Subgroups — C₅₃×#34402 #27241914

Order 13568.

2410 conjugacy classes of subgroups.

The subgroup lattice has 476 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×#2377	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2367	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2353	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2341	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2339	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2333	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2328	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2327	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2318	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2283	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×Q₈×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#391	C₂²	1	4	yes	—
26	C₅₃×#390	C₂²	1	4	yes	—
27	C₅₃×#390	C₂²	1	8	yes	—
28	C₅₃×#388	C₂²	1	8	yes	—
29	C₅₃×#388	C₂²	1	8	yes	—
30	C₅₃×#387	C₂²	1	8	yes	—
31	C₅₃×#387	C₂²	1	8	yes	—
32	C₅₃×#386	C₂²	1	8	yes	—
33	C₅₃×#386	C₂²	1	8	yes	—
34	C₅₃×#386	C₂²	1	8	yes	—
35	C₅₃×#385	C₂²	1	8	yes	—
36	C₅₃×#383	C₂²	1	2	yes	—
37	C₅₃×#383	C₂²	1	2	yes	—
38	C₅₃×#382	C₂²	1	2	yes	—
39	C₅₃×#382	C₂²	1	2	yes	—
40	C₅₃×#381	C₂²	1	2	yes	—
41	C₅₃×#381	C₂²	1	2	yes	—
42	C₅₃×#381	C₂²	1	4	yes	—
43	C₅₃×#380	C₂²	1	2	yes	—
44	C₅₃×#380	C₂²	1	2	yes	—
45	C₅₃×#380	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
51	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
53	C₅₃×#376	C₂²	1	2	yes	—
54	C₅₃×#376	C₂²	1	2	yes	—
55	C₂×#95×C₅₃	C₂²	1	4	yes	—
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	2	yes	—
62	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
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×Q₈×C₅₃	—	4	8	no	—
74	C₄×D₄×C₅₃	—	4	8	no	—
75	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
76	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	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	C₂×C₄	1	8	yes	complemented
91	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	#95×C₅₃	C₂³	1	2	yes	—
100	#95×C₅₃	C₂³	1	4	yes	—
101	#95×C₅₃	C₂³	1	8	yes	—
102	#95×C₅₃	C₂³	1	4	yes	—
103	#95×C₅₃	C₂³	1	4	yes	—
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	2	yes	—
106	#95×C₅₃	C₂³	1	4	yes	—
107	#95×C₅₃	C₂³	1	8	yes	—
108	#95×C₅₃	C₂³	1	4	yes	—
109	#95×C₅₃	C₂³	1	4	yes	—
110	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
112	#41×C₅₃	—	4	32	no	—
113	C₂×Q₈×C₅₃	—	2	8	no	—
114	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	4	4	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	4	8	no	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
120	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	4	2	no	—
125	C₂²×C₄×C₅₃	—	4	2	no	—
126	C₂²×C₄×C₅₃	—	4	8	no	—
127	C₂²×C₄×C₅₃	—	4	8	no	—
128	C₂²×C₄×C₅₃	—	4	8	no	—
129	C₂²×C₄×C₅₃	—	4	8	no	—
130	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₅₃	—	2	8	no	—
145	C₄⋊C₄×C₅₃	—	4	4	no	—
146	C₄⋊C₄×C₅₃	—	4	8	no	—
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
148	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
149	C₂²⋊C₄×C₅₃	—	4	2	no	—
150	C₂²⋊C₄×C₅₃	—	4	8	no	—
151	C₂²⋊C₄×C₅₃	—	4	4	no	—
152	C₂²⋊C₄×C₅₃	—	4	8	no	—
153	C₂²⋊C₄×C₅₃	—	4	2	no	—
154	C₂²⋊C₄×C₅₃	—	4	8	no	—
155	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
157	C₄²×C₅₃	—	2	4	no	—
158	C₄²×C₅₃	—	2	4	no	—
159	C₄²×C₅₃	—	4	4	no	—
160	C₄²×C₅₃	—	4	8	no	—
161	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
162	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
163	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
164	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	2	8	no	—
167	C₂³×C₅₃	—	2	4	no	—
168	C₂³×C₅₃	—	2	8	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	4	no	—
172	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
173	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
174	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
175	Q₈×C₅₃	—	2	16	no	—
176	D₄×C₅₃	—	2	16	no	—
177	D₄×C₅₃	—	4	8	no	—
178	D₄×C₅₃	—	4	8	no	—
179	D₄×C₅₃	—	4	16	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	8	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	8	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	8	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	8	no	—
193	C₂×C₄×C₅₃	—	2	8	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	4	4	no	—
197	C₂×C₄×C₅₃	—	4	4	no	—
198	C₂×C₄×C₅₃	—	4	16	no	—
199	C₂×C₄×C₅₃	—	8	16	no	—
200	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
201	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
202	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
204	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
207	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
208	#34402	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂²×C₅₃	—	2	2	no	—
210	C₂²×C₅₃	—	2	8	no	—
211	C₂²×C₅₃	—	2	2	no	—
212	C₂²×C₅₃	—	2	2	no	—
213	C₂²×C₅₃	—	2	8	no	—
214	C₂²×C₅₃	—	2	8	no	—
215	C₂²×C₅₃	—	4	8	no	—
216	C₂²×C₅₃	—	4	4	no	—
217	C₂²×C₅₃	—	4	8	no	—
218	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
219	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
220	C₂²×C₅₃	#585	1	2	yes	—
221	C₂²×C₅₃	#585	1	2	yes	—
222	C₄×C₅₃	—	2	2	no	—
223	C₄×C₅₃	—	2	8	no	—
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	2	no	—
226	C₄×C₅₃	—	2	2	no	—
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	4	4	no	—
229	C₄×C₅₃	—	4	8	no	—
230	C₄×C₅₃	—	4	4	no	—
231	C₄×C₅₃	#518	1	4	yes	—
232	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
233	#2377	C₂×C₅₃	1	4	yes	complemented
234	#2367	C₂×C₅₃	1	4	yes	complemented
235	#2353	C₂×C₅₃	1	4	yes	complemented
236	#2341	C₂×C₅₃	1	2	yes	complemented
237	#2339	C₂×C₅₃	1	2	yes	complemented
238	#2333	C₂×C₅₃	1	4	yes	complemented
239	#2328	C₂×C₅₃	1	2	yes	complemented
240	#2327	C₂×C₅₃	1	2	yes	complemented
241	#2318	C₂×C₅₃	1	4	yes	complemented
242	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×C₅₃	—	4	2	no	—
245	C₂×C₅₃	—	4	8	no	—
246	C₂×C₅₃	C₂×#518	1	1	yes	characteristic
247	C₂×C₅₃	#3428	1	1	yes	characteristic
248	C₂×C₅₃	#3428	1	1	yes	characteristic
249	C₂×C₅₃	#3562	1	4	yes	—
250	C₂²×#41	C₂²×C₅₃	1	2	yes	—
251	#532	C₄×C₅₃	1	16	yes	complemented
252	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
253	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×#124	C₂²×C₅₃	1	2	yes	—
255	C₂×#121	C₂²×C₅₃	1	2	yes	—
256	C₂×#121	C₂²×C₅₃	1	2	yes	—
257	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
258	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
259	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
260	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
261	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
262	#391	C₂²×C₅₃	1	4	yes	—
263	#390	C₂²×C₅₃	1	4	yes	—
264	#390	C₂²×C₅₃	1	8	yes	—
265	#388	C₂²×C₅₃	1	8	yes	—
266	#388	C₂²×C₅₃	1	8	yes	—
267	#387	C₂²×C₅₃	1	8	yes	—
268	#387	C₂²×C₅₃	1	8	yes	—
269	#386	C₂²×C₅₃	1	8	yes	—
270	#386	C₂²×C₅₃	1	8	yes	—
271	#386	C₂²×C₅₃	1	8	yes	—
272	#385	C₂²×C₅₃	1	8	yes	—
273	#383	C₂²×C₅₃	1	2	yes	—
274	#383	C₂²×C₅₃	1	2	yes	—
275	#382	C₂²×C₅₃	1	2	yes	—
276	#382	C₂²×C₅₃	1	2	yes	—
277	#381	C₂²×C₅₃	1	4	yes	—
278	#381	C₂²×C₅₃	1	2	yes	—
279	#381	C₂²×C₅₃	1	2	yes	—
280	#380	C₂²×C₅₃	1	4	yes	—
281	#380	C₂²×C₅₃	1	2	yes	—
282	#380	C₂²×C₅₃	1	2	yes	—
283	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
284	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
285	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
286	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
288	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
290	#376	C₂²×C₅₃	1	2	yes	—
291	#376	C₂²×C₅₃	1	2	yes	—
292	C₂×#95	C₂²×C₅₃	1	4	yes	—
293	C₅₃	#34402	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×#41	—	4	8	no	—
295	C₂×#41	C₂×C₄×C₅₃	1	16	yes	complemented
296	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
297	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
298	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
299	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
300	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
301	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
303	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
304	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
305	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
306	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
307	#127	C₂×C₄×C₅₃	1	8	yes	—
308	#124	C₂×C₄×C₅₃	1	16	yes	complemented
309	#121	C₂×C₄×C₅₃	1	16	yes	complemented
310	#121	C₂×C₄×C₅₃	1	16	yes	complemented
311	C₄×Q₈	—	4	8	no	—
312	C₄×D₄	—	4	8	no	—
313	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
314	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
315	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
316	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
317	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
318	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
320	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
328	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
329	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
330	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
331	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
332	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
333	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
337	#95	C₂³×C₅₃	1	2	yes	—
338	#95	C₂³×C₅₃	1	4	yes	—
339	#95	C₂³×C₅₃	1	8	yes	—
340	#95	C₂³×C₅₃	1	4	yes	—
341	#95	C₂³×C₅₃	1	4	yes	—
342	#95	C₂³×C₅₃	1	4	yes	—
343	#95	C₂³×C₅₃	1	2	yes	—
344	#95	C₂³×C₅₃	1	4	yes	—
345	#95	C₂³×C₅₃	1	8	yes	—
346	#95	C₂³×C₅₃	1	4	yes	—
347	#95	C₂³×C₅₃	1	4	yes	—
348	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
349	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
350	#41	—	4	32	no	—
351	C₂×Q₈	—	2	8	no	—
352	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
353	C₂×D₄	—	2	8	no	—
354	C₂×D₄	—	4	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	8	no	—
357	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
358	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
359	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
360	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	8	no	—
365	C₂²×C₄	—	4	8	no	—
366	C₂²×C₄	—	4	8	no	—
367	C₂²×C₄	—	4	8	no	—
368	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
369	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
370	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
371	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
372	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
374	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	4	8	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
386	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
387	C₂²⋊C₄	—	4	8	no	—
388	C₂²⋊C₄	—	4	8	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	8	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	2	no	—
393	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
394	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
395	C₄²	—	2	4	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	4	8	no	—
398	C₄²	—	4	4	no	—
399	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
400	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
401	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
402	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
403	C₂³	—	2	8	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	4	no	—
408	C₂³	—	2	8	no	—
409	C₂³	—	2	4	no	—
410	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
411	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
412	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
413	Q₈	—	2	16	no	—
414	D₄	—	2	16	no	—
415	D₄	—	4	8	no	—
416	D₄	—	4	8	no	—
417	D₄	—	4	16	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	8	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	4	no	—
436	C₂×C₄	—	4	16	no	—
437	C₂×C₄	—	8	16	no	—
438	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
440	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
442	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
443	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
444	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
445	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	8	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	8	no	—
451	C₂²	—	2	8	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	8	no	—
454	C₂²	—	4	4	no	—
455	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
456	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
457	C₂²	C₅₃×#585	1	2	yes	—
458	C₂²	C₅₃×#585	1	2	yes	—
459	C₄	—	2	2	no	—
460	C₄	—	2	8	no	—
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	8	no	—
467	C₄	—	4	4	no	—
468	C₄	C₅₃×#518	1	4	yes	—
469	C₂	—	4	2	no	—
470	C₂	—	4	8	no	—
471	C₂	C₂×C₅₃×#518	1	1	yes	characteristic
472	C₂	C₅₃×#3428	1	1	yes	characteristic
473	C₂	C₅₃×#3428	1	1	yes	characteristic
474	C₂	C₅₃×#3562	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2377	C₂	semidirect
3. C₅₃×#2367	C₂	semidirect
4. C₅₃×#2353	C₂	semidirect
5. C₅₃×#2341	C₂	semidirect
6. C₅₃×#2339	C₂	semidirect
7. C₅₃×#2333	C₂	semidirect
8. C₅₃×#2328	C₂	semidirect
9. C₅₃×#2327	C₂	semidirect
10. C₅₃×#2318	C₂	semidirect
11. C₅₃×#2283	C₂	semidirect
12. C₅₃×#2283	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
75. C₄×D₄×C₅₃	C₂×C₄	semidirect
76. C₄×D₄×C₅₃	C₂×C₄	semidirect
90. C₂×C₄²×C₅₃	C₂×C₄	semidirect
208. #34402	C₅₃	direct (N × H)
233. #2377	C₂×C₅₃	semidirect
234. #2367	C₂×C₅₃	semidirect
235. #2353	C₂×C₅₃	semidirect
236. #2341	C₂×C₅₃	semidirect
237. #2339	C₂×C₅₃	semidirect
238. #2333	C₂×C₅₃	semidirect
239. #2328	C₂×C₅₃	semidirect
240. #2327	C₂×C₅₃	semidirect
241. #2318	C₂×C₅₃	semidirect
242. #2283	C₂×C₅₃	semidirect
243. #2283	C₂×C₅₃	semidirect
251. #532	C₄×C₅₃	semidirect
293. C₅₃	#34402	direct (N × H)
295. C₂×#41	C₂×C₄×C₅₃	semidirect
306. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
308. #124	C₂×C₄×C₅₃	semidirect
309. #121	C₂×C₄×C₅₃	semidirect
310. #121	C₂×C₄×C₅₃	semidirect
313. C₄×D₄	C₂×C₄×C₅₃	semidirect
314. C₄×D₄	C₂×C₄×C₅₃	semidirect
328. C₂×C₄²	C₂×C₄×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×#2377	2
C₅₃×#2367	2
C₅₃×#2353	2
C₅₃×#2341	2
C₅₃×#2339	2
C₅₃×#2333	2
C₅₃×#2328	2
C₅₃×#2327	2
C₅₃×#2318	2
C₅₃×#2283	2
#34402	53

Subgroups — C53×#34402 #27241914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34402 #27241914