Subgroups — C₅₃×#34086 #27241598

Order 13568.

2638 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2337	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2333	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2331	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2328	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2324	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2297	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2293	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2290	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
12	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#532	C₄	1	16	yes	complemented
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#124×C₅₃	C₂²	1	2	yes	—
18	C₂×#121×C₅₃	C₂²	1	2	yes	—
19	C₂×#121×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂×#117×C₅₃	C₂²	1	2	yes	—
24	C₂×#117×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#391	C₂²	1	8	yes	—
28	C₅₃×#390	C₂²	1	4	yes	—
29	C₅₃×#390	C₂²	1	8	yes	—
30	C₅₃×#389	C₂²	1	4	yes	—
31	C₅₃×#388	C₂²	1	8	yes	—
32	C₅₃×#388	C₂²	1	8	yes	—
33	C₅₃×#387	C₂²	1	8	yes	—
34	C₅₃×#386	C₂²	1	8	yes	—
35	C₅₃×#386	C₂²	1	8	yes	—
36	C₅₃×#386	C₂²	1	8	yes	—
37	C₅₃×#385	C₂²	1	8	yes	—
38	C₅₃×#383	C₂²	1	2	yes	—
39	C₅₃×#382	C₂²	1	2	yes	—
40	C₅₃×#381	C₂²	1	2	yes	—
41	C₅₃×#380	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
46	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
47	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
52	C₅₃×#376	C₂²	1	2	yes	—
53	C₅₃×#376	C₂²	1	2	yes	—
54	C₄³×C₅₃	C₂²	1	1	yes	characteristic
55	C₄³×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#41×C₅₃	—	4	8	no	—
57	C₂×#41×C₅₃	C₂×C₄	1	16	yes	complemented
58	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
62	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
69	#127×C₅₃	C₂×C₄	1	8	yes	—
70	#124×C₅₃	C₂×C₄	1	16	yes	complemented
71	#121×C₅₃	C₂×C₄	1	16	yes	complemented
72	#121×C₅₃	C₂×C₄	1	16	yes	complemented
73	C₄×D₄×C₅₃	—	2	16	no	—
74	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
75	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
76	#117×C₅₃	—	2	8	no	—
77	#117×C₅₃	—	4	8	no	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₄²×C₅₃	—	4	8	no	—
91	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
92	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
99	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	4	yes	—
106	#95×C₅₃	C₂³	1	4	yes	—
107	#95×C₅₃	C₂³	1	4	yes	—
108	#95×C₅₃	C₂³	1	4	yes	—
109	#95×C₅₃	C₂³	1	8	yes	—
110	#95×C₅₃	C₂³	1	4	yes	—
111	#95×C₅₃	C₂³	1	8	yes	—
112	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
113	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
114	#41×C₅₃	—	4	32	no	—
115	C₂×Q₈×C₅₃	—	4	4	no	—
116	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
117	C₂×D₄×C₅₃	—	2	8	no	—
118	C₂×D₄×C₅₃	—	4	4	no	—
119	C₂×D₄×C₅₃	—	4	4	no	—
120	C₂×D₄×C₅₃	—	4	4	no	—
121	C₂×D₄×C₅₃	—	4	4	no	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
123	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
125	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	8	no	—
128	C₂²×C₄×C₅₃	—	2	8	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	16	no	—
131	C₂²×C₄×C₅₃	—	4	8	no	—
132	C₂²×C₄×C₅₃	—	4	8	no	—
133	C₂²×C₄×C₅₃	—	4	2	no	—
134	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
137	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₄⋊C₄×C₅₃	—	2	8	no	—
149	C₄⋊C₄×C₅₃	—	2	8	no	—
150	C₄⋊C₄×C₅₃	—	4	8	no	—
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
152	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
153	C₂²⋊C₄×C₅₃	—	2	8	no	—
154	C₂²⋊C₄×C₅₃	—	2	16	no	—
155	C₂²⋊C₄×C₅₃	—	4	4	no	—
156	C₂²⋊C₄×C₅₃	—	4	8	no	—
157	C₂²⋊C₄×C₅₃	—	4	8	no	—
158	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
159	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
160	C₄²×C₅₃	—	2	4	no	—
161	C₄²×C₅₃	—	2	4	no	—
162	C₄²×C₅₃	—	2	8	no	—
163	C₄²×C₅₃	—	2	4	no	—
164	C₄²×C₅₃	—	4	8	no	—
165	C₄²×C₅₃	—	4	2	no	—
166	C₄²×C₅₃	—	4	2	no	—
167	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
168	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
169	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
170	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
171	C₄²×C₅₃	#41	1	8	yes	complemented
172	C₂³×C₅₃	—	2	8	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	2	4	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	2	4	no	—
177	C₂³×C₅₃	—	2	8	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
180	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
181	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
182	Q₈×C₅₃	—	4	8	no	—
183	D₄×C₅₃	—	2	16	no	—
184	D₄×C₅₃	—	4	8	no	—
185	D₄×C₅₃	—	4	8	no	—
186	D₄×C₅₃	—	4	16	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	4	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	4	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	8	no	—
196	C₂×C₄×C₅₃	—	2	32	no	—
197	C₂×C₄×C₅₃	—	2	8	no	—
198	C₂×C₄×C₅₃	—	2	8	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	8	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	8	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	4	4	no	—
205	C₂×C₄×C₅₃	—	4	32	no	—
206	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
207	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
209	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
211	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
214	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
215	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
216	#34086	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	8	no	—
219	C₂²×C₅₃	—	2	2	no	—
220	C₂²×C₅₃	—	2	8	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	2	8	no	—
223	C₂²×C₅₃	—	4	8	no	—
224	C₂²×C₅₃	—	4	8	no	—
225	C₂²×C₅₃	—	4	4	no	—
226	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
227	C₂²×C₅₃	C₂²×#41	1	2	yes	—
228	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
229	C₂²×C₅₃	#585	1	2	yes	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	8	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	2	2	no	—
235	C₄×C₅₃	—	2	8	no	—
236	C₄×C₅₃	—	4	4	no	—
237	C₄×C₅₃	—	4	8	no	—
238	C₄×C₅₃	C₄×#41	1	4	yes	—
239	C₄×C₅₃	#518	1	4	yes	—
240	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
241	#2337	C₂×C₅₃	1	4	yes	complemented
242	#2333	C₂×C₅₃	1	4	yes	complemented
243	#2331	C₂×C₅₃	1	2	yes	complemented
244	#2328	C₂×C₅₃	1	2	yes	complemented
245	#2324	C₂×C₅₃	1	4	yes	complemented
246	#2297	C₂×C₅₃	1	4	yes	complemented
247	#2293	C₂×C₅₃	1	2	yes	complemented
248	#2290	C₂×C₅₃	1	4	yes	complemented
249	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
250	C₄²×D₄	C₂×C₅₃	1	2	yes	complemented
251	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	—	4	8	no	—
254	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
255	C₂×C₅₃	C₂×#518	1	1	yes	characteristic
256	C₂×C₅₃	#3428	1	1	yes	characteristic
257	C₂×C₅₃	#3460	1	4	yes	—
258	C₂²×#41	C₂²×C₅₃	1	2	yes	—
259	#532	C₄×C₅₃	1	16	yes	complemented
260	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#124	C₂²×C₅₃	1	2	yes	—
263	C₂×#121	C₂²×C₅₃	1	2	yes	—
264	C₂×#121	C₂²×C₅₃	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
266	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
267	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
268	C₂×#117	C₂²×C₅₃	1	2	yes	—
269	C₂×#117	C₂²×C₅₃	1	4	yes	—
270	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
271	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
272	#391	C₂²×C₅₃	1	8	yes	—
273	#390	C₂²×C₅₃	1	4	yes	—
274	#390	C₂²×C₅₃	1	8	yes	—
275	#389	C₂²×C₅₃	1	4	yes	—
276	#388	C₂²×C₅₃	1	8	yes	—
277	#388	C₂²×C₅₃	1	8	yes	—
278	#387	C₂²×C₅₃	1	8	yes	—
279	#386	C₂²×C₅₃	1	8	yes	—
280	#386	C₂²×C₅₃	1	8	yes	—
281	#386	C₂²×C₅₃	1	8	yes	—
282	#385	C₂²×C₅₃	1	8	yes	—
283	#383	C₂²×C₅₃	1	2	yes	—
284	#382	C₂²×C₅₃	1	2	yes	—
285	#381	C₂²×C₅₃	1	2	yes	—
286	#380	C₂²×C₅₃	1	2	yes	—
287	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
288	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
289	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
290	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
291	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
292	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
293	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
294	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
295	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
296	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
297	#376	C₂²×C₅₃	1	2	yes	—
298	#376	C₂²×C₅₃	1	2	yes	—
299	C₄³	C₂²×C₅₃	1	1	yes	characteristic
300	C₄³	C₂²×C₅₃	1	1	yes	characteristic
301	C₅₃	#34086	1	1	yes	characteristic, direct product factor, Sylow
302	C₂×#41	—	4	8	no	—
303	C₂×#41	C₂×C₄×C₅₃	1	16	yes	complemented
304	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
305	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
306	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
307	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
308	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
309	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
311	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
313	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
314	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
315	#127	C₂×C₄×C₅₃	1	8	yes	—
316	#124	C₂×C₄×C₅₃	1	16	yes	complemented
317	#121	C₂×C₄×C₅₃	1	16	yes	complemented
318	#121	C₂×C₄×C₅₃	1	16	yes	complemented
319	C₄×D₄	—	2	16	no	—
320	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
321	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
322	#117	—	2	8	no	—
323	#117	—	4	8	no	—
324	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
327	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
336	C₂×C₄²	—	4	8	no	—
337	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
338	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
339	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
340	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
341	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
342	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
344	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
345	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
350	#95	C₂³×C₅₃	1	4	yes	—
351	#95	C₂³×C₅₃	1	4	yes	—
352	#95	C₂³×C₅₃	1	4	yes	—
353	#95	C₂³×C₅₃	1	4	yes	—
354	#95	C₂³×C₅₃	1	4	yes	—
355	#95	C₂³×C₅₃	1	8	yes	—
356	#95	C₂³×C₅₃	1	4	yes	—
357	#95	C₂³×C₅₃	1	8	yes	—
358	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
359	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
360	#41	—	4	32	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	—	4	4	no	—
367	C₂×D₄	—	4	4	no	—
368	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
369	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
370	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
371	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	2	8	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	16	no	—
377	C₂²×C₄	—	4	8	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	8	no	—
380	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
381	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
382	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
383	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
385	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
394	C₄⋊C₄	—	2	8	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	4	8	no	—
397	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
398	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	16	no	—
401	C₂²⋊C₄	—	4	8	no	—
402	C₂²⋊C₄	—	4	8	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
405	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	4	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	2	4	no	—
410	C₄²	—	4	8	no	—
411	C₄²	—	4	2	no	—
412	C₄²	—	4	2	no	—
413	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
414	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
415	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
416	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
417	C₄²	#41×C₅₃	1	8	yes	complemented
418	C₂³	—	2	2	no	—
419	C₂³	—	2	8	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	8	no	—
425	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
426	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
427	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
428	Q₈	—	4	8	no	—
429	D₄	—	2	16	no	—
430	D₄	—	4	8	no	—
431	D₄	—	4	8	no	—
432	D₄	—	4	16	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	32	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	32	no	—
452	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
453	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
454	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
455	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
457	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
459	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
460	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
461	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	8	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	2	8	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	8	no	—
471	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
472	C₂²	C₂²×#41×C₅₃	1	2	yes	—
473	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
474	C₂²	C₅₃×#585	1	2	yes	—
475	C₄	—	2	8	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	8	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	8	no	—
482	C₄	—	4	4	no	—
483	C₄	C₄×#41×C₅₃	1	4	yes	—
484	C₄	C₅₃×#518	1	4	yes	—
485	C₂	—	4	2	no	—
486	C₂	—	4	8	no	—
487	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
488	C₂	C₂×C₅₃×#518	1	1	yes	characteristic
489	C₂	C₅₃×#3428	1	1	yes	characteristic
490	C₂	C₅₃×#3460	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×C₅₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2337	C₂	semidirect
3. C₅₃×#2333	C₂	semidirect
4. C₅₃×#2331	C₂	semidirect
5. C₅₃×#2328	C₂	semidirect
6. C₅₃×#2324	C₂	semidirect
7. C₅₃×#2297	C₂	semidirect
8. C₅₃×#2293	C₂	semidirect
9. C₅₃×#2290	C₂	semidirect
10. C₅₃×#2283	C₂	semidirect
11. C₄²×D₄×C₅₃	C₂	semidirect
12. C₄×#117×C₅₃	C₂	semidirect
14. C₅₃×#532	C₄	semidirect
57. C₂×#41×C₅₃	C₂×C₄	semidirect
68. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
70. #124×C₅₃	C₂×C₄	semidirect
71. #121×C₅₃	C₂×C₄	semidirect
72. #121×C₅₃	C₂×C₄	semidirect
74. C₄×D₄×C₅₃	C₂×C₄	semidirect
75. C₄×D₄×C₅₃	C₂×C₄	semidirect
91. C₂×C₄²×C₅₃	C₂×C₄	semidirect
171. C₄²×C₅₃	#41	semidirect
216. #34086	C₅₃	direct (N × H)
241. #2337	C₂×C₅₃	semidirect
242. #2333	C₂×C₅₃	semidirect
243. #2331	C₂×C₅₃	semidirect
244. #2328	C₂×C₅₃	semidirect
245. #2324	C₂×C₅₃	semidirect
246. #2297	C₂×C₅₃	semidirect
247. #2293	C₂×C₅₃	semidirect
248. #2290	C₂×C₅₃	semidirect
249. #2283	C₂×C₅₃	semidirect
250. C₄²×D₄	C₂×C₅₃	semidirect
251. C₄×#117	C₂×C₅₃	semidirect
259. #532	C₄×C₅₃	semidirect
301. C₅₃	#34086	direct (N × H)
303. C₂×#41	C₂×C₄×C₅₃	semidirect
314. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
316. #124	C₂×C₄×C₅₃	semidirect
317. #121	C₂×C₄×C₅₃	semidirect
318. #121	C₂×C₄×C₅₃	semidirect
320. C₄×D₄	C₂×C₄×C₅₃	semidirect
321. C₄×D₄	C₂×C₄×C₅₃	semidirect
337. C₂×C₄²	C₂×C₄×C₅₃	semidirect
417. C₄²	#41×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×#532	2
C₅₃×#2337	2
C₅₃×#2333	2
C₅₃×#2331	2
C₅₃×#2328	2
C₅₃×#2324	2
C₅₃×#2297	2
C₅₃×#2293	2
C₅₃×#2290	2
C₅₃×#2283	2
C₄²×D₄×C₅₃	2
C₄×#117×C₅₃	2
#34086	53

Subgroups — C53×#34086 #27241598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34086 #27241598