Subgroups — C₅₃×#8828 #27216340

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#108×C₅₃	—	2	2	no	—
9	C₅₃×#423	C₂²	1	1	yes	characteristic
10	C₅₃×#423	C₂²	1	1	yes	characteristic
11	C₅₃×#423	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#341	—	2	2	no	—
17	C₅₃×#335	—	2	2	no	—
18	C₅₃×#335	—	2	2	no	—
19	C₅₃×#334	—	2	2	no	—
20	C₅₃×#334	—	2	2	no	—
21	C₅₃×#322	C₄	1	2	yes	—
22	C₅₃×#322	C₄	1	2	yes	—
23	C₂×M₁₆×C₅₃	—	2	1	no	—
24	C₂×M₁₆×C₅₃	—	2	1	no	—
25	C₂×M₁₆×C₅₃	—	2	1	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#108×C₅₃	—	4	4	no	—
32	C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₈×C₅₃	—	2	1	no	—
34	C₄⋊C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₈×C₅₃	—	2	1	no	—
36	C₄⋊C₈×C₅₃	—	2	1	no	—
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	Q₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₅₃	—	4	2	no	—
51	M₁₆×C₅₃	—	4	2	no	—
52	M₁₆×C₅₃	—	4	2	no	—
53	M₁₆×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	1	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₃	#137	1	2	yes	—
81	C₈×C₅₃	—	2	4	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	—	8	1	no	—
85	C₈×C₅₃	—	8	1	no	—
86	C₈×C₅₃	—	8	1	no	—
87	C₈×C₅₃	—	8	1	no	—
88	C₈×C₅₃	#108	1	4	yes	complemented
89	#8828	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#384	1	1	yes	characteristic
93	C₂²×C₅₃	#501	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	#428	1	2	yes	—
97	C₄×C₅₃	C₂×#108	1	2	yes	—
98	C₄×C₅₃	#430	1	2	yes	—
99	C₄×C₅₃	#496	1	1	yes	characteristic
100	C₄×C₅₃	#497	1	1	yes	characteristic
101	#1848	C₂×C₅₃	1	1	yes	characteristic
102	#1837	C₂×C₅₃	1	1	yes	characteristic
103	#1567	C₂×C₅₃	1	1	yes	characteristic
104	#1567	C₂×C₅₃	1	1	yes	characteristic
105	#1566	C₂×C₅₃	1	1	yes	characteristic
106	#1566	C₂×C₅₃	1	1	yes	characteristic
107	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	#1847	1	1	yes	characteristic
110	C₂×C₅₃	#1848	1	1	yes	characteristic
111	C₂×C₅₃	#1849	1	1	yes	characteristic
112	C₂×#108	—	2	2	no	—
113	#423	C₂²×C₅₃	1	1	yes	characteristic
114	#423	C₂²×C₅₃	1	1	yes	characteristic
115	#423	C₂²×C₅₃	1	1	yes	characteristic
116	#423	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
120	#341	—	2	2	no	—
121	#335	—	2	2	no	—
122	#335	—	2	2	no	—
123	#334	—	2	2	no	—
124	#334	—	2	2	no	—
125	#322	C₄×C₅₃	1	2	yes	—
126	#322	C₄×C₅₃	1	2	yes	—
127	C₅₃	#8828	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	1	no	—
131	C₂×M₁₆	—	2	1	no	—
132	C₂²×C₈	D₄×C₅₃	1	2	yes	—
133	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
134	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
135	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
136	#108	—	4	4	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	1	no	—
143	C₄⋊C₈	—	2	1	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
147	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
148	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
149	C₄×C₈	D₄×C₅₃	1	2	yes	—
150	C₄×C₈	D₄×C₅₃	1	2	yes	—
151	C₄×C₈	Q₈×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
155	M₁₆	—	4	2	no	—
156	M₁₆	—	4	2	no	—
157	M₁₆	—	4	2	no	—
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
168	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
169	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
170	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
171	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
173	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
174	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
175	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
183	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
184	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
185	C₂×C₄	#137×C₅₃	1	2	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	4	no	—
189	C₈	—	8	1	no	—
190	C₈	—	8	1	no	—
191	C₈	—	8	1	no	—
192	C₈	—	8	1	no	—
193	C₈	#108×C₅₃	1	4	yes	complemented
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₅₃×#384	1	1	yes	characteristic
197	C₂²	C₅₃×#501	1	2	yes	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	C₅₃×#428	1	2	yes	—
201	C₄	C₂×#108×C₅₃	1	2	yes	—
202	C₄	C₅₃×#430	1	2	yes	—
203	C₄	C₅₃×#496	1	1	yes	characteristic
204	C₄	C₅₃×#497	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	C₅₃×#1847	1	1	yes	characteristic
207	C₂	C₅₃×#1848	1	1	yes	characteristic
208	C₂	C₅₃×#1849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
88. C₈×C₅₃	#108	semidirect
89. #8828	C₅₃	direct (N × H)
127. C₅₃	#8828	direct (N × H)
193. C₈	#108×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1848	2
C₅₃×#1837	2
C₅₃×#1567	2
C₅₃×#1566	2
C₂×C₅₃×#322	2
#8828	53

Subgroups — C53×#8828 #27216340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8828 #27216340