Subgroups — #27354657

Order 13568.

734 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	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283426	C₂	1	2	yes	maximal, complemented
3	#20273628	C₂	1	4	yes	maximal, complemented
4	#20273538	C₂	1	4	yes	maximal, complemented
5	#20273165	C₂	1	2	yes	maximal, complemented
6	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133225	—	2	2	no	—
9	#16811173	C₂²	1	2	yes	—
10	#16811028	C₄	1	1	yes	characteristic
11	#16811000	C₄	1	1	yes	characteristic
12	#16810970	C₄	1	2	yes	—
13	#16810859	C₄	1	4	yes	—
14	C₄×D₄₂₄	C₂²	1	4	yes	—
15	#16810700	—	2	2	no	—
16	#16810700	C₂²	1	4	yes	—
17	#16810688	—	2	2	no	—
18	#16810688	C₂²	1	4	yes	—
19	#16810591	—	2	2	no	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	#16810350	C₂²	1	8	yes	—
23	#16810344	C₂²	1	2	yes	—
24	#16810344	C₂²	1	2	yes	—
25	C₅₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#334	C₂²	1	4	yes	—
28	C₂×#11719945	—	2	2	no	—
29	C₂×D₄₂₄	C₂×C₄	1	4	yes	—
30	#13133225	—	4	8	no	—
31	C₂×#11719912	—	2	4	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	D₄	1	2	yes	—
34	C₂×C₈×D₅₃	—	2	4	no	—
35	#13133217	C₂×C₄	1	2	yes	—
36	#13133214	C₂×C₄	1	2	yes	—
37	#13133205	—	2	2	no	—
38	#13133191	—	2	2	no	—
39	#13133191	C₂×C₄	1	4	yes	—
40	C₄×D₂₁₂	—	2	2	no	—
41	C₄×D₂₁₂	C₂³	1	4	yes	—
42	#13133151	C₂×C₄	1	4	yes	—
43	#13133139	C₂×C₄	1	8	yes	—
44	#13133129	—	2	4	no	—
45	#13133129	—	2	4	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	D₄	1	2	yes	—
48	C₄×#88322	C₂³	1	2	yes	—
49	C₄×#88322	C₂³	1	1	yes	characteristic
50	C₄×#88322	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	—	2	4	no	—
63	#11719945	—	4	4	no	—
64	#11719945	—	4	4	no	—
65	C₂×D₂₁₂	—	2	2	no	—
66	C₂×D₂₁₂	—	2	2	no	—
67	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₅₃	—	2	4	no	—
69	C₂×Dic₁₀₆	—	2	2	no	—
70	M₁₆×C₅₃	—	4	4	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
79	#11719921	—	2	4	no	—
80	#11719921	—	2	4	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	C₂×D₄	1	2	yes	—
87	C₂×#88322	C₂×D₄	1	2	yes	—
88	D₄₂₄	—	2	8	no	—
89	#11719912	—	4	8	no	—
90	C₈×D₅₃	—	4	8	no	—
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂²×D₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₂²⋊D₅₃	—	4	8	no	—
103	C₂×Dic₅₃	—	2	4	no	—
104	D₂₁₂	—	2	8	no	—
105	D₂₁₂	—	4	4	no	—
106	D₂₁₂	—	4	4	no	—
107	C₄×D₅₃	—	4	4	no	—
108	C₄×D₅₃	—	4	4	no	—
109	Dic₁₀₆	—	4	4	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	—	4	4	no	—
112	C₈×C₅₃	#131	1	4	yes	complemented
113	#88322	—	2	2	no	—
114	#88322	—	2	2	no	—
115	#88322	—	4	2	no	—
116	#17962	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	#442	1	2	yes	—
120	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×D₅₃	—	4	4	no	—
122	C₂×D₅₃	—	4	4	no	—
123	C₂×D₅₃	—	4	4	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	#442	1	2	yes	—
127	C₄×C₅₃	#444	1	2	yes	—
128	C₄×C₅₃	C₂×#131	1	2	yes	—
129	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	Dic₅₃	—	4	4	no	—
131	#3169	—	53	1	no	—
132	#2931	—	53	2	no	—
133	#1594	—	53	4	no	—
134	#1580	—	53	4	no	—
135	#1566	D₅₃	1	1	yes	characteristic, complemented
136	#1492	—	53	2	no	—
137	C₂×#322	—	53	1	no	—
138	C₂×C₅₃	—	2	2	no	—
139	C₂×C₅₃	#2931	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
142	D₅₃	—	8	4	no	—
143	C₂×#131	—	106	2	no	—
144	#532	—	53	2	no	—
145	#500	—	53	1	no	—
146	#493	—	53	1	no	—
147	#486	—	53	2	no	—
148	#466	—	53	4	no	—
149	C₄×D₈	—	53	4	no	—
150	#436	—	53	4	no	—
151	#436	—	106	2	no	—
152	C₈×D₄	—	53	4	no	—
153	C₈×D₄	—	106	2	no	—
154	#423	C₂×D₅₃	1	2	yes	—
155	#408	—	106	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	#334	C₂×D₅₃	1	4	yes	—
160	D₄⋊C₈	—	53	8	no	—
161	#322	—	53	2	no	—
162	#322	—	53	2	no	—
163	C₅₃	#17962	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	—	106	2	no	—
165	C₂×D₈	—	53	4	no	—
166	#131	—	212	8	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	106	4	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	106	4	no	—
171	C₂²×C₈	—	106	1	no	—
172	C₂²×C₈	C₄×D₅₃	1	2	yes	—
173	C₄⋊Q₈	—	53	2	no	—
174	#127	—	53	2	no	—
175	#124	—	106	2	no	—
176	#121	—	53	4	no	—
177	#121	—	106	2	no	—
178	C₄×D₄	—	53	4	no	—
179	C₄×D₄	—	106	2	no	—
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	#107	—	53	4	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
184	#102	—	53	8	no	—
185	C₂²⋊C₈	—	106	4	no	—
186	C₂²⋊C₈	—	106	4	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	1	no	—
189	C₄×C₈	—	53	1	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
193	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅₃	1	2	yes	—
195	#41	—	212	4	no	—
196	#41	—	212	4	no	—
197	C₂×Q₈	—	106	2	no	—
198	C₂×D₄	—	53	4	no	—
199	C₂×D₄	—	106	4	no	—
200	C₂×D₄	—	106	2	no	—
201	C₂×D₄	—	106	2	no	—
202	C₂²×C₄	—	106	4	no	—
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	#11719949	1	1	yes	characteristic
206	D₈	—	106	8	no	—
207	M₁₆	—	4	4	no	—
208	M₁₆	—	212	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	212	8	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
219	C₄⋊C₄	—	53	4	no	—
220	C₄⋊C₄	—	106	2	no	—
221	C₂²⋊C₄	—	106	4	no	—
222	C₂²⋊C₄	—	106	4	no	—
223	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₄²	D₄×D₅₃	1	1	yes	characteristic
225	C₄²	#11719949	1	1	yes	characteristic
226	C₄²	C₂³×D₅₃	1	1	yes	characteristic
227	C₂³	—	106	4	no	—
228	C₂³	#13133179	1	1	yes	characteristic
229	Q₈	—	212	4	no	—
230	D₄	—	106	8	no	—
231	D₄	—	212	8	no	—
232	D₄	—	212	4	no	—
233	D₄	—	212	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	106	4	no	—
237	C₂×C₄	—	212	4	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	#13133179	1	1	yes	characteristic
240	C₂×C₄	#13133179	1	1	yes	characteristic
241	C₂×C₄	#13133179	1	1	yes	characteristic
242	C₂×C₄	#13133244	1	4	yes	—
243	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	4	4	no	—
248	C₈	—	106	2	no	—
249	C₈	—	106	2	no	—
250	C₈	—	212	2	no	—
251	C₈	#13133225	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	212	4	no	—
255	C₂²	—	212	4	no	—
256	C₂²	—	212	4	no	—
257	C₂²	#16810734	1	2	yes	—
258	C₂²	C₂×#13133179	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	212	4	no	—
262	C₄	#16810734	1	2	yes	—
263	C₄	#16810744	1	2	yes	—
264	C₄	C₂×#13133225	1	2	yes	—
265	C₄	C₂×#13133244	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	424	4	no	—
268	C₂	#20283426	1	1	yes	characteristic
269	C₂	C₂×#16810734	1	1	yes	characteristic
270	C₂	C₂×#16810744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#17962
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #20283426	C₂	semidirect
3. #20273628	C₂	semidirect
4. #20273538	C₂	semidirect
5. #20273165	C₂	semidirect
6. C₂×#16810344	C₂	semidirect
7. C₅₃×#1566	C₂	semidirect
112. C₈×C₅₃	#131	semidirect
135. #1566	D₅₃	semidirect
163. C₅₃	#17962	semidirect
251. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20283426	2
#20273628	2
#20273538	2
#20273165	2
C₂×#16810344	2
C₅₃×#1566	2
#17962	53