Subgroups — #27272557

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272538	C₂	1	4	yes	maximal
4	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×#13133128	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₂²	1	2	yes	—
10	#16810378	C₂²	1	2	yes	—
11	C₂×#125×C₅₃	—	2	1	no	—
12	C₅₃×#389	—	2	1	no	—
13	C₅₃×#389	—	2	1	no	—
14	C₅₃×#389	C₄	1	2	yes	—
15	C₅₃×#389	C₂²	1	2	yes	—
16	C₅₃×#384	C₄	1	1	yes	characteristic
17	C₅₃×#384	C₄	1	1	yes	characteristic
18	C₅₃×#382	—	2	1	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	#13133128	—	4	2	no	—
24	C₄×#88322	—	2	4	no	—
25	#125×C₅₃	—	4	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#95×C₅₃	—	2	2	no	—
42	#95×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	4	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	4	1	no	—
66	C₄²×C₅₃	SD₁₆	1	2	yes	—
67	C₄²×C₅₃	Q₁₆	1	2	yes	—
68	C₂×#88322	—	2	4	no	—
69	C₂×#88322	—	2	4	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	#101	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#103	1	2	yes	—
97	C₂×C₄×C₅₃	#104	1	2	yes	—
98	C₂×C₄×C₅₃	#104	1	2	yes	—
99	C₂×C₄×C₅₃	#131	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	#88322	—	4	4	no	—
104	#88322	—	4	4	no	—
105	#7683	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#408	1	1	yes	characteristic
107	C₂²×C₅₃	#410	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
110	C₂²×C₅₃	#418	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	—	8	2	no	—
121	C₄×C₅₃	—	8	2	no	—
122	C₄×C₅₃	#330	1	4	yes	—
123	#2859	D₅₃	1	1	yes	characteristic
124	#1767	—	53	1	no	—
125	#1744	—	53	1	no	—
126	#1271	—	53	4	no	—
127	C₂×C₅₃	#1470	1	1	yes	characteristic
128	C₂×C₅₃	#1484	1	1	yes	characteristic
129	C₂×C₅₃	#1493	1	1	yes	characteristic
130	C₂×C₅₃	#1503	1	1	yes	characteristic
131	C₂×C₅₃	#1505	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#330	1	1	yes	characteristic
133	C₂×C₅₃	#1549	1	1	yes	characteristic
134	C₂×#125	—	2	1	no	—
135	C₂×C₄⋊C₈	—	106	1	no	—
136	C₂×#97	—	106	1	no	—
137	C₂×C₄×C₈	—	106	1	no	—
138	#389	—	2	1	no	—
139	#389	—	2	1	no	—
140	#389	Dic₅₃	1	2	yes	—
141	#389	C₂×D₅₃	1	2	yes	—
142	#384	Dic₅₃	1	1	yes	characteristic
143	#384	Dic₅₃	1	1	yes	characteristic
144	#382	—	2	1	no	—
145	C₄³	C₂×D₅₃	1	1	yes	characteristic
146	#336	—	53	2	no	—
147	#336	—	53	2	no	—
148	#336	—	106	1	no	—
149	C₅₃	#7683	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₈	—	106	2	no	—
151	C₂²×C₈	—	106	2	no	—
152	#125	—	4	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	C₄⋊C₈	—	106	4	no	—
169	#97	—	212	2	no	—
170	C₄×C₈	—	106	4	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	#11719924	1	2	yes	—
203	C₄²	#11719925	1	2	yes	—
204	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	#13133136	1	2	yes	—
225	C₂×C₄	#13133141	1	2	yes	—
226	C₂×C₄	#13133141	1	2	yes	—
227	C₂×C₄	#13133144	1	2	yes	—
228	C₂×C₄	#13133144	1	2	yes	—
229	C₂×C₄	#13133224	1	2	yes	—
230	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
232	C₂×C₄	#13133243	1	2	yes	—
233	C₈	—	212	4	no	—
234	C₈	—	212	4	no	—
235	C₂²	#16810590	1	1	yes	characteristic
236	C₂²	#16810597	1	1	yes	characteristic
237	C₂²	C₂×#13133136	1	1	yes	characteristic
238	C₂²	C₂×#13133141	1	1	yes	characteristic
239	C₂²	#16810628	1	1	yes	characteristic
240	C₂²	C₂×#13133144	1	1	yes	characteristic
241	C₂²	C₂×#13133144	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	8	2	no	—
250	C₄	—	8	2	no	—
251	C₄	#16810365	1	4	yes	—
252	C₂	#20273071	1	1	yes	characteristic
253	C₂	#20273132	1	1	yes	characteristic
254	C₂	#20273170	1	1	yes	characteristic
255	C₂	#20273214	1	1	yes	characteristic
256	C₂	#20273223	1	1	yes	characteristic
257	C₂	C₂×#16810365	1	1	yes	characteristic
258	C₂	#20273400	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#7683
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
149. C₅₃	#7683	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
#20274461	2
#20272538	2
C₅₃×#2859	2
#7683	53