Subgroups — C₂×C₅₃×#1279 #27215401

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1279	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₈×C₅₃	—	2	1	no	—
6	C₂³×C₈×C₅₃	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₅₃	—	2	1	no	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×#95×C₅₃	—	2	1	no	—
23	C₂³×C₄×C₅₃	—	2	2	no	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	8	no	—
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	4	2	no	—
33	C₂²×C₈×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₅₃	C₈	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	—	4	2	no	—
41	C₂×C₄²×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₄×C₈×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	—	2	2	no	—
55	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	8	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
74	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
75	C₂²×C₄×C₅₃	D₈	1	2	yes	—
76	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	—	2	16	no	—
83	C₂×C₈×C₅₃	—	2	8	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	4	8	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₄⋊C₄×C₅₃	—	2	8	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
92	C₄²×C₅₃	—	4	4	no	—
93	C₄²×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	C₄²	1	2	yes	—
96	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
99	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
105	C₂³×C₅₃	#95	1	2	yes	—
106	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
107	C₂³×C₅₃	#102	1	2	yes	—
108	C₂³×C₅₃	#103	1	2	yes	—
109	C₂³×C₅₃	#104	1	4	yes	—
110	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	8	no	—
116	C₂×C₄×C₅₃	—	4	16	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	#95	1	2	yes	—
120	C₂×C₄×C₅₃	#95	1	2	yes	—
121	C₂×C₄×C₅₃	#95	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
123	C₂×C₄×C₅₃	#97	1	4	yes	—
124	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
125	C₂×C₄×C₅₃	#102	1	2	yes	—
126	C₂×C₄×C₅₃	#102	1	4	yes	—
127	C₂×C₄×C₅₃	#103	1	2	yes	—
128	C₂×C₄×C₅₃	#103	1	4	yes	—
129	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
130	C₂×C₄×C₅₃	#106	1	4	yes	—
131	C₂×C₄×C₅₃	#107	1	4	yes	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
144	C₈×C₅₃	—	2	8	no	—
145	C₈×C₅₃	—	4	4	no	—
146	C₂×#1279	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₅₃	D₄⋊C₈	1	8	yes	—
148	C₂²×C₅₃	Q₈⋊C₈	1	8	yes	—
149	C₂²×C₅₃	#336	1	4	yes	—
150	C₂²×C₅₃	#339	1	4	yes	—
151	C₂²×C₅₃	#340	1	4	yes	—
152	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
153	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
154	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
155	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
156	C₂²×C₅₃	C₂×#104	1	2	yes	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	4	8	no	—
161	C₄×C₅₃	#336	1	4	yes	—
162	C₄×C₅₃	#340	1	4	yes	—
163	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
166	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
167	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
168	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
169	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
170	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
171	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
172	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
173	#1279	C₂×C₅₃	1	4	yes	direct product factor
174	C₂×C₅₃	#1279	1	8	yes	direct product factor
175	C₂×C₅₃	C₂×D₄⋊C₈	1	2	yes	—
176	C₂×C₅₃	C₂×Q₈⋊C₈	1	2	yes	—
177	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
180	C₂³×C₈	—	2	1	no	—
181	C₂³×C₈	—	2	1	no	—
182	C₂²×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
183	C₂²×C₄²	—	2	1	no	—
184	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	—	2	2	no	—
186	C₂×C₄⋊C₈	—	2	1	no	—
187	C₂×C₄⋊C₈	C₄×C₅₃	1	4	yes	complemented
188	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
189	C₂×C₄×C₈	—	2	2	no	—
190	C₂×C₄×C₈	—	2	1	no	—
191	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
192	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
193	C₄×C₄⋊C₄	—	2	1	no	—
194	C₄×C₄⋊C₄	—	2	1	no	—
195	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
196	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
197	C₂×#95	—	2	1	no	—
198	C₅₃	C₂×#1279	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₈	—	2	8	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	—	4	2	no	—
209	C₂²×C₈	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	C₈×C₅₃	1	8	yes	complemented
213	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	4	2	no	—
217	C₂×C₄²	—	4	2	no	—
218	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
219	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
220	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₅₃	1	2	yes	—
222	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
225	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
246	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
247	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
248	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
249	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
250	C₂²×C₄	M₁₆×C₅₃	1	4	yes	—
251	C₂²×C₄	D₈×C₅₃	1	2	yes	—
252	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
253	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
254	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
258	C₂×C₈	—	2	16	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	8	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	complemented
268	C₄²	—	4	4	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₄²×C₅₃	1	2	yes	—
272	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
273	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
274	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
275	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
276	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
277	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
278	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
281	C₂³	#95×C₅₃	1	2	yes	—
282	C₂³	C₂²⋊C₈×C₅₃	1	4	yes	—
283	C₂³	#102×C₅₃	1	2	yes	—
284	C₂³	#103×C₅₃	1	2	yes	—
285	C₂³	#104×C₅₃	1	4	yes	—
286	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	4	16	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	#95×C₅₃	1	2	yes	—
296	C₂×C₄	#95×C₅₃	1	2	yes	—
297	C₂×C₄	#95×C₅₃	1	2	yes	—
298	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
299	C₂×C₄	#97×C₅₃	1	4	yes	—
300	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
301	C₂×C₄	#102×C₅₃	1	2	yes	—
302	C₂×C₄	#102×C₅₃	1	4	yes	—
303	C₂×C₄	#103×C₅₃	1	2	yes	—
304	C₂×C₄	#103×C₅₃	1	4	yes	—
305	C₂×C₄	C₄⋊C₈×C₅₃	1	8	yes	—
306	C₂×C₄	#106×C₅₃	1	4	yes	—
307	C₂×C₄	#107×C₅₃	1	4	yes	—
308	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
315	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
320	C₈	—	2	8	no	—
321	C₈	—	4	4	no	—
322	C₂²	C₅₃×D₄⋊C₈	1	8	yes	—
323	C₂²	C₅₃×Q₈⋊C₈	1	8	yes	—
324	C₂²	C₅₃×#336	1	4	yes	—
325	C₂²	C₅₃×#339	1	4	yes	—
326	C₂²	C₅₃×#340	1	4	yes	—
327	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
328	C₂²	C₂×C₂²⋊C₈×C₅₃	1	2	yes	—
329	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
330	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
331	C₂²	C₂×#104×C₅₃	1	2	yes	—
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	8	no	—
336	C₄	C₅₃×#336	1	4	yes	—
337	C₄	C₅₃×#340	1	4	yes	—
338	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
339	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
340	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
341	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
342	C₄	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
343	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
344	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
345	C₂	C₅₃×#1279	1	8	yes	direct product factor
346	C₂	C₂×C₅₃×D₄⋊C₈	1	2	yes	—
347	C₂	C₂×C₅₃×Q₈⋊C₈	1	2	yes	—
348	C₂	C₂×C₅₃×#336	1	1	yes	characteristic
349	C₂	C₂×C₅₃×#339	1	1	yes	characteristic
350	C₂	C₂×C₅₃×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅₃	D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1279	C₂	direct (N × H)
12. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
16. C₂×C₄×C₈×C₅₃	C₄	semidirect
36. C₂×C₄⋊C₄×C₅₃	C₈	semidirect
52. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
55. C₄×C₈×C₅₃	C₂×C₄	semidirect
91. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
146. C₂×#1279	C₅₃	direct (N × H)
173. #1279	C₂×C₅₃	direct (N × H)
174. C₂×C₅₃	#1279	direct (N × H)
187. C₂×C₄⋊C₈	C₄×C₅₃	semidirect
191. C₂×C₄×C₈	C₄×C₅₃	semidirect
198. C₅₃	C₂×#1279	direct (N × H)
212. C₂×C₄⋊C₄	C₈×C₅₃	semidirect
228. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
231. C₄×C₈	C₂×C₄×C₅₃	semidirect
267. C₄⋊C₄	C₂×C₈×C₅₃	semidirect
345. C₂	C₅₃×#1279	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⋊C₈×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₅₃×#1279	2
C₂×#1279	53

Subgroups — C2×C53×#1279 #27215401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1279 #27215401