Subgroups — #27300478

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 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	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272807	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	—	2	1	no	—
6	C₂×#13133151	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₂×#13133149	—	2	1	no	—
9	#16810493	—	2	1	no	—
10	#16810489	—	2	1	no	—
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#346	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	4	2	no	—
17	#13133151	—	2	4	no	—
18	#13133151	—	2	4	no	—
19	#13133151	—	4	2	no	—
20	#13133149	—	4	2	no	—
21	#13133124	—	4	2	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	D₈	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	4	2	no	—
33	C₂²×Dic₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₅₃	Q₁₆	1	2	yes	—
43	C₄⋊Dic₅₃	—	4	2	no	—
44	C₄⋊Dic₅₃	—	4	2	no	—
45	C₄⋊Dic₅₃	—	4	2	no	—
46	C₄⋊Dic₅₃	—	4	2	no	—
47	C₄⋊Dic₅₃	—	8	4	no	—
48	C₁₆×C₅₃	—	2	4	no	—
49	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	#107	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₅₃	—	8	2	no	—
60	C₂×Dic₅₃	—	8	2	no	—
61	C₂×Dic₅₃	—	8	2	no	—
62	C₂×Dic₅₃	—	8	4	no	—
63	C₂×Dic₅₃	—	8	2	no	—
64	C₈×C₅₃	#106	1	4	yes	—
65	C₈×C₅₃	#117	1	2	yes	—
66	C₈×C₅₃	#125	1	2	yes	—
67	#12890	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
69	C₂²×C₅₃	#486	1	1	yes	characteristic
70	C₂²×C₅₃	#486	1	1	yes	characteristic
71	C₂²×C₅₃	#509	1	2	yes	—
72	C₂²×C₅₃	#510	1	2	yes	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	#382	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
76	C₄×C₅₃	#487	1	1	yes	characteristic
77	C₄×C₅₃	#487	1	1	yes	characteristic
78	Dic₅₃	—	16	2	no	—
79	Dic₅₃	—	16	2	no	—
80	C₂×#346	D₅₃	1	1	yes	characteristic
81	#1846	—	53	1	no	—
82	#1845	—	53	1	no	—
83	#1380	—	53	4	no	—
84	C₂×C₅₃	#1834	1	1	yes	characteristic
85	C₂×C₅₃	#2159	1	2	yes	—
86	C₂×C₅₃	#2243	1	2	yes	—
87	C₂×C₅₃	#2244	1	1	yes	characteristic
88	C₂×C₅₃	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
90	C₂×#107	—	53	2	no	—
91	C₂×#107	—	53	2	no	—
92	C₂×#107	—	106	1	no	—
93	C₂×#106	—	106	1	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	#384	—	106	1	no	—
96	#382	—	106	1	no	—
97	#346	Dic₅₃	1	4	yes	complemented
98	C₅₃	#12890	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	106	2	no	—
102	C₂×C₄⋊C₄	—	106	2	no	—
103	C₂×C₄⋊C₄	—	212	2	no	—
104	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
108	#107	—	106	4	no	—
109	#107	—	106	4	no	—
110	#107	—	212	2	no	—
111	#106	—	212	2	no	—
112	C₄×C₈	Dic₁₀₆	1	2	yes	—
113	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
114	#95	—	212	2	no	—
115	C₂²×C₄	—	212	2	no	—
116	C₂²×C₄	—	212	2	no	—
117	C₂²×C₄	D₄₂₄	1	2	yes	—
118	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₈	#11719913	1	2	yes	—
120	C₂×C₈	#11719913	1	2	yes	—
121	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₈	#11719945	1	2	yes	—
124	C₂×C₈	#11719945	1	2	yes	—
125	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
126	C₄⋊C₄	—	424	4	no	—
127	C₄⋊C₄	—	212	2	no	—
128	C₄⋊C₄	—	212	2	no	—
129	C₄⋊C₄	—	212	2	no	—
130	C₄⋊C₄	—	212	2	no	—
131	C₄²	Dic₂₁₂	1	2	yes	—
132	C₄²	C₄⋊Dic₅₃	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	424	4	no	—
138	C₂×C₄	—	424	2	no	—
139	C₂×C₄	—	424	2	no	—
140	C₂×C₄	—	424	2	no	—
141	C₂×C₄	—	424	2	no	—
142	C₂×C₄	#13133151	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
144	C₂×C₄	#13133205	1	1	yes	characteristic
145	C₂×C₄	#13133205	1	1	yes	characteristic
146	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
147	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
149	C₈	#13133149	1	4	yes	—
150	C₈	#13133176	1	2	yes	—
151	C₈	#13133209	1	2	yes	—
152	C₂²	C₂×#13133151	1	1	yes	characteristic
153	C₂²	#16810970	1	1	yes	characteristic
154	C₂²	#16810970	1	1	yes	characteristic
155	C₂²	#16811061	1	2	yes	—
156	C₂²	#16811066	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	848	2	no	—
159	C₄	—	848	2	no	—
160	C₄	#16810489	1	1	yes	characteristic
161	C₄	C₂×#13133149	1	1	yes	characteristic
162	C₄	#16810974	1	1	yes	characteristic
163	C₄	#16810974	1	1	yes	characteristic
164	C₂	#20274843	1	1	yes	characteristic
165	C₂	#20276503	1	2	yes	—
166	C₂	#20276952	1	2	yes	—
167	C₂	#20276957	1	1	yes	characteristic
168	C₂	#20276961	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12890
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. C₅₃×#346	C₄	semidirect
97. #346	Dic₅₃	semidirect
98. C₅₃	#12890	semidirect

Maximal subgroups

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

subgroup	index
#20274879	2
#20274875	2
#20272807	2
C₂×C₅₃×#346	2
#12890	53