Subgroups — C₅₃×#9657 #27217169

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 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×C₈²×C₅₃	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	C₈²×C₅₃	—	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	2	no	—
24	C₂×D₈×C₅₃	C₈	1	4	yes	complemented
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₄×D₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#107×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	#102×C₅₃	—	2	4	no	—
41	C₂²⋊C₈×C₅₃	—	4	8	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂⁴×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	—	2	2	no	—
54	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	D₈×C₅₃	—	2	8	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	4	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
76	C₂×C₈×C₅₃	D₈	1	4	yes	complemented
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	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₂³×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	—	4	4	no	—
89	C₂³×C₅₃	—	4	4	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	2	4	no	—
93	D₄×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	8	4	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	—	8	2	no	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₈×C₅₃	—	2	4	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	2	4	no	—
116	C₈×C₅₃	—	8	2	no	—
117	C₈×C₅₃	C₂²⋊C₈	1	4	yes	complemented
118	#9657	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₅₃	—	4	8	no	—
120	C₂²×C₅₃	—	4	4	no	—
121	C₂²×C₅₃	—	8	4	no	—
122	C₂²×C₅₃	—	8	2	no	—
123	C₂²×C₅₃	—	8	2	no	—
124	C₂²×C₅₃	#390	1	1	yes	characteristic
125	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
126	C₂²×C₅₃	#436	1	1	yes	characteristic
127	C₂²×C₅₃	C₄×D₈	1	2	yes	—
128	C₂²×C₅₃	#443	1	2	yes	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅₃	#408	1	1	yes	characteristic
134	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
135	C₄×C₅₃	#436	1	1	yes	characteristic
136	C₄×C₅₃	#466	1	2	yes	—
137	C₄×C₅₃	#493	1	1	yes	characteristic
138	C₄×C₅₃	#495	1	1	yes	characteristic
139	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
140	#1954	C₂×C₅₃	1	2	yes	complemented
141	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
142	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
143	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
144	C₂×C₅₃	—	8	4	no	—
145	C₂×C₅₃	C₈×D₈	1	2	yes	—
146	C₂×C₅₃	#1589	1	2	yes	—
147	C₂×C₅₃	#1954	1	1	yes	characteristic
148	C₂×C₅₃	#1970	1	1	yes	characteristic
149	C₂×C₅₃	#1974	1	1	yes	characteristic
150	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
152	C₄×D₈	—	2	2	no	—
153	C₄×D₈	—	2	2	no	—
154	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
156	C₂×#102	C₄×C₅₃	1	2	yes	—
157	C₂×C₂²⋊C₈	—	2	4	no	—
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	#335	—	2	2	no	—
162	D₄⋊C₈	—	2	4	no	—
163	C₈²	—	2	2	no	—
164	C₅₃	#9657	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	C₈×C₅₃	1	4	yes	complemented
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₅₃	1	2	yes	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
182	#107	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	2	2	no	—
185	#102	—	2	4	no	—
186	C₂²⋊C₈	—	4	8	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	D₄×C₅₃	1	2	yes	—
191	C₄×C₈	D₄×C₅₃	1	2	yes	—
192	C₄×C₈	D₄×C₅₃	1	2	yes	—
193	C₂⁴	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
208	D₈	—	2	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
218	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
219	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
220	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
221	C₂×C₈	D₈×C₅₃	1	4	yes	complemented
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₄²	#41×C₅₃	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
249	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
250	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
251	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
254	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
255	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
257	C₂×C₄	#135×C₅₃	1	2	yes	—
258	C₈	—	2	4	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	4	no	—
261	C₈	—	8	2	no	—
262	C₈	C₂²⋊C₈×C₅₃	1	4	yes	complemented
263	C₂²	—	4	4	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	C₅₃×#390	1	1	yes	characteristic
269	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
270	C₂²	C₅₃×#436	1	1	yes	characteristic
271	C₂²	C₄×D₈×C₅₃	1	2	yes	—
272	C₂²	C₅₃×#443	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	2	no	—
276	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
277	C₄	C₅₃×#408	1	1	yes	characteristic
278	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
279	C₄	C₅₃×#436	1	1	yes	characteristic
280	C₄	C₅₃×#466	1	2	yes	—
281	C₄	C₅₃×#493	1	1	yes	characteristic
282	C₄	C₅₃×#495	1	1	yes	characteristic
283	C₂	—	8	4	no	—
284	C₂	C₈×D₈×C₅₃	1	2	yes	—
285	C₂	C₅₃×#1589	1	2	yes	—
286	C₂	C₅₃×#1954	1	1	yes	characteristic
287	C₂	C₅₃×#1970	1	1	yes	characteristic
288	C₂	C₅₃×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₂×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×C₈²×C₅₃	C₂	semidirect
24. C₂×D₈×C₅₃	C₈	semidirect
76. C₂×C₈×C₅₃	D₈	semidirect
117. C₈×C₅₃	C₂²⋊C₈	semidirect
118. #9657	C₅₃	direct (N × H)
140. #1954	C₂×C₅₃	semidirect
141. C₂×#335	C₂×C₅₃	semidirect
142. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
143. C₂×C₈²	C₂×C₅₃	semidirect
164. C₅₃	#9657	direct (N × H)
169. C₂×D₈	C₈×C₅₃	semidirect
221. C₂×C₈	D₈×C₅₃	semidirect
262. C₈	C₂²⋊C₈×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×C₈²×C₅₃	2
#9657	53

Subgroups — C53×#9657 #27217169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9657 #27217169