Subgroups — C₅₃×#9876 #27217388

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 212 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	C₄×#97×C₅₃	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×#97×C₅₃	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#389	C₄	1	2	yes	—
11	C₅₃×#387	—	2	2	no	—
12	C₅₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	#125×C₅₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₄⋊C₈×C₅₃	—	4	4	no	—
31	#97×C₅₃	—	2	8	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	8	no	—
45	C₂×C₈×C₅₃	—	2	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	M₁₆	1	4	yes	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
61	C₄²×C₅₃	#41	1	2	yes	—
62	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	4	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	#104	1	4	yes	—
76	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	#131	1	2	yes	—
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₅₃	#137	1	2	yes	—
84	C₈×C₅₃	—	2	8	no	—
85	C₈×C₅₃	—	8	2	no	—
86	#9876	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	#391	1	1	yes	characteristic
88	C₂²×C₅₃	#407	1	1	yes	characteristic
89	C₂²×C₅₃	#408	1	1	yes	characteristic
90	C₂²×C₅₃	#417	1	2	yes	—
91	C₂²×C₅₃	C₂×#104	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	4	2	no	—
95	C₄×C₅₃	—	4	4	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	#446	1	4	yes	—
98	C₄×C₅₃	#490	1	1	yes	characteristic
99	C₄×C₅₃	#491	1	1	yes	characteristic
100	C₄×C₅₃	#501	1	2	yes	—
101	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
102	#1842	C₂×C₅₃	1	1	yes	characteristic
103	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
104	#1279	C₂×C₅₃	1	4	yes	—
105	C₂×C₅₃	#1483	1	2	yes	—
106	C₂×C₅₃	#1489	1	2	yes	—
107	C₂×C₅₃	#1985	1	1	yes	characteristic
108	C₂×C₅₃	#1992	1	1	yes	characteristic
109	C₂×C₅₃	#1994	1	1	yes	characteristic
110	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
113	C₂×#97	—	2	2	no	—
114	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
115	#389	C₄×C₅₃	1	2	yes	—
116	#387	—	2	2	no	—
117	#382	C₄×C₅₃	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
120	C₄³	C₂²×C₅₃	1	1	yes	characteristic
121	C₅₃	#9876	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	#125	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
132	C₂×C₄²	D₄×C₅₃	1	2	yes	—
133	C₂×C₄²	D₄×C₅₃	1	2	yes	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	4	4	no	—
137	#97	—	2	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	Q₈×C₅₃	1	4	yes	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	#95	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
167	C₄²	#41×C₅₃	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#104×C₅₃	1	4	yes	—
182	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
183	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
184	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
187	C₂×C₄	#131×C₅₃	1	2	yes	—
188	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
189	C₂×C₄	#137×C₅₃	1	2	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	8	2	no	—
192	C₂²	C₅₃×#391	1	1	yes	characteristic
193	C₂²	C₅₃×#407	1	1	yes	characteristic
194	C₂²	C₅₃×#408	1	1	yes	characteristic
195	C₂²	C₅₃×#417	1	2	yes	—
196	C₂²	C₂×#104×C₅₃	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	C₅₃×#446	1	4	yes	—
203	C₄	C₅₃×#490	1	1	yes	characteristic
204	C₄	C₅₃×#491	1	1	yes	characteristic
205	C₄	C₅₃×#501	1	2	yes	—
206	C₂	C₅₃×#1483	1	2	yes	—
207	C₂	C₅₃×#1489	1	2	yes	—
208	C₂	C₅₃×#1985	1	1	yes	characteristic
209	C₂	C₅₃×#1992	1	1	yes	characteristic
210	C₂	C₅₃×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×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
86. #9876	C₅₃	direct (N × H)
121. C₅₃	#9876	direct (N × H)

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
C₄×#97×C₅₃	2
C₅₃×#1279	2
#9876	53

Subgroups — C53×#9876 #27217388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9876 #27217388