Subgroups — C₂×C₅₃×#3153 #27268150

Order 13568.

3390 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×AGL(1,ℤ/8)×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3153	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅₃×#497	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#496	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#495	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	2	yes	—
11	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#137×C₅₃	—	2	16	no	—
14	C₂×#137×C₅₃	C₂²	1	16	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
16	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	16	yes	complemented
17	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
18	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	—
19	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
20	C₂²×D₈×C₅₃	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#532	—	2	4	no	—
23	C₅₃×#532	C₂²	1	8	yes	complemented
24	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×#124×C₅₃	—	2	2	no	—
27	C₂×#124×C₅₃	—	2	4	no	—
28	C₂×#124×C₅₃	C₂²	1	2	yes	—
29	C₂×#124×C₅₃	C₂²	1	2	yes	—
30	C₂×#124×C₅₃	C₂²	1	8	yes	complemented
31	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
32	C₂×#121×C₅₃	—	2	2	no	—
33	C₂×C₄×D₄×C₅₃	—	2	2	no	—
34	C₂²×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#497	C₂²	1	16	yes	complemented
37	C₅₃×#496	C₂²	1	16	yes	complemented
38	C₅₃×#495	C₂²	1	32	yes	complemented
39	C₄×M₁₆×C₅₃	C₂²	1	8	yes	complemented
40	C₂×#97×C₅₃	C₂²	1	2	yes	complemented
41	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
42	C₂⁵×C₅₃	—	2	1	no	—
43	C₂×#41×C₅₃	—	2	4	no	—
44	C₂×#41×C₅₃	—	2	8	no	—
45	C₂×#41×C₅₃	—	2	8	no	—
46	C₂×#41×C₅₃	C₂³	1	8	yes	—
47	C₂²×Q₈×C₅₃	—	2	2	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	8	no	—
50	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
52	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₅₃	—	2	2	no	—
54	C₂²×D₄×C₅₃	—	2	8	no	—
55	C₂²×D₄×C₅₃	—	2	2	no	—
56	C₂²×D₄×C₅₃	—	2	2	no	—
57	C₂²×D₄×C₅₃	—	2	1	no	—
58	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
60	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	—	2	1	no	—
62	C₂³×C₄×C₅₃	—	2	2	no	—
63	C₂³×C₄×C₅₃	D₄	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	#137×C₅₃	—	2	64	no	—
66	AGL(1,ℤ/8)×C₅₃	—	2	64	no	—
67	C₂×Q₁₆×C₅₃	—	2	16	no	—
68	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	complemented
69	C₂×SD₁₆×C₅₃	—	2	16	no	—
70	C₂×SD₁₆×C₅₃	—	2	16	no	—
71	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	—
72	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	complemented
73	C₂×D₈×C₅₃	—	2	16	no	—
74	C₂×D₈×C₅₃	C₂³	1	16	yes	—
75	C₂×M₁₆×C₅₃	D₄	1	16	yes	complemented
76	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
77	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
78	C₄⋊Q₈×C₅₃	—	2	2	no	—
79	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	complemented
80	#127×C₅₃	—	2	2	no	—
81	#127×C₅₃	C₂³	1	4	yes	complemented
82	#124×C₅₃	—	2	4	no	—
83	#124×C₅₃	—	2	4	no	—
84	#124×C₅₃	—	2	16	no	—
85	#124×C₅₃	—	2	4	no	—
86	#124×C₅₃	C₂³	1	8	yes	—
87	#124×C₅₃	C₂³	1	8	yes	—
88	#124×C₅₃	C₂³	1	8	yes	complemented
89	#121×C₅₃	—	2	16	no	—
90	C₄×D₄×C₅₃	—	2	16	no	—
91	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
94	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
95	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
96	C₂×C₄²×C₅₃	D₄	1	2	yes	—
97	C₂×C₄²×C₅₃	D₄	1	4	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
101	#97×C₅₃	C₂³	1	8	yes	complemented
102	C₄×C₈×C₅₃	C₂³	1	8	yes	complemented
103	C₂⁴×C₅₃	—	2	4	no	—
104	C₂⁴×C₅₃	—	2	2	no	—
105	C₂⁴×C₅₃	—	2	2	no	—
106	C₂⁴×C₅₃	—	4	8	no	—
107	C₂⁴×C₅₃	—	4	4	no	—
108	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
109	#41×C₅₃	—	2	16	no	—
110	#41×C₅₃	—	2	16	no	—
111	#41×C₅₃	—	4	16	no	—
112	C₂×Q₈×C₅₃	—	2	4	no	—
113	C₂×Q₈×C₅₃	—	2	2	no	—
114	C₂×Q₈×C₅₃	—	2	2	no	—
115	C₂×Q₈×C₅₃	—	2	4	no	—
116	C₂×Q₈×C₅₃	—	2	8	no	—
117	C₂×Q₈×C₅₃	—	2	4	no	—
118	C₂×Q₈×C₅₃	—	2	16	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×Q₈×C₅₃	—	4	8	no	—
121	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
122	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
123	C₂×D₄×C₅₃	—	2	8	no	—
124	C₂×D₄×C₅₃	—	2	4	no	—
125	C₂×D₄×C₅₃	—	2	8	no	—
126	C₂×D₄×C₅₃	—	2	4	no	—
127	C₂×D₄×C₅₃	—	2	16	no	—
128	C₂×D₄×C₅₃	—	2	8	no	—
129	C₂×D₄×C₅₃	—	2	4	no	—
130	C₂×D₄×C₅₃	—	2	2	no	—
131	C₂×D₄×C₅₃	—	2	2	no	—
132	C₂×D₄×C₅₃	—	4	8	no	—
133	C₂×D₄×C₅₃	—	4	8	no	—
134	C₂×D₄×C₅₃	—	4	4	no	—
135	C₂×D₄×C₅₃	—	4	8	no	—
136	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
137	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₅₃	—	2	4	no	—
139	C₂²×C₄×C₅₃	—	2	2	no	—
140	C₂²×C₄×C₅₃	—	2	2	no	—
141	C₂²×C₄×C₅₃	—	2	8	no	—
142	C₂²×C₄×C₅₃	—	4	4	no	—
143	C₂²×C₄×C₅₃	—	4	4	no	—
144	C₂²×C₄×C₅₃	—	4	8	no	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
148	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₅₃	—	2	32	no	—
154	SD₁₆×C₅₃	—	2	32	no	—
155	SD₁₆×C₅₃	—	2	32	no	—
156	D₈×C₅₃	—	2	32	no	—
157	M₁₆×C₅₃	C₂×D₄	1	32	yes	complemented
158	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	—
159	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
160	C₄⋊C₄×C₅₃	—	2	8	no	—
161	C₂²⋊C₄×C₅₃	—	2	16	no	—
162	C₂²⋊C₄×C₅₃	—	2	16	no	—
163	C₂²⋊C₄×C₅₃	—	2	16	no	—
164	C₄²×C₅₃	C₂×D₄	1	8	yes	—
165	C₄²×C₅₃	C₂×D₄	1	2	yes	—
166	C₄²×C₅₃	C₂×D₄	1	2	yes	—
167	C₄²×C₅₃	C₂⁴	1	2	yes	—
168	C₄²×C₅₃	C₂⁴	1	2	yes	—
169	C₂³×C₅₃	—	2	4	no	—
170	C₂³×C₅₃	—	2	4	no	—
171	C₂³×C₅₃	—	2	4	no	—
172	C₂³×C₅₃	—	4	16	no	—
173	C₂³×C₅₃	—	4	4	no	—
174	C₂³×C₅₃	—	4	8	no	—
175	C₂³×C₅₃	—	4	4	no	—
176	C₂³×C₅₃	—	4	8	no	—
177	C₂³×C₅₃	—	4	8	no	—
178	C₂³×C₅₃	—	8	16	no	—
179	C₂³×C₅₃	#127	1	4	yes	—
180	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
181	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₅₃	—	2	8	no	—
183	Q₈×C₅₃	—	2	4	no	—
184	Q₈×C₅₃	—	2	4	no	—
185	Q₈×C₅₃	—	4	8	no	—
186	Q₈×C₅₃	—	4	4	no	—
187	Q₈×C₅₃	—	4	4	no	—
188	D₄×C₅₃	—	2	8	no	—
189	D₄×C₅₃	—	2	4	no	—
190	D₄×C₅₃	—	2	4	no	—
191	D₄×C₅₃	—	4	16	no	—
192	D₄×C₅₃	—	4	16	no	—
193	D₄×C₅₃	—	4	8	no	—
194	D₄×C₅₃	—	4	4	no	—
195	D₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	4	8	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	8	no	—
203	C₂×C₄×C₅₃	—	4	4	no	—
204	C₂×C₄×C₅₃	—	4	16	no	—
205	C₂×C₄×C₅₃	#127	1	4	yes	—
206	C₂×C₄×C₅₃	#127	1	8	yes	—
207	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
209	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
213	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
214	C₈×C₅₃	C₂²×D₄	1	16	yes	—
215	C₂×#3153	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	8	no	—
217	C₂²×C₅₃	—	2	4	no	—
218	C₂²×C₅₃	—	4	4	no	—
219	C₂²×C₅₃	—	4	4	no	—
220	C₂²×C₅₃	—	4	8	no	—
221	C₂²×C₅₃	—	4	8	no	—
222	C₂²×C₅₃	—	8	16	no	—
223	C₂²×C₅₃	—	8	8	no	—
224	C₂²×C₅₃	—	8	8	no	—
225	C₂²×C₅₃	C₂×#127	1	4	yes	—
226	C₂²×C₅₃	C₂×#127	1	2	yes	—
227	C₂²×C₅₃	#575	1	4	yes	—
228	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
229	C₄×C₅₃	—	2	4	no	—
230	C₄×C₅₃	—	2	4	no	—
231	C₄×C₅₃	—	4	4	no	—
232	C₄×C₅₃	—	4	4	no	—
233	C₄×C₅₃	C₂×#127	1	4	yes	—
234	C₄×C₅₃	C₂×#127	1	2	yes	—
235	C₄×C₅₃	C₂³×D₄	1	2	yes	—
236	C₂²×#137	C₂×C₅₃	1	2	yes	complemented
237	C₂²×AGL(1,ℤ/8)	C₂×C₅₃	1	2	yes	complemented
238	C₂×#532	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂²×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
240	#3153	C₂×C₅₃	1	16	yes	direct product factor
241	C₂×#497	C₂×C₅₃	1	2	yes	complemented
242	C₂×#496	C₂×C₅₃	1	2	yes	complemented
243	C₂×#495	C₂×C₅₃	1	4	yes	complemented
244	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
245	C₂×C₅₃	—	2	4	no	—
246	C₂×C₅₃	—	8	4	no	—
247	C₂×C₅₃	—	8	4	no	—
248	C₂×C₅₃	#3153	1	4	yes	direct product factor
249	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
250	C₂×C₅₃	C₂×#575	1	2	yes	—
251	C₂²×#41	C₂²×C₅₃	1	2	yes	—
252	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
253	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×#137	—	2	16	no	—
255	C₂×#137	C₂²×C₅₃	1	16	yes	complemented
256	C₂×AGL(1,ℤ/8)	—	2	16	no	—
257	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	16	yes	complemented
258	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
259	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	—
260	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
261	C₂²×D₈	C₂²×C₅₃	1	4	yes	—
262	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
263	#532	—	2	4	no	—
264	#532	C₂²×C₅₃	1	8	yes	complemented
265	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
266	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
267	C₂×#124	—	2	2	no	—
268	C₂×#124	—	2	4	no	—
269	C₂×#124	C₂²×C₅₃	1	2	yes	—
270	C₂×#124	C₂²×C₅₃	1	2	yes	—
271	C₂×#124	C₂²×C₅₃	1	8	yes	complemented
272	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
273	C₂×#121	—	2	2	no	—
274	C₂×C₄×D₄	—	2	2	no	—
275	C₂²×C₂²⋊C₄	—	2	2	no	—
276	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
277	#497	C₂²×C₅₃	1	16	yes	complemented
278	#496	C₂²×C₅₃	1	16	yes	complemented
279	#495	C₂²×C₅₃	1	32	yes	complemented
280	C₄×M₁₆	C₂²×C₅₃	1	8	yes	complemented
281	C₂×#97	C₂²×C₅₃	1	2	yes	complemented
282	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
283	C₅₃	C₂×#3153	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁵	—	2	1	no	—
285	C₂×#41	—	2	4	no	—
286	C₂×#41	—	2	8	no	—
287	C₂×#41	—	2	8	no	—
288	C₂×#41	C₂³×C₅₃	1	8	yes	—
289	C₂²×Q₈	—	2	1	no	—
290	C₂²×Q₈	—	2	2	no	—
291	C₂²×Q₈	—	2	8	no	—
292	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
293	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
294	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	8	no	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
301	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
302	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	2	no	—
305	C₂³×C₄	D₄×C₅₃	1	2	yes	—
306	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
307	#137	—	2	64	no	—
308	AGL(1,ℤ/8)	—	2	64	no	—
309	C₂×Q₁₆	—	2	16	no	—
310	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	complemented
311	C₂×SD₁₆	—	2	16	no	—
312	C₂×SD₁₆	—	2	16	no	—
313	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	—
314	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	complemented
315	C₂×D₈	—	2	16	no	—
316	C₂×D₈	C₂³×C₅₃	1	16	yes	—
317	C₂×M₁₆	D₄×C₅₃	1	16	yes	complemented
318	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
319	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	complemented
322	#127	—	2	2	no	—
323	#127	C₂³×C₅₃	1	4	yes	complemented
324	#124	—	2	4	no	—
325	#124	—	2	4	no	—
326	#124	—	2	16	no	—
327	#124	—	2	4	no	—
328	#124	C₂³×C₅₃	1	8	yes	—
329	#124	C₂³×C₅₃	1	8	yes	—
330	#124	C₂³×C₅₃	1	8	yes	complemented
331	#121	—	2	16	no	—
332	C₄×D₄	—	2	16	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	4	no	—
335	C₂×C₂²⋊C₄	—	2	4	no	—
336	C₂×C₂²⋊C₄	—	2	16	no	—
337	C₂×C₂²⋊C₄	—	2	4	no	—
338	C₂×C₄²	D₄×C₅₃	1	2	yes	—
339	C₂×C₄²	D₄×C₅₃	1	4	yes	—
340	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
343	#97	C₂³×C₅₃	1	8	yes	complemented
344	C₄×C₈	C₂³×C₅₃	1	8	yes	complemented
345	C₂⁴	—	2	4	no	—
346	C₂⁴	—	2	2	no	—
347	C₂⁴	—	2	2	no	—
348	C₂⁴	—	4	8	no	—
349	C₂⁴	—	4	4	no	—
350	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
351	#41	—	2	16	no	—
352	#41	—	2	16	no	—
353	#41	—	4	16	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	4	no	—
358	C₂×Q₈	—	2	8	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	16	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	8	no	—
363	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
364	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
365	C₂×D₄	—	2	8	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	8	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	16	no	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	4	8	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
379	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
390	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
393	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
395	Q₁₆	—	2	32	no	—
396	SD₁₆	—	2	32	no	—
397	SD₁₆	—	2	32	no	—
398	D₈	—	2	32	no	—
399	M₁₆	C₂×D₄×C₅₃	1	32	yes	complemented
400	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	—
401	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
402	C₄⋊C₄	—	2	8	no	—
403	C₂²⋊C₄	—	2	16	no	—
404	C₂²⋊C₄	—	2	16	no	—
405	C₂²⋊C₄	—	2	16	no	—
406	C₄²	C₂×D₄×C₅₃	1	8	yes	—
407	C₄²	C₂×D₄×C₅₃	1	2	yes	—
408	C₄²	C₂×D₄×C₅₃	1	2	yes	—
409	C₄²	C₂⁴×C₅₃	1	2	yes	—
410	C₄²	C₂⁴×C₅₃	1	2	yes	—
411	C₂³	—	2	4	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	4	16	no	—
415	C₂³	—	4	4	no	—
416	C₂³	—	4	8	no	—
417	C₂³	—	4	4	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	8	no	—
420	C₂³	—	8	16	no	—
421	C₂³	#127×C₅₃	1	4	yes	—
422	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
423	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
424	Q₈	—	2	4	no	—
425	Q₈	—	2	8	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	4	8	no	—
428	Q₈	—	4	4	no	—
429	Q₈	—	4	4	no	—
430	D₄	—	2	4	no	—
431	D₄	—	2	8	no	—
432	D₄	—	2	4	no	—
433	D₄	—	4	16	no	—
434	D₄	—	4	16	no	—
435	D₄	—	4	8	no	—
436	D₄	—	4	4	no	—
437	D₄	—	4	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	16	no	—
447	C₂×C₄	#127×C₅₃	1	4	yes	—
448	C₂×C₄	#127×C₅₃	1	8	yes	—
449	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
450	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
451	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
454	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
455	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
456	C₈	C₂²×D₄×C₅₃	1	16	yes	—
457	C₂²	—	2	4	no	—
458	C₂²	—	2	8	no	—
459	C₂²	—	4	8	no	—
460	C₂²	—	4	8	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	8	16	no	—
464	C₂²	—	8	8	no	—
465	C₂²	—	8	8	no	—
466	C₂²	C₂×#127×C₅₃	1	4	yes	—
467	C₂²	C₂×#127×C₅₃	1	2	yes	—
468	C₂²	C₅₃×#575	1	4	yes	—
469	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
470	C₄	—	2	4	no	—
471	C₄	—	2	4	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	4	no	—
474	C₄	C₂×#127×C₅₃	1	4	yes	—
475	C₄	C₂×#127×C₅₃	1	2	yes	—
476	C₄	C₂³×D₄×C₅₃	1	2	yes	—
477	C₂	—	2	4	no	—
478	C₂	—	8	4	no	—
479	C₂	—	8	4	no	—
480	C₂	C₅₃×#3153	1	4	yes	direct product factor
481	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
482	C₂	C₂×C₅₃×#575	1	2	yes	—

	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 ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₅₃	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
3. C₂×C₅₃×#532	C₂	semidirect
4. C₂²×#124×C₅₃	C₂	semidirect
5. C₅₃×#3153	C₂	direct (N × H)
6. C₂×C₅₃×#497	C₂	semidirect
7. C₂×C₅₃×#496	C₂	semidirect
8. C₂×C₅₃×#495	C₂	semidirect
9. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
14. C₂×#137×C₅₃	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
17. C₂²×Q₁₆×C₅₃	C₂²	semidirect
19. C₂²×SD₁₆×C₅₃	C₂²	semidirect
23. C₅₃×#532	C₂²	semidirect
24. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
25. C₂×#127×C₅₃	C₂²	semidirect
30. C₂×#124×C₅₃	C₂²	semidirect
31. C₂×#124×C₅₃	C₂²	semidirect
36. C₅₃×#497	C₂²	semidirect
37. C₅₃×#496	C₂²	semidirect
38. C₅₃×#495	C₂²	semidirect
39. C₄×M₁₆×C₅₃	C₂²	semidirect
40. C₂×#97×C₅₃	C₂²	semidirect
41. C₂×C₄×C₈×C₅₃	C₂²	semidirect
68. C₂×Q₁₆×C₅₃	C₂³	semidirect
72. C₂×SD₁₆×C₅₃	C₂³	semidirect
75. C₂×M₁₆×C₅₃	D₄	semidirect
79. C₄⋊Q₈×C₅₃	C₂³	semidirect
81. #127×C₅₃	C₂³	semidirect
88. #124×C₅₃	C₂³	semidirect
101. #97×C₅₃	C₂³	semidirect
102. C₄×C₈×C₅₃	C₂³	semidirect
157. M₁₆×C₅₃	C₂×D₄	semidirect
215. C₂×#3153	C₅₃	direct (N × H)
236. C₂²×#137	C₂×C₅₃	semidirect
237. C₂²×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
238. C₂×#532	C₂×C₅₃	semidirect
239. C₂²×#124	C₂×C₅₃	semidirect
240. #3153	C₂×C₅₃	direct (N × H)
241. C₂×#497	C₂×C₅₃	semidirect
242. C₂×#496	C₂×C₅₃	semidirect
243. C₂×#495	C₂×C₅₃	semidirect
244. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
248. C₂×C₅₃	#3153	direct (N × H)
255. C₂×#137	C₂²×C₅₃	semidirect
257. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
258. C₂²×Q₁₆	C₂²×C₅₃	semidirect
260. C₂²×SD₁₆	C₂²×C₅₃	semidirect
264. #532	C₂²×C₅₃	semidirect
265. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
266. C₂×#127	C₂²×C₅₃	semidirect
271. C₂×#124	C₂²×C₅₃	semidirect
272. C₂×#124	C₂²×C₅₃	semidirect
277. #497	C₂²×C₅₃	semidirect
278. #496	C₂²×C₅₃	semidirect
279. #495	C₂²×C₅₃	semidirect
280. C₄×M₁₆	C₂²×C₅₃	semidirect
281. C₂×#97	C₂²×C₅₃	semidirect
282. C₂×C₄×C₈	C₂²×C₅₃	semidirect
283. C₅₃	C₂×#3153	direct (N × H)
310. C₂×Q₁₆	C₂³×C₅₃	semidirect
314. C₂×SD₁₆	C₂³×C₅₃	semidirect
317. C₂×M₁₆	D₄×C₅₃	semidirect
321. C₄⋊Q₈	C₂³×C₅₃	semidirect
323. #127	C₂³×C₅₃	semidirect
330. #124	C₂³×C₅₃	semidirect
343. #97	C₂³×C₅₃	semidirect
344. C₄×C₈	C₂³×C₅₃	semidirect
399. M₁₆	C₂×D₄×C₅₃	semidirect
480. C₂	C₅₃×#3153	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×#137×C₅₃	2
C₂²×AGL(1,ℤ/8)×C₅₃	2
C₂×C₅₃×#532	2
C₂²×#124×C₅₃	2
C₅₃×#3153	2
C₂×C₅₃×#497	2
C₂×C₅₃×#496	2
C₂×C₅₃×#495	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#3153	53

Subgroups — C2×C53×#3153 #27268150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3153 #27268150