Subgroups — C₅₃×#36595 #27244107

Order 13568.

2706 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#559	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2867	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2858	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2548	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2548	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2544	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2445	C₂	1	8	yes	maximal, complemented
8	C₅₃×#2366	C₂	1	4	yes	maximal, complemented
9	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#127×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#559	—	2	16	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#127×C₅₃	C₂²	1	2	yes	—
15	C₂×#127×C₅₃	C₂²	1	4	yes	—
16	C₂×#127×C₅₃	C₂²	1	2	yes	—
17	C₂×#127×C₅₃	C₂²	1	4	yes	—
18	C₂×#127×C₅₃	C₂²	1	2	yes	—
19	C₂×#124×C₅₃	C₂²	1	4	yes	—
20	C₂×#124×C₅₃	C₂²	1	4	yes	—
21	C₂×#124×C₅₃	C₂²	1	4	yes	—
22	C₂×#123×C₅₃	C₂²	1	8	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
29	C₅₃×#394	C₂²	1	8	yes	—
30	C₅₃×#394	C₂²	1	16	yes	—
31	C₅₃×#391	C₂²	1	2	yes	—
32	C₅₃×#390	C₂²	1	8	yes	—
33	C₅₃×#390	C₂²	1	8	yes	—
34	C₅₃×#390	C₂²	1	8	yes	—
35	C₅₃×#390	C₂²	1	4	yes	—
36	C₅₃×#390	C₂²	1	8	yes	—
37	C₅₃×#389	C₂²	1	4	yes	—
38	C₅₃×#386	C₂²	1	16	yes	—
39	C₅₃×#384	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
45	C₄³×C₅₃	C₂²	1	2	yes	—
46	C₄³×C₅₃	C₂²	1	1	yes	characteristic
47	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
50	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
51	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
52	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
55	C₄⋊Q₈×C₅₃	—	2	4	no	—
56	#127×C₅₃	—	2	8	no	—
57	#127×C₅₃	—	2	8	no	—
58	#127×C₅₃	—	2	16	no	—
59	#127×C₅₃	—	4	8	no	—
60	#127×C₅₃	—	4	4	no	—
61	#124×C₅₃	—	2	16	no	—
62	#124×C₅₃	—	2	16	no	—
63	#124×C₅₃	—	4	8	no	—
64	#123×C₅₃	—	2	32	no	—
65	C₄×Q₈×C₅₃	—	2	8	no	—
66	C₄×Q₈×C₅₃	—	2	8	no	—
67	C₄×Q₈×C₅₃	D₄	1	8	yes	complemented
68	C₄×D₄×C₅₃	—	2	16	no	—
69	C₄×D₄×C₅₃	—	2	16	no	—
70	C₄×D₄×C₅₃	—	4	16	no	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	#95×C₅₃	C₂³	1	4	yes	—
95	#95×C₅₃	C₂³	1	2	yes	—
96	#95×C₅₃	C₂³	1	8	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
98	C₂×Q₈×C₅₃	—	2	4	no	—
99	C₂×Q₈×C₅₃	—	2	4	no	—
100	C₂×Q₈×C₅₃	—	2	4	no	—
101	C₂×Q₈×C₅₃	—	2	4	no	—
102	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
103	C₂×D₄×C₅₃	—	2	8	no	—
104	C₂×D₄×C₅₃	—	2	8	no	—
105	C₂×D₄×C₅₃	—	2	8	no	—
106	C₂×D₄×C₅₃	—	2	16	no	—
107	C₂×D₄×C₅₃	—	2	8	no	—
108	C₂×D₄×C₅₃	—	2	16	no	—
109	C₂×D₄×C₅₃	—	2	16	no	—
110	C₂×D₄×C₅₃	—	2	16	no	—
111	C₂×D₄×C₅₃	—	2	16	no	—
112	C₂×D₄×C₅₃	—	2	8	no	—
113	C₂×D₄×C₅₃	—	4	8	no	—
114	C₂×D₄×C₅₃	—	4	8	no	—
115	C₂×D₄×C₅₃	—	4	8	no	—
116	C₂²×C₄×C₅₃	—	2	16	no	—
117	C₂²×C₄×C₅₃	—	4	16	no	—
118	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	2	8	no	—
130	C₄⋊C₄×C₅₃	—	2	8	no	—
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	2	8	no	—
133	C₄⋊C₄×C₅₃	—	2	8	no	—
134	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
135	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
136	C₂²⋊C₄×C₅₃	—	2	16	no	—
137	C₂²⋊C₄×C₅₃	—	2	16	no	—
138	C₂²⋊C₄×C₅₃	—	2	16	no	—
139	C₂²⋊C₄×C₅₃	—	4	16	no	—
140	C₂²⋊C₄×C₅₃	—	4	16	no	—
141	C₄²×C₅₃	—	2	8	no	—
142	C₄²×C₅₃	—	2	8	no	—
143	C₄²×C₅₃	—	2	8	no	—
144	C₄²×C₅₃	—	2	4	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	4	no	—
147	C₄²×C₅₃	C₂×D₄	1	8	yes	—
148	C₄²×C₅₃	C₂×D₄	1	4	yes	—
149	C₄²×C₅₃	#41	1	8	yes	—
150	C₄²×C₅₃	#41	1	4	yes	—
151	C₂³×C₅₃	—	2	8	no	—
152	C₂³×C₅₃	—	2	16	no	—
153	C₂³×C₅₃	—	2	8	no	—
154	C₂³×C₅₃	—	2	16	no	—
155	C₂³×C₅₃	—	2	8	no	—
156	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
157	Q₈×C₅₃	—	2	8	no	—
158	Q₈×C₅₃	—	4	8	no	—
159	D₄×C₅₃	—	4	16	no	—
160	D₄×C₅₃	—	4	16	no	—
161	D₄×C₅₃	—	4	32	no	—
162	D₄×C₅₃	—	8	8	no	—
163	D₄×C₅₃	—	8	16	no	—
164	C₂×C₄×C₅₃	—	2	4	no	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	2	8	no	—
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	4	no	—
177	C₂×C₄×C₅₃	—	8	16	no	—
178	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
181	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
184	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
185	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
187	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
188	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
189	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
190	#36595	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂²×C₅₃	—	4	8	no	—
192	C₂²×C₅₃	—	4	8	no	—
193	C₂²×C₅₃	—	4	16	no	—
194	C₂²×C₅₃	—	4	8	no	—
195	C₂²×C₅₃	—	4	16	no	—
196	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
197	C₂²×C₅₃	C₂²×#41	1	2	yes	—
198	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
199	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
200	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
201	C₄×C₅₃	—	2	4	no	—
202	C₄×C₅₃	—	2	4	no	—
203	C₄×C₅₃	—	2	2	no	—
204	C₄×C₅₃	—	2	4	no	—
205	C₄×C₅₃	—	4	4	no	—
206	C₄×C₅₃	—	4	4	no	—
207	C₄×C₅₃	—	4	4	no	—
208	C₄×C₅₃	#532	1	4	yes	—
209	C₄×C₅₃	#550	1	4	yes	—
210	C₄×C₅₃	#559	1	8	yes	complemented
211	C₂×#559	C₂×C₅₃	1	2	yes	complemented
212	#2867	C₂×C₅₃	1	1	yes	characteristic
213	#2858	C₂×C₅₃	1	2	yes	complemented
214	#2548	C₂×C₅₃	1	2	yes	complemented
215	#2548	C₂×C₅₃	1	4	yes	complemented
216	#2544	C₂×C₅₃	1	4	yes	complemented
217	#2445	C₂×C₅₃	1	8	yes	complemented
218	#2366	C₂×C₅₃	1	4	yes	complemented
219	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₄×#127	C₂×C₅₃	1	2	yes	complemented
221	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×C₅₃	—	8	8	no	—
223	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
224	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#559	1	2	yes	—
226	C₂×C₅₃	#3498	1	1	yes	characteristic
227	C₂×C₅₃	#3508	1	2	yes	—
228	#559	—	2	16	no	—
229	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×#127	C₂²×C₅₃	1	2	yes	—
231	C₂×#127	C₂²×C₅₃	1	4	yes	—
232	C₂×#127	C₂²×C₅₃	1	2	yes	—
233	C₂×#127	C₂²×C₅₃	1	4	yes	—
234	C₂×#127	C₂²×C₅₃	1	2	yes	—
235	C₂×#124	C₂²×C₅₃	1	4	yes	—
236	C₂×#124	C₂²×C₅₃	1	4	yes	—
237	C₂×#124	C₂²×C₅₃	1	4	yes	—
238	C₂×#123	C₂²×C₅₃	1	8	yes	—
239	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
240	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
241	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
243	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
244	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
245	#394	C₂²×C₅₃	1	8	yes	—
246	#394	C₂²×C₅₃	1	16	yes	—
247	#391	C₂²×C₅₃	1	2	yes	—
248	#390	C₂²×C₅₃	1	8	yes	—
249	#390	C₂²×C₅₃	1	4	yes	—
250	#390	C₂²×C₅₃	1	8	yes	—
251	#390	C₂²×C₅₃	1	8	yes	—
252	#390	C₂²×C₅₃	1	8	yes	—
253	#389	C₂²×C₅₃	1	4	yes	—
254	#386	C₂²×C₅₃	1	16	yes	—
255	#384	C₂²×C₅₃	1	1	yes	characteristic
256	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
257	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
258	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
259	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
260	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
261	C₄³	C₂²×C₅₃	1	2	yes	—
262	C₄³	C₂²×C₅₃	1	1	yes	characteristic
263	C₅₃	#36595	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
265	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
266	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
267	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
268	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
269	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
270	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
271	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
272	C₄⋊Q₈	—	2	4	no	—
273	#127	—	2	8	no	—
274	#127	—	2	16	no	—
275	#127	—	2	8	no	—
276	#127	—	4	8	no	—
277	#127	—	4	4	no	—
278	#124	—	2	16	no	—
279	#124	—	2	16	no	—
280	#124	—	4	8	no	—
281	#123	—	2	32	no	—
282	C₄×Q₈	—	2	8	no	—
283	C₄×Q₈	—	2	8	no	—
284	C₄×Q₈	D₄×C₅₃	1	8	yes	complemented
285	C₄×D₄	—	2	16	no	—
286	C₄×D₄	—	2	16	no	—
287	C₄×D₄	—	4	16	no	—
288	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
300	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
301	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
302	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
303	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
304	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
305	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
306	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
308	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
310	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
311	#95	C₂³×C₅₃	1	4	yes	—
312	#95	C₂³×C₅₃	1	2	yes	—
313	#95	C₂³×C₅₃	1	8	yes	—
314	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	—	2	4	no	—
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	2	8	no	—
322	C₂×D₄	—	2	8	no	—
323	C₂×D₄	—	2	16	no	—
324	C₂×D₄	—	2	8	no	—
325	C₂×D₄	—	2	16	no	—
326	C₂×D₄	—	2	16	no	—
327	C₂×D₄	—	2	16	no	—
328	C₂×D₄	—	2	16	no	—
329	C₂×D₄	—	2	8	no	—
330	C₂×D₄	—	4	8	no	—
331	C₂×D₄	—	4	8	no	—
332	C₂×D₄	—	4	8	no	—
333	C₂²×C₄	—	2	16	no	—
334	C₂²×C₄	—	4	16	no	—
335	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
337	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
338	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
339	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
342	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	8	no	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	8	no	—
350	C₄⋊C₄	—	2	8	no	—
351	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
352	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
353	C₂²⋊C₄	—	2	16	no	—
354	C₂²⋊C₄	—	2	16	no	—
355	C₂²⋊C₄	—	2	16	no	—
356	C₂²⋊C₄	—	4	16	no	—
357	C₂²⋊C₄	—	4	16	no	—
358	C₄²	—	2	8	no	—
359	C₄²	—	2	8	no	—
360	C₄²	—	2	8	no	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	4	no	—
364	C₄²	C₂×D₄×C₅₃	1	8	yes	—
365	C₄²	C₂×D₄×C₅₃	1	4	yes	—
366	C₄²	#41×C₅₃	1	8	yes	—
367	C₄²	#41×C₅₃	1	4	yes	—
368	C₂³	—	2	8	no	—
369	C₂³	—	2	8	no	—
370	C₂³	—	2	16	no	—
371	C₂³	—	2	8	no	—
372	C₂³	—	2	16	no	—
373	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
374	Q₈	—	2	8	no	—
375	Q₈	—	4	8	no	—
376	D₄	—	4	16	no	—
377	D₄	—	4	16	no	—
378	D₄	—	4	32	no	—
379	D₄	—	8	8	no	—
380	D₄	—	8	16	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	8	16	no	—
395	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
396	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
397	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
399	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
400	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
401	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
402	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
403	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
404	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
405	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
406	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
407	C₂²	—	4	8	no	—
408	C₂²	—	4	16	no	—
409	C₂²	—	4	8	no	—
410	C₂²	—	4	16	no	—
411	C₂²	—	4	8	no	—
412	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
413	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
414	C₂²	C₂²×#41×C₅₃	1	2	yes	—
415	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
416	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
417	C₄	—	2	4	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	4	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	4	no	—
424	C₄	C₅₃×#532	1	4	yes	—
425	C₄	C₅₃×#550	1	4	yes	—
426	C₄	C₅₃×#559	1	8	yes	complemented
427	C₂	—	8	8	no	—
428	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
429	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
430	C₂	C₂×C₅₃×#559	1	2	yes	—
431	C₂	C₅₃×#3498	1	1	yes	characteristic
432	C₂	C₅₃×#3508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₅₃×#2867	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#559	C₂	semidirect
3. C₅₃×#2858	C₂	semidirect
4. C₅₃×#2548	C₂	semidirect
5. C₅₃×#2548	C₂	semidirect
6. C₅₃×#2544	C₂	semidirect
7. C₅₃×#2445	C₂	semidirect
8. C₅₃×#2366	C₂	semidirect
9. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
10. C₄×#127×C₅₃	C₂	semidirect
11. C₄²×Q₈×C₅₃	C₂	semidirect
67. C₄×Q₈×C₅₃	D₄	semidirect
190. #36595	C₅₃	direct (N × H)
210. C₄×C₅₃	#559	semidirect
211. C₂×#559	C₂×C₅₃	semidirect
213. #2858	C₂×C₅₃	semidirect
214. #2548	C₂×C₅₃	semidirect
215. #2548	C₂×C₅₃	semidirect
216. #2544	C₂×C₅₃	semidirect
217. #2445	C₂×C₅₃	semidirect
218. #2366	C₂×C₅₃	semidirect
219. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
220. C₄×#127	C₂×C₅₃	semidirect
221. C₄²×Q₈	C₂×C₅₃	semidirect
263. C₅₃	#36595	direct (N × H)
284. C₄×Q₈	D₄×C₅₃	semidirect
426. C₄	C₅₃×#559	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#559	2
C₅₃×#2867	2
C₅₃×#2858	2
C₅₃×#2548	2
C₅₃×#2544	2
C₅₃×#2445	2
C₅₃×#2366	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#127×C₅₃	2
C₄²×Q₈×C₅₃	2
#36595	53

Subgroups — C53×#36595 #27244107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36595 #27244107