Subgroups — C₅₃×#32703 #27240215

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2103	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2080	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2050	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1938	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1881	C₂	1	2	yes	maximal
7	C₅₃×#1861	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1858	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#575	C₂²	1	1	yes	characteristic
11	C₅₃×#568	C₂²	1	2	yes	—
12	C₅₃×#430	C₂²	1	1	yes	characteristic
13	C₅₃×#430	C₂²	1	2	yes	—
14	C₅₃×#424	C₂²	1	2	yes	—
15	C₅₃×#423	C₂²	1	1	yes	characteristic
16	C₅₃×#423	C₂²	1	1	yes	characteristic
17	C₅₃×#421	C₂²	1	2	yes	—
18	C₂×#104×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#416	C₂²	1	2	yes	—
20	C₅₃×#413	C₂²	1	2	yes	—
21	C₅₃×#408	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#344	C₂²	1	2	yes	—
24	C₅₃×#344	C₂²	1	2	yes	—
25	C₅₃×#341	C₂²	1	2	yes	—
26	C₅₃×#341	C₂²	1	2	yes	—
27	C₅₃×#341	C₂²	1	2	yes	—
28	C₅₃×#339	C₂²	1	2	yes	—
29	C₅₃×#338	C₂²	1	1	yes	characteristic
30	C₅₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41×C₅₃	C₂³	1	2	yes	—
32	#137×C₅₃	—	2	2	no	—
33	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
34	#135×C₅₃	—	4	1	no	—
35	#131×C₅₃	—	4	2	no	—
36	#131×C₅₃	—	4	2	no	—
37	C₂×M₁₆×C₅₃	—	4	2	no	—
38	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
40	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	#117×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#108×C₅₃	—	4	2	no	—
47	#108×C₅₃	—	4	1	no	—
48	C₄⋊C₈×C₅₃	—	4	1	no	—
49	C₄⋊C₈×C₅₃	—	4	1	no	—
50	C₄⋊C₈×C₅₃	—	4	2	no	—
51	#104×C₅₃	—	2	4	no	—
52	#104×C₅₃	—	4	2	no	—
53	#103×C₅₃	—	2	2	no	—
54	#102×C₅₃	—	2	2	no	—
55	#101×C₅₃	—	2	2	no	—
56	#100×C₅₃	—	2	2	no	—
57	C₂²⋊C₈×C₅₃	—	4	2	no	—
58	#97×C₅₃	—	2	2	no	—
59	#41×C₅₃	—	2	2	no	—
60	#41×C₅₃	—	2	2	no	—
61	#41×C₅₃	—	2	2	no	—
62	#41×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	#41	1	2	yes	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₅₃	—	4	1	no	—
70	SD₁₆×C₅₃	—	4	2	no	—
71	D₈×C₅₃	—	4	1	no	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	—	2	2	no	—
76	M₁₆×C₅₃	—	2	2	no	—
77	M₁₆×C₅₃	—	2	2	no	—
78	M₁₆×C₅₃	—	2	2	no	—
79	M₁₆×C₅₃	—	2	2	no	—
80	M₁₆×C₅₃	—	4	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	4	4	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	#41	1	2	yes	—
95	C₂³×C₅₃	—	2	2	no	—
96	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
97	Q₈×C₅₃	—	2	2	no	—
98	Q₈×C₅₃	—	2	2	no	—
99	D₄×C₅₃	—	2	2	no	—
100	D₄×C₅₃	—	2	2	no	—
101	D₄×C₅₃	—	4	2	no	—
102	D₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	8	1	no	—
109	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
113	C₈×C₅₃	—	4	1	no	—
114	C₈×C₅₃	—	4	1	no	—
115	C₈×C₅₃	—	4	2	no	—
116	C₈×C₅₃	—	4	2	no	—
117	C₈×C₅₃	—	4	2	no	—
118	C₈×C₅₃	—	4	2	no	—
119	#32703	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	4	2	no	—
122	C₂²×C₅₃	—	8	2	no	—
123	C₂²×C₅₃	#539	1	1	yes	characteristic
124	C₂²×C₅₃	#539	1	2	yes	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	#551	1	1	yes	characteristic
129	C₄×C₅₃	#553	1	2	yes	—
130	C₄×C₅₃	H(3,𝔽₄)	1	1	yes	characteristic
131	#2103	C₂×C₅₃	1	2	yes	complemented
132	#2102	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#2080	C₂×C₅₃	1	2	yes	complemented
134	#2050	C₂×C₅₃	1	1	yes	characteristic
135	#1938	C₂×C₅₃	1	2	yes	complemented
136	#1881	C₂×C₅₃	1	2	yes	—
137	#1861	C₂×C₅₃	1	2	yes	complemented
138	#1858	C₂×C₅₃	1	2	yes	complemented
139	#1838	C₂×C₅₃	1	1	yes	characteristic, complemented
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	8	2	no	—
143	C₂×C₅₃	#2782	1	1	yes	characteristic
144	#575	C₂²×C₅₃	1	1	yes	characteristic
145	#568	C₂²×C₅₃	1	2	yes	—
146	#430	C₂²×C₅₃	1	1	yes	characteristic
147	#430	C₂²×C₅₃	1	2	yes	—
148	#424	C₂²×C₅₃	1	2	yes	—
149	#423	C₂²×C₅₃	1	1	yes	characteristic
150	#423	C₂²×C₅₃	1	1	yes	characteristic
151	#421	C₂²×C₅₃	1	2	yes	—
152	C₂×#104	C₂²×C₅₃	1	2	yes	—
153	#416	C₂²×C₅₃	1	2	yes	—
154	#413	C₂²×C₅₃	1	2	yes	—
155	#408	C₂²×C₅₃	1	2	yes	—
156	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
157	#344	C₂²×C₅₃	1	2	yes	—
158	#344	C₂²×C₅₃	1	2	yes	—
159	#341	C₂²×C₅₃	1	2	yes	—
160	#341	C₂²×C₅₃	1	2	yes	—
161	#341	C₂²×C₅₃	1	2	yes	—
162	#339	C₂²×C₅₃	1	2	yes	—
163	#338	C₂²×C₅₃	1	1	yes	characteristic
164	H(3,ℤ/4)	C₂²×C₅₃	1	1	yes	characteristic
165	C₅₃	#32703	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#41	C₂³×C₅₃	1	2	yes	—
167	#137	—	2	2	no	—
168	AGL(1,ℤ/8)	—	2	2	no	—
169	#135	—	4	1	no	—
170	#131	—	4	2	no	—
171	#131	—	4	2	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
174	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
175	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
178	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
179	#117	C₂³×C₅₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
181	#108	—	4	2	no	—
182	#108	—	4	1	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄⋊C₈	—	4	1	no	—
185	C₄⋊C₈	—	4	2	no	—
186	#104	—	2	4	no	—
187	#104	—	4	2	no	—
188	#103	—	2	2	no	—
189	#102	—	2	2	no	—
190	#101	—	2	2	no	—
191	#100	—	2	2	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	#97	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#41×C₅₃	1	2	yes	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	#41×C₅₃	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
204	Q₁₆	—	4	1	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	1	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	2	no	—
229	C₄²	#41×C₅₃	1	2	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
245	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
247	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	C₅₃×#539	1	1	yes	characteristic
258	C₂²	C₅₃×#539	1	2	yes	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	2	no	—
262	C₄	C₅₃×#551	1	1	yes	characteristic
263	C₄	C₅₃×#553	1	2	yes	—
264	C₄	C₅₃×H(3,𝔽₄)	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₅₃×#2782	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	#2782
prime radical	C₅₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2103	C₂	semidirect
2. C₅₃×#2102	C₂	semidirect
3. C₅₃×#2080	C₂	semidirect
5. C₅₃×#1938	C₂	semidirect
7. C₅₃×#1861	C₂	semidirect
8. C₅₃×#1858	C₂	semidirect
9. C₅₃×#1838	C₂	semidirect
119. #32703	C₅₃	direct (N × H)
131. #2103	C₂×C₅₃	semidirect
132. #2102	C₂×C₅₃	semidirect
133. #2080	C₂×C₅₃	semidirect
135. #1938	C₂×C₅₃	semidirect
137. #1861	C₂×C₅₃	semidirect
138. #1858	C₂×C₅₃	semidirect
139. #1838	C₂×C₅₃	semidirect
165. C₅₃	#32703	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2103	2
C₅₃×#2102	2
C₅₃×#2080	2
C₅₃×#2050	2
C₅₃×#1938	2
C₅₃×#1881	2
C₅₃×#1861	2
C₅₃×#1858	2
C₅₃×#1838	2
#32703	53

Subgroups — C53×#32703 #27240215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#32703 #27240215