Subgroups — C₅₃×#8676 #27216188

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1517	C₂	1	2	yes	maximal
5	C₅₃×#1286	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₅₃	—	2	1	no	—
7	C₅₃×#532	C₂²	1	1	yes	characteristic
8	C₅₃×#436	—	2	1	no	—
9	C₅₃×#436	C₄	1	2	yes	complemented
10	C₈×D₄×C₅₃	—	2	1	no	—
11	C₈×D₄×C₅₃	C₄	1	2	yes	complemented
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₅₃×#423	C₂²	1	1	yes	characteristic
16	C₅₃×#408	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#382	C₂²	1	2	yes	—
19	C₅₃×#335	—	2	1	no	—
20	C₅₃×#334	—	2	1	no	—
21	C₅₃×#327	—	2	4	no	—
22	C₂×#41×C₅₃	—	2	1	no	—
23	#131×C₅₃	—	4	2	no	—
24	#131×C₅₃	—	4	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
31	#127×C₅₃	Q₈	1	1	yes	characteristic
32	#124×C₅₃	—	2	1	no	—
33	#121×C₅₃	—	2	1	no	—
34	#121×C₅₃	D₄	1	2	yes	—
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	4	2	no	—
40	#106×C₅₃	—	4	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₂²⋊C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	#95×C₅₃	—	2	2	no	—
53	#95×C₅₃	—	2	2	no	—
54	#41×C₅₃	—	4	1	no	—
55	#41×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	#41×C₅₃	—	4	1	no	—
58	C₂×Q₈×C₅₃	—	2	1	no	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	M₁₆×C₅₃	—	4	1	no	—
69	M₁₆×C₅₃	—	4	1	no	—
70	M₁₆×C₅₃	—	4	4	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	4	4	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	D₈	1	2	yes	—
81	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
82	C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#121	1	1	yes	characteristic
94	Q₈×C₅₃	—	4	1	no	—
95	Q₈×C₅₃	—	4	1	no	—
96	D₄×C₅₃	—	4	2	no	—
97	D₄×C₅₃	—	4	2	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	4	2	no	—
100	D₄×C₅₃	—	4	1	no	—
101	D₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	—	8	2	no	—
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	—	8	2	no	—
124	C₈×C₅₃	#102	1	2	yes	—
125	C₈×C₅₃	#102	1	2	yes	—
126	#8676	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	#380	1	1	yes	characteristic
133	C₂²×C₅₃	#459	1	1	yes	characteristic
134	C₂²×C₅₃	#461	1	1	yes	characteristic
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	16	2	no	—
141	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
142	C₄×C₅₃	#417	1	1	yes	characteristic
143	C₄×C₅₃	#474	1	1	yes	characteristic
144	C₄×C₅₃	#476	1	1	yes	characteristic
145	#2931	C₂×C₅₃	1	1	yes	characteristic
146	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
147	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1517	C₂×C₅₃	1	2	yes	—
149	#1286	C₂×C₅₃	1	2	yes	complemented
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	8	2	no	—
153	C₂×C₅₃	#1802	1	1	yes	characteristic
154	C₂×C₅₃	#1812	1	1	yes	characteristic
155	C₂×C₅₃	#1814	1	1	yes	characteristic
156	C₂×#131	—	2	1	no	—
157	#532	C₂²×C₅₃	1	1	yes	characteristic
158	#436	—	2	1	no	—
159	#436	C₄×C₅₃	1	2	yes	complemented
160	C₈×D₄	—	2	1	no	—
161	C₈×D₄	C₄×C₅₃	1	2	yes	complemented
162	#431	C₂²×C₅₃	1	2	yes	—
163	C₂×#107	—	2	1	no	—
164	C₂×#106	—	2	1	no	—
165	#423	C₂²×C₅₃	1	1	yes	characteristic
166	#408	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
168	#382	C₂²×C₅₃	1	2	yes	—
169	#335	—	2	1	no	—
170	#334	—	2	1	no	—
171	#327	—	2	4	no	—
172	C₅₃	#8676	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#41	—	2	1	no	—
174	#131	—	4	2	no	—
175	#131	—	4	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
181	C₄⋊Q₈	Q₈×C₅₃	1	1	yes	characteristic
182	#127	Q₈×C₅₃	1	1	yes	characteristic
183	#124	—	2	1	no	—
184	#121	—	2	1	no	—
185	#121	D₄×C₅₃	1	2	yes	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
190	#107	—	4	2	no	—
191	#106	—	4	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
202	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#41	—	4	1	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₈	D₈×C₅₃	1	2	yes	—
232	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
242	C₄²	#41×C₅₃	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	#121×C₅₃	1	1	yes	characteristic
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
264	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
269	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
270	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	8	2	no	—
275	C₈	#102×C₅₃	1	2	yes	—
276	C₈	#102×C₅₃	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₅₃×#380	1	1	yes	characteristic
283	C₂²	C₅₃×#459	1	1	yes	characteristic
284	C₂²	C₅₃×#461	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	16	2	no	—
291	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
292	C₄	C₅₃×#417	1	1	yes	characteristic
293	C₄	C₅₃×#474	1	1	yes	characteristic
294	C₄	C₅₃×#476	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	8	2	no	—
298	C₂	C₅₃×#1802	1	1	yes	characteristic
299	C₂	C₅₃×#1812	1	1	yes	characteristic
300	C₂	C₅₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1846	C₂	semidirect
3. C₅₃×#1564	C₂	semidirect
5. C₅₃×#1286	C₂	semidirect
9. C₅₃×#436	C₄	semidirect
11. C₈×D₄×C₅₃	C₄	semidirect
126. #8676	C₅₃	direct (N × H)
146. #1846	C₂×C₅₃	semidirect
147. #1564	C₂×C₅₃	semidirect
149. #1286	C₂×C₅₃	semidirect
159. #436	C₄×C₅₃	semidirect
161. C₈×D₄	C₄×C₅₃	semidirect
172. C₅₃	#8676	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2931	2
C₅₃×#1846	2
C₅₃×#1564	2
C₅₃×#1517	2
C₅₃×#1286	2
#8676	53

Subgroups — C53×#8676 #27216188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8676 #27216188