Subgroups — C₅₃×#61089 #27268601

Order 13568.

2922 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#576	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3498	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3287	C₂	1	4	yes	maximal, complemented
4	D₄×D₈×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3274	C₂	1	8	yes	maximal, complemented
6	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3058	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3038	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3008	C₂	1	4	yes	maximal, complemented
10	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	complemented
13	C₅₃×#576	—	2	16	no	—
14	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	8	yes	complemented
16	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#135×C₅₃	C₂²	1	4	yes	complemented
18	C₂²×D₈×C₅₃	C₂²	1	4	yes	complemented
19	C₂²×D₈×C₅₃	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₅₃	C₂²	1	2	yes	complemented
21	C₂×#131×C₅₃	C₂²	1	2	yes	complemented
22	C₅₃×#550	C₂²	1	8	yes	complemented
23	D₄²×C₅₃	—	2	8	no	—
24	D₄²×C₅₃	C₂²	1	8	yes	complemented
25	C₅₃×#535	—	2	4	no	—
26	C₅₃×#535	C₂²	1	4	yes	complemented
27	C₅₃×#532	—	2	2	no	—
28	C₅₃×#532	C₂²	1	1	yes	characteristic
29	C₂×#127×C₅₃	—	2	2	no	—
30	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
31	C₄×#41×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#493	C₂²	1	4	yes	complemented
33	C₅₃×#493	C₂²	1	4	yes	complemented
34	C₅₃×#469	C₂²	1	8	yes	complemented
35	C₅₃×#466	C₂²	1	8	yes	complemented
36	C₅₃×#460	C₂²	1	8	yes	complemented
37	C₅₃×#459	C₂²	1	8	yes	complemented
38	C₅₃×#449	C₂²	1	16	yes	complemented
39	C₅₃×#447	C₂²	1	16	yes	complemented
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
41	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
42	C₄×D₈×C₅₃	C₂²	1	2	yes	complemented
43	C₅₃×#436	C₂²	1	4	yes	complemented
44	C₈×D₄×C₅₃	C₂²	1	4	yes	complemented
45	C₅₃×#427	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
47	C₅₃×#418	C₂²	1	4	yes	complemented
48	C₅₃×#416	C₂²	1	2	yes	complemented
49	C₅₃×#408	C₂²	1	2	yes	complemented
50	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	H(5,𝔽₂)×C₅₃	—	2	16	no	—
52	H(5,𝔽₂)×C₅₃	—	4	8	no	—
53	C₂×#41×C₅₃	—	2	4	no	—
54	C₂×#41×C₅₃	C₂³	1	8	yes	—
55	C₂×#41×C₅₃	C₂³	1	2	yes	—
56	C₂×#41×C₅₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₅₃	—	2	4	no	—
58	C₂²×D₄×C₅₃	—	2	8	no	—
59	C₂²×D₄×C₅₃	—	2	8	no	—
60	C₂²×D₄×C₅₃	—	2	4	no	—
61	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
62	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
63	AGL(1,ℤ/8)×C₅₃	—	2	32	no	—
64	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
65	#135×C₅₃	—	2	16	no	—
66	#135×C₅₃	D₄	1	8	yes	complemented
67	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
69	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
70	C₂×D₈×C₅₃	—	2	16	no	—
71	C₂×D₈×C₅₃	—	2	8	no	—
72	C₂×D₈×C₅₃	—	4	4	no	—
73	C₂×D₈×C₅₃	C₂³	1	4	yes	—
74	C₂×D₈×C₅₃	C₂³	1	4	yes	—
75	C₂×D₈×C₅₃	C₂³	1	8	yes	—
76	C₂×D₈×C₅₃	C₂³	1	8	yes	—
77	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
78	#131×C₅₃	—	2	8	no	—
79	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
80	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
81	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
85	#127×C₅₃	—	2	2	no	—
86	#127×C₅₃	—	2	2	no	—
87	#127×C₅₃	—	2	4	no	—
88	#127×C₅₃	—	2	8	no	—
89	#127×C₅₃	D₄	1	1	yes	characteristic
90	#127×C₅₃	C₂³	1	8	yes	—
91	#124×C₅₃	—	2	4	no	—
92	#124×C₅₃	D₄	1	2	yes	—
93	#121×C₅₃	—	2	8	no	—
94	#121×C₅₃	—	2	16	no	—
95	#121×C₅₃	D₄	1	4	yes	—
96	#121×C₅₃	C₂³	1	16	yes	—
97	#120×C₅₃	—	2	16	no	—
98	#120×C₅₃	—	2	8	no	—
99	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
100	C₄×D₄×C₅₃	—	2	8	no	—
101	C₄×D₄×C₅₃	—	2	4	no	—
102	C₄×D₄×C₅₃	C₂³	1	4	yes	—
103	C₄×D₄×C₅₃	C₂³	1	4	yes	—
104	#117×C₅₃	—	2	2	no	—
105	#117×C₅₃	C₂³	1	2	yes	—
106	C₂×C₄²×C₅₃	—	2	2	no	—
107	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
108	#107×C₅₃	C₂³	1	2	yes	—
109	#106×C₅₃	C₂³	1	2	yes	—
110	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
111	#103×C₅₃	C₂³	1	4	yes	—
112	#102×C₅₃	C₂³	1	4	yes	—
113	#102×C₅₃	C₂³	1	16	yes	—
114	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
115	C₄×C₈×C₅₃	C₂³	1	2	yes	—
116	C₄×C₈×C₅₃	C₂³	1	2	yes	—
117	C₂⁴×C₅₃	—	2	8	no	—
118	C₂⁴×C₅₃	—	2	4	no	—
119	#41×C₅₃	—	2	4	no	—
120	#41×C₅₃	—	2	4	no	—
121	#41×C₅₃	—	2	4	no	—
122	#41×C₅₃	—	2	16	no	—
123	#41×C₅₃	—	4	8	no	—
124	#41×C₅₃	—	4	16	no	—
125	#41×C₅₃	C₂×D₄	1	8	yes	complemented
126	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
128	C₂×D₄×C₅₃	—	2	2	no	—
129	C₂×D₄×C₅₃	—	2	16	no	—
130	C₂×D₄×C₅₃	—	2	8	no	—
131	C₂×D₄×C₅₃	—	2	16	no	—
132	C₂×D₄×C₅₃	—	2	8	no	—
133	C₂×D₄×C₅₃	—	2	8	no	—
134	C₂×D₄×C₅₃	—	2	8	no	—
135	C₂×D₄×C₅₃	—	2	4	no	—
136	C₂×D₄×C₅₃	—	2	8	no	—
137	C₂×D₄×C₅₃	—	2	8	no	—
138	C₂×D₄×C₅₃	—	4	8	no	—
139	C₂×D₄×C₅₃	—	4	16	no	—
140	C₂×D₄×C₅₃	—	4	32	no	—
141	C₂×D₄×C₅₃	—	8	8	no	—
142	C₂×D₄×C₅₃	—	8	16	no	—
143	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
144	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
147	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	4	no	—
150	C₂²×C₄×C₅₃	—	2	8	no	—
151	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₅₃	C₂×D₄	1	4	yes	complemented
155	SD₁₆×C₅₃	—	2	16	no	—
156	SD₁₆×C₅₃	C₂×D₄	1	8	yes	complemented
157	D₈×C₅₃	—	2	16	no	—
158	D₈×C₅₃	—	2	8	no	—
159	D₈×C₅₃	—	2	8	no	—
160	D₈×C₅₃	—	4	8	no	—
161	D₈×C₅₃	C₂×D₄	1	4	yes	complemented
162	M₁₆×C₅₃	—	2	4	no	—
163	M₁₆×C₅₃	—	2	8	no	—
164	C₂×C₈×C₅₃	—	2	2	no	—
165	C₂×C₈×C₅₃	—	2	8	no	—
166	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	complemented
167	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
168	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
169	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
171	C₄⋊C₄×C₅₃	—	2	4	no	—
172	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
174	C₂²⋊C₄×C₅₃	—	2	2	no	—
175	C₂²⋊C₄×C₅₃	—	2	4	no	—
176	C₂²⋊C₄×C₅₃	—	2	16	no	—
177	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
178	C₄²×C₅₃	—	2	2	no	—
179	C₄²×C₅₃	—	2	4	no	—
180	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂⁴	1	2	yes	—
183	C₂³×C₅₃	—	2	2	no	—
184	C₂³×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	4	16	no	—
186	C₂³×C₅₃	—	4	8	no	—
187	C₂³×C₅₃	—	8	8	no	—
188	C₂³×C₅₃	—	8	16	no	—
189	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
190	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₅₃	—	2	4	no	—
192	Q₈×C₅₃	—	2	2	no	—
193	Q₈×C₅₃	C₂²×D₄	1	4	yes	—
194	D₄×C₅₃	—	2	8	no	—
195	D₄×C₅₃	—	2	8	no	—
196	D₄×C₅₃	—	2	4	no	—
197	D₄×C₅₃	—	2	8	no	—
198	D₄×C₅₃	—	2	8	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	2	8	no	—
201	D₄×C₅₃	—	4	16	no	—
202	D₄×C₅₃	—	4	16	no	—
203	D₄×C₅₃	—	4	8	no	—
204	D₄×C₅₃	—	4	16	no	—
205	D₄×C₅₃	—	8	8	no	—
206	D₄×C₅₃	—	8	16	no	—
207	D₄×C₅₃	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	8	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	4	16	no	—
215	C₂×C₄×C₅₃	—	4	8	no	—
216	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
222	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
223	C₈×C₅₃	—	2	4	no	—
224	C₈×C₅₃	—	2	2	no	—
225	C₈×C₅₃	C₂²×D₄	1	2	yes	—
226	C₈×C₅₃	C₂²×D₄	1	2	yes	—
227	#61089	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂²×C₅₃	—	2	2	no	—
229	C₂²×C₅₃	—	2	4	no	—
230	C₂²×C₅₃	—	2	4	no	—
231	C₂²×C₅₃	—	2	8	no	—
232	C₂²×C₅₃	—	4	4	no	—
233	C₂²×C₅₃	—	4	8	no	—
234	C₂²×C₅₃	—	4	16	no	—
235	C₂²×C₅₃	—	16	8	no	—
236	C₂²×C₅₃	—	16	16	no	—
237	C₂²×C₅₃	D₄²	1	2	yes	—
238	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₅₃	—	2	4	no	—
240	C₄×C₅₃	—	2	4	no	—
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	4	4	no	—
243	C₄×C₅₃	D₄²	1	2	yes	—
244	C₄×C₅₃	#576	1	4	yes	complemented
245	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×#576	C₂×C₅₃	1	2	yes	complemented
248	#3498	C₂×C₅₃	1	2	yes	complemented
249	#3287	C₂×C₅₃	1	4	yes	complemented
250	D₄×D₈	C₂×C₅₃	1	4	yes	complemented
251	#3274	C₂×C₅₃	1	8	yes	complemented
252	C₂×#493	C₂×C₅₃	1	1	yes	characteristic, complemented
253	#3058	C₂×C₅₃	1	2	yes	complemented
254	#3038	C₂×C₅₃	1	2	yes	complemented
255	#3008	C₂×C₅₃	1	4	yes	complemented
256	C₄×#135	C₂×C₅₃	1	1	yes	characteristic, complemented
257	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
258	C₂×C₅₃	—	2	2	no	—
259	C₂×C₅₃	—	4	4	no	—
260	C₂×C₅₃	—	4	4	no	—
261	C₂×C₅₃	—	8	8	no	—
262	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
263	C₂×C₅₃	C₂×#576	1	2	yes	—
264	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	4	yes	complemented
265	#576	—	2	16	no	—
266	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	complemented
267	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	8	yes	complemented
268	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×#135	C₂²×C₅₃	1	4	yes	complemented
270	C₂²×D₈	C₂²×C₅₃	1	2	yes	complemented
271	C₂²×D₈	C₂²×C₅₃	1	4	yes	complemented
272	C₂²×D₈	C₂²×C₅₃	1	2	yes	complemented
273	C₂×#131	C₂²×C₅₃	1	2	yes	complemented
274	#550	C₂²×C₅₃	1	8	yes	complemented
275	D₄²	—	2	8	no	—
276	D₄²	C₂²×C₅₃	1	8	yes	complemented
277	#535	—	2	4	no	—
278	#535	C₂²×C₅₃	1	4	yes	complemented
279	#532	—	2	2	no	—
280	#532	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×#127	—	2	2	no	—
282	C₂×#127	C₂²×C₅₃	1	2	yes	complemented
283	C₄×#41	C₂²×C₅₃	1	2	yes	complemented
284	#493	C₂²×C₅₃	1	4	yes	complemented
285	#493	C₂²×C₅₃	1	4	yes	complemented
286	#469	C₂²×C₅₃	1	8	yes	complemented
287	#466	C₂²×C₅₃	1	8	yes	complemented
288	#460	C₂²×C₅₃	1	8	yes	complemented
289	#459	C₂²×C₅₃	1	8	yes	complemented
290	#449	C₂²×C₅₃	1	16	yes	complemented
291	#447	C₂²×C₅₃	1	16	yes	complemented
292	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	complemented
293	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
294	C₄×D₈	C₂²×C₅₃	1	2	yes	complemented
295	#436	C₂²×C₅₃	1	4	yes	complemented
296	C₈×D₄	C₂²×C₅₃	1	4	yes	complemented
297	#427	C₂²×C₅₃	1	1	yes	characteristic, complemented
298	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
299	#418	C₂²×C₅₃	1	4	yes	complemented
300	#416	C₂²×C₅₃	1	2	yes	complemented
301	#408	C₂²×C₅₃	1	2	yes	complemented
302	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
303	C₅₃	#61089	1	1	yes	characteristic, direct product factor, Sylow
304	H(5,𝔽₂)	—	2	16	no	—
305	H(5,𝔽₂)	—	4	8	no	—
306	C₂×#41	—	2	4	no	—
307	C₂×#41	C₂³×C₅₃	1	8	yes	—
308	C₂×#41	C₂³×C₅₃	1	2	yes	—
309	C₂×#41	C₂³×C₅₃	1	2	yes	—
310	C₂²×D₄	—	2	4	no	—
311	C₂²×D₄	—	2	8	no	—
312	C₂²×D₄	—	2	8	no	—
313	C₂²×D₄	—	2	4	no	—
314	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
315	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
316	AGL(1,ℤ/8)	—	2	16	no	—
317	AGL(1,ℤ/8)	—	2	32	no	—
318	#135	—	2	16	no	—
319	#135	D₄×C₅₃	1	8	yes	complemented
320	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
321	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
322	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
323	C₂×D₈	—	2	16	no	—
324	C₂×D₈	—	2	8	no	—
325	C₂×D₈	—	4	4	no	—
326	C₂×D₈	C₂³×C₅₃	1	4	yes	—
327	C₂×D₈	C₂³×C₅₃	1	8	yes	—
328	C₂×D₈	C₂³×C₅₃	1	4	yes	—
329	C₂×D₈	C₂³×C₅₃	1	8	yes	—
330	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
331	#131	—	2	8	no	—
332	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
333	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
334	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
335	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
337	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
338	#127	—	2	4	no	—
339	#127	—	2	8	no	—
340	#127	—	2	2	no	—
341	#127	—	2	2	no	—
342	#127	D₄×C₅₃	1	1	yes	characteristic
343	#127	C₂³×C₅₃	1	8	yes	—
344	#124	—	2	4	no	—
345	#124	D₄×C₅₃	1	2	yes	—
346	#121	—	2	8	no	—
347	#121	—	2	16	no	—
348	#121	D₄×C₅₃	1	4	yes	—
349	#121	C₂³×C₅₃	1	16	yes	—
350	#120	—	2	8	no	—
351	#120	—	2	16	no	—
352	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	C₂³×C₅₃	1	4	yes	—
356	C₄×D₄	C₂³×C₅₃	1	4	yes	—
357	#117	—	2	2	no	—
358	#117	C₂³×C₅₃	1	2	yes	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
361	#107	C₂³×C₅₃	1	2	yes	—
362	#106	C₂³×C₅₃	1	2	yes	—
363	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
364	#103	C₂³×C₅₃	1	4	yes	—
365	#102	C₂³×C₅₃	1	4	yes	—
366	#102	C₂³×C₅₃	1	16	yes	—
367	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
368	C₄×C₈	C₂³×C₅₃	1	2	yes	—
369	C₄×C₈	C₂³×C₅₃	1	2	yes	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	4	no	—
372	#41	—	2	4	no	—
373	#41	—	2	4	no	—
374	#41	—	2	4	no	—
375	#41	—	2	16	no	—
376	#41	—	4	8	no	—
377	#41	—	4	16	no	—
378	#41	C₂×D₄×C₅₃	1	8	yes	complemented
379	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
380	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	16	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	4	8	no	—
392	C₂×D₄	—	4	16	no	—
393	C₂×D₄	—	4	32	no	—
394	C₂×D₄	—	8	8	no	—
395	C₂×D₄	—	8	16	no	—
396	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
397	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
398	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
399	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
400	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
407	Q₁₆	C₂×D₄×C₅₃	1	4	yes	complemented
408	SD₁₆	—	2	16	no	—
409	SD₁₆	C₂×D₄×C₅₃	1	8	yes	complemented
410	D₈	—	2	16	no	—
411	D₈	—	2	8	no	—
412	D₈	—	2	8	no	—
413	D₈	—	4	8	no	—
414	D₈	C₂×D₄×C₅₃	1	4	yes	complemented
415	M₁₆	—	2	4	no	—
416	M₁₆	—	2	8	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	8	no	—
419	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	complemented
420	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
421	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
422	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
423	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
426	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	4	no	—
433	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
434	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
435	C₄²	C₂⁴×C₅₃	1	2	yes	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	8	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	8	no	—
440	C₂³	—	8	8	no	—
441	C₂³	—	8	16	no	—
442	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
443	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
444	Q₈	—	2	4	no	—
445	Q₈	—	2	2	no	—
446	Q₈	C₂²×D₄×C₅₃	1	4	yes	—
447	D₄	—	2	8	no	—
448	D₄	—	2	8	no	—
449	D₄	—	2	4	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	8	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	8	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	16	no	—
456	D₄	—	4	8	no	—
457	D₄	—	4	16	no	—
458	D₄	—	8	8	no	—
459	D₄	—	8	16	no	—
460	D₄	C₂²×D₄×C₅₃	1	4	yes	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	4	16	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
474	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
475	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	2	2	no	—
478	C₈	C₂²×D₄×C₅₃	1	2	yes	—
479	C₈	C₂²×D₄×C₅₃	1	2	yes	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	8	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	8	no	—
486	C₂²	—	4	16	no	—
487	C₂²	—	16	8	no	—
488	C₂²	—	16	16	no	—
489	C₂²	D₄²×C₅₃	1	2	yes	—
490	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
491	C₄	—	2	4	no	—
492	C₄	—	2	4	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	4	no	—
495	C₄	D₄²×C₅₃	1	2	yes	—
496	C₄	C₅₃×#576	1	4	yes	complemented
497	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
498	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
499	C₂	—	2	2	no	—
500	C₂	—	4	4	no	—
501	C₂	—	4	4	no	—
502	C₂	—	8	8	no	—
503	C₂	C₂×D₄²×C₅₃	1	1	yes	characteristic
504	C₂	C₂×C₅₃×#576	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 ◁ C₅₃×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#576	C₂	semidirect
2. C₅₃×#3498	C₂	semidirect
3. C₅₃×#3287	C₂	semidirect
4. D₄×D₈×C₅₃	C₂	semidirect
5. C₅₃×#3274	C₂	semidirect
6. C₂×C₅₃×#493	C₂	semidirect
7. C₅₃×#3058	C₂	semidirect
8. C₅₃×#3038	C₂	semidirect
9. C₅₃×#3008	C₂	semidirect
10. C₄×#135×C₅₃	C₂	semidirect
11. C₅₃×#2931	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
15. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
17. C₂×#135×C₅₃	C₂²	semidirect
18. C₂²×D₈×C₅₃	C₂²	semidirect
19. C₂²×D₈×C₅₃	C₂²	semidirect
20. C₂²×D₈×C₅₃	C₂²	semidirect
21. C₂×#131×C₅₃	C₂²	semidirect
22. C₅₃×#550	C₂²	semidirect
24. D₄²×C₅₃	C₂²	semidirect
26. C₅₃×#535	C₂²	semidirect
30. C₂×#127×C₅₃	C₂²	semidirect
31. C₄×#41×C₅₃	C₂²	semidirect
32. C₅₃×#493	C₂²	semidirect
33. C₅₃×#493	C₂²	semidirect
34. C₅₃×#469	C₂²	semidirect
35. C₅₃×#466	C₂²	semidirect
36. C₅₃×#460	C₂²	semidirect
37. C₅₃×#459	C₂²	semidirect
38. C₅₃×#449	C₂²	semidirect
39. C₅₃×#447	C₂²	semidirect
40. C₄×Q₁₆×C₅₃	C₂²	semidirect
41. C₄×SD₁₆×C₅₃	C₂²	semidirect
42. C₄×D₈×C₅₃	C₂²	semidirect
43. C₅₃×#436	C₂²	semidirect
44. C₈×D₄×C₅₃	C₂²	semidirect
45. C₅₃×#427	C₂²	semidirect
46. C₅₃×#423	C₂²	semidirect
47. C₅₃×#418	C₂²	semidirect
48. C₅₃×#416	C₂²	semidirect
49. C₅₃×#408	C₂²	semidirect
50. C₂×C₄×C₈×C₅₃	C₂²	semidirect
66. #135×C₅₃	D₄	semidirect
125. #41×C₅₃	C₂×D₄	semidirect
154. Q₁₆×C₅₃	C₂×D₄	semidirect
156. SD₁₆×C₅₃	C₂×D₄	semidirect
161. D₈×C₅₃	C₂×D₄	semidirect
166. C₂×C₈×C₅₃	C₂×D₄	semidirect
227. #61089	C₅₃	direct (N × H)
244. C₄×C₅₃	#576	semidirect
247. C₂×#576	C₂×C₅₃	semidirect
248. #3498	C₂×C₅₃	semidirect
249. #3287	C₂×C₅₃	semidirect
250. D₄×D₈	C₂×C₅₃	semidirect
251. #3274	C₂×C₅₃	semidirect
252. C₂×#493	C₂×C₅₃	semidirect
253. #3058	C₂×C₅₃	semidirect
254. #3038	C₂×C₅₃	semidirect
255. #3008	C₂×C₅₃	semidirect
256. C₄×#135	C₂×C₅₃	semidirect
257. #2931	C₂×C₅₃	semidirect
264. C₂×H(5,𝔽₂)	C₂²×C₅₃	semidirect
266. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
267. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
269. C₂×#135	C₂²×C₅₃	semidirect
270. C₂²×D₈	C₂²×C₅₃	semidirect
271. C₂²×D₈	C₂²×C₅₃	semidirect
272. C₂²×D₈	C₂²×C₅₃	semidirect
273. C₂×#131	C₂²×C₅₃	semidirect
274. #550	C₂²×C₅₃	semidirect
276. D₄²	C₂²×C₅₃	semidirect
278. #535	C₂²×C₅₃	semidirect
282. C₂×#127	C₂²×C₅₃	semidirect
283. C₄×#41	C₂²×C₅₃	semidirect
284. #493	C₂²×C₅₃	semidirect
285. #493	C₂²×C₅₃	semidirect
286. #469	C₂²×C₅₃	semidirect
287. #466	C₂²×C₅₃	semidirect
288. #460	C₂²×C₅₃	semidirect
289. #459	C₂²×C₅₃	semidirect
290. #449	C₂²×C₅₃	semidirect
291. #447	C₂²×C₅₃	semidirect
292. C₄×Q₁₆	C₂²×C₅₃	semidirect
293. C₄×SD₁₆	C₂²×C₅₃	semidirect
294. C₄×D₈	C₂²×C₅₃	semidirect
295. #436	C₂²×C₅₃	semidirect
296. C₈×D₄	C₂²×C₅₃	semidirect
297. #427	C₂²×C₅₃	semidirect
298. #423	C₂²×C₅₃	semidirect
299. #418	C₂²×C₅₃	semidirect
300. #416	C₂²×C₅₃	semidirect
301. #408	C₂²×C₅₃	semidirect
302. C₂×C₄×C₈	C₂²×C₅₃	semidirect
303. C₅₃	#61089	direct (N × H)
319. #135	D₄×C₅₃	semidirect
378. #41	C₂×D₄×C₅₃	semidirect
407. Q₁₆	C₂×D₄×C₅₃	semidirect
409. SD₁₆	C₂×D₄×C₅₃	semidirect
414. D₈	C₂×D₄×C₅₃	semidirect
419. C₂×C₈	C₂×D₄×C₅₃	semidirect
496. C₄	C₅₃×#576	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#576	2
C₅₃×#3498	2
C₅₃×#3287	2
D₄×D₈×C₅₃	2
C₅₃×#3274	2
C₂×C₅₃×#493	2
C₅₃×#3058	2
C₅₃×#3038	2
C₅₃×#3008	2
C₄×#135×C₅₃	2
C₅₃×#2931	2
#61089	53

Subgroups — C53×#61089 #27268601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61089 #27268601