Subgroups — C₅₃×#8340 #27215852

Order 13568.

1030 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#332	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₅₃	—	2	1	no	—
6	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×#332	—	2	8	no	—
21	C₂²×Q₈×C₅₃	—	2	8	no	—
22	C₂²×Q₈×C₅₃	—	2	2	no	—
23	C₂²×Q₈×C₅₃	—	2	2	no	—
24	C₂²×Q₈×C₅₃	—	4	4	no	—
25	C₂²×Q₈×C₅₃	—	4	2	no	—
26	C₂²×Q₈×C₅₃	—	4	2	no	—
27	C₂³×C₄×C₅₃	—	2	2	no	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	8	no	—
34	C₄⋊Q₈×C₅₃	—	2	8	no	—
35	C₄⋊Q₈×C₅₃	—	2	8	no	—
36	C₄⋊Q₈×C₅₃	—	2	4	no	—
37	C₄⋊Q₈×C₅₃	D₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	4	no	—
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₂²⋊C₈×C₅₃	—	4	2	no	—
52	C₂²⋊C₈×C₅₃	—	4	2	no	—
53	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₅₃	—	2	8	no	—
55	C₂×Q₈×C₅₃	—	2	8	no	—
56	C₂×Q₈×C₅₃	—	4	4	no	—
57	C₂×Q₈×C₅₃	—	4	16	no	—
58	C₂×Q₈×C₅₃	—	4	16	no	—
59	C₂×Q₈×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	4	2	no	—
61	C₂×Q₈×C₅₃	—	4	2	no	—
62	C₂×Q₈×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	8	no	—
71	C₂²×C₄×C₅₃	—	2	4	no	—
72	C₂²×C₄×C₅₃	—	4	4	no	—
73	C₂²×C₄×C₅₃	—	4	4	no	—
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	D₈	1	2	yes	—
78	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₂×C₈×C₅₃	—	4	2	no	—
90	C₄⋊C₄×C₅₃	—	2	8	no	—
91	C₄⋊C₄×C₅₃	—	2	8	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	2	8	no	—
94	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	—	2	1	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	—	2	1	no	—
102	C₂³×C₅₃	—	2	1	no	—
103	C₂³×C₅₃	#101	1	2	yes	—
104	C₂³×C₅₃	#102	1	2	yes	—
105	C₂³×C₅₃	#103	1	2	yes	—
106	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₅₃	—	4	16	no	—
108	Q₈×C₅₃	—	4	8	no	—
109	Q₈×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	—	2	8	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	—	2	8	no	—
124	C₂×C₄×C₅₃	—	4	16	no	—
125	C₂×C₄×C₅₃	—	4	4	no	—
126	C₂×C₄×C₅₃	—	4	4	no	—
127	C₂×C₄×C₅₃	#101	1	2	yes	—
128	C₂×C₄×C₅₃	#102	1	2	yes	—
129	C₂×C₄×C₅₃	#103	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₅₃	#120	1	2	yes	—
132	C₂×C₄×C₅₃	#120	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₅₃	#137	1	2	yes	—
139	C₈×C₅₃	—	8	2	no	—
140	C₈×C₅₃	—	8	2	no	—
141	#8340	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	#332	1	4	yes	complemented
149	C₂²×C₅₃	#379	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
152	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
153	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
154	C₂²×C₅₃	#418	1	1	yes	characteristic
155	C₂²×C₅₃	#419	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	4	8	no	—
161	C₄×C₅₃	#447	1	2	yes	—
162	C₄×C₅₃	#448	1	2	yes	—
163	C₄×C₅₃	#450	1	2	yes	—
164	C₄×C₅₃	#451	1	2	yes	—
165	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
166	#1797	C₂×C₅₃	1	1	yes	characteristic
167	#1797	C₂×C₅₃	1	1	yes	characteristic
168	C₂×#332	C₂×C₅₃	1	4	yes	complemented
169	C₂×C₅₃	—	2	4	no	—
170	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
171	C₂×C₅₃	#1546	1	1	yes	characteristic
172	C₂×C₅₃	#1549	1	1	yes	characteristic
173	C₂×C₅₃	#1556	1	1	yes	characteristic
174	C₂×C₅₃	#1784	1	1	yes	characteristic
175	C₂×C₅₃	#1786	1	1	yes	characteristic
176	C₂×C₅₃	#1787	1	1	yes	characteristic
177	C₂³×Q₈	—	2	1	no	—
178	C₂×C₄⋊Q₈	—	2	8	no	—
179	C₂×C₄⋊Q₈	—	2	1	no	—
180	C₂×C₄⋊Q₈	—	2	1	no	—
181	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	—
182	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
183	C₂²×C₄⋊C₄	—	2	1	no	—
184	C₂²×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
185	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
188	C₂×C₂²⋊C₈	—	2	1	no	—
189	C₂×C₂²⋊C₈	—	2	1	no	—
190	#336	—	2	1	no	—
191	#336	—	2	1	no	—
192	#332	—	2	8	no	—
193	C₅₃	#8340	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×Q₈	—	2	8	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	4	4	no	—
198	C₂²×Q₈	—	4	2	no	—
199	C₂²×Q₈	—	4	2	no	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	8	no	—
207	C₄⋊Q₈	—	2	8	no	—
208	C₄⋊Q₈	—	2	8	no	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	D₄×C₅₃	1	8	yes	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	2	4	no	—
224	C₂²⋊C₈	—	4	2	no	—
225	C₂²⋊C₈	—	4	2	no	—
226	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	16	no	—
231	C₂×Q₈	—	4	16	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₅₃	1	2	yes	—
251	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
252	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
253	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂×D₄×C₅₃	1	4	yes	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	#101×C₅₃	1	2	yes	—
277	C₂³	#102×C₅₃	1	2	yes	—
278	C₂³	#103×C₅₃	1	2	yes	—
279	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
280	Q₈	—	4	16	no	—
281	Q₈	—	4	8	no	—
282	Q₈	—	4	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#101×C₅₃	1	2	yes	—
301	C₂×C₄	#102×C₅₃	1	2	yes	—
302	C₂×C₄	#103×C₅₃	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
304	C₂×C₄	#120×C₅₃	1	2	yes	—
305	C₂×C₄	#120×C₅₃	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
310	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
311	C₂×C₄	#137×C₅₃	1	2	yes	—
312	C₈	—	8	2	no	—
313	C₈	—	8	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	C₅₃×#332	1	4	yes	complemented
321	C₂²	C₅₃×#379	1	1	yes	characteristic
322	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
323	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
324	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
325	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
326	C₂²	C₅₃×#418	1	1	yes	characteristic
327	C₂²	C₅₃×#419	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	8	no	—
333	C₄	C₅₃×#447	1	2	yes	—
334	C₄	C₅₃×#448	1	2	yes	—
335	C₄	C₅₃×#450	1	2	yes	—
336	C₄	C₅₃×#451	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	C₂×C₅₃×#332	1	1	yes	characteristic
339	C₂	C₅₃×#1546	1	1	yes	characteristic
340	C₂	C₅₃×#1549	1	1	yes	characteristic
341	C₂	C₅₃×#1556	1	1	yes	characteristic
342	C₂	C₅₃×#1784	1	1	yes	characteristic
343	C₂	C₅₃×#1786	1	1	yes	characteristic
344	C₂	C₅₃×#1787	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
4. C₂×C₅₃×#332	C₂	semidirect
141. #8340	C₅₃	direct (N × H)
148. C₂²×C₅₃	#332	semidirect
168. C₂×#332	C₂×C₅₃	semidirect
193. C₅₃	#8340	direct (N × H)
320. C₂²	C₅₃×#332	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₅₃	2
C₅₃×#1797	2
C₂×C₅₃×#332	2
#8340	53

Subgroups — C53×#8340 #27215852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8340 #27215852