Subgroups — C₅₃×#34274 #27241786

Order 13568.

3154 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#537	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#520	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2363	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2362	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2361	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2360	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2325	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#121×C₅₃	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2297	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#537	C₄	1	16	yes	complemented
14	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
15	C₂×#121×C₅₃	C₂²	1	4	yes	—
16	C₂×#121×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#520	—	2	16	no	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₂×#117×C₅₃	C₂²	1	2	yes	—
23	C₂×#117×C₅₃	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#391	C₂²	1	2	yes	—
30	C₅₃×#391	C₂²	1	2	yes	—
31	C₅₃×#390	C₂²	1	2	yes	—
32	C₅₃×#390	C₂²	1	2	yes	—
33	C₅₃×#390	C₂²	1	4	yes	—
34	C₅₃×#389	C₂²	1	4	yes	—
35	C₅₃×#389	C₂²	1	4	yes	—
36	C₅₃×#389	C₂²	1	4	yes	—
37	C₅₃×#388	C₂²	1	8	yes	—
38	C₅₃×#388	C₂²	1	8	yes	—
39	C₅₃×#387	C₂²	1	8	yes	—
40	C₅₃×#386	C₂²	1	4	yes	—
41	C₅₃×#386	C₂²	1	4	yes	—
42	C₅₃×#386	C₂²	1	8	yes	—
43	C₅₃×#385	C₂²	1	8	yes	—
44	C₅₃×#380	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
51	C₂×#95×C₅₃	C₂²	1	2	yes	—
52	C₂×#41×C₅₃	—	2	8	no	—
53	C₂×#41×C₅₃	—	2	8	no	—
54	C₂×#41×C₅₃	C₂×C₄	1	16	yes	complemented
55	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²⋊Q₈×C₅₃	C₂×C₄	1	32	yes	complemented
67	#121×C₅₃	C₂×C₄	1	32	yes	complemented
68	#121×C₅₃	C₂×C₄	1	32	yes	complemented
69	C₄×Q₈×C₅₃	—	2	16	no	—
70	C₄×D₄×C₅₃	—	2	16	no	—
71	C₄×D₄×C₅₃	—	2	32	no	—
72	C₄×D₄×C₅₃	—	4	8	no	—
73	#117×C₅₃	—	2	16	no	—
74	#117×C₅₃	—	2	8	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
76	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
77	C₂×C₄⋊C₄×C₅₃	D₄	1	8	yes	complemented
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	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×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
93	C₂×C₄²×C₅₃	—	2	4	no	—
94	C₂×C₄²×C₅₃	—	4	4	no	—
95	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
100	C₂×C₄²×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	8	yes	—
105	#95×C₅₃	C₂³	1	4	yes	—
106	#95×C₅₃	C₂³	1	4	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
108	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
109	#41×C₅₃	—	2	64	no	—
110	C₂×Q₈×C₅₃	—	2	4	no	—
111	C₂×Q₈×C₅₃	—	2	4	no	—
112	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
113	C₂×D₄×C₅₃	—	2	8	no	—
114	C₂×D₄×C₅₃	—	2	8	no	—
115	C₂×D₄×C₅₃	—	2	4	no	—
116	C₂×D₄×C₅₃	—	2	4	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	4	4	no	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
120	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	8	no	—
124	C₂²×C₄×C₅₃	—	2	8	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	16	no	—
129	C₂²×C₄×C₅₃	—	4	2	no	—
130	C₂²×C₄×C₅₃	—	4	2	no	—
131	C₂²×C₄×C₅₃	—	4	2	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	—	4	1	no	—
134	C₂²×C₄×C₅₃	—	4	1	no	—
135	C₂²×C₄×C₅₃	—	4	8	no	—
136	C₂²×C₄×C₅₃	—	4	8	no	—
137	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
138	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
139	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
141	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₅₃	#41	1	4	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
152	C₄⋊C₄×C₅₃	—	2	4	no	—
153	C₄⋊C₄×C₅₃	—	2	16	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	4	4	no	—
156	C₄⋊C₄×C₅₃	—	4	4	no	—
157	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
158	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
159	C₄⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
160	C₂²⋊C₄×C₅₃	—	2	8	no	—
161	C₂²⋊C₄×C₅₃	—	2	16	no	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	4	4	no	—
165	C₂²⋊C₄×C₅₃	—	4	4	no	—
166	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
167	C₄²×C₅₃	—	2	16	no	—
168	C₄²×C₅₃	—	2	8	no	—
169	C₄²×C₅₃	—	2	4	no	—
170	C₄²×C₅₃	—	2	4	no	—
171	C₄²×C₅₃	—	2	8	no	—
172	C₄²×C₅₃	—	4	4	no	—
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	—	2	1	no	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	4	no	—
178	C₂³×C₅₃	—	2	4	no	—
179	C₂³×C₅₃	—	2	4	no	—
180	C₂³×C₅₃	—	2	4	no	—
181	C₂³×C₅₃	—	2	4	no	—
182	C₂³×C₅₃	—	2	4	no	—
183	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
184	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
185	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
186	C₂³×C₅₃	C₂×#41	1	2	yes	—
187	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
188	Q₈×C₅₃	—	2	16	no	—
189	D₄×C₅₃	—	2	32	no	—
190	D₄×C₅₃	—	2	16	no	—
191	D₄×C₅₃	—	4	16	no	—
192	C₂×C₄×C₅₃	—	2	8	no	—
193	C₂×C₄×C₅₃	—	2	4	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	32	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	4	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	4	8	no	—
214	C₂×C₄×C₅₃	—	4	8	no	—
215	C₂×C₄×C₅₃	—	4	4	no	—
216	C₂×C₄×C₅₃	—	8	16	no	—
217	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	complemented
218	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
219	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
220	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
221	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
224	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
225	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
226	#34274	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂²×C₅₃	—	2	2	no	—
228	C₂²×C₅₃	—	2	2	no	—
229	C₂²×C₅₃	—	2	2	no	—
230	C₂²×C₅₃	—	2	2	no	—
231	C₂²×C₅₃	—	2	2	no	—
232	C₂²×C₅₃	—	2	2	no	—
233	C₂²×C₅₃	—	2	8	no	—
234	C₂²×C₅₃	—	2	8	no	—
235	C₂²×C₅₃	—	2	8	no	—
236	C₂²×C₅₃	—	4	4	no	—
237	C₂²×C₅₃	—	4	4	no	—
238	C₂²×C₅₃	—	4	4	no	—
239	C₂²×C₅₃	—	4	4	no	—
240	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
241	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
242	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
243	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
244	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
246	C₂²×C₅₃	#585	1	1	yes	characteristic
247	C₂²×C₅₃	#585	1	1	yes	characteristic
248	C₄×C₅₃	—	2	8	no	—
249	C₄×C₅₃	—	2	8	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	8	no	—
253	C₄×C₅₃	—	4	2	no	—
254	C₄×C₅₃	—	4	4	no	—
255	C₄×C₅₃	—	4	4	no	—
256	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	—
257	C₂×#537	C₂×C₅₃	1	1	yes	characteristic
258	C₂×#520	C₂×C₅₃	1	2	yes	complemented
259	#2363	C₂×C₅₃	1	4	yes	complemented
260	#2362	C₂×C₅₃	1	4	yes	complemented
261	#2361	C₂×C₅₃	1	4	yes	complemented
262	#2360	C₂×C₅₃	1	4	yes	complemented
263	#2325	C₂×C₅₃	1	4	yes	complemented
264	#2316	C₂×C₅₃	1	1	yes	characteristic, complemented
265	C₄×#121	C₂×C₅₃	1	4	yes	complemented
266	#2297	C₂×C₅₃	1	2	yes	complemented
267	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
268	C₂×C₅₃	—	2	4	no	—
269	C₂×C₅₃	—	4	8	no	—
270	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
271	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₅₃	C₄×2₋¹⁺⁴	1	1	yes	characteristic
273	C₂×C₅₃	#3492	1	1	yes	characteristic
274	C₂×C₅₃	#3493	1	1	yes	characteristic
275	C₂×C₅₃	#3496	1	1	yes	characteristic
276	C₂×C₅₃	#3497	1	1	yes	characteristic
277	C₂²×#41	C₂²×C₅₃	1	2	yes	—
278	#537	C₄×C₅₃	1	16	yes	complemented
279	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
280	C₂×#121	C₂²×C₅₃	1	4	yes	—
281	C₂×#121	C₂²×C₅₃	1	4	yes	—
282	#520	—	2	16	no	—
283	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
284	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
285	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
286	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
287	C₂×#117	C₂²×C₅₃	1	2	yes	—
288	C₂×#117	C₂²×C₅₃	1	4	yes	—
289	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
290	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
291	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
292	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
293	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
294	#391	C₂²×C₅₃	1	2	yes	—
295	#391	C₂²×C₅₃	1	2	yes	—
296	#390	C₂²×C₅₃	1	2	yes	—
297	#390	C₂²×C₅₃	1	2	yes	—
298	#390	C₂²×C₅₃	1	4	yes	—
299	#389	C₂²×C₅₃	1	4	yes	—
300	#389	C₂²×C₅₃	1	4	yes	—
301	#389	C₂²×C₅₃	1	4	yes	—
302	#388	C₂²×C₅₃	1	8	yes	—
303	#388	C₂²×C₅₃	1	8	yes	—
304	#387	C₂²×C₅₃	1	8	yes	—
305	#386	C₂²×C₅₃	1	4	yes	—
306	#386	C₂²×C₅₃	1	8	yes	—
307	#386	C₂²×C₅₃	1	4	yes	—
308	#385	C₂²×C₅₃	1	8	yes	—
309	#380	C₂²×C₅₃	1	4	yes	—
310	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
311	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
312	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
313	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
315	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
316	C₂×#95	C₂²×C₅₃	1	2	yes	—
317	C₅₃	#34274	1	1	yes	characteristic, direct product factor, Sylow
318	C₂×#41	—	2	8	no	—
319	C₂×#41	—	2	8	no	—
320	C₂×#41	C₂×C₄×C₅₃	1	16	yes	complemented
321	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
322	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
323	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
324	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
325	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
327	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
328	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
330	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
331	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂²⋊Q₈	C₂×C₄×C₅₃	1	32	yes	complemented
333	#121	C₂×C₄×C₅₃	1	32	yes	complemented
334	#121	C₂×C₄×C₅₃	1	32	yes	complemented
335	C₄×Q₈	—	2	16	no	—
336	C₄×D₄	—	2	32	no	—
337	C₄×D₄	—	2	16	no	—
338	C₄×D₄	—	4	8	no	—
339	#117	—	2	8	no	—
340	#117	—	2	16	no	—
341	C₂×C₄⋊C₄	—	2	16	no	—
342	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
343	C₂×C₄⋊C₄	D₄×C₅₃	1	8	yes	complemented
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
349	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
350	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₄²	—	2	4	no	—
360	C₂×C₄²	—	4	4	no	—
361	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
362	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
363	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
364	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
365	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
366	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
367	#95	C₂³×C₅₃	1	8	yes	—
368	#95	C₂³×C₅₃	1	4	yes	—
369	#95	C₂³×C₅₃	1	4	yes	—
370	#95	C₂³×C₅₃	1	8	yes	—
371	#95	C₂³×C₅₃	1	4	yes	—
372	#95	C₂³×C₅₃	1	4	yes	—
373	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
374	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
375	#41	—	2	64	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	2	4	no	—
378	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
386	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
387	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	8	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	16	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	8	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
404	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
405	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
406	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
407	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
408	C₂²×C₄	#41×C₅₃	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	16	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₄⋊C₄	—	4	4	no	—
423	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
424	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
425	C₄⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	16	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	—
433	C₄²	—	2	16	no	—
434	C₄²	—	2	8	no	—
435	C₄²	—	2	4	no	—
436	C₄²	—	2	4	no	—
437	C₄²	—	2	8	no	—
438	C₄²	—	4	4	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
450	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
451	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
452	C₂³	C₂×#41×C₅₃	1	2	yes	—
453	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
454	Q₈	—	2	16	no	—
455	D₄	—	2	32	no	—
456	D₄	—	2	16	no	—
457	D₄	—	4	16	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	32	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	4	8	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	8	16	no	—
483	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	complemented
484	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
485	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
486	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
488	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
489	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
490	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
491	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
506	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
507	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
508	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
509	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
510	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
511	C₂²	C₅₃×#585	1	1	yes	characteristic
512	C₂²	C₅₃×#585	1	1	yes	characteristic
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	2	8	no	—
516	C₄	—	2	8	no	—
517	C₄	—	2	8	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	—
522	C₂	—	2	4	no	—
523	C₂	—	4	8	no	—
524	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
525	C₂	C₄×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
526	C₂	C₄×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
527	C₂	C₅₃×#3492	1	1	yes	characteristic
528	C₂	C₅₃×#3493	1	1	yes	characteristic
529	C₂	C₅₃×#3496	1	1	yes	characteristic
530	C₂	C₅₃×#3497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#537	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. C₂×C₅₃×#520	C₂	semidirect
3. C₅₃×#2363	C₂	semidirect
4. C₅₃×#2362	C₂	semidirect
5. C₅₃×#2361	C₂	semidirect
6. C₅₃×#2360	C₂	semidirect
7. C₅₃×#2325	C₂	semidirect
8. C₅₃×#2316	C₂	semidirect
9. C₄×#121×C₅₃	C₂	semidirect
10. C₅₃×#2297	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
13. C₅₃×#537	C₄	semidirect
54. C₂×#41×C₅₃	C₂×C₄	semidirect
66. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
67. #121×C₅₃	C₂×C₄	semidirect
68. #121×C₅₃	C₂×C₄	semidirect
76. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
77. C₂×C₄⋊C₄×C₅₃	D₄	semidirect
217. C₂×C₄×C₅₃	C₄×D₄	semidirect
226. #34274	C₅₃	direct (N × H)
258. C₂×#520	C₂×C₅₃	semidirect
259. #2363	C₂×C₅₃	semidirect
260. #2362	C₂×C₅₃	semidirect
261. #2361	C₂×C₅₃	semidirect
262. #2360	C₂×C₅₃	semidirect
263. #2325	C₂×C₅₃	semidirect
264. #2316	C₂×C₅₃	semidirect
265. C₄×#121	C₂×C₅₃	semidirect
266. #2297	C₂×C₅₃	semidirect
267. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
278. #537	C₄×C₅₃	semidirect
317. C₅₃	#34274	direct (N × H)
320. C₂×#41	C₂×C₄×C₅₃	semidirect
332. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
333. #121	C₂×C₄×C₅₃	semidirect
334. #121	C₂×C₄×C₅₃	semidirect
342. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
343. C₂×C₄⋊C₄	D₄×C₅₃	semidirect
483. C₂×C₄	C₄×D₄×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×#537	2
C₂×C₅₃×#520	2
C₅₃×#2363	2
C₅₃×#2362	2
C₅₃×#2361	2
C₅₃×#2360	2
C₅₃×#2325	2
C₅₃×#2316	2
C₄×#121×C₅₃	2
C₅₃×#2297	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#34274	53

Subgroups — C53×#34274 #27241786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34274 #27241786