Subgroups — C₅₃×#13137 #27220649

Order 13568.

316 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2148	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1974	C₂	1	1	yes	maximal, characteristic
5	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×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	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#104×C₅₃	—	2	1	no	—
15	C₅₃×#408	—	2	1	no	—
16	C₄×M₁₆×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#368	—	2	2	no	—
19	C₅₃×#362	C₄	1	2	yes	complemented
20	C₅₃×#361	C₄	1	2	yes	complemented
21	C₅₃×#359	—	2	2	no	—
22	C₅₃×#341	—	2	1	no	—
23	C₂×#41×C₅₃	—	2	1	no	—
24	#135×C₅₃	—	2	1	no	—
25	#135×C₅₃	—	2	1	no	—
26	#135×C₅₃	—	2	1	no	—
27	#135×C₅₃	—	2	1	no	—
28	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₅₃	—	2	1	no	—
30	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	—	2	1	no	—
32	C₂×M₁₆×C₅₃	—	2	1	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	M₃₂×C₅₃	—	2	1	no	—
38	M₃₂×C₅₃	—	2	1	no	—
39	M₃₂×C₅₃	—	2	1	no	—
40	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₁₆×C₅₃	—	2	1	no	—
42	C₂×C₁₆×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	—	2	1	no	—
47	#108×C₅₃	C₂×C₄	1	2	yes	—
48	#108×C₅₃	D₄	1	2	yes	—
49	#104×C₅₃	—	4	1	no	—
50	#104×C₅₃	—	4	1	no	—
51	C₂²⋊C₈×C₅₃	—	4	2	no	—
52	#97×C₅₃	—	4	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	#41×C₅₃	—	4	1	no	—
58	#41×C₅₃	—	4	1	no	—
59	#41×C₅₃	—	4	1	no	—
60	C₂×Q₈×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₅₃	—	2	1	no	—
67	Q₁₆×C₅₃	—	2	1	no	—
68	SD₁₆×C₅₃	—	4	1	no	—
69	SD₁₆×C₅₃	—	4	1	no	—
70	D₈×C₅₃	—	2	1	no	—
71	D₈×C₅₃	—	2	1	no	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	4	1	no	—
75	M₁₆×C₅₃	—	4	1	no	—
76	M₁₆×C₅₃	—	4	1	no	—
77	M₁₆×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
90	C₄²×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	D₈	1	1	yes	characteristic
92	C₄²×C₅₃	D₈	1	1	yes	characteristic
93	C₁₆×C₅₃	—	2	2	no	—
94	C₁₆×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	—	4	1	no	—
96	C₂³×C₅₃	#137	1	1	yes	characteristic
97	Q₈×C₅₃	—	4	1	no	—
98	Q₈×C₅₃	—	4	1	no	—
99	D₄×C₅₃	—	4	1	no	—
100	D₄×C₅₃	—	4	1	no	—
101	D₄×C₅₃	—	8	1	no	—
102	D₄×C₅₃	—	8	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	8	1	no	—
108	C₂×C₄×C₅₃	—	8	1	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₈×C₅₃	—	4	2	no	—
116	C₈×C₅₃	—	4	2	no	—
117	C₈×C₅₃	—	4	1	no	—
118	C₈×C₅₃	—	8	1	no	—
119	C₈×C₅₃	#117	1	1	yes	characteristic
120	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₅₃	#124	1	1	yes	characteristic
122	C₈×C₅₃	#126	1	1	yes	characteristic
123	#13137	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	—	8	1	no	—
127	C₂²×C₅₃	#440	1	1	yes	characteristic
128	C₂²×C₅₃	#464	1	1	yes	characteristic
129	C₂²×C₅₃	#480	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	4	1	no	—
132	C₄×C₅₃	—	8	1	no	—
133	C₄×C₅₃	#388	1	1	yes	characteristic
134	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
135	C₄×C₅₃	#459	1	1	yes	characteristic
136	C₄×C₅₃	#482	1	1	yes	characteristic
137	#2161	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#2148	C₂×C₅₃	1	1	yes	characteristic
139	#2143	C₂×C₅₃	1	1	yes	characteristic
140	#1974	C₂×C₅₃	1	1	yes	characteristic
141	C₄×#108	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#1388	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#1376	C₂×C₅₃	1	1	yes	characteristic, complemented
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	16	1	no	—
148	C₂×C₅₃	#1925	1	1	yes	characteristic
149	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×#104	—	2	1	no	—
156	#408	—	2	1	no	—
157	C₄×M₁₆	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	#368	—	2	2	no	—
160	#362	C₄×C₅₃	1	2	yes	complemented
161	#361	C₄×C₅₃	1	2	yes	complemented
162	#359	—	2	2	no	—
163	#341	—	2	1	no	—
164	C₅₃	#13137	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#41	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	—	2	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
171	C₂×SD₁₆	—	2	1	no	—
172	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	C₂×C₄×C₅₃	1	2	yes	—
190	#108	D₄×C₅₃	1	2	yes	—
191	#104	—	4	1	no	—
192	#104	—	4	1	no	—
193	C₂²⋊C₈	—	4	2	no	—
194	#97	—	4	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	4	1	no	—
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
208	Q₁₆	—	2	1	no	—
209	Q₁₆	—	2	1	no	—
210	SD₁₆	—	4	1	no	—
211	SD₁₆	—	4	1	no	—
212	D₈	—	2	1	no	—
213	D₈	—	2	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
229	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
230	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
231	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	D₈×C₅₃	1	1	yes	characteristic
234	C₄²	D₈×C₅₃	1	1	yes	characteristic
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	#137×C₅₃	1	1	yes	characteristic
239	Q₈	—	4	1	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	1	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
253	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
255	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
256	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	1	no	—
260	C₈	—	8	1	no	—
261	C₈	#117×C₅₃	1	1	yes	characteristic
262	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
263	C₈	#124×C₅₃	1	1	yes	characteristic
264	C₈	#126×C₅₃	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₅₃×#440	1	1	yes	characteristic
269	C₂²	C₅₃×#464	1	1	yes	characteristic
270	C₂²	C₅₃×#480	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₅₃×#388	1	1	yes	characteristic
275	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
276	C₄	C₅₃×#459	1	1	yes	characteristic
277	C₄	C₅₃×#482	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	16	1	no	—
282	C₂	C₅₃×#1925	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#459
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1925
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
1. C₅₃×#2161	C₂	semidirect
5. C₄×#108×C₅₃	C₂	semidirect
6. C₅₃×#1388	C₂	semidirect
7. C₅₃×#1376	C₂	semidirect
19. C₅₃×#362	C₄	semidirect
20. C₅₃×#361	C₄	semidirect
123. #13137	C₅₃	direct (N × H)
137. #2161	C₂×C₅₃	semidirect
141. C₄×#108	C₂×C₅₃	semidirect
142. #1388	C₂×C₅₃	semidirect
143. #1376	C₂×C₅₃	semidirect
160. #362	C₄×C₅₃	semidirect
161. #361	C₄×C₅₃	semidirect
164. C₅₃	#13137	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×#2161	2
C₅₃×#2148	2
C₅₃×#2143	2
C₅₃×#1974	2
C₄×#108×C₅₃	2
C₅₃×#1388	2
C₅₃×#1376	2
#13137	53

Subgroups — C53×#13137 #27220649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13137 #27220649