Subgroups — #27277762

Order 13568.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal
4	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	C₄	1	2	yes	—
10	#16810378	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	2	no	—
16	C₂²×#88322	C₂×C₄	1	4	yes	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	—	4	2	no	—
19	C₄⋊#88322	—	4	2	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄⋊C₈×C₅₃	—	4	2	no	—
32	C₄⋊C₈×C₅₃	—	4	2	no	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	4	1	no	—
46	C₄²×C₅₃	—	4	1	no	—
47	C₄²×C₅₃	D₈	1	2	yes	—
48	C₄²×C₅₃	SD₁₆	1	2	yes	—
49	C₄²×C₅₃	SD₁₆	1	2	yes	—
50	C₄²×C₅₃	Q₁₆	1	2	yes	—
51	C₄²×C₅₃	#41	1	2	yes	—
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	#106	1	2	yes	—
71	C₂×C₄×C₅₃	#106	1	2	yes	—
72	C₂×C₄×C₅₃	#107	1	2	yes	—
73	C₂×C₄×C₅₃	#107	1	2	yes	—
74	C₂×C₄×C₅₃	#117	1	2	yes	—
75	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#125	1	2	yes	—
78	C₂×C₄×C₅₃	#131	1	2	yes	—
79	C₂×C₄×C₅₃	#131	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₅₃	—	8	2	no	—
85	#88322	—	4	4	no	—
86	#88322	—	8	2	no	—
87	#8787	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	#382	1	1	yes	characteristic
89	C₂²×C₅₃	#424	1	1	yes	characteristic
90	C₂²×C₅₃	#424	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
93	C₂²×C₅₃	#430	1	1	yes	characteristic
94	C₂²×C₅₃	#430	1	1	yes	characteristic
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	#433	1	4	yes	—
101	C₄×C₅₃	#486	1	1	yes	characteristic
102	C₄×C₅₃	#486	1	1	yes	characteristic
103	C₄×C₅₃	#487	1	1	yes	characteristic
104	C₄×C₅₃	#487	1	1	yes	characteristic
105	#1842	—	53	1	no	—
106	#1842	D₅₃	1	1	yes	characteristic
107	#1744	—	53	1	no	—
108	#1277	—	53	4	no	—
109	C₂×C₅₃	#1570	1	1	yes	characteristic
110	C₂×C₅₃	#1570	1	1	yes	characteristic
111	C₂×C₅₃	#1571	1	1	yes	characteristic
112	C₂×C₅₃	#1571	1	1	yes	characteristic
113	C₂×C₅₃	#1832	1	1	yes	characteristic
114	C₂×C₅₃	#1834	1	1	yes	characteristic
115	C₂×C₅₃	#1838	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	53	2	no	—
119	C₂×C₄⋊C₈	—	106	1	no	—
120	C₂×C₄⋊C₈	—	106	1	no	—
121	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
122	C₂×C₄×C₈	—	53	2	no	—
123	C₄³	C₂×D₅₃	1	1	yes	characteristic
124	#336	—	53	2	no	—
125	#336	—	53	2	no	—
126	C₅₃	#8787	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	53	4	no	—
129	C₂²×C₈	—	106	2	no	—
130	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
131	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	—	106	4	no	—
141	C₄⋊C₈	—	212	2	no	—
142	C₄⋊C₈	—	212	2	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	—	106	2	no	—
145	C₂²×C₄	#11719919	1	1	yes	characteristic
146	C₂²×C₄	#11719919	1	1	yes	characteristic
147	C₂²×C₄	#11719919	1	1	yes	characteristic
148	C₂²×C₄	#11719919	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	106	4	no	—
156	C₂×C₈	—	106	4	no	—
157	C₂×C₈	—	106	4	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#11719922	1	2	yes	—
166	C₄²	#11719923	1	2	yes	—
167	C₄²	#11719924	1	2	yes	—
168	C₄²	#11719925	1	2	yes	—
169	C₄²	#11719947	1	2	yes	—
170	C₄²	#11719949	1	2	yes	—
171	C₂³	C₂×#11719919	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	#13133148	1	2	yes	—
183	C₂×C₄	#13133148	1	2	yes	—
184	C₂×C₄	#13133150	1	2	yes	—
185	C₂×C₄	#13133150	1	2	yes	—
186	C₂×C₄	#13133175	1	2	yes	—
187	C₂×C₄	#13133204	1	1	yes	characteristic
188	C₂×C₄	#13133206	1	1	yes	characteristic
189	C₂×C₄	#13133208	1	2	yes	—
190	C₂×C₄	#13133223	1	2	yes	—
191	C₂×C₄	#13133225	1	2	yes	—
192	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
194	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	424	2	no	—
198	C₈	—	212	4	no	—
199	C₂²	#16810488	1	1	yes	characteristic
200	C₂²	#16810649	1	1	yes	characteristic
201	C₂²	#16810651	1	1	yes	characteristic
202	C₂²	C₂×#13133148	1	1	yes	characteristic
203	C₂²	C₂×#13133150	1	1	yes	characteristic
204	C₂²	#16810671	1	1	yes	characteristic
205	C₂²	#16810671	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#16810683	1	4	yes	—
212	C₄	#16810969	1	1	yes	characteristic
213	C₄	#16810971	1	1	yes	characteristic
214	C₄	#16810973	1	1	yes	characteristic
215	C₄	#16810975	1	1	yes	characteristic
216	C₂	#20273485	1	1	yes	characteristic
217	C₂	#20273487	1	1	yes	characteristic
218	C₂	#20273489	1	1	yes	characteristic
219	C₂	#20273491	1	1	yes	characteristic
220	C₂	#20274834	1	1	yes	characteristic
221	C₂	#20274842	1	1	yes	characteristic
222	C₂	#20274855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8787
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
126. C₅₃	#8787	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
#20274461	2
#20272549	2
C₅₃×#1842	2
#8787	53