Subgroups — C₅₃×#8234 #27215746

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 356 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×#331	C₂	1	2	yes	maximal
5	C₅₃×#1296	C₂	1	2	yes	maximal, complemented
6	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₃	—	2	1	no	—
8	C₂×C₄×D₄×C₅₃	—	2	1	no	—
9	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#97×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	—	2	1	no	—
14	C₅₃×#390	—	2	1	no	—
15	C₅₃×#390	—	2	1	no	—
16	C₅₃×#390	C₄	1	2	yes	—
17	C₅₃×#384	C₂²	1	1	yes	characteristic
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×#336	—	2	1	no	—
21	C₅₃×#331	C₄	1	8	yes	complemented
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	—	2	1	no	—
24	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	#127×C₅₃	—	2	2	no	—
29	#127×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×D₄×C₅₃	—	4	4	no	—
31	C₄×D₄×C₅₃	—	4	4	no	—
32	C₄×D₄×C₅₃	—	4	2	no	—
33	C₄×D₄×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄⋊C₈×C₅₃	—	2	2	no	—
49	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
50	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	#97×C₅₃	—	4	2	no	—
52	C₄×C₈×C₅₃	—	4	2	no	—
53	C₂⁴×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	2	2	no	—
55	C₂×D₄×C₅₃	—	2	4	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	1	no	—
60	C₂×D₄×C₅₃	—	4	1	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	4	1	no	—
64	C₂×D₄×C₅₃	—	4	1	no	—
65	C₂×D₄×C₅₃	C₄²	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	4	4	no	—
68	C₂²×C₄×C₅₃	—	4	4	no	—
69	C₂²×C₄×C₅₃	—	4	4	no	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	4	2	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄⋊C₄×C₅₃	—	4	1	no	—
85	C₄⋊C₄×C₅₃	—	4	1	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₂²⋊C₄×C₅₃	—	4	4	no	—
89	C₂²⋊C₄×C₅₃	—	4	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	4	1	no	—
96	C₄²×C₅₃	D₈	1	2	yes	—
97	C₄²×C₅₃	SD₁₆	1	2	yes	—
98	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	#41	1	2	yes	—
100	C₂³×C₅₃	—	2	4	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	—	4	2	no	—
105	C₂³×C₅₃	—	4	2	no	—
106	C₂³×C₅₃	—	4	2	no	—
107	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	D₄×C₅₃	—	4	4	no	—
109	D₄×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	4	4	no	—
111	D₄×C₅₃	—	4	4	no	—
112	D₄×C₅₃	—	4	4	no	—
113	D₄×C₅₃	—	4	4	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	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	8	8	no	—
129	C₂×C₄×C₅₃	#100	1	2	yes	—
130	C₂×C₄×C₅₃	#102	1	2	yes	—
131	C₂×C₄×C₅₃	#102	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#135	1	2	yes	—
139	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
142	C₈×C₅₃	—	4	4	no	—
143	C₈×C₅₃	—	4	4	no	—
144	#8234	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₅₃	—	4	4	no	—
146	C₂²×C₅₃	—	4	4	no	—
147	C₂²×C₅₃	—	4	4	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	—	8	2	no	—
150	C₂²×C₅₃	—	8	2	no	—
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
154	C₂²×C₅₃	#413	1	1	yes	characteristic
155	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
156	C₂²×C₅₃	#416	1	1	yes	characteristic
157	C₂²×C₅₃	#417	1	1	yes	characteristic
158	C₂²×C₅₃	#418	1	1	yes	characteristic
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	4	1	no	—
165	C₄×C₅₃	—	8	2	no	—
166	C₄×C₅₃	C₄×D₈	1	2	yes	—
167	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
168	C₄×C₅₃	#440	1	2	yes	—
169	C₄×C₅₃	#442	1	2	yes	—
170	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
171	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
173	C₂×#331	C₂×C₅₃	1	2	yes	—
174	#1296	C₂×C₅₃	1	2	yes	complemented
175	C₂×C₅₃	—	8	4	no	—
176	C₂×C₅₃	#1545	1	1	yes	characteristic
177	C₂×C₅₃	#1547	1	1	yes	characteristic
178	C₂×C₅₃	#1551	1	1	yes	characteristic
179	C₂×C₅₃	#1553	1	1	yes	characteristic
180	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
181	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
182	C₂×C₅₃	#1762	1	1	yes	characteristic
183	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×C₄×D₄	—	2	1	no	—
185	C₂×C₄×D₄	—	2	1	no	—
186	C₂×C₄×D₄	C₄×C₅₃	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
188	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
189	C₂×#97	—	2	1	no	—
190	C₂×C₄×C₈	—	2	1	no	—
191	#390	—	2	1	no	—
192	#390	—	2	1	no	—
193	#390	C₄×C₅₃	1	2	yes	—
194	#384	C₂²×C₅₃	1	1	yes	characteristic
195	C₄³	C₂²×C₅₃	1	1	yes	characteristic
196	#336	—	2	1	no	—
197	#336	—	2	1	no	—
198	#331	C₄×C₅₃	1	8	yes	complemented
199	C₅₃	#8234	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
203	C₂³×C₄	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	#127	—	2	2	no	—
207	#127	C₂×C₄×C₅₃	1	4	yes	—
208	C₄×D₄	—	4	4	no	—
209	C₄×D₄	—	4	4	no	—
210	C₄×D₄	—	4	2	no	—
211	C₄×D₄	—	4	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
228	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
229	#97	—	4	2	no	—
230	C₄×C₈	—	4	2	no	—
231	C₂⁴	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	C₄²×C₅₃	1	4	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
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	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	D₈×C₅₃	1	2	yes	—
275	C₄²	SD₁₆×C₅₃	1	2	yes	—
276	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
277	C₄²	#41×C₅₃	1	2	yes	—
278	C₂³	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	8	no	—
307	C₂×C₄	#100×C₅₃	1	2	yes	—
308	C₂×C₄	#102×C₅₃	1	2	yes	—
309	C₂×C₄	#102×C₅₃	1	2	yes	—
310	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
311	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
312	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
313	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
314	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
316	C₂×C₄	#135×C₅₃	1	2	yes	—
317	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
318	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
319	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
320	C₈	—	4	4	no	—
321	C₈	—	4	4	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
330	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
331	C₂²	C₅₃×#413	1	1	yes	characteristic
332	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
333	C₂²	C₅₃×#416	1	1	yes	characteristic
334	C₂²	C₅₃×#417	1	1	yes	characteristic
335	C₂²	C₅₃×#418	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	8	2	no	—
343	C₄	C₄×D₈×C₅₃	1	2	yes	—
344	C₄	C₄×SD₁₆×C₅₃	1	2	yes	—
345	C₄	C₅₃×#440	1	2	yes	—
346	C₄	C₅₃×#442	1	2	yes	—
347	C₂	—	8	4	no	—
348	C₂	C₅₃×#1545	1	1	yes	characteristic
349	C₂	C₅₃×#1547	1	1	yes	characteristic
350	C₂	C₅₃×#1551	1	1	yes	characteristic
351	C₂	C₅₃×#1553	1	1	yes	characteristic
352	C₂	C₄×#100×C₅₃	1	1	yes	characteristic
353	C₂	C₄×#102×C₅₃	1	1	yes	characteristic
354	C₂	C₅₃×#1762	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×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
5. C₅₃×#1296	C₂	semidirect
21. C₅₃×#331	C₄	semidirect
49. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
50. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
144. #8234	C₅₃	direct (N × H)
171. #1767	C₂×C₅₃	semidirect
172. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
174. #1296	C₂×C₅₃	semidirect
198. #331	C₄×C₅₃	semidirect
199. C₅₃	#8234	direct (N × H)
227. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
228. C₄⋊C₈	C₂×C₄×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×#331	2
C₅₃×#1296	2
#8234	53

Subgroups — C53×#8234 #27215746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8234 #27215746