Subgroups — C₂×C₅₃×#1637 #27226001

Order 13568.

1174 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1637	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₅₃	—	2	1	no	—
10	D₄×Q₈×C₅₃	—	2	2	no	—
11	D₄×Q₈×C₅₃	—	2	2	no	—
12	D₄×Q₈×C₅₃	—	2	2	no	—
13	D₄×Q₈×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#487	—	2	2	no	—
23	C₅₃×#487	C₂²	1	4	yes	complemented
24	C₅₃×#477	—	2	2	no	—
25	C₅₃×#477	C₂²	1	4	yes	complemented
26	C₅₃×#476	—	2	2	no	—
27	C₅₃×#476	C₂²	1	4	yes	—
28	C₂×#107×C₅₃	—	2	1	no	—
29	C₂×#106×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#103×C₅₃	—	2	1	no	—
33	C₂×#103×C₅₃	—	2	2	no	—
34	C₂×#102×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#332	C₂²	1	4	yes	complemented
37	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
38	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
39	C₂²×Q₈×C₅₃	—	2	2	no	—
40	C₂²×Q₈×C₅₃	—	2	4	no	—
41	C₂²×Q₈×C₅₃	—	2	4	no	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂²×Q₈×C₅₃	—	4	4	no	—
45	C₂²×Q₈×C₅₃	—	4	4	no	—
46	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	2	no	—
49	C₂³×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	—	2	1	no	—
51	C₂²×C₈×C₅₃	—	2	1	no	—
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	4	no	—
54	C₄⋊Q₈×C₅₃	—	2	4	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₂²⋊Q₈×C₅₃	—	2	8	no	—
64	C₂²⋊Q₈×C₅₃	—	2	4	no	—
65	C₂²⋊Q₈×C₅₃	—	2	8	no	—
66	C₂²⋊Q₈×C₅₃	—	2	4	no	—
67	C₄×Q₈×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	—	2	1	no	—
69	C₄×Q₈×C₅₃	D₄	1	2	yes	—
70	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
71	C₄×D₄×C₅₃	—	2	4	no	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	4	no	—
75	C₄×D₄×C₅₃	D₄	1	2	yes	—
76	C₄×D₄×C₅₃	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	—	2	1	no	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	#107×C₅₃	—	2	4	no	—
90	#106×C₅₃	—	2	4	no	—
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	—	2	1	no	—
93	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
94	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
95	#103×C₅₃	—	2	8	no	—
96	#103×C₅₃	—	2	4	no	—
97	#102×C₅₃	—	2	4	no	—
98	C₄×C₈×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₂⁴×C₅₃	—	2	1	no	—
101	C₂×Q₈×C₅₃	—	2	8	no	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	—	4	8	no	—
108	C₂×Q₈×C₅₃	—	4	4	no	—
109	C₂×Q₈×C₅₃	—	4	8	no	—
110	C₂×Q₈×C₅₃	—	4	16	no	—
111	C₂×Q₈×C₅₃	—	4	8	no	—
112	C₂×Q₈×C₅₃	—	4	1	no	—
113	C₂×Q₈×C₅₃	—	4	2	no	—
114	C₂×Q₈×C₅₃	—	4	2	no	—
115	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
116	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	2	no	—
119	C₂×D₄×C₅₃	Q₁₆	1	4	yes	—
120	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	8	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	4	8	no	—
129	C₂²×C₄×C₅₃	—	4	2	no	—
130	C₂²×C₄×C₅₃	—	4	2	no	—
131	C₂²×C₄×C₅₃	—	4	1	no	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	4	2	no	—
139	C₂×C₈×C₅₃	—	4	2	no	—
140	C₂×C₈×C₅₃	—	4	2	no	—
141	C₄⋊C₄×C₅₃	—	2	8	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₄⋊C₄×C₅₃	—	4	1	no	—
152	C₄⋊C₄×C₅₃	—	4	1	no	—
153	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
154	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
156	C₂²⋊C₄×C₅₃	—	2	8	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₂²⋊C₄×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	2	no	—
161	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	2	4	no	—
167	C₂³×C₅₃	—	4	2	no	—
168	C₂³×C₅₃	—	4	2	no	—
169	C₂³×C₅₃	#120	1	1	yes	characteristic
170	Q₈×C₅₃	—	2	2	no	—
171	Q₈×C₅₃	—	2	4	no	—
172	Q₈×C₅₃	—	4	16	no	—
173	Q₈×C₅₃	—	4	4	no	—
174	Q₈×C₅₃	—	4	2	no	—
175	Q₈×C₅₃	—	4	2	no	—
176	Q₈×C₅₃	—	4	4	no	—
177	Q₈×C₅₃	—	8	2	no	—
178	Q₈×C₅₃	C₂×SD₁₆	1	4	yes	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	4	no	—
181	D₄×C₅₃	C₂×Q₁₆	1	4	yes	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	4	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	4	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	4	16	no	—
189	C₂×C₄×C₅₃	—	4	4	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	4	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	8	2	no	—
197	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
198	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
199	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
201	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
202	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₅₃	#137	1	2	yes	—
204	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
207	C₈×C₅₃	—	4	2	no	—
208	C₈×C₅₃	—	4	2	no	—
209	C₈×C₅₃	—	4	2	no	—
210	C₂×#1637	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	2	4	no	—
212	C₂²×C₅₃	—	4	8	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	#450	1	2	yes	—
216	C₂²×C₅₃	#451	1	2	yes	—
217	C₂²×C₅₃	#454	1	2	yes	—
218	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
219	C₄×C₅₃	—	2	4	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	—	4	4	no	—
223	C₄×C₅₃	—	4	2	no	—
224	C₄×C₅₃	—	4	2	no	—
225	C₄×C₅₃	—	8	2	no	—
226	C₄×C₅₃	#450	1	2	yes	—
227	C₄×C₅₃	#451	1	2	yes	—
228	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
229	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
230	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
232	C₂×D₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
233	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×#477	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₂×#476	C₂×C₅₃	1	1	yes	characteristic
236	#1637	C₂×C₅₃	1	8	yes	direct product factor
237	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
238	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
240	C₂×C₅₃	—	4	4	no	—
241	C₂×C₅₃	#1637	1	4	yes	direct product factor
242	C₂×C₅₃	C₂×#450	1	1	yes	characteristic
243	C₂×C₅₃	C₂×#451	1	1	yes	characteristic
244	C₂×C₅₃	C₂×#454	1	1	yes	characteristic
245	C₂³×Q₈	—	2	1	no	—
246	D₄×Q₈	—	2	2	no	—
247	D₄×Q₈	—	2	2	no	—
248	D₄×Q₈	—	2	2	no	—
249	D₄×Q₈	C₂²×C₅₃	1	4	yes	—
250	C₂×C₄⋊Q₈	—	2	1	no	—
251	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
253	C₂×C₂²⋊Q₈	—	2	1	no	—
254	C₂×C₂²⋊Q₈	—	2	2	no	—
255	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×C₄×D₄	—	2	1	no	—
257	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
258	#487	—	2	2	no	—
259	#487	C₂²×C₅₃	1	4	yes	complemented
260	#477	—	2	2	no	—
261	#477	C₂²×C₅₃	1	4	yes	complemented
262	#476	—	2	2	no	—
263	#476	C₂²×C₅₃	1	4	yes	—
264	C₂×#107	—	2	1	no	—
265	C₂×#106	—	2	1	no	—
266	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×#103	—	2	1	no	—
269	C₂×#103	—	2	2	no	—
270	C₂×#102	—	2	1	no	—
271	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
272	#332	C₂²×C₅₃	1	4	yes	complemented
273	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
274	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
275	C₅₃	C₂×#1637	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²×Q₈	—	2	2	no	—
277	C₂²×Q₈	—	2	4	no	—
278	C₂²×Q₈	—	2	4	no	—
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×Q₈	—	2	1	no	—
281	C₂²×Q₈	—	4	4	no	—
282	C₂²×Q₈	—	4	4	no	—
283	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
284	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	—	2	1	no	—
287	C₂²×C₈	—	2	1	no	—
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₄⋊Q₈	—	2	4	no	—
291	C₄⋊Q₈	—	2	4	no	—
292	C₄⋊Q₈	—	2	1	no	—
293	C₄⋊Q₈	—	2	1	no	—
294	C₄⋊Q₈	—	2	1	no	—
295	C₄⋊Q₈	—	2	1	no	—
296	C₄⋊Q₈	—	2	1	no	—
297	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
298	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
299	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
300	C₂²⋊Q₈	—	2	4	no	—
301	C₂²⋊Q₈	—	2	8	no	—
302	C₂²⋊Q₈	—	2	4	no	—
303	C₂²⋊Q₈	—	2	8	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×Q₈	D₄×C₅₃	1	2	yes	—
307	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	4	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	D₄×C₅₃	1	2	yes	—
313	C₄×D₄	C₂³×C₅₃	1	2	yes	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
321	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
326	#107	—	2	4	no	—
327	#106	—	2	4	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
331	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
332	#103	—	2	4	no	—
333	#103	—	2	8	no	—
334	#102	—	2	4	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	C₂³×C₅₃	1	2	yes	—
337	C₂⁴	—	2	1	no	—
338	C₂×Q₈	—	2	8	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	4	8	no	—
345	C₂×Q₈	—	4	4	no	—
346	C₂×Q₈	—	4	8	no	—
347	C₂×Q₈	—	4	16	no	—
348	C₂×Q₈	—	4	8	no	—
349	C₂×Q₈	—	4	1	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	SD₁₆×C₅₃	1	4	yes	—
353	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	Q₁₆×C₅₃	1	4	yes	—
357	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	8	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	8	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	4	2	no	—
377	C₂×C₈	—	4	2	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	4	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
391	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
392	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
399	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
400	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
401	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
402	C₂³	—	2	2	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	#120×C₅₃	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	Q₈	—	2	4	no	—
409	Q₈	—	4	16	no	—
410	Q₈	—	4	4	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	4	no	—
414	Q₈	—	8	2	no	—
415	Q₈	C₂×SD₁₆×C₅₃	1	4	yes	—
416	D₄	—	2	2	no	—
417	D₄	—	2	4	no	—
418	D₄	C₂×Q₁₆×C₅₃	1	4	yes	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	4	16	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	8	2	no	—
434	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
435	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
436	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
437	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
438	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
439	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
440	C₂×C₄	#137×C₅₃	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
444	C₈	—	4	2	no	—
445	C₈	—	4	2	no	—
446	C₈	—	4	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	8	no	—
451	C₂²	C₅₃×#450	1	2	yes	—
452	C₂²	C₅₃×#451	1	2	yes	—
453	C₂²	C₅₃×#454	1	2	yes	—
454	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
455	C₄	—	2	4	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	4	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₅₃×#450	1	2	yes	—
463	C₄	C₅₃×#451	1	2	yes	—
464	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
465	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
466	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
467	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
468	C₂	—	4	4	no	—
469	C₂	C₅₃×#1637	1	4	yes	direct product factor
470	C₂	C₂×C₅₃×#450	1	1	yes	characteristic
471	C₂	C₂×C₅₃×#451	1	1	yes	characteristic
472	C₂	C₂×C₅₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#487	C₂	semidirect
3. C₂×C₅₃×#477	C₂	semidirect
5. C₅₃×#1637	C₂	direct (N × H)
6. C₂×C₅₃×#332	C₂	semidirect
7. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
23. C₅₃×#487	C₂²	semidirect
25. C₅₃×#477	C₂²	semidirect
36. C₅₃×#332	C₂²	semidirect
37. C₅₃×Q₈⋊C₈	C₂²	semidirect
210. C₂×#1637	C₅₃	direct (N × H)
233. C₂×#487	C₂×C₅₃	semidirect
234. C₂×#477	C₂×C₅₃	semidirect
236. #1637	C₂×C₅₃	direct (N × H)
237. C₂×#332	C₂×C₅₃	semidirect
238. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
241. C₂×C₅₃	#1637	direct (N × H)
259. #487	C₂²×C₅₃	semidirect
261. #477	C₂²×C₅₃	semidirect
272. #332	C₂²×C₅₃	semidirect
273. Q₈⋊C₈	C₂²×C₅₃	semidirect
275. C₅₃	C₂×#1637	direct (N × H)
469. C₂	C₅₃×#1637	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×D₄×Q₈×C₅₃	2
C₂×C₅₃×#487	2
C₂×C₅₃×#477	2
C₂×C₅₃×#476	2
C₅₃×#1637	2
C₂×C₅₃×#332	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×D₄⋊C₈	2
C₂×#1637	53

Subgroups — C2×C53×#1637 #27226001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1637 #27226001