Subgroups — C₅₃×#9872 #27217384

Order 13568.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1842	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×#125×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#389	C₄	1	2	yes	—
10	C₅₃×#387	—	2	2	no	—
11	C₅₃×#382	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	#125×C₅₃	—	2	2	no	—
19	#125×C₅₃	C₈	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄⋊C₈×C₅₃	—	4	4	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	Q₈	1	4	yes	—
34	#95×C₅₃	—	2	2	no	—
35	#95×C₅₃	—	2	2	no	—
36	#95×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	4	4	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	2	8	no	—
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	C₂×C₈	1	2	yes	—
60	C₄²×C₅₃	M₁₆	1	2	yes	—
61	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
62	C₄²×C₅₃	#41	1	2	yes	—
63	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#131	1	2	yes	—
84	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₅₃	#137	1	2	yes	—
86	C₈×C₅₃	—	4	4	no	—
87	C₈×C₅₃	—	8	2	no	—
88	#9872	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	#391	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₅₃	#408	1	1	yes	characteristic
92	C₂²×C₅₃	#417	1	2	yes	—
93	C₂²×C₅₃	#421	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	4	no	—
97	C₄×C₅₃	—	4	2	no	—
98	C₄×C₅₃	—	8	2	no	—
99	C₄×C₅₃	C₈×Q₈	1	2	yes	—
100	C₄×C₅₃	#446	1	2	yes	—
101	C₄×C₅₃	#490	1	1	yes	characteristic
102	C₄×C₅₃	#491	1	1	yes	characteristic
103	C₄×C₅₃	#501	1	2	yes	—
104	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
105	#1842	C₂×C₅₃	1	1	yes	characteristic
106	#1744	C₂×C₅₃	1	1	yes	characteristic
107	#1279	C₂×C₅₃	1	4	yes	—
108	C₂×C₅₃	#1477	1	2	yes	—
109	C₂×C₅₃	#1495	1	2	yes	—
110	C₂×C₅₃	#1984	1	1	yes	characteristic
111	C₂×C₅₃	#1992	1	1	yes	characteristic
112	C₂×C₅₃	#1995	1	1	yes	characteristic
113	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₈	—	2	2	no	—
115	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
117	#389	C₄×C₅₃	1	2	yes	—
118	#387	—	2	2	no	—
119	#382	C₄×C₅₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	2	2	no	—
121	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
122	C₄³	C₂²×C₅₃	1	1	yes	characteristic
123	#336	—	2	2	no	—
124	C₅₃	#9872	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	#125	—	2	2	no	—
128	#125	C₈×C₅₃	1	4	yes	complemented
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
136	C₂×C₄²	D₄×C₅₃	1	2	yes	—
137	C₂×C₄²	D₄×C₅₃	1	2	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	4	4	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	Q₈×C₅₃	1	4	yes	—
143	#95	—	2	2	no	—
144	#95	—	2	2	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₈×C₅₃	1	2	yes	—
169	C₄²	M₁₆×C₅₃	1	2	yes	—
170	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
171	C₄²	#41×C₅₃	1	2	yes	—
172	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
186	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
187	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
188	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
192	C₂×C₄	#131×C₅₃	1	2	yes	—
193	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
194	C₂×C₄	#137×C₅₃	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	2	no	—
197	C₂²	C₅₃×#391	1	1	yes	characteristic
198	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
199	C₂²	C₅₃×#408	1	1	yes	characteristic
200	C₂²	C₅₃×#417	1	2	yes	—
201	C₂²	C₅₃×#421	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	8	2	no	—
207	C₄	C₈×Q₈×C₅₃	1	2	yes	—
208	C₄	C₅₃×#446	1	2	yes	—
209	C₄	C₅₃×#490	1	1	yes	characteristic
210	C₄	C₅₃×#491	1	1	yes	characteristic
211	C₄	C₅₃×#501	1	2	yes	—
212	C₂	C₅₃×#1477	1	2	yes	—
213	C₂	C₅₃×#1495	1	2	yes	—
214	C₂	C₅₃×#1984	1	1	yes	characteristic
215	C₂	C₅₃×#1992	1	1	yes	characteristic
216	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
19. #125×C₅₃	C₈	semidirect
88. #9872	C₅₃	direct (N × H)
124. C₅₃	#9872	direct (N × H)
128. #125	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₅₃	2
C₅₃×#1842	2
C₅₃×#1744	2
C₅₃×#1279	2
#9872	53

Subgroups — C53×#9872 #27217384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9872 #27217384