Subgroups — C₅₃×#13158 #27220670

Order 13568.

316 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1978	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#430	—	2	1	no	—
12	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#418	—	2	1	no	—
16	C₂×#100×C₅₃	—	2	1	no	—
17	C₅₃×#405	C₂²	1	1	yes	characteristic
18	C₅₃×#361	—	2	2	no	—
19	C₅₃×#361	C₄	1	4	yes	complemented
20	C₅₃×#347	—	2	2	no	—
21	C₅₃×#344	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂×D₈×C₅₃	—	2	1	no	—
24	C₂×D₈×C₅₃	—	2	1	no	—
25	C₂×D₈×C₅₃	—	2	1	no	—
26	C₂×D₈×C₅₃	—	2	1	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂×M₁₆×C₅₃	—	2	1	no	—
32	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	#117×C₅₃	D₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	1	no	—
36	M₃₂×C₅₃	—	2	1	no	—
37	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
38	M₃₂×C₅₃	D₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	—	2	1	no	—
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	#108×C₅₃	—	2	1	no	—
42	#108×C₅₃	—	2	1	no	—
43	#108×C₅₃	—	2	1	no	—
44	#108×C₅₃	—	2	1	no	—
45	#108×C₅₃	—	2	1	no	—
46	#108×C₅₃	—	4	2	no	—
47	#108×C₅₃	C₂×C₄	1	2	yes	—
48	#102×C₅₃	—	4	2	no	—
49	#100×C₅₃	—	4	2	no	—
50	#97×C₅₃	—	2	1	no	—
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₂⁴×C₅₃	—	4	1	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	4	1	no	—
55	C₂×D₄×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	D₈×C₅₃	—	2	2	no	—
61	D₈×C₅₃	—	2	2	no	—
62	D₈×C₅₃	—	4	1	no	—
63	D₈×C₅₃	—	4	1	no	—
64	M₁₆×C₅₃	—	2	2	no	—
65	M₁₆×C₅₃	—	2	1	no	—
66	M₁₆×C₅₃	—	2	1	no	—
67	M₁₆×C₅₃	—	4	1	no	—
68	M₁₆×C₅₃	—	4	1	no	—
69	M₁₆×C₅₃	—	4	1	no	—
70	M₁₆×C₅₃	—	4	1	no	—
71	M₁₆×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂²⋊C₄×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₁₆×C₅₃	—	2	2	no	—
86	C₁₆×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	—	8	1	no	—
89	C₂³×C₅₃	—	8	1	no	—
90	C₂³×C₅₃	—	16	2	no	—
91	C₂³×C₅₃	#137	1	1	yes	characteristic
92	D₄×C₅₃	—	4	2	no	—
93	D₄×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	8	1	no	—
95	D₄×C₅₃	—	8	1	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₈×C₅₃	—	4	2	no	—
104	C₈×C₅₃	—	4	1	no	—
105	C₈×C₅₃	—	8	1	no	—
106	C₈×C₅₃	—	8	1	no	—
107	C₈×C₅₃	#117	1	1	yes	characteristic
108	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₈×C₅₃	#123	1	1	yes	characteristic
110	C₈×C₅₃	#126	1	1	yes	characteristic
111	#13158	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₅₃	—	4	1	no	—
113	C₂²×C₅₃	—	8	2	no	—
114	C₂²×C₅₃	—	16	2	no	—
115	C₂²×C₅₃	—	16	2	no	—
116	C₂²×C₅₃	—	16	2	no	—
117	C₂²×C₅₃	#440	1	1	yes	characteristic
118	C₂²×C₅₃	#468	1	1	yes	characteristic
119	C₂²×C₅₃	#489	1	1	yes	characteristic
120	C₄×C₅₃	—	8	1	no	—
121	C₄×C₅₃	#388	1	1	yes	characteristic
122	C₄×C₅₃	#442	1	1	yes	characteristic
123	C₄×C₅₃	#469	1	1	yes	characteristic
124	C₄×C₅₃	#490	1	1	yes	characteristic
125	C₂×#361	C₂×C₅₃	1	1	yes	characteristic
126	C₂×#361	C₂×C₅₃	1	1	yes	characteristic
127	#2110	C₂×C₅₃	1	1	yes	characteristic, complemented
128	#1978	C₂×C₅₃	1	1	yes	characteristic
129	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
130	#1388	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#1388	C₂×C₅₃	1	1	yes	characteristic, complemented
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	16	2	no	—
136	C₂×C₅₃	#1936	1	1	yes	characteristic
137	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
140	#430	—	2	1	no	—
141	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
144	#418	—	2	1	no	—
145	C₂×#100	—	2	1	no	—
146	#405	C₂²×C₅₃	1	1	yes	characteristic
147	#361	—	2	2	no	—
148	#361	C₄×C₅₃	1	4	yes	complemented
149	#347	—	2	2	no	—
150	#344	—	2	1	no	—
151	C₅₃	#13158	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×D₄	—	2	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
164	#117	D₄×C₅₃	1	1	yes	characteristic
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
168	M₃₂	D₄×C₅₃	1	2	yes	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	—	2	1	no	—
175	#108	—	2	1	no	—
176	#108	—	4	2	no	—
177	#108	C₂×C₄×C₅₃	1	2	yes	—
178	#102	—	4	2	no	—
179	#100	—	4	2	no	—
180	#97	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₂⁴	—	4	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
190	D₈	—	2	2	no	—
191	D₈	—	2	2	no	—
192	D₈	—	4	1	no	—
193	D₈	—	4	1	no	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
209	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
210	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
211	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	16	2	no	—
221	C₂³	#137×C₅₃	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	1	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
229	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
230	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
231	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
232	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
233	C₈	—	4	2	no	—
234	C₈	—	4	1	no	—
235	C₈	—	8	1	no	—
236	C₈	—	8	1	no	—
237	C₈	#117×C₅₃	1	1	yes	characteristic
238	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
239	C₈	#123×C₅₃	1	1	yes	characteristic
240	C₈	#126×C₅₃	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₅₃×#440	1	1	yes	characteristic
247	C₂²	C₅₃×#468	1	1	yes	characteristic
248	C₂²	C₅₃×#489	1	1	yes	characteristic
249	C₄	—	8	1	no	—
250	C₄	C₅₃×#388	1	1	yes	characteristic
251	C₄	C₅₃×#442	1	1	yes	characteristic
252	C₄	C₅₃×#469	1	1	yes	characteristic
253	C₄	C₅₃×#490	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₅₃×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1936
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2110	C₂	semidirect
5. C₅₃×#1953	C₂	semidirect
6. C₅₃×#1388	C₂	semidirect
7. C₅₃×#1388	C₂	semidirect
19. C₅₃×#361	C₄	semidirect
111. #13158	C₅₃	direct (N × H)
127. #2110	C₂×C₅₃	semidirect
129. #1953	C₂×C₅₃	semidirect
130. #1388	C₂×C₅₃	semidirect
131. #1388	C₂×C₅₃	semidirect
148. #361	C₄×C₅₃	semidirect
151. C₅₃	#13158	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×#361	2
C₅₃×#2110	2
C₅₃×#1978	2
C₅₃×#1953	2
C₅₃×#1388	2
#13158	53

Subgroups — C53×#13158 #27220670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13158 #27220670