Subgroups — #27277786

Order 13568.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274879	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	#20272575	C₂	1	4	yes	maximal
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133149	—	2	1	no	—
8	#16810489	C₂²	1	2	yes	—
9	#16810344	C₄	1	2	yes	complemented
10	C₈×#88322	C₄	1	2	yes	complemented
11	C₂×#108×C₅₃	—	2	1	no	—
12	C₂×#108×C₅₃	—	2	1	no	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	#13133151	—	4	2	no	—
17	#13133149	—	4	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	2	no	—
20	C₄×#88322	C₂×C₄	1	4	yes	—
21	#13133124	—	2	4	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	#108×C₅₃	—	4	2	no	—
27	#108×C₅₃	—	4	2	no	—
28	C₄⋊C₈×C₅₃	—	2	4	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	#41	1	2	yes	—
35	C₂²×Dic₅₃	—	4	2	no	—
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	D₈	1	2	yes	—
44	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂³×C₅₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#117	1	2	yes	—
57	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#125	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	—	8	4	no	—
64	C₂×Dic₅₃	—	8	2	no	—
65	C₈×C₅₃	—	8	2	no	—
66	C₈×C₅₃	#106	1	2	yes	—
67	C₈×C₅₃	#106	1	2	yes	—
68	C₈×C₅₃	#107	1	2	yes	—
69	C₈×C₅₃	#107	1	2	yes	—
70	#88322	—	4	4	no	—
71	#8793	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	#382	1	1	yes	characteristic
74	C₂²×C₅₃	#486	1	1	yes	characteristic
75	C₂²×C₅₃	#487	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₄×C₅₃	#486	1	1	yes	characteristic
81	C₄×C₅₃	#487	1	1	yes	characteristic
82	Dic₅₃	—	16	2	no	—
83	#1848	D₅₃	1	1	yes	characteristic
84	#1846	—	53	1	no	—
85	#1450	—	53	1	no	—
86	#1286	—	53	4	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1833	1	2	yes	—
90	C₂×C₅₃	#1834	1	1	yes	characteristic
91	#431	—	53	2	no	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	—	2	1	no	—
94	C₂×#107	—	106	1	no	—
95	C₂×#106	—	106	1	no	—
96	#423	C₂×D₅₃	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	#382	—	53	2	no	—
99	#322	—	53	2	no	—
100	C₈²	—	53	2	no	—
101	C₅₃	#8793	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	106	2	no	—
106	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
107	#108	—	4	2	no	—
108	#108	—	4	2	no	—
109	#107	—	212	2	no	—
110	#106	—	212	2	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	106	2	no	—
113	C₂²⋊C₈	—	106	2	no	—
114	C₄×C₈	—	53	4	no	—
115	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
116	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
117	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
118	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
119	#95	—	106	4	no	—
120	C₂²×C₄	—	212	2	no	—
121	C₂²×C₄	#11719949	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	106	4	no	—
127	C₂×C₈	#11719919	1	1	yes	characteristic
128	C₂×C₈	#11719919	1	1	yes	characteristic
129	C₂×C₈	#11719919	1	1	yes	characteristic
130	C₂×C₈	#11719919	1	1	yes	characteristic
131	C₂×C₈	#11719922	1	2	yes	—
132	C₂×C₈	#11719923	1	2	yes	—
133	C₂×C₈	#11719924	1	2	yes	—
134	C₂×C₈	#11719925	1	2	yes	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄⋊C₄	—	212	2	no	—
137	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
138	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
139	C₄²	#11719947	1	2	yes	—
140	C₂³	#13133206	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	424	4	no	—
143	C₂×C₄	—	424	2	no	—
144	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
145	C₂×C₄	#13133175	1	2	yes	—
146	C₂×C₄	#13133204	1	1	yes	characteristic
147	C₂×C₄	#13133208	1	2	yes	—
148	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
149	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
151	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	212	4	no	—
154	C₈	#13133148	1	2	yes	—
155	C₈	#13133148	1	2	yes	—
156	C₈	#13133150	1	2	yes	—
157	C₈	#13133150	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#16810488	1	1	yes	characteristic
160	C₂²	#16810971	1	1	yes	characteristic
161	C₂²	#16810975	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	848	2	no	—
165	C₄	C₂×#13133148	1	1	yes	characteristic
166	C₄	C₂×#13133150	1	1	yes	characteristic
167	C₄	#16810969	1	1	yes	characteristic
168	C₄	#16810973	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#20274840	1	2	yes	—
172	C₂	#20274842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810488
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#8793
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #16810344	C₄	semidirect
10. C₈×#88322	C₄	semidirect
101. C₅₃	#8793	semidirect

Maximal subgroups

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

subgroup	index
#20274879	2
#20272998	2
#20272575	2
C₅₃×#1848	2
#8793	53