Subgroups — C₂×C₅₃×#3330 #27268662

Order 13568.

2374 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3330	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#484	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#449	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅₃×#417	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₅₃	—	2	1	no	—
14	C₂²×#41×C₅₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#547	—	2	4	no	—
19	C₅₃×#547	—	2	4	no	—
20	C₅₃×#547	C₂²	1	8	yes	complemented
21	C₅₃×#547	C₂²	1	8	yes	complemented
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#123×C₅₃	—	2	2	no	—
25	C₂×#123×C₅₃	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
28	C₂×#121×C₅₃	—	2	1	no	—
29	C₂×#121×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₅₃×#487	C₂²	1	8	yes	complemented
36	C₅₃×#484	C₂²	1	16	yes	complemented
37	C₅₃×#474	C₂²	1	8	yes	complemented
38	C₅₃×#468	C₂²	1	16	yes	complemented
39	C₅₃×#461	C₂²	1	8	yes	complemented
40	C₅₃×#449	C₂²	1	16	yes	complemented
41	C₄×D₈×C₅₃	C₂²	1	8	yes	complemented
42	C₅₃×#436	C₂²	1	8	yes	complemented
43	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#107×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#417	C₂²	1	16	yes	complemented
47	C₂×#103×C₅₃	C₂²	1	2	yes	—
48	C₂×#103×C₅₃	C₂²	1	2	yes	—
49	C₂×#103×C₅₃	C₂²	1	2	yes	—
50	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
51	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×#102×C₅₃	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#41×C₅₃	—	2	8	no	—
56	C₂×#41×C₅₃	—	2	4	no	—
57	C₂×#41×C₅₃	—	2	8	no	—
58	C₂×#41×C₅₃	—	4	4	no	—
59	C₂×#41×C₅₃	—	4	8	no	—
60	C₂×#41×C₅₃	C₂³	1	8	yes	—
61	C₂²×Q₈×C₅₃	—	2	1	no	—
62	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
63	C₂²×D₄×C₅₃	—	2	2	no	—
64	C₂²×D₄×C₅₃	—	2	2	no	—
65	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	—	2	2	no	—
69	C₂³×C₄×C₅₃	—	2	1	no	—
70	C₂³×C₄×C₅₃	—	2	2	no	—
71	C₂³×C₄×C₅₃	—	2	1	no	—
72	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₅₃	—	4	4	no	—
74	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
75	C₂×D₈×C₅₃	—	2	4	no	—
76	C₂×D₈×C₅₃	C₂³	1	4	yes	—
77	C₂×M₁₆×C₅₃	—	4	4	no	—
78	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
81	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₅₃	—	2	2	no	—
83	C₄⋊Q₈×C₅₃	—	2	2	no	—
84	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
85	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
86	#123×C₅₃	—	2	16	no	—
87	#123×C₅₃	—	2	16	no	—
88	C₂²⋊Q₈×C₅₃	—	2	8	no	—
89	C₂²⋊Q₈×C₅₃	—	2	4	no	—
90	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
91	#121×C₅₃	—	2	8	no	—
92	#121×C₅₃	—	2	4	no	—
93	#121×C₅₃	C₂³	1	8	yes	—
94	C₄×D₄×C₅₃	—	2	2	no	—
95	C₄×D₄×C₅₃	—	2	2	no	—
96	C₄×D₄×C₅₃	D₄	1	4	yes	—
97	C₄×D₄×C₅₃	C₂³	1	4	yes	—
98	C₄×D₄×C₅₃	C₂³	1	4	yes	—
99	C₄×D₄×C₅₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
101	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	#107×C₅₃	C₂³	1	4	yes	—
119	#107×C₅₃	C₂³	1	8	yes	—
120	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
121	#103×C₅₃	C₂³	1	8	yes	—
122	#103×C₅₃	C₂³	1	8	yes	—
123	#103×C₅₃	C₂³	1	8	yes	—
124	#102×C₅₃	C₂³	1	4	yes	—
125	#102×C₅₃	C₂³	1	8	yes	—
126	#102×C₅₃	C₂³	1	4	yes	—
127	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
128	C₄×C₈×C₅₃	C₂³	1	4	yes	—
129	C₂⁴×C₅₃	—	2	1	no	—
130	C₂⁴×C₅₃	—	2	1	no	—
131	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
132	#41×C₅₃	—	2	16	no	—
133	#41×C₅₃	—	4	8	no	—
134	#41×C₅₃	—	4	16	no	—
135	#41×C₅₃	—	4	32	no	—
136	C₂×Q₈×C₅₃	—	2	2	no	—
137	C₂×Q₈×C₅₃	—	2	4	no	—
138	C₂×Q₈×C₅₃	—	2	4	no	—
139	C₂×Q₈×C₅₃	—	2	4	no	—
140	C₂×Q₈×C₅₃	—	4	4	no	—
141	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
142	C₂×D₄×C₅₃	—	2	8	no	—
143	C₂×D₄×C₅₃	—	2	8	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	2	no	—
149	C₂×D₄×C₅₃	—	2	2	no	—
150	C₂×D₄×C₅₃	—	4	8	no	—
151	C₂×D₄×C₅₃	—	4	8	no	—
152	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
153	C₂×D₄×C₅₃	#41	1	4	yes	complemented
154	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	8	no	—
159	C₂²×C₄×C₅₃	—	2	1	no	—
160	C₂²×C₄×C₅₃	—	2	2	no	—
161	C₂²×C₄×C₅₃	—	2	1	no	—
162	C₂²×C₄×C₅₃	—	2	2	no	—
163	C₂²×C₄×C₅₃	—	2	4	no	—
164	C₂²×C₄×C₅₃	—	2	8	no	—
165	C₂²×C₄×C₅₃	—	2	4	no	—
166	C₂²×C₄×C₅₃	—	4	8	no	—
167	C₂²×C₄×C₅₃	—	4	8	no	—
168	C₂²×C₄×C₅₃	—	4	4	no	—
169	C₂²×C₄×C₅₃	—	4	4	no	—
170	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
176	SD₁₆×C₅₃	—	4	8	no	—
177	D₈×C₅₃	—	2	8	no	—
178	M₁₆×C₅₃	—	4	8	no	—
179	C₂×C₈×C₅₃	—	2	2	no	—
180	C₂×C₈×C₅₃	—	2	2	no	—
181	C₂×C₈×C₅₃	—	2	4	no	—
182	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
183	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
184	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	4	no	—
187	C₄⋊C₄×C₅₃	—	2	8	no	—
188	C₄⋊C₄×C₅₃	—	2	2	no	—
189	C₄⋊C₄×C₅₃	—	2	8	no	—
190	C₄⋊C₄×C₅₃	—	2	4	no	—
191	C₄⋊C₄×C₅₃	—	2	4	no	—
192	C₄⋊C₄×C₅₃	—	2	1	no	—
193	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₅₃	—	2	8	no	—
198	C₂²⋊C₄×C₅₃	—	2	4	no	—
199	C₂²⋊C₄×C₅₃	—	2	4	no	—
200	C₂²⋊C₄×C₅₃	—	2	8	no	—
201	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
202	C₄²×C₅₃	C₂×D₄	1	2	yes	—
203	C₄²×C₅₃	C₂⁴	1	2	yes	—
204	C₂³×C₅₃	—	2	4	no	—
205	C₂³×C₅₃	—	2	4	no	—
206	C₂³×C₅₃	—	2	4	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	2	2	no	—
209	C₂³×C₅₃	—	2	2	no	—
210	C₂³×C₅₃	—	4	2	no	—
211	C₂³×C₅₃	—	4	2	no	—
212	C₂³×C₅₃	—	4	2	no	—
213	C₂³×C₅₃	—	4	2	no	—
214	C₂³×C₅₃	—	4	2	no	—
215	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
216	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₅₃	—	2	4	no	—
218	Q₈×C₅₃	—	2	4	no	—
219	Q₈×C₅₃	—	4	4	no	—
220	Q₈×C₅₃	—	4	8	no	—
221	D₄×C₅₃	—	2	4	no	—
222	D₄×C₅₃	—	2	8	no	—
223	D₄×C₅₃	—	2	2	no	—
224	D₄×C₅₃	—	2	2	no	—
225	D₄×C₅₃	—	2	2	no	—
226	D₄×C₅₃	—	4	16	no	—
227	D₄×C₅₃	—	4	16	no	—
228	D₄×C₅₃	—	4	2	no	—
229	D₄×C₅₃	C₂×#41	1	4	yes	complemented
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	8	no	—
233	C₂×C₄×C₅₃	—	2	4	no	—
234	C₂×C₄×C₅₃	—	2	4	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	4	16	no	—
240	C₂×C₄×C₅₃	—	4	16	no	—
241	C₂×C₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	2	no	—
244	C₂×C₄×C₅₃	—	4	4	no	—
245	C₂×C₄×C₅₃	—	4	8	no	—
246	C₂×C₄×C₅₃	—	4	8	no	—
247	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
248	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
253	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
254	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
255	C₈×C₅₃	—	2	2	no	—
256	C₈×C₅₃	—	2	2	no	—
257	C₈×C₅₃	—	2	4	no	—
258	C₂×#3330	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂²×C₅₃	—	2	4	no	—
260	C₂²×C₅₃	—	2	4	no	—
261	C₂²×C₅₃	—	2	4	no	—
262	C₂²×C₅₃	—	2	4	no	—
263	C₂²×C₅₃	—	4	8	no	—
264	C₂²×C₅₃	—	4	8	no	—
265	C₂²×C₅₃	—	4	2	no	—
266	C₂²×C₅₃	—	4	2	no	—
267	C₂²×C₅₃	—	4	2	no	—
268	C₂²×C₅₃	—	4	2	no	—
269	C₂²×C₅₃	—	4	2	no	—
270	C₂²×C₅₃	—	8	4	no	—
271	C₂²×C₅₃	#546	1	2	yes	—
272	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
273	C₂²×C₅₃	#578	1	2	yes	—
274	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₅₃	—	2	4	no	—
276	C₄×C₅₃	—	2	2	no	—
277	C₄×C₅₃	—	4	4	no	—
278	C₄×C₅₃	—	4	2	no	—
279	C₄×C₅₃	—	4	2	no	—
280	C₄×C₅₃	—	4	4	no	—
281	C₄×C₅₃	#546	1	2	yes	—
282	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
283	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
284	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×#547	C₂×C₅₃	1	1	yes	characteristic, complemented
286	C₂×#547	C₂×C₅₃	1	1	yes	characteristic, complemented
287	#3330	C₂×C₅₃	1	16	yes	direct product factor
288	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
289	C₂×#484	C₂×C₅₃	1	2	yes	complemented
290	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
291	C₂×#468	C₂×C₅₃	1	2	yes	complemented
292	C₂×#461	C₂×C₅₃	1	1	yes	characteristic, complemented
293	C₂×#449	C₂×C₅₃	1	2	yes	complemented
294	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
295	C₂×#436	C₂×C₅₃	1	1	yes	characteristic, complemented
296	C₂×#417	C₂×C₅₃	1	2	yes	complemented
297	C₂×C₅₃	—	4	4	no	—
298	C₂×C₅₃	—	4	4	no	—
299	C₂×C₅₃	—	8	2	no	—
300	C₂×C₅₃	#3330	1	4	yes	direct product factor
301	C₂×C₅₃	C₂×#546	1	1	yes	characteristic
302	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
304	C₂²×#41	—	2	1	no	—
305	C₂²×#41	C₂²×C₅₃	1	2	yes	—
306	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
307	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
308	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
309	#547	—	2	4	no	—
310	#547	—	2	4	no	—
311	#547	C₂²×C₅₃	1	8	yes	complemented
312	#547	C₂²×C₅₃	1	8	yes	complemented
313	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
315	C₂×#123	—	2	2	no	—
316	C₂×#123	—	2	2	no	—
317	C₂×C₂²⋊Q₈	—	2	1	no	—
318	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
319	C₂×#121	—	2	1	no	—
320	C₂×#121	C₂²×C₅₃	1	2	yes	—
321	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
324	C₂²×C₄⋊C₄	—	2	1	no	—
325	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
326	#487	C₂²×C₅₃	1	8	yes	complemented
327	#484	C₂²×C₅₃	1	16	yes	complemented
328	#474	C₂²×C₅₃	1	8	yes	complemented
329	#468	C₂²×C₅₃	1	16	yes	complemented
330	#461	C₂²×C₅₃	1	8	yes	complemented
331	#449	C₂²×C₅₃	1	16	yes	complemented
332	C₄×D₈	C₂²×C₅₃	1	8	yes	complemented
333	#436	C₂²×C₅₃	1	8	yes	complemented
334	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
335	C₂×#107	C₂²×C₅₃	1	2	yes	—
336	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
337	#417	C₂²×C₅₃	1	16	yes	complemented
338	C₂×#103	C₂²×C₅₃	1	2	yes	—
339	C₂×#103	C₂²×C₅₃	1	2	yes	—
340	C₂×#103	C₂²×C₅₃	1	2	yes	—
341	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
342	C₂×#102	C₂²×C₅₃	1	2	yes	—
343	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
345	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
346	C₅₃	C₂×#3330	1	1	yes	characteristic, direct product factor, Sylow
347	C₂×#41	—	2	8	no	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	8	no	—
350	C₂×#41	—	4	4	no	—
351	C₂×#41	—	4	8	no	—
352	C₂×#41	C₂³×C₅₃	1	8	yes	—
353	C₂²×Q₈	—	2	1	no	—
354	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
360	C₂³×C₄	—	2	2	no	—
361	C₂³×C₄	—	2	1	no	—
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
365	C₂×SD₁₆	—	4	4	no	—
366	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
367	C₂×D₈	—	2	4	no	—
368	C₂×D₈	C₂³×C₅₃	1	4	yes	—
369	C₂×M₁₆	—	4	4	no	—
370	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
371	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
373	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	—	2	2	no	—
376	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
377	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
378	#123	—	2	16	no	—
379	#123	—	2	16	no	—
380	C₂²⋊Q₈	—	2	4	no	—
381	C₂²⋊Q₈	—	2	8	no	—
382	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
383	#121	—	2	8	no	—
384	#121	—	2	4	no	—
385	#121	C₂³×C₅₃	1	8	yes	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	D₄×C₅₃	1	4	yes	—
389	C₄×D₄	C₂³×C₅₃	1	4	yes	—
390	C₄×D₄	C₂³×C₅₃	1	4	yes	—
391	C₄×D₄	C₂³×C₅₃	1	4	yes	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄⋊C₄	—	2	4	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	8	no	—
399	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
403	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	—	2	2	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
409	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
410	#107	C₂³×C₅₃	1	4	yes	—
411	#107	C₂³×C₅₃	1	8	yes	—
412	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
413	#103	C₂³×C₅₃	1	8	yes	—
414	#103	C₂³×C₅₃	1	8	yes	—
415	#103	C₂³×C₅₃	1	8	yes	—
416	#102	C₂³×C₅₃	1	4	yes	—
417	#102	C₂³×C₅₃	1	8	yes	—
418	#102	C₂³×C₅₃	1	4	yes	—
419	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
420	C₄×C₈	C₂³×C₅₃	1	4	yes	—
421	C₂⁴	—	2	1	no	—
422	C₂⁴	—	2	1	no	—
423	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
424	#41	—	2	16	no	—
425	#41	—	4	8	no	—
426	#41	—	4	16	no	—
427	#41	—	4	32	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	4	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
445	C₂×D₄	#41×C₅₃	1	4	yes	complemented
446	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
447	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	8	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	8	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	4	8	no	—
459	C₂²×C₄	—	4	8	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	—	4	4	no	—
462	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
465	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
468	SD₁₆	—	4	8	no	—
469	D₈	—	2	8	no	—
470	M₁₆	—	4	8	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	4	no	—
474	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
475	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
476	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	8	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
486	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
487	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
488	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	8	no	—
493	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
494	C₄²	C₂×D₄×C₅₃	1	2	yes	—
495	C₄²	C₂⁴×C₅₃	1	2	yes	—
496	C₂³	—	2	4	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	2	no	—
507	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
508	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
509	Q₈	—	2	4	no	—
510	Q₈	—	2	4	no	—
511	Q₈	—	4	4	no	—
512	Q₈	—	4	8	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	8	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	4	16	no	—
519	D₄	—	4	16	no	—
520	D₄	—	4	2	no	—
521	D₄	C₂×#41×C₅₃	1	4	yes	complemented
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	8	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	4	16	no	—
532	C₂×C₄	—	4	16	no	—
533	C₂×C₄	—	4	4	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	—	4	8	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
540	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
542	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
544	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
545	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
546	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
547	C₈	—	2	2	no	—
548	C₈	—	2	2	no	—
549	C₈	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	8	no	—
557	C₂²	—	4	8	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	8	4	no	—
562	C₂²	C₅₃×#546	1	2	yes	—
563	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
564	C₂²	C₅₃×#578	1	2	yes	—
565	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
566	C₄	—	2	4	no	—
567	C₄	—	2	2	no	—
568	C₄	—	4	4	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	4	no	—
571	C₄	—	4	2	no	—
572	C₄	C₅₃×#546	1	2	yes	—
573	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
574	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
575	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
576	C₂	—	4	4	no	—
577	C₂	—	4	4	no	—
578	C₂	—	8	2	no	—
579	C₂	C₅₃×#3330	1	4	yes	direct product factor
580	C₂	C₂×C₅₃×#546	1	1	yes	characteristic
581	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
582	C₂	C₂×C₅₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#547	C₂	semidirect
2. C₂×C₅₃×#547	C₂	semidirect
3. C₅₃×#3330	C₂	direct (N × H)
4. C₂×C₅₃×#487	C₂	semidirect
5. C₂×C₅₃×#484	C₂	semidirect
6. C₂×C₅₃×#474	C₂	semidirect
7. C₂×C₅₃×#468	C₂	semidirect
8. C₂×C₅₃×#461	C₂	semidirect
9. C₂×C₅₃×#449	C₂	semidirect
10. C₂×C₄×D₈×C₅₃	C₂	semidirect
11. C₂×C₅₃×#436	C₂	semidirect
12. C₂×C₅₃×#417	C₂	semidirect
20. C₅₃×#547	C₂²	semidirect
21. C₅₃×#547	C₂²	semidirect
35. C₅₃×#487	C₂²	semidirect
36. C₅₃×#484	C₂²	semidirect
37. C₅₃×#474	C₂²	semidirect
38. C₅₃×#468	C₂²	semidirect
39. C₅₃×#461	C₂²	semidirect
40. C₅₃×#449	C₂²	semidirect
41. C₄×D₈×C₅₃	C₂²	semidirect
42. C₅₃×#436	C₂²	semidirect
46. C₅₃×#417	C₂²	semidirect
153. C₂×D₄×C₅₃	#41	semidirect
229. D₄×C₅₃	C₂×#41	semidirect
258. C₂×#3330	C₅₃	direct (N × H)
285. C₂×#547	C₂×C₅₃	semidirect
286. C₂×#547	C₂×C₅₃	semidirect
287. #3330	C₂×C₅₃	direct (N × H)
288. C₂×#487	C₂×C₅₃	semidirect
289. C₂×#484	C₂×C₅₃	semidirect
290. C₂×#474	C₂×C₅₃	semidirect
291. C₂×#468	C₂×C₅₃	semidirect
292. C₂×#461	C₂×C₅₃	semidirect
293. C₂×#449	C₂×C₅₃	semidirect
294. C₂×C₄×D₈	C₂×C₅₃	semidirect
295. C₂×#436	C₂×C₅₃	semidirect
296. C₂×#417	C₂×C₅₃	semidirect
300. C₂×C₅₃	#3330	direct (N × H)
311. #547	C₂²×C₅₃	semidirect
312. #547	C₂²×C₅₃	semidirect
326. #487	C₂²×C₅₃	semidirect
327. #484	C₂²×C₅₃	semidirect
328. #474	C₂²×C₅₃	semidirect
329. #468	C₂²×C₅₃	semidirect
330. #461	C₂²×C₅₃	semidirect
331. #449	C₂²×C₅₃	semidirect
332. C₄×D₈	C₂²×C₅₃	semidirect
333. #436	C₂²×C₅₃	semidirect
337. #417	C₂²×C₅₃	semidirect
346. C₅₃	C₂×#3330	direct (N × H)
445. C₂×D₄	#41×C₅₃	semidirect
521. D₄	C₂×#41×C₅₃	semidirect
579. C₂	C₅₃×#3330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#547	2
C₅₃×#3330	2
C₂×C₅₃×#487	2
C₂×C₅₃×#484	2
C₂×C₅₃×#474	2
C₂×C₅₃×#468	2
C₂×C₅₃×#461	2
C₂×C₅₃×#449	2
C₂×C₄×D₈×C₅₃	2
C₂×C₅₃×#436	2
C₂×C₅₃×#417	2
C₂×#3330	53

Subgroups — C2×C53×#3330 #27268662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3330 #27268662