Subgroups — C₅₃×#9873 #27217385

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1279	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#391	—	2	1	no	—
10	C₅₃×#389	—	2	1	no	—
11	C₅₃×#389	C₄	1	2	yes	—
12	C₅₃×#384	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
25	C₄×Q₈×C₅₃	—	4	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	4	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
43	#95×C₅₃	—	2	2	no	—
44	#95×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	—	4	4	no	—
76	C₄²×C₅₃	C₂×C₈	1	2	yes	—
77	C₄²×C₅₃	M₁₆	1	2	yes	—
78	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
79	C₄²×C₅₃	#41	1	2	yes	—
80	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₅₃	—	4	8	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#131	1	2	yes	—
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₅₃	#137	1	2	yes	—
111	C₈×C₅₃	—	4	4	no	—
112	C₈×C₅₃	—	8	2	no	—
113	#9873	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	#391	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	#418	1	1	yes	characteristic
118	C₂²×C₅₃	#419	1	1	yes	characteristic
119	C₂²×C₅₃	#421	1	1	yes	characteristic
120	C₂²×C₅₃	#421	1	1	yes	characteristic
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	4	4	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	C₈×Q₈	1	2	yes	—
129	C₄×C₅₃	#446	1	2	yes	—
130	C₄×C₅₃	#489	1	2	yes	—
131	C₄×C₅₃	#501	1	2	yes	—
132	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
133	#1831	C₂×C₅₃	1	1	yes	characteristic
134	#1744	C₂×C₅₃	1	1	yes	characteristic
135	#1279	C₂×C₅₃	1	4	yes	—
136	C₂×C₅₃	#1478	1	1	yes	characteristic
137	C₂×C₅₃	#1479	1	1	yes	characteristic
138	C₂×C₅₃	#1496	1	1	yes	characteristic
139	C₂×C₅₃	#1497	1	1	yes	characteristic
140	C₂×C₅₃	#1984	1	1	yes	characteristic
141	C₂×C₅₃	#1993	1	1	yes	characteristic
142	C₂×C₅₃	#1995	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
147	#391	—	2	1	no	—
148	#389	—	2	1	no	—
149	#389	C₄×C₅₃	1	2	yes	—
150	#384	C₄×C₅₃	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
153	C₄³	C₂²×C₅₃	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	2	1	no	—
156	#336	—	2	1	no	—
157	#336	—	2	1	no	—
158	C₅₃	#9873	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₈×C₅₃	1	4	yes	complemented
164	C₄×Q₈	—	4	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂×C₈×C₅₃	1	2	yes	—
216	C₄²	M₁₆×C₅₃	1	2	yes	—
217	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
218	C₄²	#41×C₅₃	1	2	yes	—
219	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
240	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
241	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
242	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
243	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
247	C₂×C₄	#131×C₅₃	1	2	yes	—
248	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
249	C₂×C₄	#137×C₅₃	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₈	—	8	2	no	—
252	C₂²	C₅₃×#391	1	1	yes	characteristic
253	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
254	C₂²	C₅₃×#408	1	1	yes	characteristic
255	C₂²	C₅₃×#418	1	1	yes	characteristic
256	C₂²	C₅₃×#419	1	1	yes	characteristic
257	C₂²	C₅₃×#421	1	1	yes	characteristic
258	C₂²	C₅₃×#421	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	2	no	—
266	C₄	C₈×Q₈×C₅₃	1	2	yes	—
267	C₄	C₅₃×#446	1	2	yes	—
268	C₄	C₅₃×#489	1	2	yes	—
269	C₄	C₅₃×#501	1	2	yes	—
270	C₂	C₅₃×#1478	1	1	yes	characteristic
271	C₂	C₅₃×#1479	1	1	yes	characteristic
272	C₂	C₅₃×#1496	1	1	yes	characteristic
273	C₂	C₅₃×#1497	1	1	yes	characteristic
274	C₂	C₅₃×#1984	1	1	yes	characteristic
275	C₂	C₅₃×#1993	1	1	yes	characteristic
276	C₂	C₅₃×#1995	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, prime radical	C₂³×C₅₃	C₂×C₂²⋊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
24. C₄⋊Q₈×C₅₃	C₈	semidirect
42. C₄×C₈×C₅₃	Q₈	semidirect
113. #9873	C₅₃	direct (N × H)
158. C₅₃	#9873	direct (N × H)
163. C₄⋊Q₈	C₈×C₅₃	semidirect
181. C₄×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1831	2
C₅₃×#1744	2
C₅₃×#1279	2
#9873	53

Subgroups — C53×#9873 #27217385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9873 #27217385