Subgroups — C₅₃×#10265 #27217777

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×C₅₃×#462	C₂	1	2	yes	maximal
3	C₅₃×#1984	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	C₂²×Q₁₆×C₅₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#500	—	2	4	no	—
10	C₅₃×#462	—	2	4	no	—
11	C₅₃×#462	—	2	4	no	—
12	C₂×#107×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#103×C₅₃	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₅₃	—	2	2	no	—
20	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₅₃	—	2	8	no	—
22	C₂×Q₁₆×C₅₃	—	4	4	no	—
23	C₂×Q₁₆×C₅₃	—	4	4	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	2	yes	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	2	8	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
43	#103×C₅₃	—	2	8	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	D₈	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	4	16	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₄⋊C₄×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₄²×C₅₃	C₂×D₄	1	2	yes	—
80	C₂³×C₅₃	#121	1	1	yes	characteristic
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	Q₈×C₅₃	—	4	4	no	—
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	8	2	no	—
86	Q₈×C₅₃	—	8	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	#120	1	2	yes	—
97	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	2	yes	—
99	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
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	2	yes	—
105	C₈×C₅₃	—	2	4	no	—
106	C₈×C₅₃	—	4	4	no	—
107	C₈×C₅₃	—	8	1	no	—
108	#10265	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₅₃	#392	1	1	yes	characteristic
110	C₂²×C₅₃	#462	1	2	yes	—
111	C₂²×C₅₃	#466	1	1	yes	characteristic
112	C₂²×C₅₃	#469	1	1	yes	characteristic
113	C₂²×C₅₃	#473	1	2	yes	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	8	4	no	—
117	C₄×C₅₃	—	8	1	no	—
118	C₄×C₅₃	#447	1	2	yes	—
119	C₄×C₅₃	#449	1	2	yes	—
120	C₄×C₅₃	#451	1	2	yes	—
121	C₄×C₅₃	#494	1	1	yes	characteristic
122	C₄×C₅₃	#497	1	1	yes	characteristic
123	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
124	C₂×#462	C₂×C₅₃	1	2	yes	—
125	#1984	C₂×C₅₃	1	1	yes	characteristic
126	C₂×#332	C₂×C₅₃	1	2	yes	—
127	#1277	C₂×C₅₃	1	1	yes	characteristic
128	C₂×C₅₃	#1674	1	2	yes	—
129	C₂×C₅₃	#1694	1	2	yes	—
130	C₂×C₅₃	#1999	1	1	yes	characteristic
131	C₂×C₅₃	#2016	1	1	yes	characteristic
132	C₂×C₅₃	#2020	1	1	yes	characteristic
133	C₂²×Q₁₆	—	2	2	no	—
134	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
136	#500	—	2	4	no	—
137	#462	—	2	4	no	—
138	#462	—	2	4	no	—
139	C₂×#107	—	2	2	no	—
140	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
142	C₂×#103	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
144	#336	—	2	2	no	—
145	#332	—	2	4	no	—
146	C₅₃	#10265	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
149	C₂×Q₁₆	—	2	8	no	—
150	C₂×Q₁₆	—	4	4	no	—
151	C₂×Q₁₆	—	4	4	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	D₄×C₅₃	1	2	yes	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
171	#103	—	2	8	no	—
172	C₄×C₈	—	2	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	D₈×C₅₃	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
189	Q₁₆	—	4	16	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	Q₁₆×C₅₃	1	4	yes	complemented
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	C₂×D₄×C₅₃	1	2	yes	—
207	C₄²	C₂×D₄×C₅₃	1	2	yes	—
208	C₂³	#121×C₅₃	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	#120×C₅₃	1	2	yes	—
225	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#121×C₅₃	1	2	yes	—
227	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
230	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
231	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
232	C₂×C₄	#137×C₅₃	1	2	yes	—
233	C₈	—	2	4	no	—
234	C₈	—	4	4	no	—
235	C₈	—	8	1	no	—
236	C₂²	C₅₃×#392	1	1	yes	characteristic
237	C₂²	C₅₃×#462	1	2	yes	—
238	C₂²	C₅₃×#466	1	1	yes	characteristic
239	C₂²	C₅₃×#469	1	1	yes	characteristic
240	C₂²	C₅₃×#473	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	4	no	—
244	C₄	—	8	1	no	—
245	C₄	C₅₃×#447	1	2	yes	—
246	C₄	C₅₃×#449	1	2	yes	—
247	C₄	C₅₃×#451	1	2	yes	—
248	C₄	C₅₃×#494	1	1	yes	characteristic
249	C₄	C₅₃×#497	1	1	yes	characteristic
250	C₂	C₅₃×#1674	1	2	yes	—
251	C₂	C₅₃×#1694	1	2	yes	—
252	C₂	C₅₃×#1999	1	1	yes	characteristic
253	C₂	C₅₃×#2016	1	1	yes	characteristic
254	C₂	C₅₃×#2020	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
69. C₂×C₈×C₅₃	Q₁₆	semidirect
108. #10265	C₅₃	direct (N × H)
146. C₅₃	#10265	direct (N × H)
197. C₂×C₈	Q₁₆×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×#500	2
C₂×C₅₃×#462	2
C₅₃×#1984	2
C₂×C₅₃×#332	2
C₅₃×#1277	2
#10265	53

Subgroups — C53×#10265 #27217777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10265 #27217777