Subgroups — C₅₃×#60650 #27268162

Order 13568.

3274 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#577	C₂	1	2	yes	maximal, complemented
2	C₂×C₅₃×#576	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3155	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3154	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3153	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3151	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3150	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#496	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
14	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#577	—	2	16	no	—
16	C₅₃×#576	—	2	16	no	—
17	C₂×#137×C₅₃	C₂²	1	6	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	complemented
21	C₂×#135×C₅₃	C₂²	1	6	yes	complemented
22	C₂×#135×C₅₃	C₂²	1	6	yes	complemented
23	C₂²×SD₁₆×C₅₃	C₂²	1	6	yes	complemented
24	C₂²×D₈×C₅₃	C₂²	1	6	yes	complemented
25	C₂×#131×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#550	—	2	2	no	—
27	C₅₃×#548	—	2	2	no	—
28	C₅₃×#546	—	2	6	no	—
29	D₄²×C₅₃	—	2	6	no	—
30	C₅₃×#536	C₂²	1	3	yes	complemented
31	C₅₃×#535	C₂²	1	6	yes	complemented
32	C₅₃×#535	C₂²	1	3	yes	complemented
33	C₅₃×#532	C₂²	1	3	yes	complemented
34	C₅₃×#532	C₂²	1	3	yes	complemented
35	C₂×#127×C₅₃	C₂²	1	3	yes	complemented
36	C₂×#124×C₅₃	C₂²	1	3	yes	complemented
37	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#497	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#497	C₂²	1	3	yes	complemented
40	C₅₃×#497	C₂²	1	6	yes	complemented
41	C₅₃×#496	C₂²	1	3	yes	complemented
42	C₅₃×#496	C₂²	1	6	yes	complemented
43	C₅₃×#496	C₂²	1	12	yes	complemented
44	C₅₃×#496	C₂²	1	3	yes	complemented
45	C₅₃×#496	C₂²	1	6	yes	complemented
46	C₅₃×#495	C₂²	1	12	yes	complemented
47	C₅₃×#493	C₂²	1	6	yes	complemented
48	C₅₃×#492	C₂²	1	6	yes	complemented
49	C₅₃×#405	C₂²	1	6	yes	complemented
50	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
51	C₂×#97×C₅₃	C₂²	1	3	yes	complemented
52	2₋¹⁺⁴×C₅₃	—	2	8	no	—
53	H(5,𝔽₂)×C₅₃	—	2	8	no	—
54	H(5,𝔽₂)×C₅₃	—	2	8	no	—
55	H(5,𝔽₂)×C₅₃	—	2	8	no	—
56	C₂×#41×C₅₃	—	2	1	no	—
57	C₂×#41×C₅₃	—	2	2	no	—
58	C₂×#41×C₅₃	—	2	3	no	—
59	C₂×#41×C₅₃	C₂³	1	3	yes	—
60	C₂×#41×C₅₃	C₂³	1	6	yes	—
61	C₂×#41×C₅₃	C₂³	1	3	yes	—
62	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₅₃	—	2	1	no	—
64	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
65	C₂²×D₄×C₅₃	—	2	6	no	—
66	C₂²×D₄×C₅₃	—	2	3	no	—
67	C₂²×D₄×C₅₃	—	2	6	no	—
68	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
70	#137×C₅₃	—	2	24	no	—
71	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
72	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
73	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
74	#135×C₅₃	—	2	24	no	—
75	#135×C₅₃	—	2	24	no	—
76	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
77	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×SD₁₆×C₅₃	—	2	24	no	—
79	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
80	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
81	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
82	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
83	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
84	C₂×D₈×C₅₃	—	2	24	no	—
85	C₂×D₈×C₅₃	C₂³	1	6	yes	—
86	C₂×D₈×C₅₃	C₂³	1	12	yes	—
87	C₂×D₈×C₅₃	C₂³	1	6	yes	—
88	#131×C₅₃	D₄	1	16	yes	complemented
89	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
90	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
91	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
92	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
93	#127×C₅₃	—	2	6	no	—
94	#127×C₅₃	C₂³	1	6	yes	—
95	#127×C₅₃	C₂³	1	3	yes	—
96	#127×C₅₃	C₂³	1	3	yes	—
97	#124×C₅₃	—	2	6	no	—
98	#124×C₅₃	C₂³	1	6	yes	—
99	#124×C₅₃	C₂³	1	6	yes	—
100	#124×C₅₃	C₂³	1	3	yes	—
101	#124×C₅₃	C₂³	1	1	yes	characteristic
102	#123×C₅₃	—	2	6	no	—
103	C₂²⋊Q₈×C₅₃	—	2	6	no	—
104	#121×C₅₃	—	2	12	no	—
105	#121×C₅₃	—	2	6	no	—
106	#121×C₅₃	—	2	6	no	—
107	#121×C₅₃	—	2	6	no	—
108	#120×C₅₃	—	2	12	no	—
109	#120×C₅₃	—	2	6	no	—
110	C₄×Q₈×C₅₃	D₄	1	2	yes	—
111	C₄×D₄×C₅₃	—	2	6	no	—
112	C₄×D₄×C₅₃	—	2	6	no	—
113	C₄×D₄×C₅₃	D₄	1	6	yes	—
114	#117×C₅₃	C₂³	1	3	yes	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
116	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
117	#97×C₅₃	C₂³	1	1	yes	characteristic
118	#97×C₅₃	C₂³	1	3	yes	—
119	#97×C₅₃	C₂³	1	6	yes	—
120	C₄×C₈×C₅₃	C₂³	1	6	yes	—
121	C₂⁴×C₅₃	—	2	3	no	—
122	C₂⁴×C₅₃	—	2	6	no	—
123	#41×C₅₃	—	2	12	no	—
124	#41×C₅₃	—	2	12	no	—
125	#41×C₅₃	—	2	12	no	—
126	#41×C₅₃	—	2	12	no	—
127	#41×C₅₃	—	4	4	no	—
128	#41×C₅₃	—	4	8	no	—
129	#41×C₅₃	—	4	12	no	—
130	#41×C₅₃	C₂×D₄	1	8	yes	—
131	C₂×Q₈×C₅₃	—	2	3	no	—
132	C₂×Q₈×C₅₃	—	2	12	no	—
133	C₂×Q₈×C₅₃	—	4	4	no	—
134	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
136	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₅₃	—	2	6	no	—
139	C₂×D₄×C₅₃	—	2	3	no	—
140	C₂×D₄×C₅₃	—	2	3	no	—
141	C₂×D₄×C₅₃	—	2	12	no	—
142	C₂×D₄×C₅₃	—	2	12	no	—
143	C₂×D₄×C₅₃	—	2	12	no	—
144	C₂×D₄×C₅₃	—	2	6	no	—
145	C₂×D₄×C₅₃	—	2	3	no	—
146	C₂×D₄×C₅₃	—	2	6	no	—
147	C₂×D₄×C₅₃	—	2	6	no	—
148	C₂×D₄×C₅₃	—	2	12	no	—
149	C₂×D₄×C₅₃	—	4	24	no	—
150	C₂×D₄×C₅₃	—	4	12	no	—
151	C₂×D₄×C₅₃	—	8	12	no	—
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	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
156	C₂²×C₄×C₅₃	—	2	3	no	—
157	C₂²×C₄×C₅₃	—	2	6	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	C₂⁴	1	3	yes	—
162	Q₁₆×C₅₃	—	2	4	no	—
163	Q₁₆×C₅₃	—	2	12	no	—
164	SD₁₆×C₅₃	—	2	12	no	—
165	SD₁₆×C₅₃	—	2	24	no	—
166	SD₁₆×C₅₃	—	2	4	no	—
167	SD₁₆×C₅₃	—	2	12	no	—
168	SD₁₆×C₅₃	—	2	12	no	—
169	D₈×C₅₃	—	2	12	no	—
170	D₈×C₅₃	—	2	24	no	—
171	D₈×C₅₃	—	2	12	no	—
172	M₁₆×C₅₃	C₂×D₄	1	24	yes	complemented
173	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	complemented
174	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
175	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	6	no	—
178	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
179	C₂²⋊C₄×C₅₃	—	2	6	no	—
180	C₂²⋊C₄×C₅₃	—	2	12	no	—
181	C₂²⋊C₄×C₅₃	—	2	2	no	—
182	C₂²⋊C₄×C₅₃	—	2	12	no	—
183	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
184	C₄²×C₅₃	C₂×D₄	1	6	yes	—
185	C₄²×C₅₃	C₂⁴	1	3	yes	—
186	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂³×C₅₃	—	2	1	no	—
188	C₂³×C₅₃	—	2	6	no	—
189	C₂³×C₅₃	—	2	3	no	—
190	C₂³×C₅₃	—	4	6	no	—
191	C₂³×C₅₃	—	4	12	no	—
192	C₂³×C₅₃	—	8	12	no	—
193	C₂³×C₅₃	—	8	6	no	—
194	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
195	Q₈×C₅₃	—	2	6	no	—
196	Q₈×C₅₃	—	2	2	no	—
197	Q₈×C₅₃	—	2	2	no	—
198	Q₈×C₅₃	—	2	2	no	—
199	Q₈×C₅₃	—	4	6	no	—
200	Q₈×C₅₃	#127	1	4	yes	—
201	D₄×C₅₃	—	2	6	no	—
202	D₄×C₅₃	—	2	6	no	—
203	D₄×C₅₃	—	2	6	no	—
204	D₄×C₅₃	—	2	2	no	—
205	D₄×C₅₃	—	4	12	no	—
206	D₄×C₅₃	—	4	12	no	—
207	D₄×C₅₃	—	4	6	no	—
208	D₄×C₅₃	—	4	6	no	—
209	D₄×C₅₃	—	4	6	no	—
210	D₄×C₅₃	—	4	12	no	—
211	D₄×C₅₃	—	8	6	no	—
212	D₄×C₅₃	—	8	12	no	—
213	D₄×C₅₃	#127	1	12	yes	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	3	no	—
217	C₂×C₄×C₅₃	—	4	6	no	—
218	C₂×C₄×C₅₃	—	4	6	no	—
219	C₂×C₄×C₅₃	—	4	12	no	—
220	C₂×C₄×C₅₃	—	4	6	no	—
221	C₂×C₄×C₅₃	—	4	6	no	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
226	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
227	C₈×C₅₃	C₂²×D₄	1	4	yes	—
228	C₈×C₅₃	C₂²×D₄	1	12	yes	—
229	#60650	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₅₃	—	2	6	no	—
231	C₂²×C₅₃	—	4	12	no	—
232	C₂²×C₅₃	—	4	2	no	—
233	C₂²×C₅₃	—	4	6	no	—
234	C₂²×C₅₃	—	4	1	no	—
235	C₂²×C₅₃	—	4	3	no	—
236	C₂²×C₅₃	—	4	6	no	—
237	C₂²×C₅₃	—	16	12	no	—
238	C₂²×C₅₃	—	16	6	no	—
239	C₂²×C₅₃	C₂×#127	1	6	yes	—
240	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	2	6	no	—
243	C₄×C₅₃	—	4	2	no	—
244	C₄×C₅₃	—	4	1	no	—
245	C₄×C₅₃	—	4	3	no	—
246	C₄×C₅₃	C₂×#127	1	6	yes	—
247	C₄×C₅₃	C₂³×D₄	1	2	yes	—
248	C₂×#577	C₂×C₅₃	1	2	yes	complemented
249	C₂×#576	C₂×C₅₃	1	2	yes	complemented
250	#3500	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#3498	C₂×C₅₃	1	1	yes	characteristic, complemented
252	#3155	C₂×C₅₃	1	6	yes	complemented
253	#3154	C₂×C₅₃	1	6	yes	complemented
254	#3153	C₂×C₅₃	1	3	yes	complemented
255	#3151	C₂×C₅₃	1	3	yes	complemented
256	#3150	C₂×C₅₃	1	3	yes	complemented
257	C₂×#496	C₂×C₅₃	1	3	yes	complemented
258	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
259	C₂×C₅₃	—	2	6	no	—
260	C₂×C₅₃	—	8	1	no	—
261	C₂×C₅₃	—	8	6	no	—
262	C₂×C₅₃	—	8	3	no	—
263	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
264	C₂×C₅₃	#3585	1	2	yes	—
265	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
267	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
268	#577	—	2	16	no	—
269	#576	—	2	16	no	—
270	C₂×#137	C₂²×C₅₃	1	6	yes	complemented
271	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	complemented
272	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	complemented
273	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	complemented
274	C₂×#135	C₂²×C₅₃	1	6	yes	complemented
275	C₂×#135	C₂²×C₅₃	1	6	yes	complemented
276	C₂²×SD₁₆	C₂²×C₅₃	1	6	yes	complemented
277	C₂²×D₈	C₂²×C₅₃	1	6	yes	complemented
278	C₂×#131	C₂²×C₅₃	1	2	yes	—
279	#550	—	2	2	no	—
280	#548	—	2	2	no	—
281	#546	—	2	6	no	—
282	D₄²	—	2	6	no	—
283	#536	C₂²×C₅₃	1	3	yes	complemented
284	#535	C₂²×C₅₃	1	6	yes	complemented
285	#535	C₂²×C₅₃	1	3	yes	complemented
286	#532	C₂²×C₅₃	1	3	yes	complemented
287	#532	C₂²×C₅₃	1	3	yes	complemented
288	C₂×#127	C₂²×C₅₃	1	3	yes	complemented
289	C₂×#124	C₂²×C₅₃	1	3	yes	complemented
290	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
291	#497	C₂²×C₅₃	1	1	yes	characteristic, complemented
292	#497	C₂²×C₅₃	1	3	yes	complemented
293	#497	C₂²×C₅₃	1	6	yes	complemented
294	#496	C₂²×C₅₃	1	3	yes	complemented
295	#496	C₂²×C₅₃	1	6	yes	complemented
296	#496	C₂²×C₅₃	1	12	yes	complemented
297	#496	C₂²×C₅₃	1	3	yes	complemented
298	#496	C₂²×C₅₃	1	6	yes	complemented
299	#495	C₂²×C₅₃	1	12	yes	complemented
300	#493	C₂²×C₅₃	1	6	yes	complemented
301	#492	C₂²×C₅₃	1	6	yes	complemented
302	#405	C₂²×C₅₃	1	6	yes	complemented
303	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
304	C₂×#97	C₂²×C₅₃	1	3	yes	complemented
305	C₅₃	#60650	1	1	yes	characteristic, direct product factor, Sylow
306	2₋¹⁺⁴	—	2	8	no	—
307	H(5,𝔽₂)	—	2	8	no	—
308	H(5,𝔽₂)	—	2	8	no	—
309	H(5,𝔽₂)	—	2	8	no	—
310	C₂×#41	—	2	1	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	3	no	—
313	C₂×#41	C₂³×C₅₃	1	3	yes	—
314	C₂×#41	C₂³×C₅₃	1	6	yes	—
315	C₂×#41	C₂³×C₅₃	1	3	yes	—
316	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
317	C₂²×Q₈	—	2	1	no	—
318	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
319	C₂²×D₄	—	2	6	no	—
320	C₂²×D₄	—	2	3	no	—
321	C₂²×D₄	—	2	6	no	—
322	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
323	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
324	#137	—	2	24	no	—
325	AGL(1,ℤ/8)	—	2	24	no	—
326	AGL(1,ℤ/8)	—	2	24	no	—
327	AGL(1,ℤ/8)	—	2	24	no	—
328	#135	—	2	24	no	—
329	#135	—	2	24	no	—
330	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
331	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
332	C₂×SD₁₆	—	2	24	no	—
333	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
334	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
335	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
336	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
337	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
338	C₂×D₈	—	2	24	no	—
339	C₂×D₈	C₂³×C₅₃	1	6	yes	—
340	C₂×D₈	C₂³×C₅₃	1	12	yes	—
341	C₂×D₈	C₂³×C₅₃	1	6	yes	—
342	#131	D₄×C₅₃	1	16	yes	complemented
343	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
344	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
345	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
347	#127	—	2	6	no	—
348	#127	C₂³×C₅₃	1	3	yes	—
349	#127	C₂³×C₅₃	1	6	yes	—
350	#127	C₂³×C₅₃	1	3	yes	—
351	#124	—	2	6	no	—
352	#124	C₂³×C₅₃	1	3	yes	—
353	#124	C₂³×C₅₃	1	6	yes	—
354	#124	C₂³×C₅₃	1	1	yes	characteristic
355	#124	C₂³×C₅₃	1	6	yes	—
356	#123	—	2	6	no	—
357	C₂²⋊Q₈	—	2	6	no	—
358	#121	—	2	6	no	—
359	#121	—	2	6	no	—
360	#121	—	2	12	no	—
361	#121	—	2	6	no	—
362	#120	—	2	12	no	—
363	#120	—	2	6	no	—
364	C₄×Q₈	D₄×C₅₃	1	2	yes	—
365	C₄×D₄	—	2	6	no	—
366	C₄×D₄	—	2	6	no	—
367	C₄×D₄	D₄×C₅₃	1	6	yes	—
368	#117	C₂³×C₅₃	1	3	yes	—
369	C₂×C₂²⋊C₄	—	2	6	no	—
370	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
371	#97	C₂³×C₅₃	1	1	yes	characteristic
372	#97	C₂³×C₅₃	1	3	yes	—
373	#97	C₂³×C₅₃	1	6	yes	—
374	C₄×C₈	C₂³×C₅₃	1	6	yes	—
375	C₂⁴	—	2	3	no	—
376	C₂⁴	—	2	6	no	—
377	#41	—	2	12	no	—
378	#41	—	2	12	no	—
379	#41	—	2	12	no	—
380	#41	—	2	12	no	—
381	#41	—	4	4	no	—
382	#41	—	4	8	no	—
383	#41	—	4	12	no	—
384	#41	C₂×D₄×C₅₃	1	8	yes	—
385	C₂×Q₈	—	2	3	no	—
386	C₂×Q₈	—	2	12	no	—
387	C₂×Q₈	—	4	4	no	—
388	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
390	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
391	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
392	C₂×D₄	—	2	6	no	—
393	C₂×D₄	—	2	3	no	—
394	C₂×D₄	—	2	3	no	—
395	C₂×D₄	—	2	12	no	—
396	C₂×D₄	—	2	12	no	—
397	C₂×D₄	—	2	12	no	—
398	C₂×D₄	—	2	6	no	—
399	C₂×D₄	—	2	3	no	—
400	C₂×D₄	—	2	6	no	—
401	C₂×D₄	—	2	6	no	—
402	C₂×D₄	—	2	12	no	—
403	C₂×D₄	—	4	24	no	—
404	C₂×D₄	—	4	12	no	—
405	C₂×D₄	—	8	12	no	—
406	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
407	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
408	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
409	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	2	6	no	—
412	C₂²×C₄	—	2	3	no	—
413	C₂²×C₄	—	2	3	no	—
414	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
416	Q₁₆	—	2	4	no	—
417	Q₁₆	—	2	12	no	—
418	SD₁₆	—	2	12	no	—
419	SD₁₆	—	2	24	no	—
420	SD₁₆	—	2	4	no	—
421	SD₁₆	—	2	12	no	—
422	SD₁₆	—	2	12	no	—
423	D₈	—	2	12	no	—
424	D₈	—	2	24	no	—
425	D₈	—	2	12	no	—
426	M₁₆	C₂×D₄×C₅₃	1	24	yes	complemented
427	C₂×C₈	C₂×D₄×C₅₃	1	24	yes	complemented
428	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
429	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	6	no	—
432	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
433	C₂²⋊C₄	—	2	6	no	—
434	C₂²⋊C₄	—	2	12	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	12	no	—
437	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
438	C₄²	C₂×D₄×C₅₃	1	6	yes	—
439	C₄²	C₂⁴×C₅₃	1	3	yes	—
440	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
441	C₂³	—	2	3	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	6	no	—
444	C₂³	—	4	6	no	—
445	C₂³	—	4	12	no	—
446	C₂³	—	8	12	no	—
447	C₂³	—	8	6	no	—
448	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
449	Q₈	—	2	6	no	—
450	Q₈	—	2	2	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	4	6	no	—
454	Q₈	#127×C₅₃	1	4	yes	—
455	D₄	—	2	6	no	—
456	D₄	—	2	6	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	6	no	—
459	D₄	—	4	12	no	—
460	D₄	—	4	12	no	—
461	D₄	—	4	6	no	—
462	D₄	—	4	6	no	—
463	D₄	—	4	6	no	—
464	D₄	—	4	12	no	—
465	D₄	—	8	6	no	—
466	D₄	—	8	12	no	—
467	D₄	#127×C₅₃	1	12	yes	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	4	6	no	—
472	C₂×C₄	—	4	6	no	—
473	C₂×C₄	—	4	12	no	—
474	C₂×C₄	—	4	6	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
477	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
478	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
479	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
480	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
481	C₈	C₂²×D₄×C₅₃	1	4	yes	—
482	C₈	C₂²×D₄×C₅₃	1	12	yes	—
483	C₂²	—	2	6	no	—
484	C₂²	—	4	12	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	6	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	3	no	—
489	C₂²	—	4	6	no	—
490	C₂²	—	16	12	no	—
491	C₂²	—	16	6	no	—
492	C₂²	C₂×#127×C₅₃	1	6	yes	—
493	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
494	C₄	—	2	2	no	—
495	C₄	—	2	6	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	3	no	—
499	C₄	C₂×#127×C₅₃	1	6	yes	—
500	C₄	C₂³×D₄×C₅₃	1	2	yes	—
501	C₂	—	2	6	no	—
502	C₂	—	8	6	no	—
503	C₂	—	8	3	no	—
504	C₂	—	8	1	no	—
505	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
506	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. C₂×C₅₃×#577	C₂	semidirect
2. C₂×C₅₃×#576	C₂	semidirect
3. C₅₃×#3500	C₂	semidirect
4. C₅₃×#3498	C₂	semidirect
5. C₅₃×#3155	C₂	semidirect
6. C₅₃×#3154	C₂	semidirect
7. C₅₃×#3153	C₂	semidirect
8. C₅₃×#3151	C₂	semidirect
9. C₅₃×#3150	C₂	semidirect
10. C₂×C₅₃×#496	C₂	semidirect
11. C₅₃×#2875	C₂	semidirect
17. C₂×#137×C₅₃	C₂²	semidirect
18. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
20. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
21. C₂×#135×C₅₃	C₂²	semidirect
22. C₂×#135×C₅₃	C₂²	semidirect
23. C₂²×SD₁₆×C₅₃	C₂²	semidirect
24. C₂²×D₈×C₅₃	C₂²	semidirect
30. C₅₃×#536	C₂²	semidirect
31. C₅₃×#535	C₂²	semidirect
32. C₅₃×#535	C₂²	semidirect
33. C₅₃×#532	C₂²	semidirect
34. C₅₃×#532	C₂²	semidirect
35. C₂×#127×C₅₃	C₂²	semidirect
36. C₂×#124×C₅₃	C₂²	semidirect
38. C₅₃×#497	C₂²	semidirect
39. C₅₃×#497	C₂²	semidirect
40. C₅₃×#497	C₂²	semidirect
41. C₅₃×#496	C₂²	semidirect
42. C₅₃×#496	C₂²	semidirect
43. C₅₃×#496	C₂²	semidirect
44. C₅₃×#496	C₂²	semidirect
45. C₅₃×#496	C₂²	semidirect
46. C₅₃×#495	C₂²	semidirect
47. C₅₃×#493	C₂²	semidirect
48. C₅₃×#492	C₂²	semidirect
49. C₅₃×#405	C₂²	semidirect
50. C₄×M₁₆×C₅₃	C₂²	semidirect
51. C₂×#97×C₅₃	C₂²	semidirect
88. #131×C₅₃	D₄	semidirect
172. M₁₆×C₅₃	C₂×D₄	semidirect
173. C₂×C₈×C₅₃	C₂×D₄	semidirect
229. #60650	C₅₃	direct (N × H)
248. C₂×#577	C₂×C₅₃	semidirect
249. C₂×#576	C₂×C₅₃	semidirect
250. #3500	C₂×C₅₃	semidirect
251. #3498	C₂×C₅₃	semidirect
252. #3155	C₂×C₅₃	semidirect
253. #3154	C₂×C₅₃	semidirect
254. #3153	C₂×C₅₃	semidirect
255. #3151	C₂×C₅₃	semidirect
256. #3150	C₂×C₅₃	semidirect
257. C₂×#496	C₂×C₅₃	semidirect
258. #2875	C₂×C₅₃	semidirect
270. C₂×#137	C₂²×C₅₃	semidirect
271. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
272. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
273. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
274. C₂×#135	C₂²×C₅₃	semidirect
275. C₂×#135	C₂²×C₅₃	semidirect
276. C₂²×SD₁₆	C₂²×C₅₃	semidirect
277. C₂²×D₈	C₂²×C₅₃	semidirect
283. #536	C₂²×C₅₃	semidirect
284. #535	C₂²×C₅₃	semidirect
285. #535	C₂²×C₅₃	semidirect
286. #532	C₂²×C₅₃	semidirect
287. #532	C₂²×C₅₃	semidirect
288. C₂×#127	C₂²×C₅₃	semidirect
289. C₂×#124	C₂²×C₅₃	semidirect
291. #497	C₂²×C₅₃	semidirect
292. #497	C₂²×C₅₃	semidirect
293. #497	C₂²×C₅₃	semidirect
294. #496	C₂²×C₅₃	semidirect
295. #496	C₂²×C₅₃	semidirect
296. #496	C₂²×C₅₃	semidirect
297. #496	C₂²×C₅₃	semidirect
298. #496	C₂²×C₅₃	semidirect
299. #495	C₂²×C₅₃	semidirect
300. #493	C₂²×C₅₃	semidirect
301. #492	C₂²×C₅₃	semidirect
302. #405	C₂²×C₅₃	semidirect
303. C₄×M₁₆	C₂²×C₅₃	semidirect
304. C₂×#97	C₂²×C₅₃	semidirect
305. C₅₃	#60650	direct (N × H)
342. #131	D₄×C₅₃	semidirect
426. M₁₆	C₂×D₄×C₅₃	semidirect
427. C₂×C₈	C₂×D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#577	2
C₂×C₅₃×#576	2
C₅₃×#3500	2
C₅₃×#3498	2
C₅₃×#3155	2
C₅₃×#3154	2
C₅₃×#3153	2
C₅₃×#3151	2
C₅₃×#3150	2
C₂×C₅₃×#496	2
C₅₃×#2875	2
#60650	53

Subgroups — C53×#60650 #27268162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60650 #27268162