Subgroups — #27271792

Order 13568.

190 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	—	2	1	no	—
5	C₂×#13133123	—	2	1	no	—
6	C₂×#13133123	—	2	1	no	—
7	C₂×#13133123	C₄	1	2	yes	complemented
8	#16810449	—	2	1	no	—
9	#16810449	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₃	—	2	1	no	—
12	C₅₃×#366	—	2	1	no	—
13	C₅₃×#366	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₅₃	—	4	1	no	—
15	#13133151	—	2	2	no	—
16	#13133151	—	4	1	no	—
17	#13133123	—	2	2	no	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	4	1	no	—
28	C₂²×C₄×C₅₃	—	8	1	no	—
29	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	8	1	no	—
31	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄⋊C₄×C₅₃	—	8	1	no	—
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	8	1	no	—
41	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
42	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	8	2	no	—
47	C₂×C₄×C₅₃	—	16	1	no	—
48	C₂×C₄×C₅₃	—	16	1	no	—
49	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
55	C₂×Dic₅₃	—	8	2	no	—
56	C₂×Dic₅₃	—	16	1	no	—
57	C₂×Dic₅₃	—	16	1	no	—
58	C₈×C₅₃	#95	1	1	yes	characteristic
59	C₈×C₅₃	#103	1	1	yes	characteristic
60	C₈×C₅₃	#103	1	1	yes	characteristic
61	C₈×C₅₃	#106	1	1	yes	characteristic
62	#7471	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#357	1	1	yes	characteristic
64	C₂²×C₅₃	#357	1	1	yes	characteristic
65	C₂²×C₅₃	#366	1	1	yes	characteristic
66	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
67	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
68	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
69	C₂²×C₅₃	Q₆₄	1	1	yes	characteristic
70	C₄×C₅₃	—	16	2	no	—
71	C₄×C₅₃	#340	1	1	yes	characteristic
72	C₄×C₅₃	#358	1	1	yes	characteristic
73	C₄×C₅₃	#358	1	1	yes	characteristic
74	C₄×C₅₃	#367	1	1	yes	characteristic
75	Dic₅₃	—	16	2	no	—
76	C₂²×C₃₂	—	53	1	no	—
77	C₂×#366	—	53	1	no	—
78	C₂×#366	D₅₃	1	1	yes	characteristic
79	C₂×C₅₃	#1380	1	1	yes	characteristic
80	C₂×C₅₃	#1415	1	1	yes	characteristic
81	C₂×C₅₃	#1415	1	1	yes	characteristic
82	C₂×C₅₃	#1416	1	1	yes	characteristic
83	C₂×C₅₃	#1416	1	1	yes	characteristic
84	C₂×C₅₃	#1423	1	1	yes	characteristic
85	C₂×C₅₃	#1424	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	106	1	no	—
89	C₂×C₃₂	—	53	2	no	—
90	C₂×C₃₂	—	106	1	no	—
91	C₂×C₃₂	—	106	1	no	—
92	#366	—	2	1	no	—
93	#366	—	53	2	no	—
94	#366	—	106	1	no	—
95	#366	Dic₅₃	1	2	yes	complemented
96	C₅₃	#7471	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	212	1	no	—
100	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
106	#107	—	2	2	no	—
107	#107	—	4	1	no	—
108	#107	—	106	2	no	—
109	#107	—	212	1	no	—
110	C₃₂	—	106	2	no	—
111	C₂²×C₄	—	8	1	no	—
112	C₂²×C₄	—	424	1	no	—
113	C₂²×C₄	#11719922	1	1	yes	characteristic
114	C₂×C₈	#11719919	1	1	yes	characteristic
115	C₂×C₈	#11719921	1	1	yes	characteristic
116	C₂×C₈	#11719923	1	1	yes	characteristic
117	C₂×C₈	#11719924	1	1	yes	characteristic
118	C₂×C₈	#11719925	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	424	1	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
125	C₁₆	#11719919	1	1	yes	characteristic
126	C₁₆	C₄⋊Dic₅₃	1	1	yes	characteristic
127	C₁₆	#11719921	1	1	yes	characteristic
128	C₂³	#13133158	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	848	1	no	—
133	C₂×C₄	—	848	1	no	—
134	C₂×C₄	—	424	2	no	—
135	C₂×C₄	#13133137	1	1	yes	characteristic
136	C₂×C₄	#13133138	1	1	yes	characteristic
137	C₂×C₄	#13133150	1	1	yes	characteristic
138	C₂×C₄	#13133160	1	1	yes	characteristic
139	C₂×C₄	#13133161	1	1	yes	characteristic
140	C₂×C₄	#13133163	1	1	yes	characteristic
141	C₈	#13133124	1	1	yes	characteristic
142	C₈	#13133140	1	1	yes	characteristic
143	C₈	#13133141	1	1	yes	characteristic
144	C₈	#13133148	1	1	yes	characteristic
145	C₂²	#16810423	1	1	yes	characteristic
146	C₂²	#16810424	1	1	yes	characteristic
147	C₂²	#16810448	1	1	yes	characteristic
148	C₂²	#16810458	1	1	yes	characteristic
149	C₂²	#16810460	1	1	yes	characteristic
150	C₂²	#16810461	1	1	yes	characteristic
151	C₂²	#16810463	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	848	2	no	—
154	C₄	#16810385	1	1	yes	characteristic
155	C₄	#16810426	1	1	yes	characteristic
156	C₄	#16810427	1	1	yes	characteristic
157	C₄	#16810450	1	1	yes	characteristic
158	C₂	#20272806	1	1	yes	characteristic
159	C₂	#20272892	1	1	yes	characteristic
160	C₂	#20272893	1	1	yes	characteristic
161	C₂	#20272895	1	1	yes	characteristic
162	C₂	#20272896	1	1	yes	characteristic
163	C₂	#20272914	1	1	yes	characteristic
164	C₂	#20272916	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133158
commutator	C₁₆×C₅₃	C₄²
Frattini	C₂²×C₁₆	C₂×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle, prime radical	C₂³×C₅₃	D₁₆
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	#7471
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
7. C₂×#13133123	C₄	semidirect
9. #16810449	C₄	semidirect
13. C₅₃×#366	C₄	semidirect
95. #366	Dic₅₃	semidirect
96. C₅₃	#7471	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133123	2
C₂×#16810449	2
C₂×C₅₃×#366	2
#7471	53