Subgroups — C₅₃×#13407 #27220919

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#357	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#357	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1374	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	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#493	—	2	4	no	—
11	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#389	—	2	1	no	—
15	C₅₃×#357	—	2	4	no	—
16	C₅₃×#357	—	2	4	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂²×D₄×C₅₃	—	2	2	no	—
19	C₂²×D₄×C₅₃	—	4	1	no	—
20	C₂²×D₄×C₅₃	—	4	1	no	—
21	C₂×D₈×C₅₃	—	2	4	no	—
22	C₂×D₈×C₅₃	—	2	2	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	—	4	2	no	—
28	C₂×D₈×C₅₃	—	4	2	no	—
29	C₂×D₈×C₅₃	D₄	1	4	yes	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#127×C₅₃	—	2	4	no	—
33	#127×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	#107×C₅₃	—	2	1	no	—
42	#107×C₅₃	—	2	2	no	—
43	#107×C₅₃	D₄	1	2	yes	complemented
44	C₄×C₈×C₅₃	—	2	2	no	—
45	#95×C₅₃	—	4	1	no	—
46	C₂⁴×C₅₃	—	4	2	no	—
47	C₂×D₄×C₅₃	—	2	4	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	—	4	4	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	8	2	no	—
56	C₂×D₄×C₅₃	—	8	2	no	—
57	C₂×D₄×C₅₃	—	8	2	no	—
58	C₂×D₄×C₅₃	—	8	2	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	D₈×C₅₃	—	2	4	no	—
65	D₈×C₅₃	—	2	4	no	—
66	D₈×C₅₃	—	4	8	no	—
67	D₈×C₅₃	—	4	2	no	—
68	D₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	4	1	no	—
78	C₄⋊C₄×C₅₃	—	4	1	no	—
79	C₄⋊C₄×C₅₃	—	4	1	no	—
80	C₄⋊C₄×C₅₃	—	8	2	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₁₆×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	—	4	4	no	—
85	C₂³×C₅₃	—	8	2	no	—
86	C₂³×C₅₃	—	8	2	no	—
87	C₂³×C₅₃	—	8	2	no	—
88	C₂³×C₅₃	—	8	2	no	—
89	C₂³×C₅₃	—	8	2	no	—
90	C₂³×C₅₃	—	8	2	no	—
91	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
92	D₄×C₅₃	—	4	4	no	—
93	D₄×C₅₃	—	4	4	no	—
94	D₄×C₅₃	—	8	8	no	—
95	D₄×C₅₃	—	8	8	no	—
96	D₄×C₅₃	—	8	2	no	—
97	D₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	—	8	1	no	—
105	C₂×C₄×C₅₃	—	8	1	no	—
106	C₂×C₄×C₅₃	—	8	1	no	—
107	C₂×C₄×C₅₃	—	8	1	no	—
108	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
110	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	#121	1	2	yes	—
118	C₈×C₅₃	#123	1	2	yes	—
119	#13407	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	8	4	no	—
121	C₂²×C₅₃	—	8	4	no	—
122	C₂²×C₅₃	—	8	4	no	—
123	C₂²×C₅₃	—	16	2	no	—
124	C₂²×C₅₃	—	16	2	no	—
125	C₂²×C₅₃	—	16	2	no	—
126	C₂²×C₅₃	—	16	2	no	—
127	C₂²×C₅₃	#447	1	1	yes	characteristic
128	C₂²×C₅₃	#459	1	1	yes	characteristic
129	C₂²×C₅₃	#486	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
132	C₂²×C₅₃	#509	1	1	yes	characteristic
133	C₂²×C₅₃	#509	1	1	yes	characteristic
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	—	16	1	no	—
138	C₄×C₅₃	#394	1	1	yes	characteristic
139	C₄×C₅₃	#450	1	1	yes	characteristic
140	C₄×C₅₃	#460	1	1	yes	characteristic
141	C₄×C₅₃	#490	1	1	yes	characteristic
142	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
143	C₂×#357	C₂×C₅₃	1	2	yes	complemented
144	C₂×#357	C₂×C₅₃	1	2	yes	complemented
145	#1942	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#1374	C₂×C₅₃	1	1	yes	characteristic, complemented
147	C₂×C₅₃	—	16	4	no	—
148	C₂×C₅₃	#2041	1	1	yes	characteristic
149	C₂×C₅₃	#2184	1	1	yes	characteristic
150	C₂×C₅₃	#2187	1	1	yes	characteristic
151	C₂×C₅₃	#2206	1	1	yes	characteristic
152	C₂×C₅₃	#2207	1	1	yes	characteristic
153	C₂×C₅₃	#2240	1	1	yes	characteristic
154	C₂×C₅₃	#2244	1	1	yes	characteristic
155	C₂²×D₈	—	2	1	no	—
156	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
157	C₂×#127	—	2	1	no	—
158	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
159	#493	—	2	4	no	—
160	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	#389	—	2	1	no	—
164	#357	—	2	4	no	—
165	#357	—	2	4	no	—
166	#340	—	2	1	no	—
167	C₅₃	#13407	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂×D₈	D₄×C₅₃	1	4	yes	—
180	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
182	#127	—	2	4	no	—
183	#127	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	2	2	no	—
191	#107	—	2	1	no	—
192	#107	—	2	2	no	—
193	#107	D₄×C₅₃	1	2	yes	complemented
194	C₄×C₈	—	2	2	no	—
195	#95	—	4	1	no	—
196	C₂⁴	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂×D₄	—	8	2	no	—
207	C₂×D₄	—	8	2	no	—
208	C₂×D₄	—	8	2	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	8	no	—
217	D₈	—	4	2	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
224	C₂×C₈	#41×C₅₃	1	2	yes	—
225	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	8	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
259	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
260	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
261	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
262	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
264	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
265	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#121×C₅₃	1	2	yes	—
268	C₈	#123×C₅₃	1	2	yes	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	2	no	—
273	C₂²	—	16	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₅₃×#447	1	1	yes	characteristic
277	C₂²	C₅₃×#459	1	1	yes	characteristic
278	C₂²	C₅₃×#486	1	1	yes	characteristic
279	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
280	C₂²	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
281	C₂²	C₅₃×#509	1	1	yes	characteristic
282	C₂²	C₅₃×#509	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	16	1	no	—
287	C₄	C₅₃×#394	1	1	yes	characteristic
288	C₄	C₅₃×#450	1	1	yes	characteristic
289	C₄	C₅₃×#460	1	1	yes	characteristic
290	C₄	C₅₃×#490	1	1	yes	characteristic
291	C₂	—	16	4	no	—
292	C₂	C₅₃×#2041	1	1	yes	characteristic
293	C₂	C₅₃×#2184	1	1	yes	characteristic
294	C₂	C₅₃×#2187	1	1	yes	characteristic
295	C₂	C₅₃×#2206	1	1	yes	characteristic
296	C₂	C₅₃×#2207	1	1	yes	characteristic
297	C₂	C₅₃×#2240	1	1	yes	characteristic
298	C₂	C₅₃×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×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×#357	C₂	semidirect
3. C₂×C₅₃×#357	C₂	semidirect
4. C₅₃×#1942	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
43. #107×C₅₃	D₄	semidirect
119. #13407	C₅₃	direct (N × H)
143. C₂×#357	C₂×C₅₃	semidirect
144. C₂×#357	C₂×C₅₃	semidirect
145. #1942	C₂×C₅₃	semidirect
146. #1374	C₂×C₅₃	semidirect
167. C₅₃	#13407	direct (N × H)
193. #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×#357	2
C₅₃×#1942	2
C₅₃×#1374	2
#13407	53

Subgroups — C53×#13407 #27220919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13407 #27220919