Subgroups — #27277206

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274843	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274715	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272599	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133138	—	2	2	no	—
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	1	no	—
10	#16810386	—	2	1	no	—
11	#16810385	—	2	2	no	—
12	#16810374	—	2	2	no	—
13	C₂²×M₁₆×C₅₃	—	2	1	no	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#436	—	2	2	no	—
16	C₅₃×#436	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133138	—	4	4	no	—
27	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	—	2	1	no	—
29	C₂×M₁₆×C₅₃	—	2	4	no	—
30	C₂×M₁₆×C₅₃	—	4	4	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×D₄×C₅₃	—	2	1	no	—
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₅₃	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂⁴×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	—	2	2	no	—
49	C₂×D₄×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	4	2	no	—
60	M₁₆×C₅₃	—	4	8	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	8	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	1	no	—
86	C₂³×C₅₃	—	4	1	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	#121	1	1	yes	characteristic
89	D₄×C₅₃	—	2	4	no	—
90	D₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#135	1	2	yes	—
104	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₅₃	#137	1	2	yes	—
107	C₂×Dic₅₃	—	8	2	no	—
108	C₂×Dic₅₃	—	8	2	no	—
109	C₂×Dic₅₃	—	8	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	4	2	no	—
113	#88322	—	8	2	no	—
114	#8656	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	8	2	no	—
119	C₂²×C₅₃	—	8	2	no	—
120	C₂²×C₅₃	#380	1	1	yes	characteristic
121	C₂²×C₅₃	#463	1	2	yes	—
122	C₂²×C₅₃	#469	1	1	yes	characteristic
123	C₂²×C₅₃	#470	1	1	yes	characteristic
124	C₂²×C₅₃	#471	1	2	yes	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	#416	1	1	yes	characteristic
129	C₄×C₅₃	#418	1	1	yes	characteristic
130	C₄×C₅₃	#428	1	2	yes	—
131	C₄×C₅₃	#468	1	2	yes	—
132	C₄×C₅₃	#478	1	2	yes	—
133	Dic₅₃	—	16	2	no	—
134	C₂×#436	D₅₃	1	1	yes	characteristic
135	#1834	—	53	1	no	—
136	#1802	—	53	2	no	—
137	C₂×#334	—	53	1	no	—
138	#1296	—	53	2	no	—
139	C₂×C₅₃	—	8	2	no	—
140	C₂×C₅₃	#1701	1	2	yes	—
141	C₂×C₅₃	#1703	1	2	yes	—
142	C₂×C₅₃	#1804	1	1	yes	characteristic
143	C₂×C₅₃	#1809	1	1	yes	characteristic
144	C₂×C₅₃	#1815	1	1	yes	characteristic
145	C₂²×M₁₆	—	2	1	no	—
146	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
147	#436	—	2	2	no	—
148	#436	Dic₅₃	1	4	yes	complemented
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₂×#102	—	106	2	no	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#384	—	53	2	no	—
155	#340	—	106	1	no	—
156	#340	—	106	1	no	—
157	#340	—	106	2	no	—
158	#334	—	106	2	no	—
159	C₅₃	#8656	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	106	2	no	—
166	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	Dic₁₀₆	1	2	yes	—
171	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
172	C₂×C₄⋊C₄	—	106	2	no	—
173	C₂×C₄⋊C₄	—	106	2	no	—
174	C₂×C₄⋊C₄	—	106	2	no	—
175	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
182	#102	—	212	4	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
185	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	212	2	no	—
197	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
198	C₂²×C₄	#11719945	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
209	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	424	2	no	—
212	C₄⋊C₄	—	212	2	no	—
213	C₄⋊C₄	—	212	2	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
220	C₄²	#11719945	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	#13133191	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	424	2	no	—
238	C₂×C₄	—	424	2	no	—
239	C₂×C₄	—	424	2	no	—
240	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	#13133176	1	1	yes	characteristic
243	C₂×C₄	#13133191	1	1	yes	characteristic
244	C₂×C₄	#13133196	1	1	yes	characteristic
245	C₂×C₄	#13133196	1	1	yes	characteristic
246	C₂×C₄	#13133237	1	2	yes	—
247	C₂×C₄	#13133243	1	2	yes	—
248	C₂×C₄	#13133245	1	2	yes	—
249	C₂×C₄	#13133250	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	424	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	#16810483	1	1	yes	characteristic
260	C₂²	#16810838	1	2	yes	—
261	C₂²	#16810876	1	1	yes	characteristic
262	C₂²	#16810881	1	1	yes	characteristic
263	C₂²	#16810886	1	2	yes	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	848	2	no	—
268	C₄	#16810619	1	1	yes	characteristic
269	C₄	#16810628	1	1	yes	characteristic
270	C₄	#16810665	1	2	yes	—
271	C₄	#16810869	1	2	yes	—
272	C₄	#16810923	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	#20274276	1	2	yes	—
275	C₂	#20274290	1	2	yes	—
276	C₂	#20274725	1	1	yes	characteristic
277	C₂	#20274748	1	1	yes	characteristic
278	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#436	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#436	D₅₃
hypocenter	C₅₃	#8656
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #20274843	C₂	semidirect
3. C₂×#16810374	C₂	semidirect
4. #20272599	C₂	semidirect
16. C₅₃×#436	C₄	semidirect
148. #436	Dic₅₃	semidirect
159. C₅₃	#8656	semidirect

Maximal subgroups

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

subgroup	index
#20274843	2
#20274715	2
C₂×#16810374	2
#20272599	2
C₂×C₅₃×#436	2
#8656	53