Subgroups — C₅₃×#26080 #27233592

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3083	C₂	1	2	yes	maximal
3	C₅₃×#2912	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2089	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2061	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1860	C₂	1	2	yes	maximal
9	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1815	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#544	C₂²	1	2	yes	—
14	C₅₃×#520	C₂²	1	2	yes	—
15	C₅₃×#501	—	2	2	no	—
16	C₅₃×#499	—	2	1	no	—
17	C₅₃×#498	—	2	1	no	—
18	C₅₃×#479	—	2	2	no	—
19	C₅₃×#478	—	2	2	no	—
20	C₅₃×#477	—	2	2	no	—
21	C₅₃×#474	—	2	2	no	—
22	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#428	—	2	2	no	—
24	C₅₃×#428	C₂²	1	2	yes	—
25	C₅₃×#428	C₂²	1	2	yes	—
26	C₅₃×#428	C₂²	1	2	yes	—
27	C₅₃×#427	—	2	1	no	—
28	C₅₃×#427	C₂²	1	1	yes	characteristic
29	C₂×#107×C₅₃	C₂²	1	2	yes	—
30	C₂×#106×C₅₃	—	2	1	no	—
31	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#424	C₂²	1	2	yes	—
34	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#417	C₂²	1	2	yes	—
36	C₅₃×#416	C₂²	1	1	yes	characteristic
37	C₅₃×#405	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#382	C₂²	1	2	yes	—
40	C₅₃×#343	C₂²	1	2	yes	—
41	C₅₃×#343	C₂²	1	2	yes	—
42	C₅₃×#341	C₂²	1	1	yes	characteristic
43	C₅₃×#339	C₂²	1	2	yes	—
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
46	#131×C₅₃	—	2	2	no	—
47	#131×C₅₃	Q₈	1	4	yes	complemented
48	C₂×M₁₆×C₅₃	—	2	1	no	—
49	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	2	no	—
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	#125×C₅₃	—	2	2	no	—
58	#125×C₅₃	—	2	2	no	—
59	C₂²⋊Q₈×C₅₃	—	4	2	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	—	2	2	no	—
62	C₄×D₄×C₅₃	—	4	2	no	—
63	#117×C₅₃	—	2	2	no	—
64	#117×C₅₃	C₂³	1	2	yes	—
65	#117×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#108×C₅₃	—	2	2	no	—
71	#107×C₅₃	—	2	2	no	—
72	#107×C₅₃	—	2	2	no	—
73	#107×C₅₃	—	2	2	no	—
74	#107×C₅₃	—	2	2	no	—
75	#107×C₅₃	—	2	2	no	—
76	#107×C₅₃	D₄	1	2	yes	—
77	#106×C₅₃	—	2	1	no	—
78	#106×C₅₃	—	2	1	no	—
79	#106×C₅₃	—	2	1	no	—
80	#106×C₅₃	—	2	1	no	—
81	#106×C₅₃	—	2	2	no	—
82	#106×C₅₃	—	2	2	no	—
83	#106×C₅₃	—	2	2	no	—
84	#106×C₅₃	D₄	1	1	yes	characteristic
85	#106×C₅₃	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	#104×C₅₃	Q₈	1	4	yes	complemented
89	#103×C₅₃	—	2	2	no	—
90	#103×C₅₃	D₄	1	2	yes	—
91	#102×C₅₃	—	2	2	no	—
92	#102×C₅₃	D₄	1	2	yes	—
93	#97×C₅₃	—	2	1	no	—
94	C₄×C₈×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	—	2	2	no	—
96	#95×C₅₃	—	4	2	no	—
97	#41×C₅₃	—	2	1	no	—
98	#41×C₅₃	—	2	1	no	—
99	#41×C₅₃	C₂×Q₈	1	2	yes	—
100	#41×C₅₃	C₂×Q₈	1	2	yes	—
101	C₂×Q₈×C₅₃	—	4	2	no	—
102	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	4	2	no	—
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₅₃	—	2	1	no	—
112	M₁₆×C₅₃	—	2	1	no	—
113	M₁₆×C₅₃	—	2	4	no	—
114	M₁₆×C₅₃	—	2	2	no	—
115	M₁₆×C₅₃	—	2	2	no	—
116	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
117	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	4	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
128	C₂×C₈×C₅₃	#41	1	2	yes	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	4	2	no	—
134	C₄⋊C₄×C₅₃	—	4	2	no	—
135	C₄⋊C₄×C₅₃	—	4	2	no	—
136	C₄⋊C₄×C₅₃	—	4	2	no	—
137	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	4	2	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	—	4	2	no	—
147	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
148	C₂³×C₅₃	—	2	1	no	—
149	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	1	no	—
151	Q₈×C₅₃	—	16	2	no	—
152	Q₈×C₅₃	C₄⋊Q₈	1	2	yes	—
153	D₄×C₅₃	—	2	2	no	—
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	1	no	—
156	D₄×C₅₃	C₄⋊Q₈	1	2	yes	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	8	2	no	—
168	C₂×C₄×C₅₃	—	8	2	no	—
169	C₂×C₄×C₅₃	—	16	2	no	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
176	C₈×C₅₃	—	2	2	no	—
177	C₈×C₅₃	—	2	2	no	—
178	C₈×C₅₃	—	2	2	no	—
179	C₈×C₅₃	—	4	2	no	—
180	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
181	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
182	#26080	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	#544	1	1	yes	characteristic
187	C₂²×C₅₃	D₄²	1	1	yes	characteristic
188	C₂²×C₅₃	#554	1	1	yes	characteristic
189	C₄×C₅₃	—	2	2	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	8	2	no	—
195	C₄×C₅₃	—	8	2	no	—
196	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
197	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
198	C₄×C₅₃	#547	1	1	yes	characteristic
199	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
200	#3163	C₂×C₅₃	1	1	yes	characteristic, complemented
201	#3083	C₂×C₅₃	1	2	yes	—
202	#2912	C₂×C₅₃	1	1	yes	characteristic
203	#2089	C₂×C₅₃	1	2	yes	complemented
204	#2061	C₂×C₅₃	1	2	yes	complemented
205	#1951	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#1945	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#1860	C₂×C₅₃	1	2	yes	—
208	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
209	#1815	C₂×C₅₃	1	1	yes	characteristic
210	#1764	C₂×C₅₃	1	1	yes	characteristic
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	4	2	no	—
214	C₂×C₅₃	#2440	1	1	yes	characteristic
215	C₂×C₅₃	#3290	1	2	yes	—
216	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
217	#544	C₂²×C₅₃	1	2	yes	—
218	#520	C₂²×C₅₃	1	2	yes	—
219	#501	—	2	2	no	—
220	#499	—	2	1	no	—
221	#498	—	2	1	no	—
222	#479	—	2	2	no	—
223	#478	—	2	2	no	—
224	#477	—	2	2	no	—
225	#474	—	2	2	no	—
226	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
227	#428	—	2	2	no	—
228	#428	C₂²×C₅₃	1	2	yes	—
229	#428	C₂²×C₅₃	1	2	yes	—
230	#428	C₂²×C₅₃	1	2	yes	—
231	#427	—	2	1	no	—
232	#427	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×#107	C₂²×C₅₃	1	2	yes	—
234	C₂×#106	—	2	1	no	—
235	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
236	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
237	#424	C₂²×C₅₃	1	2	yes	—
238	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
239	#417	C₂²×C₅₃	1	2	yes	—
240	#416	C₂²×C₅₃	1	1	yes	characteristic
241	#405	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
243	#382	C₂²×C₅₃	1	2	yes	—
244	#343	C₂²×C₅₃	1	2	yes	—
245	#343	C₂²×C₅₃	1	2	yes	—
246	#341	C₂²×C₅₃	1	1	yes	characteristic
247	#339	C₂²×C₅₃	1	1	yes	characteristic
248	#339	C₂²×C₅₃	1	2	yes	—
249	C₅₃	#26080	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
251	#131	—	2	2	no	—
252	#131	Q₈×C₅₃	1	4	yes	complemented
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
255	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
256	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	#125	—	2	2	no	—
263	#125	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	—	4	2	no	—
268	#117	—	2	2	no	—
269	#117	C₂³×C₅₃	1	2	yes	—
270	#117	C₂³×C₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
274	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
275	#108	—	2	2	no	—
276	#107	—	2	2	no	—
277	#107	—	2	2	no	—
278	#107	—	2	2	no	—
279	#107	—	2	2	no	—
280	#107	—	2	2	no	—
281	#107	D₄×C₅₃	1	2	yes	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	1	no	—
285	#106	—	2	1	no	—
286	#106	—	2	2	no	—
287	#106	—	2	2	no	—
288	#106	—	2	2	no	—
289	#106	D₄×C₅₃	1	1	yes	characteristic
290	#106	D₄×C₅₃	1	1	yes	characteristic
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	#104	Q₈×C₅₃	1	4	yes	complemented
294	#103	—	2	2	no	—
295	#103	D₄×C₅₃	1	2	yes	—
296	#102	—	2	2	no	—
297	#102	D₄×C₅₃	1	2	yes	—
298	#97	—	2	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	2	2	no	—
301	#95	—	4	2	no	—
302	#41	—	2	1	no	—
303	#41	—	2	1	no	—
304	#41	C₂×Q₈×C₅₃	1	2	yes	—
305	#41	C₂×Q₈×C₅₃	1	2	yes	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	1	no	—
318	M₁₆	—	2	4	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	C₂×Q₈×C₅₃	1	2	yes	—
322	M₁₆	C₂×Q₈×C₅₃	1	2	yes	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
330	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
333	C₂×C₈	#41×C₅₃	1	2	yes	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
343	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
344	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	4	2	no	—
352	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
353	C₂³	—	2	1	no	—
354	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
355	Q₈	—	2	1	no	—
356	Q₈	—	16	2	no	—
357	Q₈	C₄⋊Q₈×C₅₃	1	2	yes	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	2	1	no	—
361	D₄	C₄⋊Q₈×C₅₃	1	2	yes	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	8	2	no	—
373	C₂×C₄	—	8	2	no	—
374	C₂×C₄	—	16	2	no	—
375	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
380	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	—	2	2	no	—
384	C₈	—	4	2	no	—
385	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
386	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	C₅₃×#544	1	1	yes	characteristic
391	C₂²	D₄²×C₅₃	1	1	yes	characteristic
392	C₂²	C₅₃×#554	1	1	yes	characteristic
393	C₄	—	2	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
401	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
402	C₄	C₅₃×#547	1	1	yes	characteristic
403	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	4	2	no	—
407	C₂	C₅₃×#2440	1	1	yes	characteristic
408	C₂	C₅₃×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3163	C₂	semidirect
4. C₅₃×#2089	C₂	semidirect
5. C₅₃×#2061	C₂	semidirect
6. C₅₃×#1951	C₂	semidirect
7. C₅₃×#1945	C₂	semidirect
9. C₅₃×#1846	C₂	semidirect
47. #131×C₅₃	Q₈	semidirect
88. #104×C₅₃	Q₈	semidirect
182. #26080	C₅₃	direct (N × H)
200. #3163	C₂×C₅₃	semidirect
203. #2089	C₂×C₅₃	semidirect
204. #2061	C₂×C₅₃	semidirect
205. #1951	C₂×C₅₃	semidirect
206. #1945	C₂×C₅₃	semidirect
208. #1846	C₂×C₅₃	semidirect
249. C₅₃	#26080	direct (N × H)
252. #131	Q₈×C₅₃	semidirect
293. #104	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3163	2
C₅₃×#3083	2
C₅₃×#2912	2
C₅₃×#2089	2
C₅₃×#2061	2
C₅₃×#1951	2
C₅₃×#1945	2
C₅₃×#1860	2
C₅₃×#1846	2
C₅₃×#1815	2
C₅₃×#1764	2
#26080	53

Subgroups — C53×#26080 #27233592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26080 #27233592