Subgroups — C₅₃×#10545 #27218057

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1981	C₂	1	2	yes	maximal
3	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#332	C₂	1	2	yes	maximal
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#500	—	2	4	no	—
8	C₂×#107×C₅₃	—	2	1	no	—
9	C₂×#107×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#103×C₅₃	—	2	2	no	—
14	C₂×#103×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#107×C₅₃	—	2	4	no	—
35	#107×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	4	2	no	—
40	#103×C₅₃	—	4	4	no	—
41	#103×C₅₃	—	4	4	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	#95×C₅₃	—	2	1	no	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	1	no	—
66	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	4	4	no	—
81	Q₈×C₅₃	—	4	4	no	—
82	Q₈×C₅₃	—	8	2	no	—
83	Q₈×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	2	yes	—
97	C₂×C₄×C₅₃	#123	1	2	yes	—
98	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	8	1	no	—
109	C₈×C₅₃	—	8	2	no	—
110	#10545	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	#394	1	1	yes	characteristic
112	C₂²×C₅₃	#461	1	1	yes	characteristic
113	C₂²×C₅₃	#462	1	1	yes	characteristic
114	C₂²×C₅₃	#466	1	1	yes	characteristic
115	C₂²×C₅₃	#469	1	1	yes	characteristic
116	C₂²×C₅₃	#473	1	1	yes	characteristic
117	C₂²×C₅₃	#473	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	—	8	4	no	—
121	C₄×C₅₃	—	8	1	no	—
122	C₄×C₅₃	#448	1	1	yes	characteristic
123	C₄×C₅₃	#451	1	1	yes	characteristic
124	C₄×C₅₃	#459	1	2	yes	—
125	C₄×C₅₃	#463	1	2	yes	—
126	C₄×C₅₃	#487	1	1	yes	characteristic
127	C₄×C₅₃	#491	1	1	yes	characteristic
128	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
129	#1981	C₂×C₅₃	1	2	yes	—
130	#1939	C₂×C₅₃	1	1	yes	characteristic
131	C₂×#332	C₂×C₅₃	1	2	yes	—
132	#1277	C₂×C₅₃	1	1	yes	characteristic
133	C₂×C₅₃	#1673	1	1	yes	characteristic
134	C₂×C₅₃	#1674	1	1	yes	characteristic
135	C₂×C₅₃	#1693	1	1	yes	characteristic
136	C₂×C₅₃	#1694	1	1	yes	characteristic
137	C₂×C₅₃	#2043	1	1	yes	characteristic
138	C₂×C₅₃	#2047	1	1	yes	characteristic
139	C₂×C₅₃	#2054	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
141	#500	—	2	4	no	—
142	C₂×#107	—	2	1	no	—
143	C₂×#107	—	2	1	no	—
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
147	C₂×#103	—	2	2	no	—
148	C₂×#103	—	2	2	no	—
149	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	#332	—	2	4	no	—
153	C₅₃	#10545	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	#107	—	2	4	no	—
170	#107	—	2	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	4	2	no	—
175	#103	—	4	4	no	—
176	#103	—	4	4	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	2	1	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
202	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	C₂×D₄×C₅₃	1	2	yes	—
213	C₄²	#41×C₅₃	1	2	yes	—
214	C₂³	#121×C₅₃	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
231	C₂×C₄	#121×C₅₃	1	2	yes	—
232	C₂×C₄	#123×C₅₃	1	2	yes	—
233	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
237	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
238	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
239	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
240	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	1	no	—
244	C₈	—	8	2	no	—
245	C₂²	C₅₃×#394	1	1	yes	characteristic
246	C₂²	C₅₃×#461	1	1	yes	characteristic
247	C₂²	C₅₃×#462	1	1	yes	characteristic
248	C₂²	C₅₃×#466	1	1	yes	characteristic
249	C₂²	C₅₃×#469	1	1	yes	characteristic
250	C₂²	C₅₃×#473	1	1	yes	characteristic
251	C₂²	C₅₃×#473	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	8	1	no	—
255	C₄	—	8	4	no	—
256	C₄	C₅₃×#448	1	1	yes	characteristic
257	C₄	C₅₃×#451	1	1	yes	characteristic
258	C₄	C₅₃×#459	1	2	yes	—
259	C₄	C₅₃×#463	1	2	yes	—
260	C₄	C₅₃×#487	1	1	yes	characteristic
261	C₄	C₅₃×#491	1	1	yes	characteristic
262	C₂	C₅₃×#1673	1	1	yes	characteristic
263	C₂	C₅₃×#1674	1	1	yes	characteristic
264	C₂	C₅₃×#1693	1	1	yes	characteristic
265	C₂	C₅₃×#1694	1	1	yes	characteristic
266	C₂	C₅₃×#2043	1	1	yes	characteristic
267	C₂	C₅₃×#2047	1	1	yes	characteristic
268	C₂	C₅₃×#2054	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×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
110. #10545	C₅₃	direct (N × H)
153. C₅₃	#10545	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#500	2
C₅₃×#1981	2
C₅₃×#1939	2
C₂×C₅₃×#332	2
C₅₃×#1277	2
#10545	53

Subgroups — C53×#10545 #27218057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10545 #27218057