Subgroups — #27282779

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275478	C₂	1	2	yes	maximal
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133148	—	2	2	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810374	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#97×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	#13133151	—	4	2	no	—
23	#13133151	—	4	2	no	—
24	#13133148	—	4	4	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₈×C₅₃	Q₈	1	2	yes	—
36	#97×C₅₃	—	4	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	#95×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅₃	#41	1	1	yes	characteristic
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	8	2	no	—
68	C₂×#88322	—	2	4	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂³×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#137	1	2	yes	—
92	C₂×Dic₅₃	—	8	2	no	—
93	C₂×Dic₅₃	—	8	2	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	4	2	no	—
97	C₈×C₅₃	#106	1	4	yes	complemented
98	#88322	—	4	4	no	—
99	#9621	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	#389	1	1	yes	characteristic
101	C₂²×C₅₃	#460	1	1	yes	characteristic
102	C₂²×C₅₃	#461	1	1	yes	characteristic
103	C₂²×C₅₃	#469	1	1	yes	characteristic
104	C₂²×C₅₃	#470	1	1	yes	characteristic
105	C₂²×C₅₃	#472	1	1	yes	characteristic
106	C₂²×C₅₃	#473	1	1	yes	characteristic
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	8	1	no	—
110	C₄×C₅₃	—	8	1	no	—
111	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
112	C₄×C₅₃	#428	1	1	yes	characteristic
113	C₄×C₅₃	#440	1	2	yes	—
114	C₄×C₅₃	#475	1	1	yes	characteristic
115	C₄×C₅₃	#476	1	1	yes	characteristic
116	C₄×C₅₃	#501	1	2	yes	—
117	Dic₅₃	—	16	2	no	—
118	#1948	—	53	2	no	—
119	#1845	—	53	1	no	—
120	#1770	D₅₃	1	1	yes	characteristic
121	C₂×#334	—	53	1	no	—
122	#1296	—	53	2	no	—
123	C₂×C₅₃	#1688	1	1	yes	characteristic
124	C₂×C₅₃	#1690	1	1	yes	characteristic
125	C₂×C₅₃	#1693	1	1	yes	characteristic
126	C₂×C₅₃	#1695	1	1	yes	characteristic
127	C₂×C₅₃	#1944	1	1	yes	characteristic
128	C₂×C₅₃	#1948	1	1	yes	characteristic
129	C₂×C₅₃	#1952	1	1	yes	characteristic
130	C₂×#107	—	106	1	no	—
131	C₂×#107	—	106	1	no	—
132	C₂×#106	—	106	2	no	—
133	C₂×C₄⋊C₈	—	53	2	no	—
134	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×#97	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#384	—	53	2	no	—
138	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
139	#340	—	106	2	no	—
140	#336	—	2	1	no	—
141	#334	—	53	4	no	—
142	C₅₃	#9621	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	106	2	no	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	106	2	no	—
148	C₂×C₄⋊C₄	—	106	2	no	—
149	C₂×C₄⋊C₄	—	106	2	no	—
150	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
154	#107	—	212	2	no	—
155	#107	—	212	2	no	—
156	#106	—	212	4	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	53	4	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	Dic₁₀₆	1	2	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	Dic₁₀₆	1	2	yes	—
163	C₄×C₈	C₄×D₅₃	1	2	yes	—
164	#95	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	212	2	no	—
168	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂²×C₄	#11719945	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₂×C₈	#11719919	1	2	yes	—
180	C₂×C₈	#11719919	1	2	yes	—
181	C₂×C₈	#11719923	1	2	yes	—
182	C₂×C₈	#11719924	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	424	2	no	—
186	C₄⋊C₄	—	212	2	no	—
187	C₄⋊C₄	—	212	2	no	—
188	C₄⋊C₄	—	212	2	no	—
189	C₄⋊C₄	—	212	2	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₄²	#11719945	1	1	yes	characteristic
196	C₂³	#13133191	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	424	2	no	—
206	C₂×C₄	—	424	2	no	—
207	C₂×C₄	—	424	2	no	—
208	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
209	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
210	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
211	C₂×C₄	#13133196	1	1	yes	characteristic
212	C₂×C₄	#13133209	1	1	yes	characteristic
213	C₂×C₄	#13133217	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
216	C₂×C₄	#13133242	1	1	yes	characteristic
217	C₂×C₄	#13133245	1	2	yes	—
218	C₂×C₄	#13133247	1	1	yes	characteristic
219	C₂×C₄	#13133250	1	2	yes	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₈	—	212	4	no	—
223	C₈	#13133148	1	4	yes	complemented
224	C₂²	#16810521	1	1	yes	characteristic
225	C₂²	#16810823	1	1	yes	characteristic
226	C₂²	#16810828	1	1	yes	characteristic
227	C₂²	#16810876	1	1	yes	characteristic
228	C₂²	#16810881	1	1	yes	characteristic
229	C₂²	#16810893	1	1	yes	characteristic
230	C₂²	#16810898	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	1	no	—
234	C₄	—	8	1	no	—
235	C₄	—	848	2	no	—
236	C₄	C₂×#13133148	1	1	yes	characteristic
237	C₄	#16810666	1	1	yes	characteristic
238	C₄	#16810725	1	2	yes	—
239	C₄	#16810908	1	1	yes	characteristic
240	C₄	#16810913	1	1	yes	characteristic
241	C₄	#16811032	1	2	yes	—
242	C₂	#20274199	1	1	yes	characteristic
243	C₂	#20274209	1	1	yes	characteristic
244	C₂	#20274228	1	1	yes	characteristic
245	C₂	#20274238	1	1	yes	characteristic
246	C₂	#20275458	1	1	yes	characteristic
247	C₂	#20275478	1	1	yes	characteristic
248	C₂	#20275500	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1770	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1770	D₅₃
hypocenter	C₅₃	#9621
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
12. #16810374	C₄	semidirect
97. C₈×C₅₃	#106	semidirect
142. C₅₃	#9621	semidirect
223. C₈	#13133148	semidirect

Maximal subgroups

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

subgroup	index
#20275478	2
#20274875	2
C₂×#16810374	2
#20272599	2
C₅₃×#1770	2
#9621	53