Subgroups — C₅₃×#36577 #27244089

Order 13568.

2312 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2857	C₂	1	3	yes	maximal
4	C₅₃×#2582	C₂	1	3	yes	maximal, complemented
5	C₅₃×#2568	C₂	1	3	yes	maximal, complemented
6	C₅₃×#2453	C₂	1	6	yes	maximal, complemented
7	C₅₃×#2442	C₂	1	6	yes	maximal
8	C₅₃×#2351	C₂	1	2	yes	maximal, complemented
9	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	Q₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal
11	C₄×C₂²⋊Q₈×C₅₃	C₂	1	3	yes	maximal
12	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#557	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
15	C₂×#125×C₅₃	C₂²	1	6	yes	—
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	3	yes	—
17	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
27	C₅₃×#400	C₂²	1	12	yes	—
28	C₅₃×#398	C₂²	1	12	yes	—
29	C₅₃×#391	C₂²	1	6	yes	—
30	C₅₃×#391	C₂²	1	6	yes	—
31	C₅₃×#391	C₂²	1	3	yes	—
32	C₅₃×#391	C₂²	1	6	yes	—
33	C₅₃×#389	C₂²	1	6	yes	—
34	C₅₃×#389	C₂²	1	6	yes	—
35	C₅₃×#389	C₂²	1	6	yes	—
36	C₅₃×#389	C₂²	1	6	yes	—
37	C₅₃×#389	C₂²	1	3	yes	—
38	C₅₃×#387	C₂²	1	6	yes	—
39	C₅₃×#385	C₂²	1	6	yes	—
40	C₅₃×#384	C₂²	1	3	yes	—
41	C₅₃×#382	C₂²	1	6	yes	—
42	C₅₃×#380	C₂²	1	6	yes	—
43	C₅₃×#380	C₂²	1	6	yes	—
44	C₅₃×#380	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
47	C₄×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
48	C₂²×Q₈×C₅₃	—	2	4	no	—
49	C₂²×Q₈×C₅₃	—	2	4	no	—
50	C₂²×Q₈×C₅₃	Q₈	1	8	yes	complemented
51	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
52	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
53	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	—	2	12	no	—
58	#125×C₅₃	—	2	24	no	—
59	C₂²⋊Q₈×C₅₃	—	2	12	no	—
60	C₄×Q₈×C₅₃	—	2	12	no	—
61	C₄×Q₈×C₅₃	—	2	12	no	—
62	C₄×Q₈×C₅₃	—	2	12	no	—
63	C₄×Q₈×C₅₃	—	2	4	no	—
64	C₄×Q₈×C₅₃	—	2	8	no	—
65	C₄×Q₈×C₅₃	—	2	8	no	—
66	C₄×Q₈×C₅₃	D₄	1	8	yes	—
67	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄²×C₅₃	—	2	12	no	—
84	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
86	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
90	#95×C₅₃	C₂³	1	6	yes	—
91	#95×C₅₃	C₂³	1	6	yes	—
92	#95×C₅₃	C₂³	1	2	yes	—
93	#95×C₅₃	C₂³	1	6	yes	—
94	#95×C₅₃	C₂³	1	6	yes	—
95	#95×C₅₃	C₂³	1	12	yes	—
96	#95×C₅₃	C₂³	1	12	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂×Q₈×C₅₃	—	2	32	no	—
99	C₂×Q₈×C₅₃	—	2	6	no	—
100	C₂×Q₈×C₅₃	—	2	6	no	—
101	C₂×Q₈×C₅₃	—	2	6	no	—
102	C₂×Q₈×C₅₃	—	2	6	no	—
103	C₂×Q₈×C₅₃	—	2	6	no	—
104	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
105	C₂×Q₈×C₅₃	C₂×Q₈	1	12	yes	—
106	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
107	C₂×Q₈×C₅₃	#41	1	4	yes	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	6	no	—
110	C₂²×C₄×C₅₃	—	2	6	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	6	no	—
113	C₂²×C₄×C₅₃	—	2	6	no	—
114	C₂²×C₄×C₅₃	—	4	3	no	—
115	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
116	C₂²×C₄×C₅₃	#41	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	—	2	12	no	—
128	C₄⋊C₄×C₅₃	—	2	6	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	12	no	—
131	C₄⋊C₄×C₅₃	—	2	12	no	—
132	C₄⋊C₄×C₅₃	—	2	12	no	—
133	C₄⋊C₄×C₅₃	—	2	12	no	—
134	C₄⋊C₄×C₅₃	—	2	12	no	—
135	C₄⋊C₄×C₅₃	—	2	6	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	4	6	no	—
138	C₄⋊C₄×C₅₃	—	4	6	no	—
139	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
140	C₂²⋊C₄×C₅₃	—	2	12	no	—
141	C₂²⋊C₄×C₅₃	—	2	4	no	—
142	C₄²×C₅₃	—	2	6	no	—
143	C₄²×C₅₃	—	2	6	no	—
144	C₄²×C₅₃	—	2	6	no	—
145	C₄²×C₅₃	—	2	12	no	—
146	C₄²×C₅₃	—	2	12	no	—
147	C₄²×C₅₃	C₂×D₄	1	12	yes	—
148	C₂³×C₅₃	—	2	1	no	—
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	—	2	1	no	—
152	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
153	C₂³×C₅₃	C₂×#41	1	2	yes	—
154	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
155	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
156	Q₈×C₅₃	—	2	24	no	—
157	Q₈×C₅₃	—	4	12	no	—
158	Q₈×C₅₃	C₂²⋊Q₈	1	16	yes	complemented
159	C₂×C₄×C₅₃	—	2	3	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	3	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	3	no	—
164	C₂×C₄×C₅₃	—	2	3	no	—
165	C₂×C₄×C₅₃	—	2	24	no	—
166	C₂×C₄×C₅₃	—	2	8	no	—
167	C₂×C₄×C₅₃	—	2	3	no	—
168	C₂×C₄×C₅₃	—	2	6	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	6	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	6	no	—
173	C₂×C₄×C₅₃	—	2	6	no	—
174	C₂×C₄×C₅₃	—	2	3	no	—
175	C₂×C₄×C₅₃	—	2	6	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	3	no	—
180	C₂×C₄×C₅₃	—	2	6	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	4	12	no	—
183	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
185	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
187	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
188	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
189	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
190	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
191	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
192	#36577	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	2	2	no	—
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	#557	1	4	yes	complemented
200	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
201	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
202	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
203	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
204	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
205	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
206	C₂²×C₅₃	#585	1	1	yes	characteristic
207	C₄×C₅₃	—	2	6	no	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	6	no	—
210	C₄×C₅₃	—	4	6	no	—
211	C₄×C₅₃	—	4	6	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	3	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
216	C₄×C₅₃	#548	1	4	yes	—
217	C₂×#557	C₂×C₅₃	1	1	yes	characteristic, complemented
218	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
219	#2857	C₂×C₅₃	1	3	yes	—
220	#2582	C₂×C₅₃	1	3	yes	complemented
221	#2568	C₂×C₅₃	1	3	yes	complemented
222	#2453	C₂×C₅₃	1	6	yes	complemented
223	#2442	C₂×C₅₃	1	6	yes	—
224	#2351	C₂×C₅₃	1	2	yes	complemented
225	Q₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
226	Q₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	—
227	C₄×C₂²⋊Q₈	C₂×C₅₃	1	3	yes	—
228	C₂×C₅₃	—	2	4	no	—
229	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
231	C₂×C₅₃	C₂×#557	1	1	yes	characteristic
232	C₂×C₅₃	#3480	1	1	yes	characteristic
233	C₂×C₅₃	#3489	1	1	yes	characteristic
234	C₂×C₅₃	#3502	1	1	yes	characteristic
235	C₂×C₅₃	#3523	1	1	yes	characteristic
236	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
237	#557	—	2	8	no	—
238	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
239	C₂×#125	C₂²×C₅₃	1	6	yes	—
240	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	3	yes	—
241	C₂×C₄×Q₈	—	2	8	no	—
242	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
243	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
244	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
245	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
246	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
248	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
249	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
250	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
251	#400	C₂²×C₅₃	1	12	yes	—
252	#398	C₂²×C₅₃	1	12	yes	—
253	#391	C₂²×C₅₃	1	3	yes	—
254	#391	C₂²×C₅₃	1	6	yes	—
255	#391	C₂²×C₅₃	1	6	yes	—
256	#391	C₂²×C₅₃	1	6	yes	—
257	#389	C₂²×C₅₃	1	3	yes	—
258	#389	C₂²×C₅₃	1	6	yes	—
259	#389	C₂²×C₅₃	1	6	yes	—
260	#389	C₂²×C₅₃	1	6	yes	—
261	#389	C₂²×C₅₃	1	6	yes	—
262	#387	C₂²×C₅₃	1	6	yes	—
263	#385	C₂²×C₅₃	1	6	yes	—
264	#384	C₂²×C₅₃	1	3	yes	—
265	#382	C₂²×C₅₃	1	6	yes	—
266	#380	C₂²×C₅₃	1	6	yes	—
267	#380	C₂²×C₅₃	1	6	yes	—
268	#380	C₂²×C₅₃	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
270	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
271	C₄×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
272	C₅₃	#36577	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²×Q₈	—	2	4	no	—
274	C₂²×Q₈	—	2	4	no	—
275	C₂²×Q₈	Q₈×C₅₃	1	8	yes	complemented
276	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
277	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
278	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
280	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
281	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
282	C₄⋊Q₈	—	2	12	no	—
283	#125	—	2	24	no	—
284	C₂²⋊Q₈	—	2	12	no	—
285	C₄×Q₈	—	2	4	no	—
286	C₄×Q₈	—	2	8	no	—
287	C₄×Q₈	—	2	8	no	—
288	C₄×Q₈	—	2	12	no	—
289	C₄×Q₈	—	2	12	no	—
290	C₄×Q₈	—	2	12	no	—
291	C₄×Q₈	D₄×C₅₃	1	8	yes	—
292	C₂×C₄⋊C₄	—	2	12	no	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
295	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
298	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
299	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
300	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
301	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
302	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
304	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
305	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
308	C₂×C₄²	—	2	12	no	—
309	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
310	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
311	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
312	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
313	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
314	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
315	#95	C₂³×C₅₃	1	6	yes	—
316	#95	C₂³×C₅₃	1	6	yes	—
317	#95	C₂³×C₅₃	1	2	yes	—
318	#95	C₂³×C₅₃	1	6	yes	—
319	#95	C₂³×C₅₃	1	6	yes	—
320	#95	C₂³×C₅₃	1	12	yes	—
321	#95	C₂³×C₅₃	1	12	yes	—
322	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
323	C₂×Q₈	—	2	32	no	—
324	C₂×Q₈	—	2	6	no	—
325	C₂×Q₈	—	2	6	no	—
326	C₂×Q₈	—	2	6	no	—
327	C₂×Q₈	—	2	6	no	—
328	C₂×Q₈	—	2	6	no	—
329	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
330	C₂×Q₈	C₂×Q₈×C₅₃	1	12	yes	—
331	C₂×Q₈	C₂×Q₈×C₅₃	1	4	yes	—
332	C₂×Q₈	#41×C₅₃	1	4	yes	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	6	no	—
335	C₂²×C₄	—	2	6	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	6	no	—
338	C₂²×C₄	—	2	6	no	—
339	C₂²×C₄	—	4	3	no	—
340	C₂²×C₄	C₂×Q₈×C₅₃	1	12	yes	—
341	C₂²×C₄	#41×C₅₃	1	4	yes	—
342	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
343	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
344	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
345	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
346	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
347	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
348	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
349	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
350	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
351	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
352	C₄⋊C₄	—	2	12	no	—
353	C₄⋊C₄	—	2	6	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	12	no	—
356	C₄⋊C₄	—	2	12	no	—
357	C₄⋊C₄	—	2	12	no	—
358	C₄⋊C₄	—	2	12	no	—
359	C₄⋊C₄	—	2	6	no	—
360	C₄⋊C₄	—	2	12	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	6	no	—
363	C₄⋊C₄	—	4	6	no	—
364	C₄⋊C₄	C₂×D₄×C₅₃	1	12	yes	—
365	C₂²⋊C₄	—	2	12	no	—
366	C₂²⋊C₄	—	2	4	no	—
367	C₄²	—	2	6	no	—
368	C₄²	—	2	6	no	—
369	C₄²	—	2	6	no	—
370	C₄²	—	2	12	no	—
371	C₄²	—	2	12	no	—
372	C₄²	C₂×D₄×C₅₃	1	12	yes	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
378	C₂³	C₂×#41×C₅₃	1	2	yes	—
379	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
380	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
381	Q₈	—	2	24	no	—
382	Q₈	—	4	12	no	—
383	Q₈	C₂²⋊Q₈×C₅₃	1	16	yes	complemented
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	3	no	—
386	C₂×C₄	—	2	3	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	3	no	—
389	C₂×C₄	—	2	3	no	—
390	C₂×C₄	—	2	24	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	3	no	—
394	C₂×C₄	—	2	6	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	6	no	—
397	C₂×C₄	—	2	6	no	—
398	C₂×C₄	—	2	6	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	3	no	—
403	C₂×C₄	—	2	6	no	—
404	C₂×C₄	—	2	6	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	3	no	—
407	C₂×C₄	—	4	12	no	—
408	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
409	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
410	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
411	C₂×C₄	C₂²×Q₈×C₅₃	1	6	yes	—
412	C₂×C₄	C₂²×Q₈×C₅₃	1	6	yes	—
413	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
414	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
415	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
416	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
417	C₂²	—	2	2	no	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	C₅₃×#557	1	4	yes	complemented
424	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
425	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
426	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
427	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
428	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
429	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
430	C₂²	C₅₃×#585	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	6	no	—
433	C₄	—	2	6	no	—
434	C₄	—	4	6	no	—
435	C₄	—	4	6	no	—
436	C₄	—	4	3	no	—
437	C₄	—	4	2	no	—
438	C₄	—	4	2	no	—
439	C₄	C₂×C₂²⋊Q₈×C₅₃	1	12	yes	—
440	C₄	C₅₃×#548	1	4	yes	—
441	C₂	—	2	4	no	—
442	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
443	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
444	C₂	C₂×C₅₃×#557	1	1	yes	characteristic
445	C₂	C₅₃×#3480	1	1	yes	characteristic
446	C₂	C₅₃×#3489	1	1	yes	characteristic
447	C₂	C₅₃×#3502	1	1	yes	characteristic
448	C₂	C₅₃×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁴×C₅₃	C₂⁴
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×#557	C₂	semidirect
4. C₅₃×#2582	C₂	semidirect
5. C₅₃×#2568	C₂	semidirect
6. C₅₃×#2453	C₂	semidirect
8. C₅₃×#2351	C₂	semidirect
9. Q₈×C₄⋊C₄×C₅₃	C₂	semidirect
50. C₂²×Q₈×C₅₃	Q₈	semidirect
158. Q₈×C₅₃	C₂²⋊Q₈	semidirect
192. #36577	C₅₃	direct (N × H)
199. C₂²×C₅₃	#557	semidirect
217. C₂×#557	C₂×C₅₃	semidirect
220. #2582	C₂×C₅₃	semidirect
221. #2568	C₂×C₅₃	semidirect
222. #2453	C₂×C₅₃	semidirect
224. #2351	C₂×C₅₃	semidirect
225. Q₈×C₄⋊C₄	C₂×C₅₃	semidirect
272. C₅₃	#36577	direct (N × H)
275. C₂²×Q₈	Q₈×C₅₃	semidirect
383. Q₈	C₂²⋊Q₈×C₅₃	semidirect
423. C₂²	C₅₃×#557	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#557	2
C₂²×C₄×Q₈×C₅₃	2
C₅₃×#2857	2
C₅₃×#2582	2
C₅₃×#2568	2
C₅₃×#2453	2
C₅₃×#2442	2
C₅₃×#2351	2
Q₈×C₄⋊C₄×C₅₃	2
Q₈×C₂²⋊C₄×C₅₃	2
C₄×C₂²⋊Q₈×C₅₃	2
#36577	53

Subgroups — C53×#36577 #27244089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36577 #27244089