Subgroups — C₂×C₅₃×#2451 #27242202

Order 13568.

2702 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2451	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#398	C₂	1	2	yes	maximal
5	C₂×C₅₃×#395	C₂	1	2	yes	maximal
6	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#391	C₂	1	2	yes	maximal
9	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic
11	C₂×C₅₃×#387	C₂	1	2	yes	maximal
12	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	—
17	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#398	C₂²	1	16	yes	—
33	C₅₃×#395	C₂²	1	16	yes	—
34	C₅₃×#391	C₂²	1	16	yes	—
35	C₅₃×#391	C₂²	1	8	yes	—
36	C₅₃×#391	C₂²	1	8	yes	—
37	C₅₃×#389	C₂²	1	8	yes	—
38	C₅₃×#389	C₂²	1	8	yes	—
39	C₅₃×#387	C₂²	1	16	yes	—
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
41	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#95×C₅₃	C₂²	1	2	yes	—
44	C₂×#95×C₅₃	C₂²	1	2	yes	—
45	C₂×#95×C₅₃	C₂²	1	2	yes	—
46	C₂×#95×C₅₃	C₂²	1	2	yes	—
47	C₂×#95×C₅₃	C₂²	1	2	yes	—
48	C₂×#95×C₅₃	C₂²	1	2	yes	—
49	C₂²×Q₈×C₅₃	—	2	4	no	—
50	C₂²×Q₈×C₅₃	—	2	8	no	—
51	C₂²×Q₈×C₅₃	—	2	4	no	—
52	C₂²×Q₈×C₅₃	—	2	8	no	—
53	C₂²×Q₈×C₅₃	—	4	4	no	—
54	C₂²×Q₈×C₅₃	Q₈	1	8	yes	complemented
55	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
56	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	—	2	32	no	—
69	C₄×Q₈×C₅₃	—	2	32	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
76	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₂×C₄⋊C₄×C₅₃	D₄	1	8	yes	—
78	C₂×C₄⋊C₄×C₅₃	Q₈	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	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	8	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄²×C₅₃	—	2	8	no	—
89	C₂×C₄²×C₅₃	—	2	4	no	—
90	C₂×C₄²×C₅₃	—	2	4	no	—
91	C₂×C₄²×C₅₃	—	2	4	no	—
92	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
95	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
96	#95×C₅₃	C₂³	1	4	yes	—
97	#95×C₅₃	C₂³	1	4	yes	—
98	#95×C₅₃	C₂³	1	8	yes	—
99	#95×C₅₃	C₂³	1	8	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	8	yes	—
102	#95×C₅₃	C₂³	1	8	yes	—
103	#95×C₅₃	C₂³	1	8	yes	—
104	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
105	C₂×Q₈×C₅₃	—	2	32	no	—
106	C₂×Q₈×C₅₃	—	2	8	no	—
107	C₂×Q₈×C₅₃	—	2	16	no	—
108	C₂×Q₈×C₅₃	—	2	8	no	—
109	C₂×Q₈×C₅₃	—	2	16	no	—
110	C₂×Q₈×C₅₃	—	4	8	no	—
111	C₂×Q₈×C₅₃	—	4	32	no	—
112	C₂×Q₈×C₅₃	C₂×Q₈	1	16	yes	complemented
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	4	no	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	2	no	—
130	C₂²×C₄×C₅₃	—	2	4	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
138	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
141	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
142	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
143	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
144	C₂²×C₄×C₅₃	#41	1	4	yes	—
145	C₂²×C₄×C₅₃	#41	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₄⋊C₄×C₅₃	—	2	16	no	—
158	C₄⋊C₄×C₅₃	—	2	16	no	—
159	C₄⋊C₄×C₅₃	—	2	8	no	—
160	C₄⋊C₄×C₅₃	—	2	8	no	—
161	C₄⋊C₄×C₅₃	—	2	8	no	—
162	C₄⋊C₄×C₅₃	—	4	8	no	—
163	C₄⋊C₄×C₅₃	—	4	8	no	—
164	C₄⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
165	C₄⋊C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
166	C₄²×C₅₃	—	2	16	no	—
167	C₄²×C₅₃	—	2	8	no	—
168	C₄²×C₅₃	—	2	8	no	—
169	C₄²×C₅₃	—	2	8	no	—
170	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
171	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
172	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
173	C₂³×C₅₃	C₂×#41	1	2	yes	—
174	C₂³×C₅₃	C₂×#41	1	2	yes	—
175	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
176	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
177	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
178	Q₈×C₅₃	—	2	32	no	—
179	Q₈×C₅₃	—	4	32	no	—
180	C₂×C₄×C₅₃	—	2	16	no	—
181	C₂×C₄×C₅₃	—	2	8	no	—
182	C₂×C₄×C₅₃	—	2	16	no	—
183	C₂×C₄×C₅₃	—	2	8	no	—
184	C₂×C₄×C₅₃	—	2	8	no	—
185	C₂×C₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	4	4	no	—
211	C₂×C₄×C₅₃	—	4	8	no	—
212	C₂×C₄×C₅₃	—	4	4	no	—
213	C₂×C₄×C₅₃	—	4	8	no	—
214	C₂×C₄×C₅₃	C₂²⋊Q₈	1	16	yes	—
215	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
216	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
218	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
219	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
220	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
221	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
222	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
223	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
224	C₂×#2451	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
226	C₂²×C₅₃	#533	1	4	yes	—
227	C₂²×C₅₃	#542	1	4	yes	—
228	C₂²×C₅₃	#546	1	4	yes	—
229	C₂²×C₅₃	D₄×Q₈	1	4	yes	—
230	C₂²×C₅₃	#557	1	4	yes	—
231	C₂²×C₅₃	Q₈²	1	4	yes	—
232	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
233	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
234	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
235	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
236	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
237	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
239	C₄×C₅₃	—	2	8	no	—
240	C₄×C₅₃	—	2	4	no	—
241	C₄×C₅₃	—	2	8	no	—
242	C₄×C₅₃	—	2	4	no	—
243	C₄×C₅₃	—	2	4	no	—
244	C₄×C₅₃	—	2	4	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	4	no	—
247	C₄×C₅₃	—	4	2	no	—
248	C₄×C₅₃	—	4	4	no	—
249	C₄×C₅₃	C₂×C₂²⋊Q₈	1	8	yes	—
250	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
251	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
252	#2451	C₂×C₅₃	1	16	yes	direct product factor
253	C₂×#398	C₂×C₅₃	1	2	yes	—
254	C₂×#395	C₂×C₅₃	1	2	yes	—
255	C₂×#391	C₂×C₅₃	1	1	yes	characteristic
256	C₂×#391	C₂×C₅₃	1	1	yes	characteristic
257	C₂×#391	C₂×C₅₃	1	2	yes	—
258	C₂×#389	C₂×C₅₃	1	1	yes	characteristic
259	C₂×#389	C₂×C₅₃	1	1	yes	characteristic
260	C₂×#387	C₂×C₅₃	1	2	yes	—
261	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
262	C₂×C₅₃	#2451	1	8	yes	direct product factor
263	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
264	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
265	C₂×C₅₃	C₂×#542	1	1	yes	characteristic
266	C₂×C₅₃	C₂×#546	1	1	yes	characteristic
267	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
268	C₂×C₅₃	C₂×#557	1	1	yes	characteristic
269	C₂×C₅₃	C₂×Q₈²	1	1	yes	characteristic
270	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
271	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
272	C₂×C₄⋊Q₈	—	2	8	no	—
273	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	—
274	C₂×C₄×Q₈	—	2	8	no	—
275	C₂×C₄×Q₈	C₂²×C₅₃	1	8	yes	—
276	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
277	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
279	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
282	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
283	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
285	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
286	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
287	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
289	#398	C₂²×C₅₃	1	16	yes	—
290	#395	C₂²×C₅₃	1	16	yes	—
291	#391	C₂²×C₅₃	1	16	yes	—
292	#391	C₂²×C₅₃	1	8	yes	—
293	#391	C₂²×C₅₃	1	8	yes	—
294	#389	C₂²×C₅₃	1	8	yes	—
295	#389	C₂²×C₅₃	1	8	yes	—
296	#387	C₂²×C₅₃	1	16	yes	—
297	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
298	C₂×#95	C₂²×C₅₃	1	2	yes	—
299	C₂×#95	C₂²×C₅₃	1	2	yes	—
300	C₂×#95	C₂²×C₅₃	1	2	yes	—
301	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×#95	C₂²×C₅₃	1	2	yes	—
303	C₂×#95	C₂²×C₅₃	1	2	yes	—
304	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
305	C₂×#95	C₂²×C₅₃	1	2	yes	—
306	C₅₃	C₂×#2451	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²×Q₈	—	2	4	no	—
308	C₂²×Q₈	—	2	8	no	—
309	C₂²×Q₈	—	2	4	no	—
310	C₂²×Q₈	—	2	8	no	—
311	C₂²×Q₈	—	4	4	no	—
312	C₂²×Q₈	Q₈×C₅₃	1	8	yes	complemented
313	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
314	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
315	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
317	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
318	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
320	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
323	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₄⋊Q₈	—	2	32	no	—
327	C₄×Q₈	—	2	32	no	—
328	C₂×C₄⋊C₄	—	2	8	no	—
329	C₂×C₄⋊C₄	—	2	8	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	—	2	4	no	—
333	C₂×C₄⋊C₄	—	4	4	no	—
334	C₂×C₄⋊C₄	—	4	4	no	—
335	C₂×C₄⋊C₄	D₄×C₅₃	1	8	yes	—
336	C₂×C₄⋊C₄	Q₈×C₅₃	1	8	yes	complemented
337	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
338	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
341	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
346	C₂×C₄²	—	2	8	no	—
347	C₂×C₄²	—	2	4	no	—
348	C₂×C₄²	—	2	4	no	—
349	C₂×C₄²	—	2	4	no	—
350	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
351	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
352	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
353	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
354	#95	C₂³×C₅₃	1	4	yes	—
355	#95	C₂³×C₅₃	1	4	yes	—
356	#95	C₂³×C₅₃	1	8	yes	—
357	#95	C₂³×C₅₃	1	8	yes	—
358	#95	C₂³×C₅₃	1	8	yes	—
359	#95	C₂³×C₅₃	1	8	yes	—
360	#95	C₂³×C₅₃	1	8	yes	—
361	#95	C₂³×C₅₃	1	8	yes	—
362	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
363	C₂×Q₈	—	2	32	no	—
364	C₂×Q₈	—	2	8	no	—
365	C₂×Q₈	—	2	16	no	—
366	C₂×Q₈	—	2	8	no	—
367	C₂×Q₈	—	2	16	no	—
368	C₂×Q₈	—	4	8	no	—
369	C₂×Q₈	—	4	32	no	—
370	C₂×Q₈	C₂×Q₈×C₅₃	1	16	yes	complemented
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
396	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
397	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
398	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
399	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
400	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
401	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
402	C₂²×C₄	#41×C₅₃	1	4	yes	—
403	C₂²×C₄	#41×C₅₃	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
408	C₂²×C₄	C₂⁴×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	2	yes	—
412	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
414	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
415	C₄⋊C₄	—	2	16	no	—
416	C₄⋊C₄	—	2	16	no	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	4	8	no	—
421	C₄⋊C₄	—	4	8	no	—
422	C₄⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
423	C₄⋊C₄	C₂×Q₈×C₅₃	1	16	yes	complemented
424	C₄²	—	2	16	no	—
425	C₄²	—	2	8	no	—
426	C₄²	—	2	8	no	—
427	C₄²	—	2	8	no	—
428	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
429	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
430	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
431	C₂³	C₂×#41×C₅₃	1	2	yes	—
432	C₂³	C₂×#41×C₅₃	1	2	yes	—
433	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
434	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
435	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
436	Q₈	—	2	32	no	—
437	Q₈	—	4	32	no	—
438	C₂×C₄	—	2	16	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	8	no	—
472	C₂×C₄	C₂²⋊Q₈×C₅₃	1	16	yes	—
473	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
474	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
475	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
476	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
477	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
478	C₂×C₄	C₂²×Q₈×C₅₃	1	8	yes	—
479	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
480	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
481	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
482	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
483	C₂²	C₅₃×#533	1	4	yes	—
484	C₂²	C₅₃×#542	1	4	yes	—
485	C₂²	C₅₃×#546	1	4	yes	—
486	C₂²	D₄×Q₈×C₅₃	1	4	yes	—
487	C₂²	C₅₃×#557	1	4	yes	—
488	C₂²	Q₈²×C₅₃	1	4	yes	—
489	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
490	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
491	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
492	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
493	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
494	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
495	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
496	C₄	—	2	8	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	8	no	—
501	C₄	—	2	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	4	no	—
506	C₄	C₂×C₂²⋊Q₈×C₅₃	1	8	yes	—
507	C₂	C₅₃×#2451	1	8	yes	direct product factor
508	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
509	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
510	C₂	C₂×C₅₃×#542	1	1	yes	characteristic
511	C₂	C₂×C₅₃×#546	1	1	yes	characteristic
512	C₂	C₂×D₄×Q₈×C₅₃	1	1	yes	characteristic
513	C₂	C₂×C₅₃×#557	1	1	yes	characteristic
514	C₂	C₂×Q₈²×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2451	C₂	direct (N × H)
54. C₂²×Q₈×C₅₃	Q₈	semidirect
78. C₂×C₄⋊C₄×C₅₃	Q₈	semidirect
112. C₂×Q₈×C₅₃	C₂×Q₈	semidirect
165. C₄⋊C₄×C₅₃	C₂×Q₈	semidirect
224. C₂×#2451	C₅₃	direct (N × H)
252. #2451	C₂×C₅₃	direct (N × H)
262. C₂×C₅₃	#2451	direct (N × H)
306. C₅₃	C₂×#2451	direct (N × H)
312. C₂²×Q₈	Q₈×C₅₃	semidirect
336. C₂×C₄⋊C₄	Q₈×C₅₃	semidirect
370. C₂×Q₈	C₂×Q₈×C₅₃	semidirect
423. C₄⋊C₄	C₂×Q₈×C₅₃	semidirect
507. C₂	C₅₃×#2451	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₅₃×#2451	2
C₂×C₅₃×#398	2
C₂×C₅₃×#395	2
C₂×C₅₃×#391	2
C₂×C₅₃×#389	2
C₂×C₅₃×#387	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×#2451	53

Subgroups — C2×C53×#2451 #27242202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2451 #27242202