Subgroups — C₅₃×#9764 #27217276

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1530	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1505	C₂	1	2	yes	maximal, complemented
6	C₅₃×#532	C₂²	1	2	yes	—
7	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#486	C₄	1	2	yes	complemented
10	C₅₃×#466	—	2	1	no	—
11	C₅₃×#466	—	2	1	no	—
12	C₄×D₈×C₅₃	—	2	1	no	—
13	C₄×D₈×C₅₃	—	2	1	no	—
14	C₅₃×#432	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#376	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×#335	—	2	1	no	—
21	C₅₃×#334	—	2	1	no	—
22	C₅₃×#329	—	2	4	no	—
23	C₅₃×#327	—	2	4	no	—
24	C₂×#41×C₅₃	—	2	2	no	—
25	C₂×D₈×C₅₃	—	2	1	no	—
26	C₂×D₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	#127×C₅₃	C₂×C₄	1	2	yes	—
32	#124×C₅₃	—	2	2	no	—
33	#121×C₅₃	—	2	2	no	—
34	#121×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	2	2	no	—
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	2	1	no	—
40	#107×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	#102×C₅₃	—	2	2	no	—
43	#102×C₅₃	—	2	2	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	#97×C₅₃	—	2	1	no	—
47	#97×C₅₃	D₄	1	2	yes	—
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	1	no	—
53	#95×C₅₃	D₄	1	2	yes	—
54	#41×C₅₃	—	8	4	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	D₈×C₅₃	—	4	2	no	—
66	D₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	—	4	1	no	—
82	C₄²×C₅₃	—	4	1	no	—
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	#121	1	1	yes	characteristic
89	Q₈×C₅₃	—	8	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	2	no	—
92	D₄×C₅₃	—	8	2	no	—
93	D₄×C₅₃	—	8	2	no	—
94	D₄×C₅₃	—	8	4	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	4	2	no	—
115	C₈×C₅₃	—	8	2	no	—
116	#9764	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	8	2	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	#390	1	1	yes	characteristic
123	C₂²×C₅₃	#472	1	2	yes	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	#421	1	1	yes	characteristic
132	C₄×C₅₃	#421	1	1	yes	characteristic
133	#3169	C₂×C₅₃	1	1	yes	characteristic
134	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
135	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
136	#1530	C₂×C₅₃	1	2	yes	complemented
137	#1505	C₂×C₅₃	1	2	yes	complemented
138	C₂×C₅₃	—	2	2	no	—
139	C₂×C₅₃	—	16	2	no	—
140	C₂×C₅₃	#1963	1	1	yes	characteristic
141	C₂×C₅₃	#1976	1	1	yes	characteristic
142	C₂×C₅₃	#1978	1	1	yes	characteristic
143	#532	C₂²×C₅₃	1	2	yes	—
144	#500	C₄×C₅₃	1	1	yes	characteristic, complemented
145	#493	C₄×C₅₃	1	1	yes	characteristic, complemented
146	#486	C₄×C₅₃	1	2	yes	complemented
147	#466	—	2	1	no	—
148	#466	—	2	1	no	—
149	C₄×D₈	—	2	1	no	—
150	C₄×D₈	—	2	1	no	—
151	#432	C₂²×C₅₃	1	2	yes	—
152	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	#376	C₂²×C₅₃	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	2	1	no	—
157	#335	—	2	1	no	—
158	#334	—	2	1	no	—
159	#329	—	2	4	no	—
160	#327	—	2	4	no	—
161	C₅₃	#9764	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
168	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
169	#127	C₂×C₄×C₅₃	1	2	yes	—
170	#124	—	2	2	no	—
171	#121	—	2	2	no	—
172	#121	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
177	#107	—	2	1	no	—
178	#107	—	2	1	no	—
179	C₄⋊C₈	—	2	2	no	—
180	#102	—	2	2	no	—
181	#102	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	#97	—	2	2	no	—
184	#97	—	2	1	no	—
185	#97	D₄×C₅₃	1	2	yes	—
186	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
188	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
190	#95	—	2	1	no	—
191	#95	D₄×C₅₃	1	2	yes	—
192	#41	—	8	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
203	D₈	—	4	2	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
224	C₄²	#41×C₅₃	1	1	yes	characteristic
225	C₂³	—	4	2	no	—
226	C₂³	#121×C₅₃	1	1	yes	characteristic
227	Q₈	—	8	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	8	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
246	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
247	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₅₃×#390	1	1	yes	characteristic
260	C₂²	C₅₃×#472	1	2	yes	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	C₅₃×#421	1	1	yes	characteristic
269	C₄	C₅₃×#421	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₅₃×#1963	1	1	yes	characteristic
273	C₂	C₅₃×#1976	1	1	yes	characteristic
274	C₂	C₅₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#3169	C₂
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×#1746	C₂	semidirect
3. C₅₃×#1568	C₂	semidirect
4. C₅₃×#1530	C₂	semidirect
5. C₅₃×#1505	C₂	semidirect
7. C₅₃×#500	C₄	semidirect
8. C₅₃×#493	C₄	semidirect
9. C₅₃×#486	C₄	semidirect
116. #9764	C₅₃	direct (N × H)
134. #1746	C₂×C₅₃	semidirect
135. #1568	C₂×C₅₃	semidirect
136. #1530	C₂×C₅₃	semidirect
137. #1505	C₂×C₅₃	semidirect
144. #500	C₄×C₅₃	semidirect
145. #493	C₄×C₅₃	semidirect
146. #486	C₄×C₅₃	semidirect
161. C₅₃	#9764	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×#3169	2
C₅₃×#1746	2
C₅₃×#1568	2
C₅₃×#1530	2
C₅₃×#1505	2
#9764	53

Subgroups — C53×#9764 #27217276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9764 #27217276