Subgroups — C₅₃×#9063 #27216575

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1982	C₂	1	2	yes	maximal
4	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂²×SD₁₆×C₅₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#492	—	2	2	no	—
10	C₅₃×#492	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#102×C₅₃	—	2	2	no	—
13	C₂×#102×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	—	2	2	no	—
16	C₅₃×#389	—	2	2	no	—
17	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#333	—	2	2	no	—
20	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	2	no	—
23	C₂×SD₁₆×C₅₃	—	2	8	no	—
24	C₂×SD₁₆×C₅₃	—	4	4	no	—
25	C₂×SD₁₆×C₅₃	—	4	4	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
31	#127×C₅₃	—	2	1	no	—
32	#127×C₅₃	—	2	2	no	—
33	#127×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	#102×C₅₃	—	4	4	no	—
44	#102×C₅₃	—	4	4	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	#95×C₅₃	—	2	4	no	—
48	#95×C₅₃	—	2	2	no	—
49	C₂⁴×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
54	C₂×D₄×C₅₃	—	2	2	no	—
55	C₂×D₄×C₅₃	—	2	4	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	1	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	SD₁₆×C₅₃	—	4	16	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	8	2	no	—
82	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	complemented
83	C₄²×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	—	8	2	no	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₃	#41	1	2	yes	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	—	8	2	no	—
91	C₂³×C₅₃	—	8	2	no	—
92	C₂³×C₅₃	—	8	1	no	—
93	C₂³×C₅₃	—	8	1	no	—
94	C₂³×C₅₃	#120	1	1	yes	characteristic
95	Q₈×C₅₃	—	4	4	no	—
96	Q₈×C₅₃	—	8	2	no	—
97	D₄×C₅₃	—	4	4	no	—
98	D₄×C₅₃	—	4	4	no	—
99	D₄×C₅₃	—	4	4	no	—
100	D₄×C₅₃	—	8	2	no	—
101	D₄×C₅₃	—	8	2	no	—
102	D₄×C₅₃	—	8	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	#121	1	2	yes	—
117	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	#135	1	2	yes	—
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₅₃	#137	1	2	yes	—
122	C₈×C₅₃	—	2	4	no	—
123	C₈×C₅₃	—	8	2	no	—
124	#9063	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₅₃	—	8	4	no	—
126	C₂²×C₅₃	—	8	2	no	—
127	C₂²×C₅₃	—	16	2	no	—
128	C₂²×C₅₃	—	16	1	no	—
129	C₂²×C₅₃	—	16	1	no	—
130	C₂²×C₅₃	#386	1	1	yes	characteristic
131	C₂²×C₅₃	#448	1	2	yes	—
132	C₂²×C₅₃	#449	1	2	yes	—
133	C₂²×C₅₃	#453	1	1	yes	characteristic
134	C₂²×C₅₃	#453	1	1	yes	characteristic
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	—	8	2	no	—
140	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
141	C₄×C₅₃	#440	1	2	yes	—
142	C₄×C₅₃	#460	1	2	yes	—
143	C₄×C₅₃	#464	1	2	yes	—
144	C₂×#492	C₂×C₅₃	1	1	yes	characteristic
145	#2369	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#1982	C₂×C₅₃	1	2	yes	—
147	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1279	C₂×C₅₃	1	2	yes	complemented
149	C₂×C₅₃	—	16	2	no	—
150	C₂×C₅₃	#1622	1	2	yes	—
151	C₂×C₅₃	#1623	1	1	yes	characteristic
152	C₂×C₅₃	#1624	1	1	yes	characteristic
153	C₂×C₅₃	#1880	1	2	yes	—
154	C₂×C₅₃	#1887	1	1	yes	characteristic
155	C₂²×SD₁₆	—	2	2	no	—
156	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
158	#492	—	2	2	no	—
159	#492	C₄×C₅₃	1	4	yes	complemented
160	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
161	C₂×#102	—	2	2	no	—
162	C₂×#102	—	2	2	no	—
163	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	#391	—	2	2	no	—
165	#389	—	2	2	no	—
166	C₄×C₄⋊C₄	—	2	2	no	—
167	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
168	#333	—	2	2	no	—
169	C₅₃	#9063	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×SD₁₆	—	2	8	no	—
174	C₂×SD₁₆	—	4	4	no	—
175	C₂×SD₁₆	—	4	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
180	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
181	#127	—	2	1	no	—
182	#127	—	2	2	no	—
183	#127	C₂×C₄×C₅₃	1	2	yes	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
187	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	#102	—	4	4	no	—
194	#102	—	4	4	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
197	#95	—	2	4	no	—
198	#95	—	2	2	no	—
199	C₂⁴	—	4	1	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
219	SD₁₆	—	4	16	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	complemented
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	8	2	no	—
236	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₄²	#41×C₅₃	1	2	yes	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	#120×C₅₃	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
266	C₂×C₄	#121×C₅₃	1	2	yes	—
267	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
269	C₂×C₄	#135×C₅₃	1	2	yes	—
270	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
271	C₂×C₄	#137×C₅₃	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	1	no	—
278	C₂²	—	16	1	no	—
279	C₂²	C₅₃×#386	1	1	yes	characteristic
280	C₂²	C₅₃×#448	1	2	yes	—
281	C₂²	C₅₃×#449	1	2	yes	—
282	C₂²	C₅₃×#453	1	1	yes	characteristic
283	C₂²	C₅₃×#453	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₄×SD₁₆×C₅₃	1	2	yes	—
290	C₄	C₅₃×#440	1	2	yes	—
291	C₄	C₅₃×#460	1	2	yes	—
292	C₄	C₅₃×#464	1	2	yes	—
293	C₂	—	16	2	no	—
294	C₂	C₅₃×#1622	1	2	yes	—
295	C₂	C₅₃×#1623	1	1	yes	characteristic
296	C₂	C₅₃×#1624	1	1	yes	characteristic
297	C₂	C₅₃×#1880	1	2	yes	—
298	C₂	C₅₃×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2369	C₂	semidirect
4. C₂×C₅₃×#333	C₂	semidirect
5. C₅₃×#1279	C₂	semidirect
10. C₅₃×#492	C₄	semidirect
82. C₄⋊C₄×C₅₃	SD₁₆	semidirect
124. #9063	C₅₃	direct (N × H)
145. #2369	C₂×C₅₃	semidirect
147. C₂×#333	C₂×C₅₃	semidirect
148. #1279	C₂×C₅₃	semidirect
159. #492	C₄×C₅₃	semidirect
169. C₅₃	#9063	direct (N × H)
232. C₄⋊C₄	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#492	2
C₅₃×#2369	2
C₅₃×#1982	2
C₂×C₅₃×#333	2
C₅₃×#1279	2
#9063	53

Subgroups — C53×#9063 #27216575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9063 #27216575