Subgroups — C₅₃×#60390 #27267902

Order 13568.

4698 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3585	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3021	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3020	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3018	C₂	1	6	yes	maximal, complemented
7	C₂×C₅₃×#454	C₂	1	6	yes	maximal, complemented
8	C₅₃×#2901	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#585	—	2	8	no	—
11	C₅₃×#585	—	2	6	no	—
12	C₅₃×#585	C₂²	1	2	yes	—
13	C₅₃×#585	C₂²	1	6	yes	complemented
14	C₂×2₋¹⁺⁴×C₅₃	—	2	8	no	—
15	C₂×2₋¹⁺⁴×C₅₃	—	2	2	no	—
16	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	6	yes	complemented
18	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#577	—	2	8	no	—
20	C₅₃×#576	—	2	8	no	—
21	C₅₃×#575	C₂²	1	12	yes	complemented
22	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	12	yes	complemented
23	C₅₃×#568	C₂²	1	2	yes	complemented
24	C₅₃×#548	—	2	2	no	—
25	C₅₃×#546	—	2	6	no	—
26	C₅₃×#536	C₂²	1	3	yes	complemented
27	C₅₃×#535	C₂²	1	3	yes	complemented
28	C₅₃×#532	C₂²	1	3	yes	complemented
29	C₂×#124×C₅₃	C₂²	1	3	yes	complemented
30	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₅₃×#456	C₂²	1	12	yes	complemented
32	C₅₃×#455	C₂²	1	4	yes	complemented
33	C₅₃×#454	C₂²	1	12	yes	complemented
34	C₅₃×#454	C₂²	1	24	yes	complemented
35	C₅₃×#453	C₂²	1	12	yes	complemented
36	C₅₃×#421	C₂²	1	12	yes	complemented
37	C₂×#104×C₅₃	C₂²	1	12	yes	complemented
38	C₅₃×#413	C₂²	1	6	yes	complemented
39	C₂×#101×C₅₃	C₂²	1	6	yes	complemented
40	2₋¹⁺⁴×C₅₃	—	2	24	no	—
41	2₋¹⁺⁴×C₅₃	—	2	16	no	—
42	2₋¹⁺⁴×C₅₃	—	2	8	no	—
43	2₋¹⁺⁴×C₅₃	—	2	24	no	—
44	2₋¹⁺⁴×C₅₃	—	2	2	no	—
45	2₋¹⁺⁴×C₅₃	—	2	12	no	—
46	2₋¹⁺⁴×C₅₃	—	2	12	no	—
47	2₋¹⁺⁴×C₅₃	D₄	1	8	yes	complemented
48	2₋¹⁺⁴×C₅₃	C₂³	1	12	yes	—
49	H(5,𝔽₂)×C₅₃	—	2	24	no	—
50	H(5,𝔽₂)×C₅₃	—	2	6	no	—
51	H(5,𝔽₂)×C₅₃	—	2	12	no	—
52	H(5,𝔽₂)×C₅₃	—	2	12	no	—
53	H(5,𝔽₂)×C₅₃	—	2	8	no	—
54	H(5,𝔽₂)×C₅₃	D₄	1	8	yes	complemented
55	H(5,𝔽₂)×C₅₃	C₂³	1	4	yes	—
56	C₂×#41×C₅₃	—	2	8	no	—
57	C₂×#41×C₅₃	—	2	48	no	—
58	C₂×#41×C₅₃	—	2	8	no	—
59	C₂×#41×C₅₃	—	2	24	no	—
60	C₂×#41×C₅₃	—	2	1	no	—
61	C₂×#41×C₅₃	—	2	12	no	—
62	C₂×#41×C₅₃	—	2	12	no	—
63	C₂×#41×C₅₃	—	2	12	no	—
64	C₂×#41×C₅₃	D₄	1	6	yes	—
65	C₂×#41×C₅₃	C₂³	1	12	yes	—
66	C₂×#41×C₅₃	C₂³	1	3	yes	—
67	C₂×#41×C₅₃	C₂³	1	3	yes	—
68	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₅₃	—	2	8	no	—
70	C₂²×Q₈×C₅₃	—	2	12	no	—
71	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
72	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
73	C₂²×D₄×C₅₃	—	2	3	no	—
74	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
75	#137×C₅₃	—	2	12	no	—
76	#137×C₅₃	C₂³	1	8	yes	—
77	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
78	AGL(1,ℤ/8)×C₅₃	—	2	12	no	—
79	AGL(1,ℤ/8)×C₅₃	C₂³	1	24	yes	—
80	#135×C₅₃	—	2	24	no	—
81	C₂×SD₁₆×C₅₃	—	2	12	no	—
82	C₂×D₈×C₅₃	—	2	12	no	—
83	#131×C₅₃	—	2	8	no	—
84	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
85	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
86	#127×C₅₃	C₂³	1	3	yes	—
87	#124×C₅₃	—	2	6	no	—
88	#124×C₅₃	C₂³	1	3	yes	—
89	#124×C₅₃	C₂³	1	6	yes	—
90	#124×C₅₃	C₂³	1	1	yes	characteristic
91	#123×C₅₃	—	2	6	no	—
92	C₂²⋊Q₈×C₅₃	—	2	6	no	—
93	#121×C₅₃	—	2	6	no	—
94	#121×C₅₃	—	2	6	no	—
95	#120×C₅₃	—	2	6	no	—
96	C₄×Q₈×C₅₃	—	2	1	no	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	6	no	—
99	#117×C₅₃	C₂³	1	3	yes	—
100	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
101	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
102	#104×C₅₃	C₂³	1	8	yes	—
103	#104×C₅₃	C₂³	1	24	yes	—
104	#101×C₅₃	C₂³	1	12	yes	—
105	#100×C₅₃	C₂³	1	4	yes	—
106	C₂⁴×C₅₃	—	2	3	no	—
107	#41×C₅₃	—	2	48	no	—
108	#41×C₅₃	—	2	48	no	—
109	#41×C₅₃	—	2	48	no	—
110	#41×C₅₃	—	2	24	no	—
111	#41×C₅₃	—	2	24	no	—
112	#41×C₅₃	—	2	8	no	—
113	#41×C₅₃	—	2	8	no	—
114	#41×C₅₃	—	2	24	no	—
115	#41×C₅₃	—	2	24	no	—
116	#41×C₅₃	—	2	24	no	—
117	#41×C₅₃	—	2	24	no	—
118	#41×C₅₃	—	2	12	no	—
119	#41×C₅₃	—	2	12	no	—
120	#41×C₅₃	—	2	12	no	—
121	#41×C₅₃	—	2	12	no	—
122	#41×C₅₃	—	4	4	no	—
123	#41×C₅₃	C₂×D₄	1	24	yes	—
124	#41×C₅₃	C₂×D₄	1	8	yes	—
125	#41×C₅₃	C₂⁴	1	8	yes	—
126	C₂×Q₈×C₅₃	—	2	48	no	—
127	C₂×Q₈×C₅₃	—	2	12	no	—
128	C₂×Q₈×C₅₃	—	2	12	no	—
129	C₂×Q₈×C₅₃	—	2	8	no	—
130	C₂×Q₈×C₅₃	—	2	8	no	—
131	C₂×Q₈×C₅₃	—	2	24	no	—
132	C₂×Q₈×C₅₃	—	2	12	no	—
133	C₂×Q₈×C₅₃	C₂×D₄	1	12	yes	—
134	C₂×Q₈×C₅₃	C₂×D₄	1	6	yes	—
135	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
138	C₂×D₄×C₅₃	—	2	48	no	—
139	C₂×D₄×C₅₃	—	2	12	no	—
140	C₂×D₄×C₅₃	—	2	12	no	—
141	C₂×D₄×C₅₃	—	2	8	no	—
142	C₂×D₄×C₅₃	—	2	12	no	—
143	C₂×D₄×C₅₃	—	2	24	no	—
144	C₂×D₄×C₅₃	—	2	8	no	—
145	C₂×D₄×C₅₃	—	2	3	no	—
146	C₂×D₄×C₅₃	—	2	6	no	—
147	C₂×D₄×C₅₃	—	2	12	no	—
148	C₂×D₄×C₅₃	—	2	3	no	—
149	C₂×D₄×C₅₃	—	4	12	no	—
150	C₂×D₄×C₅₃	C₂×D₄	1	6	yes	—
151	C₂×D₄×C₅₃	C₂×D₄	1	12	yes	—
152	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
153	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
154	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	—	2	24	no	—
156	C₂²×C₄×C₅₃	—	2	24	no	—
157	C₂²×C₄×C₅₃	—	2	12	no	—
158	C₂²×C₄×C₅₃	—	2	3	no	—
159	C₂²×C₄×C₅₃	—	2	3	no	—
160	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
163	Q₁₆×C₅₃	—	2	8	no	—
164	SD₁₆×C₅₃	—	2	24	no	—
165	SD₁₆×C₅₃	—	2	8	no	—
166	D₈×C₅₃	—	2	24	no	—
167	M₁₆×C₅₃	—	2	12	no	—
168	M₁₆×C₅₃	C₂⁴	1	8	yes	—
169	C₂×C₈×C₅₃	—	2	12	no	—
170	C₄⋊C₄×C₅₃	—	2	3	no	—
171	C₄⋊C₄×C₅₃	—	2	6	no	—
172	C₂²⋊C₄×C₅₃	—	2	3	no	—
173	C₂²⋊C₄×C₅₃	—	2	2	no	—
174	C₂²⋊C₄×C₅₃	—	2	6	no	—
175	C₂²⋊C₄×C₅₃	—	2	12	no	—
176	C₄²×C₅₃	—	2	3	no	—
177	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₄²×C₅₃	C₂⁴	1	3	yes	—
179	C₂³×C₅₃	—	2	8	no	—
180	C₂³×C₅₃	—	2	12	no	—
181	C₂³×C₅₃	—	2	3	no	—
182	C₂³×C₅₃	—	4	6	no	—
183	C₂³×C₅₃	—	8	6	no	—
184	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
185	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
186	Q₈×C₅₃	—	2	24	no	—
187	Q₈×C₅₃	—	2	12	no	—
188	Q₈×C₅₃	—	2	12	no	—
189	Q₈×C₅₃	—	2	6	no	—
190	Q₈×C₅₃	—	2	2	no	—
191	Q₈×C₅₃	#120	1	4	yes	—
192	Q₈×C₅₃	C₂²×D₄	1	8	yes	—
193	D₄×C₅₃	—	2	16	no	—
194	D₄×C₅₃	—	2	24	no	—
195	D₄×C₅₃	—	2	24	no	—
196	D₄×C₅₃	—	2	12	no	—
197	D₄×C₅₃	—	2	8	no	—
198	D₄×C₅₃	—	2	12	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	2	6	no	—
201	D₄×C₅₃	—	4	6	no	—
202	D₄×C₅₃	—	4	6	no	—
203	D₄×C₅₃	—	8	6	no	—
204	D₄×C₅₃	#120	1	12	yes	—
205	D₄×C₅₃	C₂²×D₄	1	8	yes	—
206	C₂×C₄×C₅₃	—	2	8	no	—
207	C₂×C₄×C₅₃	—	2	48	no	—
208	C₂×C₄×C₅₃	—	2	12	no	—
209	C₂×C₄×C₅₃	—	2	12	no	—
210	C₂×C₄×C₅₃	—	2	24	no	—
211	C₂×C₄×C₅₃	—	2	12	no	—
212	C₂×C₄×C₅₃	—	2	8	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	3	no	—
216	C₂×C₄×C₅₃	—	4	6	no	—
217	C₂×C₄×C₅₃	—	8	3	no	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
219	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
220	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
221	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₅₃	—	2	8	no	—
223	#60390	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂²×C₅₃	—	2	2	no	—
225	C₂²×C₅₃	—	2	8	no	—
226	C₂²×C₅₃	—	4	3	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	—	4	6	no	—
229	C₂²×C₅₃	—	4	3	no	—
230	C₂²×C₅₃	—	8	12	no	—
231	C₂²×C₅₃	—	8	8	no	—
232	C₂²×C₅₃	—	16	6	no	—
233	C₂²×C₅₃	C₂×#120	1	6	yes	—
234	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
235	C₄×C₅₃	—	2	8	no	—
236	C₄×C₅₃	—	2	6	no	—
237	C₄×C₅₃	—	4	1	no	—
238	C₄×C₅₃	—	4	3	no	—
239	C₄×C₅₃	—	4	1	no	—
240	C₄×C₅₃	C₂×#120	1	6	yes	—
241	C₄×C₅₃	C₂³×D₄	1	2	yes	—
242	2₋¹⁺⁶	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#3585	C₂×C₅₃	1	2	yes	complemented
244	#3500	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#3021	C₂×C₅₃	1	6	yes	complemented
246	#3020	C₂×C₅₃	1	6	yes	complemented
247	#3018	C₂×C₅₃	1	6	yes	complemented
248	C₂×#454	C₂×C₅₃	1	6	yes	complemented
249	#2901	C₂×C₅₃	1	2	yes	complemented
250	#2889	C₂×C₅₃	1	1	yes	characteristic, complemented
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	2	6	no	—
253	C₂×C₅₃	—	4	2	no	—
254	C₂×C₅₃	—	4	8	no	—
255	C₂×C₅₃	—	8	3	no	—
256	C₂×C₅₃	C₂²×#120	1	1	yes	characteristic
257	#585	—	2	8	no	—
258	#585	—	2	6	no	—
259	#585	C₂²×C₅₃	1	2	yes	—
260	#585	C₂²×C₅₃	1	6	yes	complemented
261	C₂×2₋¹⁺⁴	—	2	8	no	—
262	C₂×2₋¹⁺⁴	—	2	2	no	—
263	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	6	yes	complemented
265	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic, complemented
266	#577	—	2	8	no	—
267	#576	—	2	8	no	—
268	#575	C₂²×C₅₃	1	12	yes	complemented
269	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	12	yes	complemented
270	#568	C₂²×C₅₃	1	2	yes	complemented
271	#548	—	2	2	no	—
272	#546	—	2	6	no	—
273	#536	C₂²×C₅₃	1	3	yes	complemented
274	#535	C₂²×C₅₃	1	3	yes	complemented
275	#532	C₂²×C₅₃	1	3	yes	complemented
276	C₂×#124	C₂²×C₅₃	1	3	yes	complemented
277	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic, complemented
278	#456	C₂²×C₅₃	1	12	yes	complemented
279	#455	C₂²×C₅₃	1	4	yes	complemented
280	#454	C₂²×C₅₃	1	12	yes	complemented
281	#454	C₂²×C₅₃	1	24	yes	complemented
282	#453	C₂²×C₅₃	1	12	yes	complemented
283	#421	C₂²×C₅₃	1	12	yes	complemented
284	C₂×#104	C₂²×C₅₃	1	12	yes	complemented
285	#413	C₂²×C₅₃	1	6	yes	complemented
286	C₂×#101	C₂²×C₅₃	1	6	yes	complemented
287	C₅₃	#60390	1	1	yes	characteristic, direct product factor, Sylow
288	2₋¹⁺⁴	—	2	24	no	—
289	2₋¹⁺⁴	—	2	16	no	—
290	2₋¹⁺⁴	—	2	8	no	—
291	2₋¹⁺⁴	—	2	24	no	—
292	2₋¹⁺⁴	—	2	2	no	—
293	2₋¹⁺⁴	—	2	12	no	—
294	2₋¹⁺⁴	—	2	12	no	—
295	2₋¹⁺⁴	D₄×C₅₃	1	8	yes	complemented
296	2₋¹⁺⁴	C₂³×C₅₃	1	12	yes	—
297	H(5,𝔽₂)	—	2	24	no	—
298	H(5,𝔽₂)	—	2	6	no	—
299	H(5,𝔽₂)	—	2	12	no	—
300	H(5,𝔽₂)	—	2	12	no	—
301	H(5,𝔽₂)	—	2	8	no	—
302	H(5,𝔽₂)	D₄×C₅₃	1	8	yes	complemented
303	H(5,𝔽₂)	C₂³×C₅₃	1	4	yes	—
304	C₂×#41	—	2	8	no	—
305	C₂×#41	—	2	48	no	—
306	C₂×#41	—	2	8	no	—
307	C₂×#41	—	2	24	no	—
308	C₂×#41	—	2	1	no	—
309	C₂×#41	—	2	12	no	—
310	C₂×#41	—	2	12	no	—
311	C₂×#41	—	2	12	no	—
312	C₂×#41	D₄×C₅₃	1	6	yes	—
313	C₂×#41	C₂³×C₅₃	1	12	yes	—
314	C₂×#41	C₂³×C₅₃	1	3	yes	—
315	C₂×#41	C₂³×C₅₃	1	3	yes	—
316	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
317	C₂²×Q₈	—	2	8	no	—
318	C₂²×Q₈	—	2	12	no	—
319	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
320	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
321	C₂²×D₄	—	2	3	no	—
322	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
323	#137	—	2	12	no	—
324	#137	C₂³×C₅₃	1	8	yes	—
325	AGL(1,ℤ/8)	—	2	24	no	—
326	AGL(1,ℤ/8)	—	2	12	no	—
327	AGL(1,ℤ/8)	C₂³×C₅₃	1	24	yes	—
328	#135	—	2	24	no	—
329	C₂×SD₁₆	—	2	12	no	—
330	C₂×D₈	—	2	12	no	—
331	#131	—	2	8	no	—
332	C₂×M₁₆	C₂³×C₅₃	1	12	yes	—
333	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
334	#127	C₂³×C₅₃	1	3	yes	—
335	#124	—	2	6	no	—
336	#124	C₂³×C₅₃	1	3	yes	—
337	#124	C₂³×C₅₃	1	6	yes	—
338	#124	C₂³×C₅₃	1	1	yes	characteristic
339	#123	—	2	6	no	—
340	C₂²⋊Q₈	—	2	6	no	—
341	#121	—	2	6	no	—
342	#121	—	2	6	no	—
343	#120	—	2	6	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	6	no	—
347	#117	C₂³×C₅₃	1	3	yes	—
348	C₂×C₂²⋊C₄	—	2	6	no	—
349	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
350	#104	C₂³×C₅₃	1	8	yes	—
351	#104	C₂³×C₅₃	1	24	yes	—
352	#101	C₂³×C₅₃	1	12	yes	—
353	#100	C₂³×C₅₃	1	4	yes	—
354	C₂⁴	—	2	3	no	—
355	#41	—	2	48	no	—
356	#41	—	2	48	no	—
357	#41	—	2	48	no	—
358	#41	—	2	24	no	—
359	#41	—	2	24	no	—
360	#41	—	2	8	no	—
361	#41	—	2	8	no	—
362	#41	—	2	24	no	—
363	#41	—	2	24	no	—
364	#41	—	2	24	no	—
365	#41	—	2	24	no	—
366	#41	—	2	12	no	—
367	#41	—	2	12	no	—
368	#41	—	2	12	no	—
369	#41	—	2	12	no	—
370	#41	—	4	4	no	—
371	#41	C₂×D₄×C₅₃	1	24	yes	—
372	#41	C₂×D₄×C₅₃	1	8	yes	—
373	#41	C₂⁴×C₅₃	1	8	yes	—
374	C₂×Q₈	—	2	48	no	—
375	C₂×Q₈	—	2	12	no	—
376	C₂×Q₈	—	2	12	no	—
377	C₂×Q₈	—	2	8	no	—
378	C₂×Q₈	—	2	8	no	—
379	C₂×Q₈	—	2	24	no	—
380	C₂×Q₈	—	2	12	no	—
381	C₂×Q₈	C₂×D₄×C₅₃	1	12	yes	—
382	C₂×Q₈	C₂×D₄×C₅₃	1	6	yes	—
383	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
384	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
385	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
386	C₂×D₄	—	2	48	no	—
387	C₂×D₄	—	2	12	no	—
388	C₂×D₄	—	2	12	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	12	no	—
391	C₂×D₄	—	2	24	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	—	2	3	no	—
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	—	2	12	no	—
396	C₂×D₄	—	2	3	no	—
397	C₂×D₄	—	4	12	no	—
398	C₂×D₄	C₂×D₄×C₅₃	1	6	yes	—
399	C₂×D₄	C₂×D₄×C₅₃	1	12	yes	—
400	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
401	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
402	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	24	no	—
404	C₂²×C₄	—	2	24	no	—
405	C₂²×C₄	—	2	12	no	—
406	C₂²×C₄	—	2	3	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
409	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
410	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
411	Q₁₆	—	2	8	no	—
412	SD₁₆	—	2	24	no	—
413	SD₁₆	—	2	8	no	—
414	D₈	—	2	24	no	—
415	M₁₆	—	2	12	no	—
416	M₁₆	C₂⁴×C₅₃	1	8	yes	—
417	C₂×C₈	—	2	12	no	—
418	C₄⋊C₄	—	2	3	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₂²⋊C₄	—	2	3	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₂²⋊C₄	—	2	12	no	—
424	C₄²	—	2	3	no	—
425	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
426	C₄²	C₂⁴×C₅₃	1	3	yes	—
427	C₂³	—	2	8	no	—
428	C₂³	—	2	12	no	—
429	C₂³	—	2	3	no	—
430	C₂³	—	4	6	no	—
431	C₂³	—	8	6	no	—
432	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
433	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
434	Q₈	—	2	24	no	—
435	Q₈	—	2	12	no	—
436	Q₈	—	2	12	no	—
437	Q₈	—	2	6	no	—
438	Q₈	—	2	2	no	—
439	Q₈	#120×C₅₃	1	4	yes	—
440	Q₈	C₂²×D₄×C₅₃	1	8	yes	—
441	D₄	—	2	16	no	—
442	D₄	—	2	24	no	—
443	D₄	—	2	24	no	—
444	D₄	—	2	12	no	—
445	D₄	—	2	8	no	—
446	D₄	—	2	12	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	6	no	—
449	D₄	—	4	6	no	—
450	D₄	—	4	6	no	—
451	D₄	—	8	6	no	—
452	D₄	#120×C₅₃	1	12	yes	—
453	D₄	C₂²×D₄×C₅₃	1	8	yes	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	48	no	—
456	C₂×C₄	—	2	12	no	—
457	C₂×C₄	—	2	12	no	—
458	C₂×C₄	—	2	24	no	—
459	C₂×C₄	—	2	12	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	3	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	4	6	no	—
465	C₂×C₄	—	8	3	no	—
466	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
467	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
468	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
469	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
470	C₈	—	2	8	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	4	3	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	6	no	—
476	C₂²	—	4	3	no	—
477	C₂²	—	8	12	no	—
478	C₂²	—	8	8	no	—
479	C₂²	—	16	6	no	—
480	C₂²	C₂×#120×C₅₃	1	6	yes	—
481	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
482	C₄	—	2	8	no	—
483	C₄	—	2	6	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	3	no	—
486	C₄	—	4	1	no	—
487	C₄	C₂×#120×C₅₃	1	6	yes	—
488	C₄	C₂³×D₄×C₅₃	1	2	yes	—
489	C₂	—	2	1	no	—
490	C₂	—	2	6	no	—
491	C₂	—	4	2	no	—
492	C₂	—	4	8	no	—
493	C₂	—	8	3	no	—
494	C₂	C₂²×#120×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	C₂²×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×2₋¹⁺⁶	C₂	semidirect
2. C₅₃×#3585	C₂	semidirect
3. C₅₃×#3500	C₂	semidirect
4. C₅₃×#3021	C₂	semidirect
5. C₅₃×#3020	C₂	semidirect
6. C₅₃×#3018	C₂	semidirect
7. C₂×C₅₃×#454	C₂	semidirect
8. C₅₃×#2901	C₂	semidirect
9. C₅₃×#2889	C₂	semidirect
13. C₅₃×#585	C₂²	semidirect
17. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
18. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
21. C₅₃×#575	C₂²	semidirect
22. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
23. C₅₃×#568	C₂²	semidirect
26. C₅₃×#536	C₂²	semidirect
27. C₅₃×#535	C₂²	semidirect
28. C₅₃×#532	C₂²	semidirect
29. C₂×#124×C₅₃	C₂²	semidirect
30. C₄×#41×C₅₃	C₂²	semidirect
31. C₅₃×#456	C₂²	semidirect
32. C₅₃×#455	C₂²	semidirect
33. C₅₃×#454	C₂²	semidirect
34. C₅₃×#454	C₂²	semidirect
35. C₅₃×#453	C₂²	semidirect
36. C₅₃×#421	C₂²	semidirect
37. C₂×#104×C₅₃	C₂²	semidirect
38. C₅₃×#413	C₂²	semidirect
39. C₂×#101×C₅₃	C₂²	semidirect
47. 2₋¹⁺⁴×C₅₃	D₄	semidirect
54. H(5,𝔽₂)×C₅₃	D₄	semidirect
223. #60390	C₅₃	direct (N × H)
242. 2₋¹⁺⁶	C₂×C₅₃	semidirect
243. #3585	C₂×C₅₃	semidirect
244. #3500	C₂×C₅₃	semidirect
245. #3021	C₂×C₅₃	semidirect
246. #3020	C₂×C₅₃	semidirect
247. #3018	C₂×C₅₃	semidirect
248. C₂×#454	C₂×C₅₃	semidirect
249. #2901	C₂×C₅₃	semidirect
250. #2889	C₂×C₅₃	semidirect
260. #585	C₂²×C₅₃	semidirect
264. C₂×2₋¹⁺⁴	C₂²×C₅₃	semidirect
265. C₂×H(5,𝔽₂)	C₂²×C₅₃	semidirect
268. #575	C₂²×C₅₃	semidirect
269. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
270. #568	C₂²×C₅₃	semidirect
273. #536	C₂²×C₅₃	semidirect
274. #535	C₂²×C₅₃	semidirect
275. #532	C₂²×C₅₃	semidirect
276. C₂×#124	C₂²×C₅₃	semidirect
277. C₄×#41	C₂²×C₅₃	semidirect
278. #456	C₂²×C₅₃	semidirect
279. #455	C₂²×C₅₃	semidirect
280. #454	C₂²×C₅₃	semidirect
281. #454	C₂²×C₅₃	semidirect
282. #453	C₂²×C₅₃	semidirect
283. #421	C₂²×C₅₃	semidirect
284. C₂×#104	C₂²×C₅₃	semidirect
285. #413	C₂²×C₅₃	semidirect
286. C₂×#101	C₂²×C₅₃	semidirect
287. C₅₃	#60390	direct (N × H)
295. 2₋¹⁺⁴	D₄×C₅₃	semidirect
302. H(5,𝔽₂)	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×2₋¹⁺⁶	2
C₅₃×#3585	2
C₅₃×#3500	2
C₅₃×#3021	2
C₅₃×#3020	2
C₅₃×#3018	2
C₂×C₅₃×#454	2
C₅₃×#2901	2
C₅₃×#2889	2
#60390	53

Subgroups — C53×#60390 #27267902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60390 #27267902