Subgroups — C₂×C₅₃×#3234 #27268359

Order 13568.

2050 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3234	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#501	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#490	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#481	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#480	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#478	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	—	2	1	no	—
11	C₅₃×#532	—	2	4	no	—
12	C₅₃×#532	C₂²	1	8	yes	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#125×C₅₃	—	2	1	no	—
16	C₂×#125×C₅₃	—	2	1	no	—
17	C₂×#125×C₅₃	—	2	4	no	—
18	C₂×#125×C₅₃	C₂²	1	2	yes	—
19	C₂×#125×C₅₃	C₂²	1	2	yes	—
20	C₂×#125×C₅₃	C₂²	1	8	yes	complemented
21	C₂×#124×C₅₃	—	2	1	no	—
22	C₂×#121×C₅₃	—	2	1	no	—
23	C₂×#121×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₅₃	—	2	1	no	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#501	C₂²	1	16	yes	complemented
32	C₅₃×#490	C₂²	1	16	yes	complemented
33	C₅₃×#481	C₂²	1	16	yes	complemented
34	C₅₃×#480	C₂²	1	16	yes	complemented
35	C₅₃×#478	C₂²	1	32	yes	complemented
36	C₅₃×#432	C₂²	1	8	yes	complemented
37	C₂×#107×C₅₃	C₂²	1	4	yes	—
38	C₂×#106×C₅₃	C₂²	1	4	yes	—
39	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
40	C₂×#102×C₅₃	C₂²	1	4	yes	—
41	C₂×#102×C₅₃	C₂²	1	4	yes	—
42	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
43	C₂×#97×C₅₃	C₂²	1	2	yes	—
44	C₂×#41×C₅₃	—	2	8	no	—
45	C₂×#41×C₅₃	—	4	8	no	—
46	C₂²×Q₈×C₅₃	—	2	1	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	C₂²×D₄×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₅₃	—	2	1	no	—
52	C₂³×C₄×C₅₃	—	2	1	no	—
53	C₂³×C₄×C₅₃	—	2	2	no	—
54	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
60	#127×C₅₃	—	2	2	no	—
61	#127×C₅₃	C₂³	1	4	yes	—
62	#125×C₅₃	—	2	4	no	—
63	#125×C₅₃	—	2	4	no	—
64	#125×C₅₃	—	2	8	no	—
65	#125×C₅₃	—	2	8	no	—
66	#125×C₅₃	C₂³	1	8	yes	—
67	#125×C₅₃	C₂³	1	8	yes	—
68	#124×C₅₃	—	2	8	no	—
69	#121×C₅₃	—	2	8	no	—
70	#121×C₅₃	—	2	4	no	—
71	#121×C₅₃	C₂³	1	8	yes	—
72	C₄×D₄×C₅₃	—	2	8	no	—
73	C₄×D₄×C₅₃	—	2	4	no	—
74	C₄×D₄×C₅₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	D₄	1	4	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
99	#107×C₅₃	C₂³	1	16	yes	—
100	#106×C₅₃	C₂³	1	16	yes	—
101	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
102	#102×C₅₃	C₂³	1	16	yes	—
103	#102×C₅₃	C₂³	1	16	yes	—
104	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
105	#97×C₅₃	C₂³	1	8	yes	—
106	C₂⁴×C₅₃	—	2	2	no	—
107	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
108	#41×C₅₃	—	4	32	no	—
109	C₂×Q₈×C₅₃	—	2	4	no	—
110	C₂×Q₈×C₅₃	—	4	4	no	—
111	C₂×D₄×C₅₃	—	2	8	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	4	no	—
114	C₂×D₄×C₅₃	—	2	4	no	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	4	4	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	4	8	no	—
119	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	—	2	4	no	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	8	no	—
128	C₂²×C₄×C₅₃	—	4	2	no	—
129	C₂²×C₄×C₅₃	—	4	8	no	—
130	C₂²×C₄×C₅₃	—	4	4	no	—
131	C₂²×C₄×C₅₃	—	4	4	no	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₅₃	—	2	8	no	—
143	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
144	C₄⋊C₄×C₅₃	—	2	8	no	—
145	C₄⋊C₄×C₅₃	—	2	8	no	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	2	8	no	—
148	C₄⋊C₄×C₅₃	—	2	4	no	—
149	C₄⋊C₄×C₅₃	—	2	4	no	—
150	C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	4	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
155	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂²⋊C₄×C₅₃	—	2	8	no	—
158	C₂²⋊C₄×C₅₃	—	2	8	no	—
159	C₄²×C₅₃	C₂×D₄	1	4	yes	—
160	C₄²×C₅₃	C₂×D₄	1	2	yes	—
161	C₄²×C₅₃	C₂×D₄	1	4	yes	—
162	C₄²×C₅₃	C₂×D₄	1	2	yes	—
163	C₄²×C₅₃	C₂⁴	1	2	yes	—
164	C₄²×C₅₃	C₂⁴	1	2	yes	—
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	2	2	no	—
167	C₂³×C₅₃	—	2	4	no	—
168	C₂³×C₅₃	—	4	4	no	—
169	C₂³×C₅₃	—	4	4	no	—
170	C₂³×C₅₃	—	4	4	no	—
171	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
172	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
173	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₅₃	—	4	8	no	—
175	D₄×C₅₃	—	2	8	no	—
176	D₄×C₅₃	—	4	8	no	—
177	D₄×C₅₃	—	4	8	no	—
178	D₄×C₅₃	—	4	16	no	—
179	D₄×C₅₃	—	4	4	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	8	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	4	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	4	no	—
189	C₂×C₄×C₅₃	—	2	4	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	4	4	no	—
192	C₂×C₄×C₅₃	—	4	16	no	—
193	C₂×C₄×C₅₃	—	4	8	no	—
194	C₂×C₄×C₅₃	—	4	8	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	4	no	—
197	C₂×C₄×C₅₃	—	4	4	no	—
198	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
205	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
206	C₈×C₅₃	—	2	8	no	—
207	C₂×#3234	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	2	8	no	—
210	C₂²×C₅₃	—	2	2	no	—
211	C₂²×C₅₃	—	4	4	no	—
212	C₂²×C₅₃	—	4	4	no	—
213	C₂²×C₅₃	—	4	4	no	—
214	C₂²×C₅₃	—	8	8	no	—
215	C₂²×C₅₃	#536	1	2	yes	—
216	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
217	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
218	C₂²×C₅₃	#575	1	2	yes	—
219	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	2	2	no	—
222	C₄×C₅₃	—	2	2	no	—
223	C₄×C₅₃	—	2	2	no	—
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	4	4	no	—
226	C₄×C₅₃	—	4	4	no	—
227	C₄×C₅₃	—	4	4	no	—
228	C₄×C₅₃	#536	1	2	yes	—
229	C₄×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
230	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
231	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#3234	C₂×C₅₃	1	16	yes	direct product factor
233	C₂×#501	C₂×C₅₃	1	2	yes	complemented
234	C₂×#490	C₂×C₅₃	1	2	yes	complemented
235	C₂×#481	C₂×C₅₃	1	2	yes	complemented
236	C₂×#480	C₂×C₅₃	1	2	yes	complemented
237	C₂×#478	C₂×C₅₃	1	4	yes	complemented
238	C₂×#432	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×C₅₃	—	2	4	no	—
240	C₂×C₅₃	—	8	4	no	—
241	C₂×C₅₃	#3234	1	4	yes	direct product factor
242	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
243	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×C₅₃	C₂×#575	1	1	yes	characteristic
245	C₂²×#41	—	2	1	no	—
246	#532	—	2	4	no	—
247	#532	C₂²×C₅₃	1	8	yes	—
248	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×#125	—	2	1	no	—
251	C₂×#125	—	2	1	no	—
252	C₂×#125	—	2	4	no	—
253	C₂×#125	C₂²×C₅₃	1	2	yes	—
254	C₂×#125	C₂²×C₅₃	1	2	yes	—
255	C₂×#125	C₂²×C₅₃	1	8	yes	complemented
256	C₂×#124	—	2	1	no	—
257	C₂×#121	—	2	1	no	—
258	C₂×#121	C₂²×C₅₃	1	2	yes	—
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
261	C₂²×C₄⋊C₄	—	2	1	no	—
262	C₂²×C₄⋊C₄	—	2	1	no	—
263	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
264	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
265	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
266	#501	C₂²×C₅₃	1	16	yes	complemented
267	#490	C₂²×C₅₃	1	16	yes	complemented
268	#481	C₂²×C₅₃	1	16	yes	complemented
269	#480	C₂²×C₅₃	1	16	yes	complemented
270	#478	C₂²×C₅₃	1	32	yes	complemented
271	#432	C₂²×C₅₃	1	8	yes	complemented
272	C₂×#107	C₂²×C₅₃	1	4	yes	—
273	C₂×#106	C₂²×C₅₃	1	4	yes	—
274	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
275	C₂×#102	C₂²×C₅₃	1	4	yes	—
276	C₂×#102	C₂²×C₅₃	1	4	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
278	C₂×#97	C₂²×C₅₃	1	2	yes	—
279	C₅₃	C₂×#3234	1	1	yes	characteristic, direct product factor, Sylow
280	C₂×#41	—	2	8	no	—
281	C₂×#41	—	4	8	no	—
282	C₂²×Q₈	—	2	1	no	—
283	C₂²×D₄	—	2	2	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	—	2	1	no	—
286	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	—	2	2	no	—
290	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
291	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
296	#127	—	2	2	no	—
297	#127	C₂³×C₅₃	1	4	yes	—
298	#125	—	2	4	no	—
299	#125	—	2	4	no	—
300	#125	—	2	8	no	—
301	#125	—	2	8	no	—
302	#125	C₂³×C₅₃	1	8	yes	—
303	#125	C₂³×C₅₃	1	8	yes	—
304	#124	—	2	8	no	—
305	#121	—	2	8	no	—
306	#121	—	2	4	no	—
307	#121	C₂³×C₅₃	1	8	yes	—
308	C₄×D₄	—	2	8	no	—
309	C₄×D₄	—	2	4	no	—
310	C₄×D₄	C₂³×C₅₃	1	8	yes	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	8	no	—
316	C₂×C₄⋊C₄	—	2	8	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄²	D₄×C₅₃	1	4	yes	—
332	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
335	#107	C₂³×C₅₃	1	16	yes	—
336	#106	C₂³×C₅₃	1	16	yes	—
337	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
338	#102	C₂³×C₅₃	1	16	yes	—
339	#102	C₂³×C₅₃	1	16	yes	—
340	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
341	#97	C₂³×C₅₃	1	8	yes	—
342	C₂⁴	—	2	2	no	—
343	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
344	#41	—	4	32	no	—
345	C₂×Q₈	—	2	4	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×D₄	—	2	8	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	4	4	no	—
353	C₂×D₄	—	4	4	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	8	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
375	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
378	C₂×C₈	—	2	8	no	—
379	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
380	C₄⋊C₄	—	2	8	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
391	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
392	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	8	no	—
395	C₄²	C₂×D₄×C₅₃	1	4	yes	—
396	C₄²	C₂×D₄×C₅₃	1	2	yes	—
397	C₄²	C₂×D₄×C₅₃	1	4	yes	—
398	C₄²	C₂×D₄×C₅₃	1	2	yes	—
399	C₄²	C₂⁴×C₅₃	1	2	yes	—
400	C₄²	C₂⁴×C₅₃	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	—	4	4	no	—
407	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
408	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
409	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
410	Q₈	—	4	8	no	—
411	D₄	—	2	8	no	—
412	D₄	—	4	8	no	—
413	D₄	—	4	8	no	—
414	D₄	—	4	16	no	—
415	D₄	—	4	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	16	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	8	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
435	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
437	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
441	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
442	C₈	—	2	8	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	8	8	no	—
450	C₂²	C₅₃×#536	1	2	yes	—
451	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
452	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
453	C₂²	C₅₃×#575	1	2	yes	—
454	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
455	C₄	—	2	2	no	—
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	2	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	4	no	—
462	C₄	—	4	4	no	—
463	C₄	C₅₃×#536	1	2	yes	—
464	C₄	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
465	C₂	—	2	4	no	—
466	C₂	—	8	4	no	—
467	C₂	C₅₃×#3234	1	4	yes	direct product factor
468	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
469	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
470	C₂	C₂×C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#532	C₂
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
2. C₂²×#125×C₅₃	C₂	semidirect
3. C₅₃×#3234	C₂	direct (N × H)
4. C₂×C₅₃×#501	C₂	semidirect
5. C₂×C₅₃×#490	C₂	semidirect
6. C₂×C₅₃×#481	C₂	semidirect
7. C₂×C₅₃×#480	C₂	semidirect
8. C₂×C₅₃×#478	C₂	semidirect
9. C₂×C₅₃×#432	C₂	semidirect
20. C₂×#125×C₅₃	C₂²	semidirect
31. C₅₃×#501	C₂²	semidirect
32. C₅₃×#490	C₂²	semidirect
33. C₅₃×#481	C₂²	semidirect
34. C₅₃×#480	C₂²	semidirect
35. C₅₃×#478	C₂²	semidirect
36. C₅₃×#432	C₂²	semidirect
207. C₂×#3234	C₅₃	direct (N × H)
231. C₂²×#125	C₂×C₅₃	semidirect
232. #3234	C₂×C₅₃	direct (N × H)
233. C₂×#501	C₂×C₅₃	semidirect
234. C₂×#490	C₂×C₅₃	semidirect
235. C₂×#481	C₂×C₅₃	semidirect
236. C₂×#480	C₂×C₅₃	semidirect
237. C₂×#478	C₂×C₅₃	semidirect
238. C₂×#432	C₂×C₅₃	semidirect
241. C₂×C₅₃	#3234	direct (N × H)
255. C₂×#125	C₂²×C₅₃	semidirect
266. #501	C₂²×C₅₃	semidirect
267. #490	C₂²×C₅₃	semidirect
268. #481	C₂²×C₅₃	semidirect
269. #480	C₂²×C₅₃	semidirect
270. #478	C₂²×C₅₃	semidirect
271. #432	C₂²×C₅₃	semidirect
279. C₅₃	C₂×#3234	direct (N × H)
467. C₂	C₅₃×#3234	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×#532	2
C₂²×#125×C₅₃	2
C₅₃×#3234	2
C₂×C₅₃×#501	2
C₂×C₅₃×#490	2
C₂×C₅₃×#481	2
C₂×C₅₃×#480	2
C₂×C₅₃×#478	2
C₂×C₅₃×#432	2
C₂×#3234	53

Subgroups — C2×C53×#3234 #27268359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3234 #27268359