Subgroups — C₅₃×#9678 #27217190

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₅₃	—	2	1	no	—
7	C₂×C₄×D₄×C₅₃	—	2	1	no	—
8	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#390	—	2	1	no	—
14	C₅₃×#390	—	2	1	no	—
15	C₅₃×#390	C₄	1	2	yes	—
16	C₅₃×#384	C₄	1	1	yes	characteristic
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×D₄⋊C₈	—	2	8	no	—
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	#127×C₅₃	—	2	4	no	—
28	C₄×D₄×C₅₃	—	2	4	no	—
29	C₄×D₄×C₅₃	—	2	4	no	—
30	C₄×D₄×C₅₃	—	4	4	no	—
31	C₄×D₄×C₅₃	—	4	2	no	—
32	C₄×D₄×C₅₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₃	—	2	4	no	—
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄⋊C₈×C₅₃	D₄	1	4	yes	complemented
49	#97×C₅₃	—	4	2	no	—
50	C₄×C₈×C₅₃	—	2	4	no	—
51	C₂⁴×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	—	2	4	no	—
55	C₂×D₄×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	4	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	M₁₆	1	4	yes	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	4	no	—
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	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	—	4	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	4	no	—
84	C₂²⋊C₄×C₅₃	—	4	4	no	—
85	C₂²⋊C₄×C₅₃	—	4	2	no	—
86	C₂²⋊C₄×C₅₃	—	4	2	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	4	1	no	—
92	C₄²×C₅₃	D₈	1	2	yes	—
93	C₄²×C₅₃	SD₁₆	1	2	yes	—
94	C₄²×C₅₃	C₂×D₄	1	2	yes	—
95	C₄²×C₅₃	#41	1	2	yes	—
96	C₂³×C₅₃	—	2	4	no	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	D₄×C₅₃	—	2	8	no	—
105	D₄×C₅₃	—	4	8	no	—
106	D₄×C₅₃	—	4	8	no	—
107	D₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	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	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	8	4	no	—
123	C₂×C₄×C₅₃	—	8	4	no	—
124	C₂×C₄×C₅₃	#102	1	2	yes	—
125	C₂×C₄×C₅₃	#102	1	2	yes	—
126	C₂×C₄×C₅₃	#104	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₅₃	#121	1	2	yes	—
129	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₅₃	#131	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	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₅₃	—	4	4	no	—
137	C₈×C₅₃	—	4	4	no	—
138	#9678	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₅₃	—	4	4	no	—
140	C₂²×C₅₃	—	4	4	no	—
141	C₂²×C₅₃	—	4	4	no	—
142	C₂²×C₅₃	—	8	2	no	—
143	C₂²×C₅₃	—	8	2	no	—
144	C₂²×C₅₃	—	8	2	no	—
145	C₂²×C₅₃	—	8	2	no	—
146	C₂²×C₅₃	#390	1	1	yes	characteristic
147	C₂²×C₅₃	#407	1	1	yes	characteristic
148	C₂²×C₅₃	#408	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
150	C₂²×C₅₃	#418	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
152	C₂²×C₅₃	#421	1	1	yes	characteristic
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	#435	1	4	yes	—
161	C₄×C₅₃	#459	1	2	yes	—
162	C₄×C₅₃	#460	1	2	yes	—
163	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
164	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
165	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
166	C₂×D₄⋊C₈	C₂×C₅₃	1	4	yes	complemented
167	C₂×C₅₃	—	8	4	no	—
168	C₂×C₅₃	#1486	1	1	yes	characteristic
169	C₂×C₅₃	#1490	1	1	yes	characteristic
170	C₂×C₅₃	#1492	1	1	yes	characteristic
171	C₂×C₅₃	#1496	1	1	yes	characteristic
172	C₂×C₅₃	#1955	1	1	yes	characteristic
173	C₂×C₅₃	#1980	1	1	yes	characteristic
174	C₂×C₅₃	#1982	1	1	yes	characteristic
175	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄×D₄	—	2	1	no	—
177	C₂×C₄×D₄	—	2	1	no	—
178	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
179	C₂×C₄⋊C₈	—	2	1	no	—
180	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
181	C₂×#97	—	2	1	no	—
182	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
183	#390	—	2	1	no	—
184	#390	—	2	1	no	—
185	#390	C₄×C₅₃	1	2	yes	—
186	#384	C₄×C₅₃	1	1	yes	characteristic
187	C₄³	C₂²×C₅₃	1	1	yes	characteristic
188	#336	—	2	1	no	—
189	#336	—	2	1	no	—
190	D₄⋊C₈	—	2	8	no	—
191	C₅₃	#9678	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	#127	—	2	4	no	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	—	4	4	no	—
202	C₄×D₄	—	4	2	no	—
203	C₄×D₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
212	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
213	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
215	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	D₄×C₅₃	1	4	yes	complemented
220	#97	—	4	2	no	—
221	C₄×C₈	—	2	4	no	—
222	C₂⁴	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	M₁₆×C₅₃	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	D₈×C₅₃	1	2	yes	—
264	C₄²	SD₁₆×C₅₃	1	2	yes	—
265	C₄²	C₂×D₄×C₅₃	1	2	yes	—
266	C₄²	#41×C₅₃	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
275	D₄	—	2	8	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	8	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	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	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	#102×C₅₃	1	2	yes	—
296	C₂×C₄	#102×C₅₃	1	2	yes	—
297	C₂×C₄	#104×C₅₃	1	2	yes	—
298	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
299	C₂×C₄	#121×C₅₃	1	2	yes	—
300	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
301	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
303	C₂×C₄	#131×C₅₃	1	2	yes	—
304	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
306	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
307	C₈	—	4	4	no	—
308	C₈	—	4	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₅₃×#390	1	1	yes	characteristic
317	C₂²	C₅₃×#407	1	1	yes	characteristic
318	C₂²	C₅₃×#408	1	1	yes	characteristic
319	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
320	C₂²	C₅₃×#418	1	1	yes	characteristic
321	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
322	C₂²	C₅₃×#421	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	8	2	no	—
330	C₄	C₅₃×#435	1	4	yes	—
331	C₄	C₅₃×#459	1	2	yes	—
332	C₄	C₅₃×#460	1	2	yes	—
333	C₂	—	8	4	no	—
334	C₂	C₅₃×#1486	1	1	yes	characteristic
335	C₂	C₅₃×#1490	1	1	yes	characteristic
336	C₂	C₅₃×#1492	1	1	yes	characteristic
337	C₂	C₅₃×#1496	1	1	yes	characteristic
338	C₂	C₅₃×#1955	1	1	yes	characteristic
339	C₂	C₅₃×#1980	1	1	yes	characteristic
340	C₂	C₅₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1767	C₂	semidirect
3. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
4. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
48. C₄⋊C₈×C₅₃	D₄	semidirect
138. #9678	C₅₃	direct (N × H)
164. #1767	C₂×C₅₃	semidirect
165. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
166. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
191. C₅₃	#9678	direct (N × H)
219. C₄⋊C₈	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#127×C₅₃	2
C₅₃×#1767	2
C₄×C₄⋊C₈×C₅₃	2
C₂×C₅₃×D₄⋊C₈	2
#9678	53

Subgroups — C53×#9678 #27217190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9678 #27217190