Subgroups — #27354845

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283427	C₂	1	2	yes	maximal, complemented
2	#20273580	C₂	1	8	yes	maximal, complemented
3	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273166	C₂	1	2	yes	maximal, complemented
5	C₂×#16810345	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133224	—	2	2	no	—
8	#16810701	—	2	2	no	—
9	#16810701	C₂²	1	4	yes	—
10	D₄×#88322	—	2	2	no	—
11	D₄×#88322	C₂²	1	4	yes	—
12	#16810647	C₂²	1	2	yes	—
13	#16810590	—	2	2	no	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	#16810374	C₂²	1	4	yes	—
17	D₄⋊#88322	C₂²	1	8	yes	—
18	#16810345	C₂²	1	4	yes	—
19	C₅₃×#532	C₂²	1	2	yes	—
20	C₅₃×#500	C₄	1	1	yes	characteristic
21	C₅₃×#493	C₄	1	1	yes	characteristic
22	C₅₃×#486	C₄	1	2	yes	—
23	C₅₃×#466	C₄	1	4	yes	—
24	C₄×D₈×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	#13133224	—	4	8	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂²×#88322	—	2	4	no	—
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	D₄	1	2	yes	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	C₂³	1	4	yes	—
34	C₂²⋊#88322	—	2	4	no	—
35	C₂²⋊#88322	—	2	4	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	D₄	1	1	yes	characteristic
38	C₄×#88322	D₄	1	1	yes	characteristic
39	C₄×#88322	C₂³	1	2	yes	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₂×#41×C₅₃	—	2	2	no	—
42	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
45	#127×C₅₃	C₂×C₄	1	2	yes	—
46	#124×C₅₃	—	2	2	no	—
47	#121×C₅₃	—	2	2	no	—
48	#121×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×D₄×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	C₂×C₄	1	4	yes	—
53	#102×C₅₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂³	1	2	yes	—
57	#41×C₅₃	—	4	4	no	—
58	#41×C₅₃	—	4	4	no	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	4	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	D₈×C₅₃	—	4	4	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²⋊C₄×C₅₃	—	2	4	no	—
75	C₂²⋊C₄×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	#11719917	—	4	8	no	—
81	#11719917	—	4	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	8	no	—
87	C₂×#88322	C₂×D₄	1	2	yes	—
88	C₂×#88322	C₂×D₄	1	2	yes	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₅₃	—	4	4	no	—
92	D₄×C₅₃	—	2	8	no	—
93	D₄×C₅₃	—	4	8	no	—
94	D₄×C₅₃	—	4	4	no	—
95	D₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	2	2	no	—
110	#88322	—	2	4	no	—
111	#88322	—	2	2	no	—
112	#88322	—	2	2	no	—
113	#88322	—	4	4	no	—
114	#17981	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	4	4	no	—
118	C₂²×C₅₃	—	4	4	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	#442	1	2	yes	—
121	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	4	4	no	—
125	C₄×C₅₃	#442	1	2	yes	—
126	C₄×C₅₃	#443	1	2	yes	—
127	C₄×C₅₃	#568	1	2	yes	—
128	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
129	#3169	D₅₃	1	1	yes	characteristic
130	#2931	—	53	2	no	—
131	#1586	—	53	8	no	—
132	#1566	—	53	1	no	—
133	#1492	—	53	2	no	—
134	C₂×#322	—	53	1	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	—	8	4	no	—
137	C₂×C₅₃	#2932	1	1	yes	characteristic
138	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
140	C₂×#131	—	106	2	no	—
141	#532	C₂×D₅₃	1	2	yes	—
142	#500	Dic₅₃	1	1	yes	characteristic
143	#493	Dic₅₃	1	1	yes	characteristic
144	#486	Dic₅₃	1	2	yes	—
145	#466	Dic₅₃	1	4	yes	—
146	C₄×D₈	C₂×D₅₃	1	4	yes	—
147	#436	—	53	4	no	—
148	#436	—	106	2	no	—
149	C₈×D₄	—	53	4	no	—
150	C₈×D₄	—	106	2	no	—
151	#423	—	53	2	no	—
152	#408	—	106	2	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	#334	—	53	4	no	—
157	D₄⋊C₈	—	53	8	no	—
158	#322	—	53	4	no	—
159	C₅₃	#17981	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	2	2	no	—
161	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
162	#131	—	212	8	no	—
163	C₂×M₁₆	—	106	4	no	—
164	C₂×M₁₆	—	106	2	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	106	4	no	—
167	C₂²×C₈	—	106	1	no	—
168	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
169	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
170	#127	C₂×Dic₅₃	1	2	yes	—
171	#124	—	2	2	no	—
172	#121	—	2	2	no	—
173	#121	C₂×Dic₅₃	1	4	yes	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	C₂²×D₅₃	1	4	yes	—
176	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
177	#107	C₂×Dic₅₃	1	4	yes	—
178	C₄⋊C₈	—	53	4	no	—
179	C₄⋊C₈	—	106	2	no	—
180	#102	C₂×Dic₅₃	1	8	yes	—
181	C₂²⋊C₈	—	106	4	no	—
182	C₂²⋊C₈	—	106	4	no	—
183	C₄×C₈	—	53	1	no	—
184	C₄×C₈	—	53	1	no	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	106	1	no	—
188	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
189	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₅₃	1	2	yes	—
191	#41	—	4	4	no	—
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	#11719947	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	D₈	—	4	4	no	—
203	M₁₆	—	212	8	no	—
204	M₁₆	—	212	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	212	8	no	—
213	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
214	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₄²	D₄×D₅₃	1	1	yes	characteristic
220	C₄²	#11719947	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂³×D₅₃	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	#13133241	1	4	yes	—
239	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	106	4	no	—
245	C₈	—	106	2	no	—
246	C₈	—	106	2	no	—
247	C₈	—	212	4	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	#16810735	1	2	yes	—
254	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	#16810735	1	2	yes	—
259	C₄	#16810740	1	2	yes	—
260	C₄	#16811390	1	2	yes	—
261	C₄	C₂×#13133241	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	#20283433	1	1	yes	characteristic
265	C₂	C₂×#16810735	1	1	yes	characteristic
266	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#17981
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283427	C₂	semidirect
2. #20273580	C₂	semidirect
3. #20273471	C₂	semidirect
4. #20273166	C₂	semidirect
5. C₂×#16810345	C₂	semidirect
159. C₅₃	#17981	semidirect

Maximal subgroups

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

subgroup	index
#20283427	2
#20273580	2
#20273471	2
#20273166	2
C₂×#16810345	2
C₅₃×#3169	2
#17981	53