	no	—
459	#121	—	2	4	no	—
460	#121	—	2	8	no	—
461	#121	C₂³×C₅₃	1	4	yes	—
462	#120	—	2	2	no	—
463	#120	—	2	4	no	—
464	#120	—	2	8	no	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	2	no	—
467	C₄×Q₈	D₄×C₅₃	1	2	yes	—
468	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
469	C₄×D₄	—	2	4	no	—
470	C₄×D₄	—	2	2	no	—
471	C₄×D₄	—	2	2	no	—
472	C₄×D₄	—	2	4	no	—
473	C₄×D₄	D₄×C₅₃	1	4	yes	—
474	C₄×D₄	D₄×C₅₃	1	2	yes	—
475	C₄×D₄	C₂³×C₅₃	1	4	yes	—
476	#117	—	2	2	no	—
477	#117	—	2	4	no	—
478	#117	C₂³×C₅₃	1	1	yes	characteristic
479	#117	C₂³×C₅₃	1	1	yes	characteristic
480	#117	C₂³×C₅₃	1	1	yes	characteristic
481	#117	C₂³×C₅₃	1	2	yes	—
482	C₂×C₂²⋊C₄	—	2	2	no	—
483	C₂×C₂²⋊C₄	—	2	2	no	—
484	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
486	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
487	#103	C₂³×C₅₃	1	4	yes	—
488	#103	C₂³×C₅₃	1	4	yes	—
489	#102	C₂³×C₅₃	1	4	yes	—
490	#102	C₂³×C₅₃	1	4	yes	—
491	#102	C₂³×C₅₃	1	8	yes	—
492	#102	C₂³×C₅₃	1	8	yes	—
493	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
494	#97	C₂³×C₅₃	1	2	yes	—
495	#97	C₂³×C₅₃	1	2	yes	—
496	#97	C₂³×C₅₃	1	2	yes	—
497	C₄×C₈	C₂³×C₅₃	1	2	yes	—
498	C₂⁴	—	2	1	no	—
499	C₂⁴	—	2	4	no	—
500	C₂⁴	—	2	2	no	—
501	#41	—	2	4	no	—
502	#41	—	2	4	no	—
503	#41	—	2	4	no	—
504	#41	—	4	4	no	—
505	#41	—	4	4	no	—
506	#41	—	4	8	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
511	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
512	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	4	no	—
520	C₂×D₄	—	2	2	no	—
521	C₂×D₄	—	2	8	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	4	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	4	4	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	4	16	no	—
529	C₂×D₄	—	4	8	no	—
530	C₂×D₄	—	8	4	no	—
531	C₂×D₄	—	8	8	no	—
532	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
533	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
534	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
535	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
536	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	4	no	—
541	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
543	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
545	Q₁₆	—	4	4	no	—
546	SD₁₆	—	4	8	no	—
547	SD₁₆	—	4	4	no	—
548	SD₁₆	—	4	4	no	—
549	D₈	—	4	8	no	—
550	D₈	—	4	4	no	—
551	M₁₆	—	2	4	no	—
552	C₂×C₈	—	2	4	no	—
553	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
554	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
562	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
563	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
564	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	1	no	—
567	C₂²⋊C₄	—	2	4	no	—
568	C₂²⋊C₄	—	2	4	no	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	8	no	—
573	C₂²⋊C₄	—	2	4	no	—
574	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
575	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
576	C₄²	—	2	4	no	—
577	C₄²	—	2	1	no	—
578	C₄²	—	2	1	no	—
579	C₄²	C₂×D₄×C₅₃	1	4	yes	—
580	C₄²	C₂×D₄×C₅₃	1	2	yes	—
581	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
582	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
583	C₄²	C₂⁴×C₅₃	1	2	yes	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	4	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	4	8	no	—
588	C₂³	—	4	4	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	8	4	no	—
591	C₂³	—	8	8	no	—
592	C₂³	—	8	2	no	—
593	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
594	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
595	Q₈	—	2	2	no	—
596	Q₈	—	2	4	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	4	2	no	—
600	Q₈	—	8	2	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	4	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	4	no	—
607	D₄	—	2	2	no	—
608	D₄	—	4	8	no	—
609	D₄	—	4	4	no	—
610	D₄	—	4	8	no	—
611	D₄	—	4	8	no	—
612	D₄	—	4	4	no	—
613	D₄	—	4	2	no	—
614	D₄	—	4	4	no	—
615	D₄	—	8	8	no	—
616	D₄	—	8	4	no	—
617	D₄	—	8	4	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	8	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	8	4	no	—
629	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
630	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
631	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
632	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
634	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
635	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
636	C₈	—	2	4	no	—
637	C₈	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	4	4	no	—
642	C₂²	—	4	8	no	—
643	C₂²	—	4	4	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	16	4	no	—
648	C₂²	—	16	2	no	—
649	C₂²	—	16	8	no	—
650	C₂²	C₅₃×#535	1	2	yes	—
651	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	4	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	C₅₃×#535	1	2	yes	—
661	C₄	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	2	no	—
664	C₂	—	8	4	no	—
665	C₂	—	8	2	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
668	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 ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3510	C₂	semidirect
2. C₅₃×#3498	C₂	semidirect
3. C₅₃×#3224	C₂	semidirect
4. C₅₃×#3219	C₂	semidirect
5. C₅₃×#3217	C₂	semidirect
6. C₅₃×#3153	C₂	semidirect
7. C₂×C₅₃×#496	C₂	semidirect
8. C₅₃×#3138	C₂	semidirect
9. C₅₃×#3135	C₂	semidirect
10. C₅₃×#3043	C₂	semidirect
11. C₅₃×#3041	C₂	semidirect
12. C₅₃×#3009	C₂	semidirect
13. C₅₃×#3008	C₂	semidirect
14. C₅₃×#2923	C₂	semidirect
16. C₂×#137×C₅₃	C₂²	semidirect
17. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
18. C₂²×SD₁₆×C₅₃	C₂²	semidirect
19. C₂²×D₈×C₅₃	C₂²	semidirect
26. C₅₃×#542	C₂²	semidirect
30. C₅₃×#535	C₂²	semidirect
31. C₅₃×#533	C₂²	semidirect
33. C₅₃×#532	C₂²	semidirect
35. C₂×#127×C₅₃	C₂²	semidirect
36. C₅₃×#529	C₂²	semidirect
37. C₂×#124×C₅₃	C₂²	semidirect
38. C₅₃×#519	C₂²	semidirect
39. C₅₃×#518	C₂²	semidirect
41. C₅₃×#497	C₂²	semidirect
42. C₅₃×#496	C₂²	semidirect
43. C₅₃×#496	C₂²	semidirect
44. C₅₃×#496	C₂²	semidirect
45. C₅₃×#495	C₂²	semidirect
46. C₅₃×#491	C₂²	semidirect
47. C₅₃×#490	C₂²	semidirect
48. C₅₃×#490	C₂²	semidirect
49. C₅₃×#489	C₂²	semidirect
50. C₅₃×#486	C₂²	semidirect
51. C₅₃×#464	C₂²	semidirect
52. C₅₃×#463	C₂²	semidirect
53. C₅₃×#460	C₂²	semidirect
54. C₅₃×#459	C₂²	semidirect
55. C₅₃×#452	C₂²	semidirect
56. C₅₃×#450	C₂²	semidirect
57. C₅₃×#449	C₂²	semidirect
58. C₅₃×#447	C₂²	semidirect
59. C₅₃×#433	C₂²	semidirect
60. C₅₃×#432	C₂²	semidirect
61. C₅₃×#424	C₂²	semidirect
62. C₅₃×#418	C₂²	semidirect
63. C₅₃×#418	C₂²	semidirect
64. C₅₃×#416	C₂²	semidirect
65. C₅₃×#416	C₂²	semidirect
67. C₄×M₁₆×C₅₃	C₂²	semidirect
68. C₂×#97×C₅₃	C₂²	semidirect
303. #60827	C₅₃	direct (N × H)
328. #3510	C₂×C₅₃	semidirect
329. #3498	C₂×C₅₃	semidirect
330. #3224	C₂×C₅₃	semidirect
331. #3219	C₂×C₅₃	semidirect
332. #3217	C₂×C₅₃	semidirect
333. #3153	C₂×C₅₃	semidirect
334. C₂×#496	C₂×C₅₃	semidirect
335. #3138	C₂×C₅₃	semidirect
336. #3135	C₂×C₅₃	semidirect
337. #3043	C₂×C₅₃	semidirect
338. #3041	C₂×C₅₃	semidirect
339. #3009	C₂×C₅₃	semidirect
340. #3008	C₂×C₅₃	semidirect
341. #2923	C₂×C₅₃	semidirect
350. C₂×#137	C₂²×C₅₃	semidirect
351. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
352. C₂²×SD₁₆	C₂²×C₅₃	semidirect
353. C₂²×D₈	C₂²×C₅₃	semidirect
360. #542	C₂²×C₅₃	semidirect
364. #535	C₂²×C₅₃	semidirect
365. #533	C₂²×C₅₃	semidirect
367. #532	C₂²×C₅₃	semidirect
369. C₂×#127	C₂²×C₅₃	semidirect
370. #529	C₂²×C₅₃	semidirect
371. C₂×#124	C₂²×C₅₃	semidirect
372. #519	C₂²×C₅₃	semidirect
373. #518	C₂²×C₅₃	semidirect
375. #497	C₂²×C₅₃	semidirect
376. #496	C₂²×C₅₃	semidirect
377. #496	C₂²×C₅₃	semidirect
378. #496	C₂²×C₅₃	semidirect
379. #495	C₂²×C₅₃	semidirect
380. #491	C₂²×C₅₃	semidirect
381. #490	C₂²×C₅₃	semidirect
382. #490	C₂²×C₅₃	semidirect
383. #489	C₂²×C₅₃	semidirect
384. #486	C₂²×C₅₃	semidirect
385. #464	C₂²×C₅₃	semidirect
386. #463	C₂²×C₅₃	semidirect
387. #460	C₂²×C₅₃	semidirect
388. #459	C₂²×C₅₃	semidirect
389. #452	C₂²×C₅₃	semidirect
390. #450	C₂²×C₅₃	semidirect
391. #449	C₂²×C₅₃	semidirect
392. #447	C₂²×C₅₃	semidirect
393. #433	C₂²×C₅₃	semidirect
394. #432	C₂²×C₅₃	semidirect
395. #424	C₂²×C₅₃	semidirect
396. #418	C₂²×C₅₃	semidirect
397. #418	C₂²×C₅₃	semidirect
398. #416	C₂²×C₅₃	semidirect
399. #416	C₂²×C₅₃	semidirect
401. C₄×M₁₆	C₂²×C₅₃	semidirect
402. C₂×#97	C₂²×C₅₃	semidirect
403. C₅₃	#60827	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×#3498	2
C₅₃×#3224	2
C₅₃×#3219	2
C₅₃×#3217	2
C₅₃×#3153	2
C₂×C₅₃×#496	2
C₅₃×#3138	2
C₅₃×#3135	2
C₅₃×#3043	2
C₅₃×#3041	2
C₅₃×#3009	2
C₅₃×#3008	2
C₅₃×#2923	2
#60827	53

Subgroups — C53×#60827 #27268339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60827 #27268339