Subgroups — C₅₃×#13261 #27220773

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#361	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2140	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₃	—	2	1	no	—
7	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
8	C₂×#127×C₅₃	—	2	1	no	—
9	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#493	—	2	2	no	—
11	C₅₃×#493	C₄	1	4	yes	complemented
12	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#427	C₂²	1	1	yes	characteristic
14	C₅₃×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#364	—	2	2	no	—
17	C₅₃×#361	—	2	4	no	—
18	C₅₃×#357	—	2	4	no	—
19	C₅₃×#339	—	2	1	no	—
20	C₂²×D₄×C₅₃	—	2	2	no	—
21	C₂²×D₄×C₅₃	—	4	2	no	—
22	C₂×D₈×C₅₃	—	2	4	no	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	C₂×D₈×C₅₃	—	2	2	no	—
26	C₂×D₈×C₅₃	—	2	2	no	—
27	C₂×D₈×C₅₃	—	2	2	no	—
28	C₂×D₈×C₅₃	—	4	1	no	—
29	C₂×D₈×C₅₃	—	4	1	no	—
30	C₂×D₈×C₅₃	—	4	2	no	—
31	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	#127×C₅₃	—	2	4	no	—
36	#127×C₅₃	—	4	2	no	—
37	#117×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	M₃₂×C₅₃	—	2	2	no	—
40	M₃₂×C₅₃	—	2	2	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	#108×C₅₃	—	2	1	no	—
43	#108×C₅₃	D₄	1	2	yes	complemented
44	#107×C₅₃	D₄	1	2	yes	complemented
45	#106×C₅₃	—	2	1	no	—
46	#97×C₅₃	—	4	1	no	—
47	C₄×C₈×C₅₃	—	4	1	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂⁴×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	—	4	8	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	8	8	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	D₈×C₅₃	—	2	4	no	—
62	D₈×C₅₃	—	2	4	no	—
63	D₈×C₅₃	—	4	4	no	—
64	D₈×C₅₃	—	4	4	no	—
65	D₈×C₅₃	—	4	2	no	—
66	D₈×C₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	2	2	no	—
68	M₁₆×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
73	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	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	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	1	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₁₆×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	4	4	no	—
87	C₂³×C₅₃	—	8	8	no	—
88	C₂³×C₅₃	—	8	2	no	—
89	C₂³×C₅₃	—	8	2	no	—
90	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	D₄×C₅₃	—	4	4	no	—
92	D₄×C₅₃	—	4	4	no	—
93	D₄×C₅₃	—	8	16	no	—
94	D₄×C₅₃	—	8	2	no	—
95	D₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₈×C₅₃	—	4	2	no	—
106	C₈×C₅₃	#102	1	2	yes	—
107	C₈×C₅₃	#102	1	2	yes	—
108	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₅₃	#124	1	1	yes	characteristic
111	C₈×C₅₃	#127	1	1	yes	characteristic
112	#13261	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	8	8	no	—
115	C₂²×C₅₃	—	8	4	no	—
116	C₂²×C₅₃	—	16	4	no	—
117	C₂²×C₅₃	—	16	4	no	—
118	C₂²×C₅₃	#418	1	1	yes	characteristic
119	C₂²×C₅₃	#459	1	1	yes	characteristic
120	C₂²×C₅₃	#460	1	1	yes	characteristic
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	—	8	2	no	—
123	C₄×C₅₃	#390	1	1	yes	characteristic
124	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₅₃	#459	1	1	yes	characteristic
126	C₄×C₅₃	#460	1	1	yes	characteristic
127	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
128	C₂×#364	C₂×C₅₃	1	1	yes	characteristic, complemented
129	C₂×#361	C₂×C₅₃	1	2	yes	complemented
130	#2140	C₂×C₅₃	1	2	yes	complemented
131	#1772	C₂×C₅₃	1	1	yes	characteristic, complemented
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	16	4	no	—
135	C₂×C₅₃	#1982	1	1	yes	characteristic
136	C₂×C₅₃	#2213	1	2	yes	—
137	C₂²×D₈	—	2	1	no	—
138	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
139	C₂×#127	—	2	1	no	—
140	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
141	#493	—	2	2	no	—
142	#493	C₄×C₅₃	1	4	yes	complemented
143	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
144	#427	C₂²×C₅₃	1	1	yes	characteristic
145	#405	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
147	#364	—	2	2	no	—
148	#361	—	2	4	no	—
149	#357	—	2	4	no	—
150	#339	—	2	1	no	—
151	C₅₃	#13261	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	4	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	—	4	1	no	—
161	C₂×D₈	—	4	1	no	—
162	C₂×D₈	—	4	2	no	—
163	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
167	#127	—	2	4	no	—
168	#127	—	4	2	no	—
169	#117	—	2	1	no	—
170	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	#108	—	2	1	no	—
175	#108	D₄×C₅₃	1	2	yes	complemented
176	#107	D₄×C₅₃	1	2	yes	complemented
177	#106	—	2	1	no	—
178	#97	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
181	C₄×C₈	D₄×C₅₃	1	2	yes	—
182	C₂⁴	—	4	2	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	4	8	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	8	8	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	D₈	—	2	4	no	—
194	D₈	—	2	4	no	—
195	D₈	—	4	4	no	—
196	D₈	—	4	4	no	—
197	D₈	—	4	2	no	—
198	D₈	—	4	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
203	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
204	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
205	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×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	2	no	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
217	C₁₆	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	8	8	no	—
220	C₂³	—	8	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
223	D₄	—	4	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	8	16	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
233	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
237	C₈	—	4	2	no	—
238	C₈	#102×C₅₃	1	2	yes	—
239	C₈	#102×C₅₃	1	2	yes	—
240	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
241	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
242	C₈	#124×C₅₃	1	1	yes	characteristic
243	C₈	#127×C₅₃	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	16	4	no	—
248	C₂²	—	16	4	no	—
249	C₂²	C₅₃×#418	1	1	yes	characteristic
250	C₂²	C₅₃×#459	1	1	yes	characteristic
251	C₂²	C₅₃×#460	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	8	2	no	—
254	C₄	C₅₃×#390	1	1	yes	characteristic
255	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
256	C₄	C₅₃×#459	1	1	yes	characteristic
257	C₄	C₅₃×#460	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₅₃×#1982	1	1	yes	characteristic
262	C₂	C₅₃×#2213	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#459
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#364	C₂	semidirect
3. C₂×C₅₃×#361	C₂	semidirect
4. C₅₃×#2140	C₂	semidirect
5. C₅₃×#1772	C₂	semidirect
11. C₅₃×#493	C₄	semidirect
43. #108×C₅₃	D₄	semidirect
44. #107×C₅₃	D₄	semidirect
112. #13261	C₅₃	direct (N × H)
128. C₂×#364	C₂×C₅₃	semidirect
129. C₂×#361	C₂×C₅₃	semidirect
130. #2140	C₂×C₅₃	semidirect
131. #1772	C₂×C₅₃	semidirect
142. #493	C₄×C₅₃	semidirect
151. C₅₃	#13261	direct (N × H)
175. #108	D₄×C₅₃	semidirect
176. #107	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#493	2
C₂×C₅₃×#364	2
C₂×C₅₃×#361	2
C₅₃×#2140	2
C₅₃×#1772	2
#13261	53

Subgroups — C53×#13261 #27220773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13261 #27220773