Subgroups — C₅₃×#9663 #27217175

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1954	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
8	C₄×D₈×C₅₃	—	2	2	no	—
9	C₄×D₈×C₅₃	—	2	2	no	—
10	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#102×C₅₃	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₅₃	—	2	4	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	2	no	—
18	C₅₃×D₄⋊C₈	—	2	4	no	—
19	C₅₃×#322	—	2	2	no	—
20	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂×D₈×C₅₃	—	2	2	no	—
23	C₂×D₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#107×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	#102×C₅₃	—	2	4	no	—
40	C₂²⋊C₈×C₅₃	—	4	8	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂⁴×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	—	2	2	no	—
49	C₂×D₄×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	M₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	D₈×C₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	4	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	D₈	1	4	yes	complemented
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	4	2	no	—
77	C₂²⋊C₄×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₂³×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	4	no	—
86	C₂³×C₅₃	—	4	4	no	—
87	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	4	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	8	4	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₅₃	#131	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	#135	1	2	yes	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	4	no	—
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	—	8	2	no	—
113	#9663	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	4	8	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	—	8	4	no	—
117	C₂²×C₅₃	—	8	2	no	—
118	C₂²×C₅₃	—	8	2	no	—
119	C₂²×C₅₃	#390	1	1	yes	characteristic
120	C₂²×C₅₃	#435	1	2	yes	—
121	C₂²×C₅₃	C₄×D₈	1	2	yes	—
122	C₂²×C₅₃	#444	1	2	yes	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	8	2	no	—
126	C₄×C₅₃	#407	1	1	yes	characteristic
127	C₄×C₅₃	#408	1	1	yes	characteristic
128	C₄×C₅₃	#435	1	2	yes	—
129	C₄×C₅₃	#466	1	2	yes	—
130	C₄×C₅₃	#493	1	1	yes	characteristic
131	C₄×C₅₃	#495	1	1	yes	characteristic
132	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
133	#1954	C₂×C₅₃	1	2	yes	complemented
134	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
135	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
136	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
137	C₂×C₅₃	—	8	4	no	—
138	C₂×C₅₃	#1581	1	4	yes	—
139	C₂×C₅₃	#1955	1	1	yes	characteristic
140	C₂×C₅₃	#1970	1	1	yes	characteristic
141	C₂×C₅₃	#1975	1	1	yes	characteristic
142	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
143	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
144	C₄×D₈	—	2	2	no	—
145	C₄×D₈	—	2	2	no	—
146	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
148	C₂×#102	C₄×C₅₃	1	2	yes	—
149	C₂×C₂²⋊C₈	—	2	4	no	—
150	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
153	#335	—	2	2	no	—
154	D₄⋊C₈	—	2	4	no	—
155	#322	—	2	2	no	—
156	C₅₃	#9663	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₅₃	1	2	yes	—
166	C₄×D₄	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
173	#107	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	#102	—	2	4	no	—
177	C₂²⋊C₈	—	4	8	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	D₄×C₅₃	1	2	yes	—
182	C₄×C₈	D₄×C₅₃	1	2	yes	—
183	C₄×C₈	D₄×C₅₃	1	2	yes	—
184	C₂⁴	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	M₁₆×C₅₃	1	4	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
199	D₈	—	2	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
209	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
210	C₂×C₈	D₈×C₅₃	1	4	yes	complemented
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₄²	#41×C₅₃	1	1	yes	characteristic
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
225	D₄	—	2	4	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
238	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
239	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
240	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
242	C₂×C₄	#131×C₅₃	1	2	yes	—
243	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
244	C₂×C₄	#135×C₅₃	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	4	2	no	—
249	C₈	—	8	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	4	no	—
255	C₂²	C₅₃×#390	1	1	yes	characteristic
256	C₂²	C₅₃×#435	1	2	yes	—
257	C₂²	C₄×D₈×C₅₃	1	2	yes	—
258	C₂²	C₅₃×#444	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	C₅₃×#407	1	1	yes	characteristic
263	C₄	C₅₃×#408	1	1	yes	characteristic
264	C₄	C₅₃×#435	1	2	yes	—
265	C₄	C₅₃×#466	1	2	yes	—
266	C₄	C₅₃×#493	1	1	yes	characteristic
267	C₄	C₅₃×#495	1	1	yes	characteristic
268	C₂	—	8	4	no	—
269	C₂	C₅₃×#1581	1	4	yes	—
270	C₂	C₅₃×#1955	1	1	yes	characteristic
271	C₂	C₅₃×#1970	1	1	yes	characteristic
272	C₂	C₅₃×#1975	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	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1954	C₂	semidirect
3. C₂×C₅₃×#335	C₂	semidirect
4. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
5. C₂×C₅₃×#322	C₂	semidirect
73. C₂×C₈×C₅₃	D₈	semidirect
113. #9663	C₅₃	direct (N × H)
133. #1954	C₂×C₅₃	semidirect
134. C₂×#335	C₂×C₅₃	semidirect
135. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
136. C₂×#322	C₂×C₅₃	semidirect
156. C₅₃	#9663	direct (N × H)
210. C₂×C₈	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
C₅₃×#1954	2
C₂×C₅₃×#335	2
C₂×C₅₃×D₄⋊C₈	2
C₂×C₅₃×#322	2
#9663	53

Subgroups — C53×#9663 #27217175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9663 #27217175