Subgroups — #27277193

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810702	C₂	1	1	yes	maximal, characteristic
2	#20274712	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272598	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	—	2	1	no	—
7	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
8	#16810702	—	2	2	no	—
9	#16810702	C₄	1	4	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133137	—	2	2	no	—
13	C₂×#13133129	—	2	1	no	—
14	#16810385	—	2	2	no	—
15	#16810376	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₅₃×#340	—	2	1	no	—
19	C₅₃×#340	—	2	1	no	—
20	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₅₃	—	2	1	no	—
22	C₂×#11719912	—	2	4	no	—
23	C₂×#11719912	—	4	4	no	—
24	C₂²×#88322	—	2	1	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₄×D₂₁₂	—	2	1	no	—
27	C₄×D₂₁₂	—	2	1	no	—
28	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
29	C₄×D₂₁₂	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#11719921	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133137	—	4	4	no	—
37	#13133129	—	2	4	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂³×D₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	1	no	—
52	C₂×D₂₁₂	—	2	2	no	—
53	C₂×D₂₁₂	—	2	2	no	—
54	C₂×D₂₁₂	—	2	1	no	—
55	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	—	2	2	no	—
58	C₂×C₄×D₅₃	—	4	2	no	—
59	C₂×C₄×D₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	8	2	no	—
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	#11719921	—	2	2	no	—
74	#11719921	—	2	2	no	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	#11719912	—	4	8	no	—
83	C₂³×C₅₃	#121	1	1	yes	characteristic
84	C₂²×D₅₃	—	2	2	no	—
85	C₂²×D₅₃	—	4	2	no	—
86	C₂²×D₅₃	—	4	2	no	—
87	C₂²×D₅₃	—	4	1	no	—
88	C₂²×D₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₅₃	#137	1	2	yes	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	D₂₁₂	—	2	4	no	—
107	D₂₁₂	—	4	1	no	—
108	D₂₁₂	—	4	1	no	—
109	C₄×D₅₃	—	4	2	no	—
110	C₄×D₅₃	—	4	2	no	—
111	C₄×D₅₃	—	4	4	no	—
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	—	2	2	no	—
114	#88322	—	4	2	no	—
115	#88322	—	8	2	no	—
116	#8654	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	#380	1	1	yes	characteristic
118	C₂²×C₅₃	#463	1	2	yes	—
119	C₂²×C₅₃	#468	1	2	yes	—
120	C₂²×C₅₃	#472	1	1	yes	characteristic
121	C₂²×C₅₃	#473	1	1	yes	characteristic
122	C₂×D₅₃	—	4	2	no	—
123	C₂×D₅₃	—	4	2	no	—
124	C₂×D₅₃	—	4	2	no	—
125	C₂×D₅₃	—	8	2	no	—
126	C₂×D₅₃	—	8	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	16	2	no	—
130	C₄×C₅₃	#416	1	1	yes	characteristic
131	C₄×C₅₃	#418	1	1	yes	characteristic
132	C₄×C₅₃	#428	1	2	yes	—
133	C₄×C₅₃	#469	1	1	yes	characteristic
134	C₄×C₅₃	#470	1	1	yes	characteristic
135	C₄×C₅₃	#478	1	2	yes	—
136	Dic₅₃	—	4	2	no	—
137	C₂×#436	—	53	1	no	—
138	#1834	D₅₃	1	1	yes	characteristic, complemented
139	#1802	—	53	2	no	—
140	C₂×#335	—	53	1	no	—
141	#1296	—	53	2	no	—
142	C₂×C₅₃	#1697	1	2	yes	—
143	C₂×C₅₃	#1699	1	2	yes	—
144	C₂×C₅₃	#1804	1	1	yes	characteristic
145	C₂×C₅₃	#1809	1	1	yes	characteristic
146	C₂×C₅₃	#1816	1	1	yes	characteristic
147	D₅₃	—	8	2	no	—
148	C₂²×M₁₆	—	106	1	no	—
149	C₂×C₄×D₄	—	53	1	no	—
150	#436	—	53	4	no	—
151	#436	—	106	2	no	—
152	C₂×C₄⋊C₈	—	53	1	no	—
153	C₂×C₄⋊C₈	—	53	2	no	—
154	C₂×#102	—	106	2	no	—
155	C₂×C₂²⋊C₈	—	106	1	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#384	C₂×D₅₃	1	2	yes	—
158	#340	—	2	1	no	—
159	#340	—	2	1	no	—
160	#340	—	106	2	no	—
161	#335	—	106	2	no	—
162	C₅₃	#8654	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	—	53	1	no	—
164	C₂³×C₄	—	106	1	no	—
165	C₂×M₁₆	—	106	4	no	—
166	C₂×M₁₆	—	212	4	no	—
167	C₂²×C₈	—	106	1	no	—
168	C₂²×C₈	—	106	2	no	—
169	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₄×D₄	—	53	2	no	—
172	C₄×D₄	—	53	2	no	—
173	C₄×D₄	—	106	1	no	—
174	C₄×D₄	—	106	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	53	1	no	—
179	C₂×C₂²⋊C₄	—	106	1	no	—
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	C₄⋊C₈	—	53	2	no	—
182	C₄⋊C₈	—	106	1	no	—
183	C₄⋊C₈	—	106	2	no	—
184	C₄⋊C₈	—	106	2	no	—
185	#102	—	212	4	no	—
186	C₂²⋊C₈	—	106	4	no	—
187	C₄×C₈	C₄×D₅₃	1	2	yes	—
188	C₄×C₈	D₂₁₂	1	2	yes	—
189	C₂⁴	—	106	1	no	—
190	C₂×D₄	—	53	2	no	—
191	C₂×D₄	—	106	1	no	—
192	C₂×D₄	—	106	2	no	—
193	C₂×D₄	—	106	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	106	1	no	—
196	C₂²×C₄	—	106	2	no	—
197	C₂²×C₄	—	106	2	no	—
198	C₂²×C₄	—	212	2	no	—
199	C₂²×C₄	—	212	2	no	—
200	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	#11719947	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	M₁₆	—	212	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	106	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₂×C₈	—	212	2	no	—
210	C₂×C₈	—	212	2	no	—
211	C₂×C₈	#11719921	1	2	yes	—
212	C₂×C₈	#11719921	1	2	yes	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	—	53	2	no	—
219	C₄⋊C₄	—	106	1	no	—
220	C₂²⋊C₄	—	106	2	no	—
221	C₂²⋊C₄	—	106	2	no	—
222	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
224	C₄²	Q₈×D₅₃	1	1	yes	characteristic
225	C₄²	#11719949	1	1	yes	characteristic
226	C₂³	—	106	2	no	—
227	C₂³	—	212	2	no	—
228	C₂³	—	212	2	no	—
229	C₂³	—	212	1	no	—
230	C₂³	—	212	1	no	—
231	C₂³	#13133188	1	1	yes	characteristic
232	D₄	—	106	4	no	—
233	D₄	—	212	1	no	—
234	D₄	—	212	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	106	2	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	—	212	2	no	—
243	C₂×C₄	—	212	2	no	—
244	C₂×C₄	—	212	4	no	—
245	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
246	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄	#13133175	1	1	yes	characteristic
248	C₂×C₄	#13133192	1	1	yes	characteristic
249	C₂×C₄	#13133193	1	1	yes	characteristic
250	C₂×C₄	#13133197	1	1	yes	characteristic
251	C₂×C₄	#13133237	1	2	yes	—
252	C₂×C₄	#13133242	1	2	yes	—
253	C₂×C₄	#13133244	1	2	yes	—
254	C₂×C₄	#13133246	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	424	2	no	—
258	C₈	—	212	2	no	—
259	C₂²	—	424	2	no	—
260	C₂²	—	424	2	no	—
261	C₂²	—	212	2	no	—
262	C₂²	—	212	2	no	—
263	C₂²	—	212	2	no	—
264	C₂²	#16810480	1	1	yes	characteristic
265	C₂²	#16810835	1	2	yes	—
266	C₂²	#16810866	1	2	yes	—
267	C₂²	#16810890	1	1	yes	characteristic
268	C₂²	#16810895	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	16	2	no	—
272	C₄	—	212	2	no	—
273	C₄	#16810617	1	1	yes	characteristic
274	C₄	#16810627	1	1	yes	characteristic
275	C₄	#16810662	1	2	yes	—
276	C₄	#16810877	1	1	yes	characteristic
277	C₄	#16810882	1	1	yes	characteristic
278	C₄	#16810920	1	2	yes	—
279	C₂	—	424	2	no	—
280	C₂	#20274245	1	2	yes	—
281	C₂	#20274259	1	2	yes	—
282	C₂	#20274722	1	1	yes	characteristic
283	C₂	#20274745	1	1	yes	characteristic
284	C₂	#20274780	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₂₁₂	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#8654
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. C₂×#16810376	C₂	semidirect
4. #20272598	C₂	semidirect
5. C₅₃×#1834	C₂	semidirect
9. #16810702	C₄	semidirect
138. #1834	D₅₃	semidirect
162. C₅₃	#8654	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810702	2
#20274712	2
C₂×#16810376	2
#20272598	2
C₅₃×#1834	2
#8654	53