Subgroups — C₅₃×#60644 #27268156

Order 13568.

3350 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#575	C₂	1	2	yes	maximal, complemented
2	C₂²×AGL(1,ℤ/8)×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3155	C₂	1	8	yes	maximal, complemented
6	C₅₃×#3154	C₂	1	8	yes	maximal, complemented
7	C₅₃×#3153	C₂	1	4	yes	maximal, complemented
8	C₅₃×#3151	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	2	yes	—
11	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#575	—	2	16	no	—
14	C₂×#137×C₅₃	C₂²	1	4	yes	complemented
15	C₂×#137×C₅₃	C₂²	1	4	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
17	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	8	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	8	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	complemented
21	C₂×#135×C₅₃	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₅₃	C₂²	1	4	yes	complemented
24	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#537	—	2	2	no	—
26	C₅₃×#536	C₂²	1	4	yes	complemented
27	C₅₃×#535	—	2	4	no	—
28	C₅₃×#535	C₂²	1	8	yes	complemented
29	C₅₃×#535	C₂²	1	4	yes	complemented
30	C₅₃×#534	—	2	2	no	—
31	C₅₃×#532	C₂²	1	4	yes	complemented
32	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
33	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
34	C₅₃×#525	—	2	4	no	—
35	C₂×#121×C₅₃	—	2	2	no	—
36	C₂×#120×C₅₃	—	2	2	no	—
37	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#497	C₂²	1	8	yes	complemented
39	C₅₃×#496	C₂²	1	8	yes	complemented
40	C₅₃×#496	C₂²	1	16	yes	complemented
41	C₅₃×#495	C₂²	1	16	yes	complemented
42	C₅₃×#493	C₂²	1	8	yes	complemented
43	C₅₃×#492	C₂²	1	8	yes	complemented
44	C₅₃×#405	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₅₃	C₂²	1	4	yes	complemented
46	C₂⁵×C₅₃	—	2	1	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	2	no	—
49	C₂×#41×C₅₃	—	2	2	no	—
50	C₂×#41×C₅₃	—	2	8	no	—
51	C₂×#41×C₅₃	—	2	8	no	—
52	C₂×#41×C₅₃	—	2	8	no	—
53	C₂×#41×C₅₃	C₂³	1	4	yes	—
54	C₂×#41×C₅₃	C₂³	1	4	yes	—
55	C₂×#41×C₅₃	C₂³	1	4	yes	—
56	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₅₃	—	2	1	no	—
59	C₂²×D₄×C₅₃	—	2	2	no	—
60	C₂²×D₄×C₅₃	—	2	8	no	—
61	C₂²×D₄×C₅₃	—	2	1	no	—
62	C₂²×D₄×C₅₃	—	2	2	no	—
63	C₂²×D₄×C₅₃	—	2	4	no	—
64	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
67	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
68	C₂³×C₄×C₅₃	—	2	1	no	—
69	C₂³×C₄×C₅₃	—	2	2	no	—
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	#137×C₅₃	—	2	16	no	—
72	#137×C₅₃	—	2	16	no	—
73	AGL(1,ℤ/8)×C₅₃	—	2	32	no	—
74	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
75	AGL(1,ℤ/8)×C₅₃	—	2	32	no	—
76	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
77	#135×C₅₃	—	2	32	no	—
78	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	complemented
79	C₂×SD₁₆×C₅₃	—	2	16	no	—
80	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	—
81	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	complemented
82	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	complemented
83	C₂×D₈×C₅₃	—	2	16	no	—
84	C₂×D₈×C₅₃	C₂³	1	16	yes	—
85	C₂×D₈×C₅₃	C₂³	1	8	yes	complemented
86	C₂×M₁₆×C₅₃	D₄	1	16	yes	complemented
87	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
88	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
89	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
90	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	complemented
91	#127×C₅₃	—	2	4	no	—
92	#127×C₅₃	C₂³	1	8	yes	—
93	#127×C₅₃	C₂³	1	4	yes	complemented
94	#124×C₅₃	—	2	4	no	—
95	#124×C₅₃	C₂³	1	8	yes	—
96	#124×C₅₃	C₂³	1	8	yes	complemented
97	#123×C₅₃	—	2	8	no	—
98	C₂²⋊Q₈×C₅₃	—	2	8	no	—
99	#121×C₅₃	—	2	16	no	—
100	#121×C₅₃	—	2	8	no	—
101	#121×C₅₃	—	2	8	no	—
102	#120×C₅₃	—	2	16	no	—
103	#120×C₅₃	—	2	8	no	—
104	C₄×D₄×C₅₃	—	2	8	no	—
105	#117×C₅₃	D₄	1	4	yes	—
106	#117×C₅₃	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
111	#97×C₅₃	C₂³	1	8	yes	complemented
112	C₄×C₈×C₅₃	C₂³	1	8	yes	complemented
113	C₂⁴×C₅₃	—	2	2	no	—
114	C₂⁴×C₅₃	—	2	1	no	—
115	C₂⁴×C₅₃	—	2	2	no	—
116	C₂⁴×C₅₃	—	2	4	no	—
117	C₂⁴×C₅₃	—	4	4	no	—
118	C₂⁴×C₅₃	—	8	4	no	—
119	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
120	#41×C₅₃	—	2	8	no	—
121	#41×C₅₃	—	2	8	no	—
122	#41×C₅₃	—	2	8	no	—
123	#41×C₅₃	—	2	8	no	—
124	#41×C₅₃	—	2	8	no	—
125	#41×C₅₃	—	2	8	no	—
126	#41×C₅₃	—	4	8	no	—
127	#41×C₅₃	—	4	8	no	—
128	#41×C₅₃	—	4	8	no	—
129	C₂×Q₈×C₅₃	—	2	2	no	—
130	C₂×Q₈×C₅₃	—	2	1	no	—
131	C₂×Q₈×C₅₃	—	2	4	no	—
132	C₂×Q₈×C₅₃	—	2	4	no	—
133	C₂×Q₈×C₅₃	—	2	4	no	—
134	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
135	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
136	C₂×D₄×C₅₃	—	2	4	no	—
137	C₂×D₄×C₅₃	—	2	4	no	—
138	C₂×D₄×C₅₃	—	2	4	no	—
139	C₂×D₄×C₅₃	—	2	4	no	—
140	C₂×D₄×C₅₃	—	2	8	no	—
141	C₂×D₄×C₅₃	—	2	8	no	—
142	C₂×D₄×C₅₃	—	2	8	no	—
143	C₂×D₄×C₅₃	—	2	4	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	2	8	no	—
150	C₂×D₄×C₅₃	—	2	16	no	—
151	C₂×D₄×C₅₃	—	4	4	no	—
152	C₂×D₄×C₅₃	—	4	8	no	—
153	C₂×D₄×C₅₃	—	4	4	no	—
154	C₂×D₄×C₅₃	—	4	8	no	—
155	C₂×D₄×C₅₃	—	8	8	no	—
156	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
158	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₅₃	—	2	1	no	—
160	C₂²×C₄×C₅₃	—	2	2	no	—
161	C₂²×C₄×C₅₃	—	2	4	no	—
162	C₂²×C₄×C₅₃	—	2	4	no	—
163	C₂²×C₄×C₅₃	—	2	4	no	—
164	C₂²×C₄×C₅₃	—	4	4	no	—
165	C₂²×C₄×C₅₃	—	4	8	no	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
170	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
172	Q₁₆×C₅₃	—	2	16	no	—
173	SD₁₆×C₅₃	—	2	32	no	—
174	SD₁₆×C₅₃	—	2	16	no	—
175	SD₁₆×C₅₃	—	2	16	no	—
176	D₈×C₅₃	—	2	32	no	—
177	D₈×C₅₃	—	2	16	no	—
178	M₁₆×C₅₃	C₂×D₄	1	16	yes	—
179	M₁₆×C₅₃	C₂×D₄	1	16	yes	complemented
180	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	complemented
181	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
182	C₄⋊C₄×C₅₃	—	2	8	no	—
183	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
184	C₂²⋊C₄×C₅₃	—	2	16	no	—
185	C₂²⋊C₄×C₅₃	—	2	16	no	—
186	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
187	C₄²×C₅₃	C₂×D₄	1	4	yes	—
188	C₄²×C₅₃	C₂⁴	1	4	yes	—
189	C₂³×C₅₃	—	2	2	no	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	2	4	no	—
192	C₂³×C₅₃	—	4	2	no	—
193	C₂³×C₅₃	—	4	4	no	—
194	C₂³×C₅₃	—	4	8	no	—
195	C₂³×C₅₃	—	4	2	no	—
196	C₂³×C₅₃	—	4	4	no	—
197	C₂³×C₅₃	—	4	4	no	—
198	C₂³×C₅₃	—	4	8	no	—
199	C₂³×C₅₃	—	8	4	no	—
200	C₂³×C₅₃	—	8	8	no	—
201	C₂³×C₅₃	—	8	16	no	—
202	C₂³×C₅₃	#127	1	4	yes	—
203	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
205	Q₈×C₅₃	—	2	4	no	—
206	Q₈×C₅₃	—	2	4	no	—
207	Q₈×C₅₃	—	2	4	no	—
208	Q₈×C₅₃	—	4	4	no	—
209	Q₈×C₅₃	—	4	2	no	—
210	D₄×C₅₃	—	2	8	no	—
211	D₄×C₅₃	—	2	4	no	—
212	D₄×C₅₃	—	2	4	no	—
213	D₄×C₅₃	—	2	4	no	—
214	D₄×C₅₃	—	4	8	no	—
215	D₄×C₅₃	—	4	8	no	—
216	D₄×C₅₃	—	4	8	no	—
217	D₄×C₅₃	—	4	8	no	—
218	D₄×C₅₃	—	4	4	no	—
219	D₄×C₅₃	—	4	4	no	—
220	D₄×C₅₃	—	4	2	no	—
221	D₄×C₅₃	—	8	8	no	—
222	D₄×C₅₃	—	8	16	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	4	no	—
225	C₂×C₄×C₅₃	—	4	4	no	—
226	C₂×C₄×C₅₃	—	4	2	no	—
227	C₂×C₄×C₅₃	—	4	4	no	—
228	C₂×C₄×C₅₃	—	4	8	no	—
229	C₂×C₄×C₅₃	—	4	8	no	—
230	C₂×C₄×C₅₃	—	4	8	no	—
231	C₂×C₄×C₅₃	#127	1	4	yes	—
232	C₂×C₄×C₅₃	#127	1	8	yes	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
234	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
240	C₈×C₅₃	C₂²×D₄	1	16	yes	—
241	#60644	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂²×C₅₃	—	2	2	no	—
243	C₂²×C₅₃	—	2	4	no	—
244	C₂²×C₅₃	—	4	4	no	—
245	C₂²×C₅₃	—	4	4	no	—
246	C₂²×C₅₃	—	4	8	no	—
247	C₂²×C₅₃	—	4	8	no	—
248	C₂²×C₅₃	—	4	4	no	—
249	C₂²×C₅₃	—	4	2	no	—
250	C₂²×C₅₃	—	4	4	no	—
251	C₂²×C₅₃	—	8	4	no	—
252	C₂²×C₅₃	—	8	8	no	—
253	C₂²×C₅₃	—	8	8	no	—
254	C₂²×C₅₃	—	8	16	no	—
255	C₂²×C₅₃	C₂×#127	1	4	yes	—
256	C₂²×C₅₃	C₂×#127	1	2	yes	—
257	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₅₃	—	2	8	no	—
259	C₄×C₅₃	—	4	2	no	—
260	C₄×C₅₃	—	4	4	no	—
261	C₄×C₅₃	C₂×#127	1	2	yes	—
262	C₄×C₅₃	C₂×#127	1	4	yes	—
263	C₄×C₅₃	C₂³×D₄	1	2	yes	—
264	C₂×#575	C₂×C₅₃	1	2	yes	complemented
265	C₂²×AGL(1,ℤ/8)	C₂×C₅₃	1	2	yes	complemented
266	#3449	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₂×#535	C₂×C₅₃	1	1	yes	characteristic, complemented
268	#3155	C₂×C₅₃	1	8	yes	complemented
269	#3154	C₂×C₅₃	1	8	yes	complemented
270	#3153	C₂×C₅₃	1	4	yes	complemented
271	#3151	C₂×C₅₃	1	4	yes	complemented
272	#2873	C₂×C₅₃	1	1	yes	characteristic, complemented
273	C₂×C₅₃	—	2	2	no	—
274	C₂×C₅₃	—	2	4	no	—
275	C₂×C₅₃	—	8	2	no	—
276	C₂×C₅₃	—	8	4	no	—
277	C₂×C₅₃	—	8	4	no	—
278	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
279	C₂×C₅₃	#3585	1	2	yes	—
280	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
281	C₂²×#41	C₂²×C₅₃	1	2	yes	—
282	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
283	#575	—	2	16	no	—
284	C₂×#137	C₂²×C₅₃	1	4	yes	complemented
285	C₂×#137	C₂²×C₅₃	1	4	yes	complemented
286	C₂×AGL(1,ℤ/8)	—	2	16	no	—
287	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	8	yes	complemented
288	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	8	yes	complemented
289	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	complemented
290	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	complemented
291	C₂×#135	C₂²×C₅₃	1	8	yes	complemented
292	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
293	C₂²×D₈	C₂²×C₅₃	1	4	yes	complemented
294	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
295	#537	—	2	2	no	—
296	#536	C₂²×C₅₃	1	4	yes	complemented
297	#535	—	2	4	no	—
298	#535	C₂²×C₅₃	1	8	yes	complemented
299	#535	C₂²×C₅₃	1	4	yes	complemented
300	#534	—	2	2	no	—
301	#532	C₂²×C₅₃	1	4	yes	complemented
302	C₂×#127	C₂²×C₅₃	1	2	yes	complemented
303	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
304	#525	—	2	4	no	—
305	C₂×#121	—	2	2	no	—
306	C₂×#120	—	2	2	no	—
307	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
308	#497	C₂²×C₅₃	1	8	yes	complemented
309	#496	C₂²×C₅₃	1	8	yes	complemented
310	#496	C₂²×C₅₃	1	16	yes	complemented
311	#495	C₂²×C₅₃	1	16	yes	complemented
312	#493	C₂²×C₅₃	1	8	yes	complemented
313	#492	C₂²×C₅₃	1	8	yes	complemented
314	#405	C₂²×C₅₃	1	8	yes	complemented
315	C₄×M₁₆	C₂²×C₅₃	1	4	yes	complemented
316	C₅₃	#60644	1	1	yes	characteristic, direct product factor, Sylow
317	C₂⁵	—	2	1	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	8	no	—
322	C₂×#41	—	2	8	no	—
323	C₂×#41	—	2	8	no	—
324	C₂×#41	C₂³×C₅₃	1	4	yes	—
325	C₂×#41	C₂³×C₅₃	1	4	yes	—
326	C₂×#41	C₂³×C₅₃	1	4	yes	—
327	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
328	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	—	2	8	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
336	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
338	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	2	no	—
341	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	#137	—	2	16	no	—
343	#137	—	2	16	no	—
344	AGL(1,ℤ/8)	—	2	32	no	—
345	AGL(1,ℤ/8)	—	2	16	no	—
346	AGL(1,ℤ/8)	—	2	32	no	—
347	AGL(1,ℤ/8)	—	2	16	no	—
348	#135	—	2	32	no	—
349	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	complemented
350	C₂×SD₁₆	—	2	16	no	—
351	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	—
352	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	complemented
353	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	complemented
354	C₂×D₈	—	2	16	no	—
355	C₂×D₈	C₂³×C₅₃	1	16	yes	—
356	C₂×D₈	C₂³×C₅₃	1	8	yes	complemented
357	C₂×M₁₆	D₄×C₅₃	1	16	yes	complemented
358	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
359	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
360	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
361	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	complemented
362	#127	—	2	4	no	—
363	#127	C₂³×C₅₃	1	8	yes	—
364	#127	C₂³×C₅₃	1	4	yes	complemented
365	#124	—	2	4	no	—
366	#124	C₂³×C₅₃	1	8	yes	—
367	#124	C₂³×C₅₃	1	8	yes	complemented
368	#123	—	2	8	no	—
369	C₂²⋊Q₈	—	2	8	no	—
370	#121	—	2	8	no	—
371	#121	—	2	16	no	—
372	#121	—	2	8	no	—
373	#120	—	2	16	no	—
374	#120	—	2	8	no	—
375	C₄×D₄	—	2	8	no	—
376	#117	D₄×C₅₃	1	4	yes	—
377	#117	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
379	C₂×C₂²⋊C₄	—	2	4	no	—
380	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
381	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
382	#97	C₂³×C₅₃	1	8	yes	complemented
383	C₄×C₈	C₂³×C₅₃	1	8	yes	complemented
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	1	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	4	4	no	—
389	C₂⁴	—	8	4	no	—
390	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
391	#41	—	2	8	no	—
392	#41	—	2	8	no	—
393	#41	—	2	8	no	—
394	#41	—	2	8	no	—
395	#41	—	2	8	no	—
396	#41	—	2	8	no	—
397	#41	—	4	8	no	—
398	#41	—	4	8	no	—
399	#41	—	4	8	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
406	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	16	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	8	8	no	—
427	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
428	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
429	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
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	8	yes	—
441	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
443	Q₁₆	—	2	16	no	—
444	SD₁₆	—	2	32	no	—
445	SD₁₆	—	2	16	no	—
446	SD₁₆	—	2	16	no	—
447	D₈	—	2	32	no	—
448	D₈	—	2	16	no	—
449	M₁₆	C₂×D₄×C₅₃	1	16	yes	—
450	M₁₆	C₂×D₄×C₅₃	1	16	yes	complemented
451	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	complemented
452	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	—	2	16	no	—
457	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
458	C₄²	C₂×D₄×C₅₃	1	4	yes	—
459	C₄²	C₂⁴×C₅₃	1	4	yes	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	4	no	—
465	C₂³	—	4	8	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	8	no	—
470	C₂³	—	8	4	no	—
471	C₂³	—	8	8	no	—
472	C₂³	—	8	16	no	—
473	C₂³	#127×C₅₃	1	4	yes	—
474	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
475	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	4	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	4	4	no	—
480	Q₈	—	4	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	8	no	—
483	D₄	—	2	4	no	—
484	D₄	—	2	4	no	—
485	D₄	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	8	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	8	no	—
493	D₄	—	8	16	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	#127×C₅₃	1	4	yes	—
503	C₂×C₄	#127×C₅₃	1	8	yes	—
504	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
505	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
507	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
510	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
511	C₈	C₂²×D₄×C₅₃	1	16	yes	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	8	no	—
517	C₂²	—	4	8	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	8	8	no	—
522	C₂²	—	8	4	no	—
523	C₂²	—	8	16	no	—
524	C₂²	—	8	8	no	—
525	C₂²	C₂×#127×C₅₃	1	4	yes	—
526	C₂²	C₂×#127×C₅₃	1	2	yes	—
527	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
528	C₄	—	2	8	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	4	no	—
531	C₄	C₂×#127×C₅₃	1	2	yes	—
532	C₄	C₂×#127×C₅₃	1	4	yes	—
533	C₄	C₂³×D₄×C₅₃	1	2	yes	—
534	C₂	—	2	2	no	—
535	C₂	—	2	4	no	—
536	C₂	—	8	4	no	—
537	C₂	—	8	2	no	—
538	C₂	—	8	4	no	—
539	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
540	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. C₂×C₅₃×#575	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
3. C₅₃×#3449	C₂	semidirect
4. C₂×C₅₃×#535	C₂	semidirect
5. C₅₃×#3155	C₂	semidirect
6. C₅₃×#3154	C₂	semidirect
7. C₅₃×#3153	C₂	semidirect
8. C₅₃×#3151	C₂	semidirect
9. C₅₃×#2873	C₂	semidirect
14. C₂×#137×C₅₃	C₂²	semidirect
15. C₂×#137×C₅₃	C₂²	semidirect
17. C₂×AGL(1,ℤ/8)×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. C₂²×SD₁₆×C₅₃	C₂²	semidirect
23. C₂²×D₈×C₅₃	C₂²	semidirect
26. C₅₃×#536	C₂²	semidirect
28. C₅₃×#535	C₂²	semidirect
29. C₅₃×#535	C₂²	semidirect
31. C₅₃×#532	C₂²	semidirect
32. C₂×#127×C₅₃	C₂²	semidirect
33. C₂×#124×C₅₃	C₂²	semidirect
38. C₅₃×#497	C₂²	semidirect
39. C₅₃×#496	C₂²	semidirect
40. C₅₃×#496	C₂²	semidirect
41. C₅₃×#495	C₂²	semidirect
42. C₅₃×#493	C₂²	semidirect
43. C₅₃×#492	C₂²	semidirect
44. C₅₃×#405	C₂²	semidirect
45. C₄×M₁₆×C₅₃	C₂²	semidirect
78. C₂×Q₁₆×C₅₃	C₂³	semidirect
81. C₂×SD₁₆×C₅₃	C₂³	semidirect
82. C₂×SD₁₆×C₅₃	C₂³	semidirect
85. C₂×D₈×C₅₃	C₂³	semidirect
86. C₂×M₁₆×C₅₃	D₄	semidirect
90. C₄⋊Q₈×C₅₃	C₂³	semidirect
93. #127×C₅₃	C₂³	semidirect
96. #124×C₅₃	C₂³	semidirect
111. #97×C₅₃	C₂³	semidirect
112. C₄×C₈×C₅₃	C₂³	semidirect
179. M₁₆×C₅₃	C₂×D₄	semidirect
180. C₂×C₈×C₅₃	C₂×D₄	semidirect
241. #60644	C₅₃	direct (N × H)
264. C₂×#575	C₂×C₅₃	semidirect
265. C₂²×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
266. #3449	C₂×C₅₃	semidirect
267. C₂×#535	C₂×C₅₃	semidirect
268. #3155	C₂×C₅₃	semidirect
269. #3154	C₂×C₅₃	semidirect
270. #3153	C₂×C₅₃	semidirect
271. #3151	C₂×C₅₃	semidirect
272. #2873	C₂×C₅₃	semidirect
284. C₂×#137	C₂²×C₅₃	semidirect
285. C₂×#137	C₂²×C₅₃	semidirect
287. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
288. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
289. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
290. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
291. C₂×#135	C₂²×C₅₃	semidirect
292. C₂²×SD₁₆	C₂²×C₅₃	semidirect
293. C₂²×D₈	C₂²×C₅₃	semidirect
296. #536	C₂²×C₅₃	semidirect
298. #535	C₂²×C₅₃	semidirect
299. #535	C₂²×C₅₃	semidirect
301. #532	C₂²×C₅₃	semidirect
302. C₂×#127	C₂²×C₅₃	semidirect
303. C₂×#124	C₂²×C₅₃	semidirect
308. #497	C₂²×C₅₃	semidirect
309. #496	C₂²×C₅₃	semidirect
310. #496	C₂²×C₅₃	semidirect
311. #495	C₂²×C₅₃	semidirect
312. #493	C₂²×C₅₃	semidirect
313. #492	C₂²×C₅₃	semidirect
314. #405	C₂²×C₅₃	semidirect
315. C₄×M₁₆	C₂²×C₅₃	semidirect
316. C₅₃	#60644	direct (N × H)
349. C₂×Q₁₆	C₂³×C₅₃	semidirect
352. C₂×SD₁₆	C₂³×C₅₃	semidirect
353. C₂×SD₁₆	C₂³×C₅₃	semidirect
356. C₂×D₈	C₂³×C₅₃	semidirect
357. C₂×M₁₆	D₄×C₅₃	semidirect
361. C₄⋊Q₈	C₂³×C₅₃	semidirect
364. #127	C₂³×C₅₃	semidirect
367. #124	C₂³×C₅₃	semidirect
382. #97	C₂³×C₅₃	semidirect
383. C₄×C₈	C₂³×C₅₃	semidirect
450. M₁₆	C₂×D₄×C₅₃	semidirect
451. 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×#575	2
C₂²×AGL(1,ℤ/8)×C₅₃	2
C₅₃×#3449	2
C₂×C₅₃×#535	2
C₅₃×#3155	2
C₅₃×#3154	2
C₅₃×#3153	2
C₅₃×#3151	2
C₅₃×#2873	2
#60644	53

Subgroups — C53×#60644 #27268156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60644 #27268156