Subgroups — #27277751

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274866	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272550	C₂	1	1	yes	maximal, characteristic
4	#20272549	C₂	1	2	yes	maximal
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133146	—	2	2	no	—
8	C₂×#13133146	—	2	2	no	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133146	—	4	4	no	—
29	#13133146	—	4	4	no	—
30	C₄×#88322	—	2	2	no	—
31	C₈×Dic₅₃	—	2	2	no	—
32	C₈×Dic₅₃	—	2	2	no	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	C₈	1	4	yes	complemented
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₄²×C₅₃	#41	1	2	yes	—
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₄×Dic₅₃	—	2	1	no	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	2	no	—
63	C₄×Dic₅₃	—	2	2	no	—
64	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
65	C₄×Dic₅₃	M₁₆	1	2	yes	—
66	C₄×Dic₅₃	D₈	1	2	yes	—
67	C₄×Dic₅₃	SD₁₆	1	4	yes	—
68	C₄×Dic₅₃	Q₁₆	1	2	yes	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	C₂×C₈	1	2	yes	—
77	C₂×#88322	M₁₆	1	2	yes	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#124	1	2	yes	—
84	C₂×C₄×C₅₃	#125	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×Dic₅₃	—	2	1	no	—
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	—	2	1	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
97	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
98	C₂×Dic₅₃	#106	1	2	yes	—
99	C₂×Dic₅₃	#106	1	2	yes	—
100	C₂×Dic₅₃	#107	1	2	yes	—
101	C₂×Dic₅₃	#107	1	2	yes	—
102	C₂×Dic₅₃	#108	1	2	yes	—
103	C₂×Dic₅₃	#108	1	2	yes	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	—	8	2	no	—
107	#88322	—	2	2	no	—
108	#88322	—	2	2	no	—
109	#88322	—	8	2	no	—
110	#8784	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	#382	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₅₃	#423	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	#431	1	1	yes	characteristic
120	C₄×C₅₃	#431	1	1	yes	characteristic
121	C₄×C₅₃	#432	1	1	yes	characteristic
122	C₄×C₅₃	#432	1	1	yes	characteristic
123	C₄×C₅₃	#486	1	2	yes	—
124	C₄×C₅₃	#487	1	2	yes	—
125	Dic₅₃	—	2	2	no	—
126	Dic₅₃	—	2	2	no	—
127	Dic₅₃	#334	1	4	yes	—
128	Dic₅₃	#335	1	4	yes	—
129	#1842	—	53	2	no	—
130	C₄²×C₈	—	53	1	no	—
131	#1277	—	53	2	no	—
132	#1277	—	53	1	no	—
133	#1277	D₅₃	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
136	C₂×C₅₃	#1566	1	1	yes	characteristic
137	C₂×C₅₃	#1567	1	1	yes	characteristic
138	C₂×C₅₃	#1831	1	1	yes	characteristic
139	C₂×C₅₃	#1834	1	1	yes	characteristic
140	C₂×C₅₃	#1837	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	53	2	no	—
142	C₂×C₄⋊C₈	—	106	2	no	—
143	C₂×C₄⋊C₈	—	106	2	no	—
144	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	—	53	1	no	—
149	C₂×C₄×C₈	—	53	1	no	—
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₄³	—	53	1	no	—
152	C₅₃	#8784	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	1	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄²	—	53	1	no	—
160	C₂×C₄²	—	53	1	no	—
161	C₂×C₄²	—	53	1	no	—
162	C₂×C₄²	—	53	1	no	—
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	C₄⋊C₈	—	106	2	no	—
170	C₄⋊C₈	—	212	4	no	—
171	C₄⋊C₈	—	212	4	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	53	4	no	—
174	C₄×C₈	—	106	2	no	—
175	C₄×C₈	—	106	2	no	—
176	C₄×C₈	—	106	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	1	no	—
182	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	106	1	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	1	no	—
197	C₂×C₈	—	106	1	no	—
198	C₂×C₈	—	212	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	C₈×D₅₃	1	2	yes	—
202	C₂×C₈	#11719912	1	2	yes	—
203	C₄²	—	53	2	no	—
204	C₄²	—	53	2	no	—
205	C₄²	—	53	2	no	—
206	C₄²	—	53	4	no	—
207	C₄²	—	53	2	no	—
208	C₄²	—	106	2	no	—
209	C₄²	—	106	2	no	—
210	C₄²	—	106	1	no	—
211	C₄²	—	106	1	no	—
212	C₄²	#11719945	1	2	yes	—
213	C₄²	#11719947	1	2	yes	—
214	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	53	2	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	53	2	no	—
220	C₂×C₄	—	53	2	no	—
221	C₂×C₄	—	53	2	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	53	2	no	—
225	C₂×C₄	—	106	1	no	—
226	C₂×C₄	—	106	1	no	—
227	C₂×C₄	—	106	1	no	—
228	C₂×C₄	—	106	1	no	—
229	C₂×C₄	#13133173	1	1	yes	characteristic
230	C₂×C₄	#13133174	1	1	yes	characteristic
231	C₂×C₄	#13133203	1	2	yes	—
232	C₂×C₄	#13133207	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
234	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
237	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
238	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
239	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	106	2	no	—
244	C₈	—	106	2	no	—
245	C₈	—	424	2	no	—
246	C₂²	#16810487	1	1	yes	characteristic
247	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
248	C₂²	#16810645	1	1	yes	characteristic
249	C₂²	#106×D₅₃	1	1	yes	characteristic
250	C₂²	#107×D₅₃	1	1	yes	characteristic
251	C₂²	#108×D₅₃	1	1	yes	characteristic
252	C₂²	#108×D₅₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	53	4	no	—
255	C₄	—	53	4	no	—
256	C₄	—	106	2	no	—
257	C₄	—	106	2	no	—
258	C₄	#16810673	1	1	yes	characteristic
259	C₄	#16810674	1	1	yes	characteristic
260	C₄	#16810677	1	1	yes	characteristic
261	C₄	#16810678	1	1	yes	characteristic
262	C₄	#16810968	1	2	yes	—
263	C₄	#16810972	1	2	yes	—
264	C₂	#334×D₅₃	1	1	yes	characteristic
265	C₂	#335×D₅₃	1	1	yes	characteristic
266	C₂	#20273469	1	1	yes	characteristic
267	C₂	#20273472	1	1	yes	characteristic
268	C₂	#20274830	1	1	yes	characteristic
269	C₂	#20274841	1	1	yes	characteristic
270	C₂	#20274851	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#8784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₅₃	C₈	semidirect
152. C₅₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#20274866	2
C₄×C₈×Dic₅₃	2
#20272550	2
#20272549	2
C₅₃×#1277	2
#8784	53