Subgroups — C₅₃×#9638 #27217150

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₅₃	—	2	2	no	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	—	2	2	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#390	—	2	2	no	—
13	C₅₃×#390	C₄	1	2	yes	—
14	C₅₃×#384	C₄	1	1	yes	characteristic
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×D₄⋊C₈	—	2	8	no	—
17	C₂²×D₄×C₅₃	—	2	2	no	—
18	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	#127×C₅₃	—	2	2	no	—
23	#127×C₅₃	C₈	1	4	yes	complemented
24	C₄×D₄×C₅₃	—	2	4	no	—
25	C₄×D₄×C₅₃	—	2	4	no	—
26	C₄×D₄×C₅₃	—	4	4	no	—
27	C₄×D₄×C₅₃	—	4	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	—	4	4	no	—
39	C₄×C₈×C₅₃	—	2	8	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
42	C₂⁴×C₅₃	—	2	2	no	—
43	C₂×D₄×C₅₃	—	2	8	no	—
44	C₂×D₄×C₅₃	—	2	4	no	—
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	4	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	8	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₂²⋊C₄×C₅₃	—	4	8	no	—
67	C₂²⋊C₄×C₅₃	—	4	4	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	C₂×C₈	1	2	yes	—
71	C₄²×C₅₃	M₁₆	1	2	yes	—
72	C₄²×C₅₃	D₈	1	4	yes	—
73	C₄²×C₅₃	SD₁₆	1	4	yes	—
74	C₄²×C₅₃	C₂×D₄	1	2	yes	—
75	C₄²×C₅₃	#41	1	2	yes	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	—	4	2	no	—
78	C₂³×C₅₃	—	4	2	no	—
79	C₂³×C₅₃	—	4	4	no	—
80	C₂³×C₅₃	—	4	4	no	—
81	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	D₄×C₅₃	—	2	8	no	—
83	D₄×C₅₃	—	4	16	no	—
84	D₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	8	4	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₅₃	#102	1	4	yes	—
95	C₂×C₄×C₅₃	#102	1	4	yes	—
96	C₂×C₄×C₅₃	#104	1	4	yes	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	#121	1	2	yes	—
99	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#131	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
106	C₈×C₅₃	—	2	8	no	—
107	C₈×C₅₃	—	8	2	no	—
108	#9638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₅₃	—	4	4	no	—
110	C₂²×C₅₃	—	4	8	no	—
111	C₂²×C₅₃	—	8	2	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	—	8	4	no	—
114	C₂²×C₅₃	#390	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₅₃	#408	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#102	1	2	yes	—
118	C₂²×C₅₃	C₂×#104	1	2	yes	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	—	8	2	no	—
122	C₄×C₅₃	D₄⋊C₈	1	8	yes	complemented
123	C₄×C₅₃	C₈×D₄	1	2	yes	—
124	C₄×C₅₃	#436	1	2	yes	—
125	C₄×C₅₃	#486	1	2	yes	—
126	C₄×C₅₃	#492	1	1	yes	characteristic
127	C₄×C₅₃	#493	1	1	yes	characteristic
128	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
129	#1842	C₂×C₅₃	1	1	yes	characteristic, complemented
130	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
131	C₂×D₄⋊C₈	C₂×C₅₃	1	4	yes	complemented
132	C₂×C₅₃	—	8	4	no	—
133	C₂×C₅₃	C₂×D₄⋊C₈	1	2	yes	—
134	C₂×C₅₃	#1492	1	2	yes	—
135	C₂×C₅₃	#1954	1	1	yes	characteristic
136	C₂×C₅₃	#1957	1	1	yes	characteristic
137	C₂×C₅₃	#1962	1	1	yes	characteristic
138	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₄×D₄	—	2	2	no	—
140	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
141	C₂×C₄⋊C₈	—	2	2	no	—
142	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
143	C₂×C₄×C₈	—	2	2	no	—
144	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
145	#390	—	2	2	no	—
146	#390	C₄×C₅₃	1	2	yes	—
147	#384	C₄×C₅₃	1	1	yes	characteristic
148	C₄³	C₂²×C₅₃	1	1	yes	characteristic
149	D₄⋊C₈	—	2	8	no	—
150	C₅₃	#9638	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×D₄	—	2	2	no	—
152	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
153	C₂³×C₄	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	C₈×C₅₃	1	4	yes	complemented
158	C₄×D₄	—	2	4	no	—
159	C₄×D₄	—	2	4	no	—
160	C₄×D₄	—	4	4	no	—
161	C₄×D₄	—	4	4	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
164	C₂×C₂²⋊C₄	—	2	2	no	—
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₅₃	1	2	yes	—
169	C₂×C₄²	D₄×C₅₃	1	2	yes	—
170	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₄×C₈	—	2	8	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
176	C₂⁴	—	2	2	no	—
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
194	C₂×C₈	—	2	8	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₂²⋊C₄	—	4	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×C₈×C₅₃	1	2	yes	—
205	C₄²	M₁₆×C₅₃	1	2	yes	—
206	C₄²	D₈×C₅₃	1	4	yes	—
207	C₄²	SD₁₆×C₅₃	1	4	yes	—
208	C₄²	C₂×D₄×C₅₃	1	2	yes	—
209	C₄²	#41×C₅₃	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
216	D₄	—	2	8	no	—
217	D₄	—	4	16	no	—
218	D₄	—	4	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
228	C₂×C₄	#102×C₅₃	1	4	yes	—
229	C₂×C₄	#102×C₅₃	1	4	yes	—
230	C₂×C₄	#104×C₅₃	1	4	yes	—
231	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
232	C₂×C₄	#121×C₅₃	1	2	yes	—
233	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
234	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
237	C₂×C₄	#131×C₅₃	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
239	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
240	C₈	—	2	8	no	—
241	C₈	—	8	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	8	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	C₅₃×#390	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
249	C₂²	C₅₃×#408	1	1	yes	characteristic
250	C₂²	C₂×#102×C₅₃	1	2	yes	—
251	C₂²	C₂×#104×C₅₃	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₅₃×D₄⋊C₈	1	8	yes	complemented
256	C₄	C₈×D₄×C₅₃	1	2	yes	—
257	C₄	C₅₃×#436	1	2	yes	—
258	C₄	C₅₃×#486	1	2	yes	—
259	C₄	C₅₃×#492	1	1	yes	characteristic
260	C₄	C₅₃×#493	1	1	yes	characteristic
261	C₂	—	8	4	no	—
262	C₂	C₂×C₅₃×D₄⋊C₈	1	2	yes	—
263	C₂	C₅₃×#1492	1	2	yes	—
264	C₂	C₅₃×#1954	1	1	yes	characteristic
265	C₂	C₅₃×#1957	1	1	yes	characteristic
266	C₂	C₅₃×#1962	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	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1842	C₂	semidirect
3. C₄²×C₈×C₅₃	C₂	semidirect
4. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
23. #127×C₅₃	C₈	semidirect
41. C₄×C₈×C₅₃	D₄	semidirect
108. #9638	C₅₃	direct (N × H)
122. C₄×C₅₃	D₄⋊C₈	semidirect
129. #1842	C₂×C₅₃	semidirect
130. C₄²×C₈	C₂×C₅₃	semidirect
131. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
150. C₅₃	#9638	direct (N × H)
157. #127	C₈×C₅₃	semidirect
175. C₄×C₈	D₄×C₅₃	semidirect
255. C₄	C₅₃×D₄⋊C₈	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×#1842	2
C₄²×C₈×C₅₃	2
C₂×C₅₃×D₄⋊C₈	2
#9638	53

Subgroups — C53×#9638 #27217150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9638 #27217150