Subgroups — C₅₃×#25196 #27232708

Order 13568.

1032 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3068	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3068	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3019	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2069	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2048	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1965	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1932	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1914	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1903	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1861	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#575	C₂²	1	1	yes	characteristic
18	C₅₃×#575	C₂²	1	1	yes	characteristic
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic, complemented
22	C₅₃×#568	C₂²	1	1	yes	characteristic
23	C₅₃×#568	C₂²	1	1	yes	characteristic
24	C₅₃×#568	C₂²	1	1	yes	characteristic, complemented
25	C₅₃×#535	C₂²	1	1	yes	characteristic
26	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#472	—	2	2	no	—
28	C₅₃×#472	—	2	2	no	—
29	C₅₃×#471	—	2	2	no	—
30	C₅₃×#471	—	2	2	no	—
31	C₅₃×#455	—	2	2	no	—
32	C₅₃×#453	—	2	2	no	—
33	C₅₃×#453	—	2	2	no	—
34	C₅₃×#453	—	2	2	no	—
35	C₅₃×#444	—	2	2	no	—
36	C₅₃×#443	—	2	2	no	—
37	C₅₃×#430	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#430	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#430	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#421	C₂²	1	1	yes	characteristic
42	C₅₃×#421	C₂²	1	1	yes	characteristic
43	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#418	C₂²	1	1	yes	characteristic, complemented
47	C₅₃×#413	—	2	2	no	—
48	C₅₃×#413	C₂²	1	1	yes	characteristic
49	C₅₃×#413	C₂²	1	1	yes	characteristic
50	C₅₃×#413	C₂²	1	1	yes	characteristic, complemented
51	C₂×#100×C₅₃	—	2	2	no	—
52	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic, complemented
57	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
59	C₅₃×#344	C₂²	1	1	yes	characteristic, complemented
60	C₅₃×#344	C₂²	1	1	yes	characteristic, complemented
61	C₅₃×#339	C₂²	1	1	yes	characteristic, complemented
62	C₅₃×#338	C₂²	1	1	yes	characteristic, complemented
63	H(5,𝔽₂)×C₅₃	—	2	2	no	—
64	H(5,𝔽₂)×C₅₃	—	2	2	no	—
65	H(5,𝔽₂)×C₅₃	—	4	1	no	—
66	H(5,𝔽₂)×C₅₃	—	4	1	no	—
67	C₂×#41×C₅₃	—	2	1	no	—
68	C₂×#41×C₅₃	—	2	1	no	—
69	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₅₃	—	2	2	no	—
73	C₂²×D₄×C₅₃	—	2	1	no	—
74	C₂²×D₄×C₅₃	—	2	1	no	—
75	C₂²×D₄×C₅₃	—	2	2	no	—
76	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
78	#137×C₅₃	—	2	1	no	—
79	#137×C₅₃	—	2	1	no	—
80	#137×C₅₃	—	2	1	no	—
81	#137×C₅₃	D₄	1	2	yes	complemented
82	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
83	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
84	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
85	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
86	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
87	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
88	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
89	AGL(1,ℤ/8)×C₅₃	—	4	1	no	—
90	AGL(1,ℤ/8)×C₅₃	—	4	1	no	—
91	AGL(1,ℤ/8)×C₅₃	D₄	1	2	yes	complemented
92	#135×C₅₃	—	2	2	no	—
93	#135×C₅₃	—	4	1	no	—
94	C₂×SD₁₆×C₅₃	—	2	1	no	—
95	C₂×SD₁₆×C₅₃	—	2	1	no	—
96	C₂×SD₁₆×C₅₃	—	2	1	no	—
97	C₂×D₈×C₅₃	—	2	1	no	—
98	C₂×D₈×C₅₃	—	2	1	no	—
99	C₂×D₈×C₅₃	—	2	1	no	—
100	#131×C₅₃	—	2	2	no	—
101	#131×C₅₃	—	2	2	no	—
102	#131×C₅₃	—	2	2	no	—
103	#131×C₅₃	—	2	2	no	—
104	#131×C₅₃	—	4	1	no	—
105	#131×C₅₃	—	4	1	no	—
106	C₂×M₁₆×C₅₃	—	2	2	no	—
107	C₂×M₁₆×C₅₃	—	2	2	no	—
108	C₂×M₁₆×C₅₃	—	4	1	no	—
109	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
117	#127×C₅₃	—	2	1	no	—
118	#127×C₅₃	—	2	1	no	—
119	#127×C₅₃	—	2	1	no	—
120	#127×C₅₃	—	4	1	no	—
121	#124×C₅₃	—	2	1	no	—
122	#121×C₅₃	—	4	1	no	—
123	#120×C₅₃	—	2	2	no	—
124	#117×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	#108×C₅₃	—	2	2	no	—
127	#108×C₅₃	—	2	2	no	—
128	#108×C₅₃	—	2	2	no	—
129	C₄⋊C₈×C₅₃	—	4	1	no	—
130	#104×C₅₃	—	2	2	no	—
131	#104×C₅₃	—	2	1	no	—
132	#104×C₅₃	—	2	1	no	—
133	#104×C₅₃	—	2	1	no	—
134	#104×C₅₃	—	2	1	no	—
135	#104×C₅₃	—	2	2	no	—
136	#104×C₅₃	D₄	1	2	yes	complemented
137	#104×C₅₃	D₄	1	2	yes	complemented
138	#102×C₅₃	—	4	1	no	—
139	#101×C₅₃	—	2	1	no	—
140	#101×C₅₃	—	2	1	no	—
141	#101×C₅₃	D₄	1	2	yes	complemented
142	#100×C₅₃	—	2	2	no	—
143	#100×C₅₃	—	2	2	no	—
144	#100×C₅₃	—	2	2	no	—
145	#100×C₅₃	—	2	2	no	—
146	#100×C₅₃	—	2	2	no	—
147	#100×C₅₃	—	2	2	no	—
148	#100×C₅₃	—	2	1	no	—
149	#100×C₅₃	—	2	1	no	—
150	#100×C₅₃	D₄	1	2	yes	complemented
151	#97×C₅₃	—	2	1	no	—
152	C₄×C₈×C₅₃	—	2	1	no	—
153	C₂⁴×C₅₃	—	2	2	no	—
154	C₂⁴×C₅₃	—	2	1	no	—
155	C₂⁴×C₅₃	#41	1	2	yes	—
156	#41×C₅₃	—	2	1	no	—
157	#41×C₅₃	—	2	1	no	—
158	#41×C₅₃	—	2	1	no	—
159	#41×C₅₃	—	2	1	no	—
160	#41×C₅₃	—	2	1	no	—
161	#41×C₅₃	—	2	1	no	—
162	#41×C₅₃	—	2	1	no	—
163	#41×C₅₃	—	2	1	no	—
164	#41×C₅₃	—	2	1	no	—
165	#41×C₅₃	—	2	1	no	—
166	#41×C₅₃	—	4	2	no	—
167	#41×C₅₃	—	4	2	no	—
168	#41×C₅₃	—	4	2	no	—
169	#41×C₅₃	—	4	2	no	—
170	#41×C₅₃	C₂×D₄	1	2	yes	—
171	#41×C₅₃	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
174	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
175	C₂×D₄×C₅₃	—	2	1	no	—
176	C₂×D₄×C₅₃	—	2	2	no	—
177	C₂×D₄×C₅₃	—	2	2	no	—
178	C₂×D₄×C₅₃	—	2	1	no	—
179	C₂×D₄×C₅₃	—	2	1	no	—
180	C₂×D₄×C₅₃	—	2	1	no	—
181	C₂×D₄×C₅₃	—	2	1	no	—
182	C₂×D₄×C₅₃	—	2	1	no	—
183	C₂×D₄×C₅₃	—	2	1	no	—
184	C₂×D₄×C₅₃	—	2	1	no	—
185	C₂×D₄×C₅₃	—	2	1	no	—
186	C₂×D₄×C₅₃	—	2	1	no	—
187	C₂×D₄×C₅₃	—	2	1	no	—
188	C₂×D₄×C₅₃	—	2	1	no	—
189	C₂×D₄×C₅₃	—	2	1	no	—
190	C₂×D₄×C₅₃	—	2	1	no	—
191	C₂×D₄×C₅₃	—	2	1	no	—
192	C₂×D₄×C₅₃	—	4	2	no	—
193	C₂×D₄×C₅₃	—	4	2	no	—
194	C₂×D₄×C₅₃	—	4	2	no	—
195	C₂×D₄×C₅₃	—	4	2	no	—
196	C₂×D₄×C₅₃	—	4	4	no	—
197	C₂×D₄×C₅₃	—	4	4	no	—
198	C₂×D₄×C₅₃	—	4	2	no	—
199	C₂×D₄×C₅₃	—	4	2	no	—
200	C₂×D₄×C₅₃	—	4	2	no	—
201	C₂×D₄×C₅₃	—	8	4	no	—
202	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
203	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
205	C₂²×C₄×C₅₃	—	2	1	no	—
206	C₂²×C₄×C₅₃	—	2	2	no	—
207	C₂²×C₄×C₅₃	—	2	1	no	—
208	C₂²×C₄×C₅₃	—	2	1	no	—
209	C₂²×C₄×C₅₃	—	2	1	no	—
210	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₅₃	—	2	2	no	—
212	Q₁₆×C₅₃	—	4	1	no	—
213	SD₁₆×C₅₃	—	2	2	no	—
214	SD₁₆×C₅₃	—	2	2	no	—
215	SD₁₆×C₅₃	—	4	2	no	—
216	SD₁₆×C₅₃	—	4	2	no	—
217	SD₁₆×C₅₃	—	4	1	no	—
218	SD₁₆×C₅₃	—	4	1	no	—
219	SD₁₆×C₅₃	—	4	2	no	—
220	D₈×C₅₃	—	2	2	no	—
221	D₈×C₅₃	—	4	2	no	—
222	D₈×C₅₃	—	4	2	no	—
223	D₈×C₅₃	—	4	1	no	—
224	D₈×C₅₃	—	4	2	no	—
225	M₁₆×C₅₃	—	2	1	no	—
226	M₁₆×C₅₃	—	2	1	no	—
227	M₁₆×C₅₃	—	2	1	no	—
228	M₁₆×C₅₃	—	2	1	no	—
229	M₁₆×C₅₃	—	2	1	no	—
230	M₁₆×C₅₃	—	2	1	no	—
231	M₁₆×C₅₃	—	2	1	no	—
232	M₁₆×C₅₃	—	2	1	no	—
233	M₁₆×C₅₃	—	2	1	no	—
234	M₁₆×C₅₃	—	2	1	no	—
235	M₁₆×C₅₃	—	2	1	no	—
236	M₁₆×C₅₃	—	2	1	no	—
237	M₁₆×C₅₃	—	4	2	no	—
238	M₁₆×C₅₃	—	4	2	no	—
239	M₁₆×C₅₃	—	4	2	no	—
240	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
241	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
242	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
243	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
244	C₂×C₈×C₅₃	—	2	1	no	—
245	C₂×C₈×C₅₃	—	2	1	no	—
246	C₂×C₈×C₅₃	—	2	1	no	—
247	C₂×C₈×C₅₃	—	2	1	no	—
248	C₂×C₈×C₅₃	—	2	1	no	—
249	C₂×C₈×C₅₃	—	2	1	no	—
250	C₂×C₈×C₅₃	—	2	1	no	—
251	C₂×C₈×C₅₃	—	2	1	no	—
252	C₂×C₈×C₅₃	—	4	2	no	—
253	C₂×C₈×C₅₃	—	4	2	no	—
254	C₂×C₈×C₅₃	—	4	2	no	—
255	C₄⋊C₄×C₅₃	—	2	1	no	—
256	C₂²⋊C₄×C₅₃	—	2	1	no	—
257	C₂²⋊C₄×C₅₃	—	4	2	no	—
258	C₄²×C₅₃	—	2	1	no	—
259	C₄²×C₅₃	—	2	1	no	—
260	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
262	C₂³×C₅₃	—	2	1	no	—
263	C₂³×C₅₃	—	2	2	no	—
264	C₂³×C₅₃	—	2	1	no	—
265	C₂³×C₅₃	—	2	1	no	—
266	C₂³×C₅₃	—	2	2	no	—
267	C₂³×C₅₃	—	2	2	no	—
268	C₂³×C₅₃	—	4	4	no	—
269	C₂³×C₅₃	—	4	1	no	—
270	C₂³×C₅₃	—	4	1	no	—
271	C₂³×C₅₃	—	4	1	no	—
272	C₂³×C₅₃	—	8	2	no	—
273	C₂³×C₅₃	—	8	2	no	—
274	C₂³×C₅₃	—	16	2	no	—
275	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
276	Q₈×C₅₃	—	2	1	no	—
277	Q₈×C₅₃	—	2	1	no	—
278	Q₈×C₅₃	—	2	1	no	—
279	Q₈×C₅₃	—	2	1	no	—
280	Q₈×C₅₃	—	2	1	no	—
281	Q₈×C₅₃	#121	1	2	yes	—
282	D₄×C₅₃	—	2	2	no	—
283	D₄×C₅₃	—	2	2	no	—
284	D₄×C₅₃	—	2	1	no	—
285	D₄×C₅₃	—	2	1	no	—
286	D₄×C₅₃	—	2	1	no	—
287	D₄×C₅₃	—	2	1	no	—
288	D₄×C₅₃	—	2	1	no	—
289	D₄×C₅₃	—	4	4	no	—
290	D₄×C₅₃	—	4	4	no	—
291	D₄×C₅₃	—	4	1	no	—
292	D₄×C₅₃	—	4	1	no	—
293	D₄×C₅₃	—	4	1	no	—
294	D₄×C₅₃	—	4	1	no	—
295	D₄×C₅₃	—	4	1	no	—
296	D₄×C₅₃	—	4	1	no	—
297	D₄×C₅₃	—	4	1	no	—
298	D₄×C₅₃	—	4	1	no	—
299	D₄×C₅₃	—	4	1	no	—
300	D₄×C₅₃	—	4	1	no	—
301	D₄×C₅₃	—	4	1	no	—
302	D₄×C₅₃	—	4	1	no	—
303	D₄×C₅₃	—	8	4	no	—
304	D₄×C₅₃	—	8	4	no	—
305	D₄×C₅₃	—	16	2	no	—
306	D₄×C₅₃	#121	1	2	yes	—
307	C₂×C₄×C₅₃	—	2	1	no	—
308	C₂×C₄×C₅₃	—	2	1	no	—
309	C₂×C₄×C₅₃	—	2	2	no	—
310	C₂×C₄×C₅₃	—	2	2	no	—
311	C₂×C₄×C₅₃	—	2	1	no	—
312	C₂×C₄×C₅₃	—	2	1	no	—
313	C₂×C₄×C₅₃	—	2	1	no	—
314	C₂×C₄×C₅₃	—	4	4	no	—
315	C₂×C₄×C₅₃	—	4	2	no	—
316	C₂×C₄×C₅₃	—	4	1	no	—
317	C₂×C₄×C₅₃	—	4	1	no	—
318	C₂×C₄×C₅₃	—	4	1	no	—
319	C₂×C₄×C₅₃	—	4	1	no	—
320	C₂×C₄×C₅₃	—	4	1	no	—
321	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
325	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
326	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
327	C₈×C₅₃	—	2	2	no	—
328	C₈×C₅₃	—	2	2	no	—
329	C₈×C₅₃	—	2	2	no	—
330	C₈×C₅₃	—	2	2	no	—
331	C₈×C₅₃	—	4	1	no	—
332	C₈×C₅₃	—	4	1	no	—
333	C₈×C₅₃	—	4	1	no	—
334	C₈×C₅₃	—	4	1	no	—
335	#25196	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
336	C₂²×C₅₃	—	2	2	no	—
337	C₂²×C₅₃	—	4	1	no	—
338	C₂²×C₅₃	—	4	1	no	—
339	C₂²×C₅₃	—	4	1	no	—
340	C₂²×C₅₃	—	4	2	no	—
341	C₂²×C₅₃	—	8	1	no	—
342	C₂²×C₅₃	—	8	2	no	—
343	C₂²×C₅₃	—	8	1	no	—
344	C₂²×C₅₃	—	8	2	no	—
345	C₂²×C₅₃	—	8	2	no	—
346	C₂²×C₅₃	—	8	1	no	—
347	C₂²×C₅₃	—	8	2	no	—
348	C₂²×C₅₃	—	8	1	no	—
349	C₂²×C₅₃	—	16	1	no	—
350	C₂²×C₅₃	—	16	1	no	—
351	C₂²×C₅₃	—	16	4	no	—
352	C₂²×C₅₃	#525	1	1	yes	characteristic
353	C₂²×C₅₃	#546	1	1	yes	characteristic
354	C₂²×C₅₃	#546	1	1	yes	characteristic
355	C₄×C₅₃	—	2	2	no	—
356	C₄×C₅₃	—	2	2	no	—
357	C₄×C₅₃	—	4	1	no	—
358	C₄×C₅₃	—	4	1	no	—
359	C₄×C₅₃	—	4	1	no	—
360	C₄×C₅₃	—	8	1	no	—
361	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
362	C₄×C₅₃	#538	1	1	yes	characteristic
363	C₄×C₅₃	D₄²	1	1	yes	characteristic
364	C₄×C₅₃	D₄²	1	1	yes	characteristic
365	#3068	C₂×C₅₃	1	1	yes	characteristic, complemented
366	#3068	C₂×C₅₃	1	1	yes	characteristic, complemented
367	#3019	C₂×C₅₃	1	1	yes	characteristic, complemented
368	C₂×#453	C₂×C₅₃	1	1	yes	characteristic, complemented
369	#2956	C₂×C₅₃	1	1	yes	characteristic, complemented
370	#2888	C₂×C₅₃	1	1	yes	characteristic, complemented
371	#2069	C₂×C₅₃	1	1	yes	characteristic, complemented
372	#2050	C₂×C₅₃	1	1	yes	characteristic, complemented
373	#2048	C₂×C₅₃	1	1	yes	characteristic, complemented
374	#2002	C₂×C₅₃	1	1	yes	characteristic, complemented
375	#1965	C₂×C₅₃	1	1	yes	characteristic, complemented
376	#1932	C₂×C₅₃	1	1	yes	characteristic, complemented
377	#1914	C₂×C₅₃	1	1	yes	characteristic, complemented
378	#1903	C₂×C₅₃	1	1	yes	characteristic, complemented
379	#1861	C₂×C₅₃	1	1	yes	characteristic, complemented
380	C₂×C₅₃	—	2	1	no	—
381	C₂×C₅₃	—	2	1	no	—
382	C₂×C₅₃	—	2	1	no	—
383	C₂×C₅₃	—	4	2	no	—
384	C₂×C₅₃	—	8	1	no	—
385	C₂×C₅₃	—	8	2	no	—
386	C₂×C₅₃	—	8	1	no	—
387	C₂×C₅₃	—	16	1	no	—
388	C₂×C₅₃	#2424	1	1	yes	characteristic
389	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
390	#575	C₂²×C₅₃	1	1	yes	characteristic
391	#575	C₂²×C₅₃	1	1	yes	characteristic
392	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic
393	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic
394	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic, complemented
395	#568	C₂²×C₅₃	1	1	yes	characteristic
396	#568	C₂²×C₅₃	1	1	yes	characteristic
397	#568	C₂²×C₅₃	1	1	yes	characteristic, complemented
398	#535	C₂²×C₅₃	1	1	yes	characteristic
399	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
400	#472	—	2	2	no	—
401	#472	—	2	2	no	—
402	#471	—	2	2	no	—
403	#471	—	2	2	no	—
404	#455	—	2	2	no	—
405	#453	—	2	2	no	—
406	#453	—	2	2	no	—
407	#453	—	2	2	no	—
408	#444	—	2	2	no	—
409	#443	—	2	2	no	—
410	#430	C₂²×C₅₃	1	1	yes	characteristic, complemented
411	#430	C₂²×C₅₃	1	1	yes	characteristic, complemented
412	#430	C₂²×C₅₃	1	1	yes	characteristic, complemented
413	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
414	#421	C₂²×C₅₃	1	1	yes	characteristic
415	#421	C₂²×C₅₃	1	1	yes	characteristic
416	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
417	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
418	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic, complemented
419	#418	C₂²×C₅₃	1	1	yes	characteristic, complemented
420	#413	—	2	2	no	—
421	#413	C₂²×C₅₃	1	1	yes	characteristic
422	#413	C₂²×C₅₃	1	1	yes	characteristic
423	#413	C₂²×C₅₃	1	1	yes	characteristic, complemented
424	C₂×#100	—	2	2	no	—
425	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
426	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
427	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
428	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
429	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic, complemented
430	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic, complemented
431	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
432	#344	C₂²×C₅₃	1	1	yes	characteristic, complemented
433	#344	C₂²×C₅₃	1	1	yes	characteristic, complemented
434	#339	C₂²×C₅₃	1	1	yes	characteristic, complemented
435	#338	C₂²×C₅₃	1	1	yes	characteristic, complemented
436	C₅₃	#25196	1	1	yes	characteristic, direct product factor, Sylow
437	H(5,𝔽₂)	—	2	2	no	—
438	H(5,𝔽₂)	—	2	2	no	—
439	H(5,𝔽₂)	—	4	1	no	—
440	H(5,𝔽₂)	—	4	1	no	—
441	C₂×#41	—	2	1	no	—
442	C₂×#41	—	2	1	no	—
443	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
444	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
445	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
446	C₂²×D₄	—	2	2	no	—
447	C₂²×D₄	—	2	1	no	—
448	C₂²×D₄	—	2	1	no	—
449	C₂²×D₄	—	2	2	no	—
450	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
451	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
452	#137	—	2	1	no	—
453	#137	—	2	1	no	—
454	#137	—	2	1	no	—
455	#137	D₄×C₅₃	1	2	yes	complemented
456	AGL(1,ℤ/8)	—	2	1	no	—
457	AGL(1,ℤ/8)	—	2	2	no	—
458	AGL(1,ℤ/8)	—	2	2	no	—
459	AGL(1,ℤ/8)	—	2	1	no	—
460	AGL(1,ℤ/8)	—	2	1	no	—
461	AGL(1,ℤ/8)	—	2	2	no	—
462	AGL(1,ℤ/8)	—	2	2	no	—
463	AGL(1,ℤ/8)	—	4	1	no	—
464	AGL(1,ℤ/8)	—	4	1	no	—
465	AGL(1,ℤ/8)	D₄×C₅₃	1	2	yes	complemented
466	#135	—	2	2	no	—
467	#135	—	4	1	no	—
468	C₂×SD₁₆	—	2	1	no	—
469	C₂×SD₁₆	—	2	1	no	—
470	C₂×SD₁₆	—	2	1	no	—
471	C₂×D₈	—	2	1	no	—
472	C₂×D₈	—	2	1	no	—
473	C₂×D₈	—	2	1	no	—
474	#131	—	2	2	no	—
475	#131	—	2	2	no	—
476	#131	—	2	2	no	—
477	#131	—	2	2	no	—
478	#131	—	4	1	no	—
479	#131	—	4	1	no	—
480	C₂×M₁₆	—	2	2	no	—
481	C₂×M₁₆	—	2	2	no	—
482	C₂×M₁₆	—	4	1	no	—
483	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
484	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
485	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
486	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
487	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
488	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
489	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
490	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
491	#127	—	2	1	no	—
492	#127	—	2	1	no	—
493	#127	—	2	1	no	—
494	#127	—	4	1	no	—
495	#124	—	2	1	no	—
496	#121	—	4	1	no	—
497	#120	—	2	2	no	—
498	#117	C₂³×C₅₃	1	1	yes	characteristic
499	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
500	#108	—	2	2	no	—
501	#108	—	2	2	no	—
502	#108	—	2	2	no	—
503	C₄⋊C₈	—	4	1	no	—
504	#104	—	2	2	no	—
505	#104	—	2	2	no	—
506	#104	—	2	1	no	—
507	#104	—	2	1	no	—
508	#104	—	2	1	no	—
509	#104	—	2	1	no	—
510	#104	D₄×C₅₃	1	2	yes	complemented
511	#104	D₄×C₅₃	1	2	yes	complemented
512	#102	—	4	1	no	—
513	#101	—	2	1	no	—
514	#101	—	2	1	no	—
515	#101	D₄×C₅₃	1	2	yes	complemented
516	#100	—	2	1	no	—
517	#100	—	2	1	no	—
518	#100	—	2	2	no	—
519	#100	—	2	2	no	—
520	#100	—	2	2	no	—
521	#100	—	2	2	no	—
522	#100	—	2	2	no	—
523	#100	—	2	2	no	—
524	#100	D₄×C₅₃	1	2	yes	complemented
525	#97	—	2	1	no	—
526	C₄×C₈	—	2	1	no	—
527	C₂⁴	—	2	2	no	—
528	C₂⁴	—	2	1	no	—
529	C₂⁴	#41×C₅₃	1	2	yes	—
530	#41	—	2	1	no	—
531	#41	—	2	1	no	—
532	#41	—	2	1	no	—
533	#41	—	2	1	no	—
534	#41	—	2	1	no	—
535	#41	—	2	1	no	—
536	#41	—	2	1	no	—
537	#41	—	2	1	no	—
538	#41	—	2	1	no	—
539	#41	—	2	1	no	—
540	#41	—	4	2	no	—
541	#41	—	4	2	no	—
542	#41	—	4	2	no	—
543	#41	—	4	2	no	—
544	#41	C₂×D₄×C₅₃	1	2	yes	—
545	#41	C₂×D₄×C₅₃	1	2	yes	—
546	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
547	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
548	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	—	2	1	no	—
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	—	2	1	no	—
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	4	2	no	—
567	C₂×D₄	—	4	2	no	—
568	C₂×D₄	—	4	2	no	—
569	C₂×D₄	—	4	2	no	—
570	C₂×D₄	—	4	4	no	—
571	C₂×D₄	—	4	4	no	—
572	C₂×D₄	—	4	2	no	—
573	C₂×D₄	—	4	2	no	—
574	C₂×D₄	—	4	2	no	—
575	C₂×D₄	—	8	4	no	—
576	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
577	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
578	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
579	C₂²×C₄	—	2	1	no	—
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
585	Q₁₆	—	2	2	no	—
586	Q₁₆	—	4	1	no	—
587	SD₁₆	—	2	2	no	—
588	SD₁₆	—	2	2	no	—
589	SD₁₆	—	4	2	no	—
590	SD₁₆	—	4	2	no	—
591	SD₁₆	—	4	1	no	—
592	SD₁₆	—	4	1	no	—
593	SD₁₆	—	4	2	no	—
594	D₈	—	2	2	no	—
595	D₈	—	4	2	no	—
596	D₈	—	4	2	no	—
597	D₈	—	4	1	no	—
598	D₈	—	4	2	no	—
599	M₁₆	—	2	1	no	—
600	M₁₆	—	2	1	no	—
601	M₁₆	—	2	1	no	—
602	M₁₆	—	2	1	no	—
603	M₁₆	—	2	1	no	—
604	M₁₆	—	2	1	no	—
605	M₁₆	—	2	1	no	—
606	M₁₆	—	2	1	no	—
607	M₁₆	—	2	1	no	—
608	M₁₆	—	2	1	no	—
609	M₁₆	—	2	1	no	—
610	M₁₆	—	2	1	no	—
611	M₁₆	—	4	2	no	—
612	M₁₆	—	4	2	no	—
613	M₁₆	—	4	2	no	—
614	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
615	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
616	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
617	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
618	C₂×C₈	—	2	1	no	—
619	C₂×C₈	—	2	1	no	—
620	C₂×C₈	—	2	1	no	—
621	C₂×C₈	—	2	1	no	—
622	C₂×C₈	—	2	1	no	—
623	C₂×C₈	—	2	1	no	—
624	C₂×C₈	—	2	1	no	—
625	C₂×C₈	—	2	1	no	—
626	C₂×C₈	—	4	2	no	—
627	C₂×C₈	—	4	2	no	—
628	C₂×C₈	—	4	2	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	4	2	no	—
632	C₄²	—	2	1	no	—
633	C₄²	—	2	1	no	—
634	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
635	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
636	C₂³	—	2	1	no	—
637	C₂³	—	2	2	no	—
638	C₂³	—	2	1	no	—
639	C₂³	—	2	2	no	—
640	C₂³	—	2	1	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	4	4	no	—
643	C₂³	—	4	1	no	—
644	C₂³	—	4	1	no	—
645	C₂³	—	4	1	no	—
646	C₂³	—	8	2	no	—
647	C₂³	—	8	2	no	—
648	C₂³	—	16	2	no	—
649	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
650	Q₈	—	2	1	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	1	no	—
653	Q₈	—	2	1	no	—
654	Q₈	—	2	1	no	—
655	Q₈	#121×C₅₃	1	2	yes	—
656	D₄	—	2	2	no	—
657	D₄	—	2	2	no	—
658	D₄	—	2	1	no	—
659	D₄	—	2	1	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	1	no	—
662	D₄	—	2	1	no	—
663	D₄	—	4	4	no	—
664	D₄	—	4	4	no	—
665	D₄	—	4	1	no	—
666	D₄	—	4	1	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	4	1	no	—
671	D₄	—	4	1	no	—
672	D₄	—	4	1	no	—
673	D₄	—	4	1	no	—
674	D₄	—	4	1	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	8	4	no	—
678	D₄	—	8	4	no	—
679	D₄	—	16	2	no	—
680	D₄	#121×C₅₃	1	2	yes	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	2	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	1	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
696	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
697	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
698	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
699	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
700	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
701	C₈	—	2	2	no	—
702	C₈	—	2	2	no	—
703	C₈	—	2	2	no	—
704	C₈	—	2	2	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₈	—	4	1	no	—
708	C₈	—	4	1	no	—
709	C₂²	—	2	2	no	—
710	C₂²	—	4	1	no	—
711	C₂²	—	4	1	no	—
712	C₂²	—	4	1	no	—
713	C₂²	—	4	2	no	—
714	C₂²	—	8	1	no	—
715	C₂²	—	8	2	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	2	no	—
718	C₂²	—	8	2	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	2	no	—
721	C₂²	—	8	1	no	—
722	C₂²	—	16	1	no	—
723	C₂²	—	16	1	no	—
724	C₂²	—	16	4	no	—
725	C₂²	C₅₃×#525	1	1	yes	characteristic
726	C₂²	C₅₃×#546	1	1	yes	characteristic
727	C₂²	C₅₃×#546	1	1	yes	characteristic
728	C₄	—	2	2	no	—
729	C₄	—	2	2	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	4	1	no	—
733	C₄	—	8	1	no	—
734	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
735	C₄	C₅₃×#538	1	1	yes	characteristic
736	C₄	D₄²×C₅₃	1	1	yes	characteristic
737	C₄	D₄²×C₅₃	1	1	yes	characteristic
738	C₂	—	2	1	no	—
739	C₂	—	2	1	no	—
740	C₂	—	2	1	no	—
741	C₂	—	4	2	no	—
742	C₂	—	8	2	no	—
743	C₂	—	8	1	no	—
744	C₂	—	8	1	no	—
745	C₂	—	16	1	no	—
746	C₂	C₅₃×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3068	C₂	semidirect
2. C₅₃×#3068	C₂	semidirect
3. C₅₃×#3019	C₂	semidirect
4. C₂×C₅₃×#453	C₂	semidirect
5. C₅₃×#2956	C₂	semidirect
6. C₅₃×#2888	C₂	semidirect
7. C₅₃×#2069	C₂	semidirect
8. C₅₃×#2050	C₂	semidirect
9. C₅₃×#2048	C₂	semidirect
10. C₅₃×#2002	C₂	semidirect
11. C₅₃×#1965	C₂	semidirect
12. C₅₃×#1932	C₂	semidirect
13. C₅₃×#1914	C₂	semidirect
14. C₅₃×#1903	C₂	semidirect
15. C₅₃×#1861	C₂	semidirect
21. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
24. C₅₃×#568	C₂²	semidirect
37. C₅₃×#430	C₂²	semidirect
38. C₅₃×#430	C₂²	semidirect
39. C₅₃×#430	C₂²	semidirect
40. C₅₃×#423	C₂²	semidirect
45. C₂×#104×C₅₃	C₂²	semidirect
46. C₅₃×#418	C₂²	semidirect
50. C₅₃×#413	C₂²	semidirect
56. C₂×#100×C₅₃	C₂²	semidirect
57. C₂×#100×C₅₃	C₂²	semidirect
58. C₄×M₁₆×C₅₃	C₂²	semidirect
59. C₅₃×#344	C₂²	semidirect
60. C₅₃×#344	C₂²	semidirect
61. C₅₃×#339	C₂²	semidirect
62. C₅₃×#338	C₂²	semidirect
81. #137×C₅₃	D₄	semidirect
91. AGL(1,ℤ/8)×C₅₃	D₄	semidirect
136. #104×C₅₃	D₄	semidirect
137. #104×C₅₃	D₄	semidirect
141. #101×C₅₃	D₄	semidirect
150. #100×C₅₃	D₄	semidirect
335. #25196	C₅₃	direct (N × H)
365. #3068	C₂×C₅₃	semidirect
366. #3068	C₂×C₅₃	semidirect
367. #3019	C₂×C₅₃	semidirect
368. C₂×#453	C₂×C₅₃	semidirect
369. #2956	C₂×C₅₃	semidirect
370. #2888	C₂×C₅₃	semidirect
371. #2069	C₂×C₅₃	semidirect
372. #2050	C₂×C₅₃	semidirect
373. #2048	C₂×C₅₃	semidirect
374. #2002	C₂×C₅₃	semidirect
375. #1965	C₂×C₅₃	semidirect
376. #1932	C₂×C₅₃	semidirect
377. #1914	C₂×C₅₃	semidirect
378. #1903	C₂×C₅₃	semidirect
379. #1861	C₂×C₅₃	semidirect
394. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
397. #568	C₂²×C₅₃	semidirect
410. #430	C₂²×C₅₃	semidirect
411. #430	C₂²×C₅₃	semidirect
412. #430	C₂²×C₅₃	semidirect
413. #423	C₂²×C₅₃	semidirect
418. C₂×#104	C₂²×C₅₃	semidirect
419. #418	C₂²×C₅₃	semidirect
423. #413	C₂²×C₅₃	semidirect
429. C₂×#100	C₂²×C₅₃	semidirect
430. C₂×#100	C₂²×C₅₃	semidirect
431. C₄×M₁₆	C₂²×C₅₃	semidirect
432. #344	C₂²×C₅₃	semidirect
433. #344	C₂²×C₅₃	semidirect
434. #339	C₂²×C₅₃	semidirect
435. #338	C₂²×C₅₃	semidirect
436. C₅₃	#25196	direct (N × H)
455. #137	D₄×C₅₃	semidirect
465. AGL(1,ℤ/8)	D₄×C₅₃	semidirect
510. #104	D₄×C₅₃	semidirect
511. #104	D₄×C₅₃	semidirect
515. #101	D₄×C₅₃	semidirect
524. #100	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×#3068	2
C₅₃×#3019	2
C₂×C₅₃×#453	2
C₅₃×#2956	2
C₅₃×#2888	2
C₅₃×#2069	2
C₅₃×#2050	2
C₅₃×#2048	2
C₅₃×#2002	2
C₅₃×#1965	2
C₅₃×#1932	2
C₅₃×#1914	2
C₅₃×#1903	2
C₅₃×#1861	2
#25196	53

Subgroups — C53×#25196 #27232708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25196 #27232708