Subgroups — #27300533

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810451	C₂	1	2	yes	maximal
2	C₂×#16810449	C₂	1	2	yes	maximal
3	#20274879	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	C₂²	1	4	yes	—
6	C₂×#13133149	—	2	2	no	—
7	#16810489	—	2	2	no	—
8	#16810451	—	2	4	no	—
9	#16810449	—	2	4	no	—
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₅₃	—	2	4	no	—
15	#13133151	—	2	4	no	—
16	#13133151	—	2	4	no	—
17	#13133149	—	4	4	no	—
18	#13133124	—	4	4	no	—
19	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
25	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
27	C₂×C₁₆×C₅₃	Q₈	1	2	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	4	no	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₅₃	D₈	1	2	yes	—
35	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₅₃	Q₁₆	1	2	yes	—
44	C₄⋊Dic₅₃	—	4	4	no	—
45	C₄⋊Dic₅₃	—	4	4	no	—
46	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
48	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	#107	1	4	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×Dic₅₃	—	8	4	no	—
58	C₂×Dic₅₃	—	8	8	no	—
59	C₈×C₅₃	#107	1	4	yes	—
60	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	#12907	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
66	C₂²×C₅₃	#493	1	1	yes	characteristic
67	C₂²×C₅₃	#500	1	1	yes	characteristic
68	C₂²×C₅₃	#508	1	4	yes	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	#384	1	1	yes	characteristic
71	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
72	C₄×C₅₃	#494	1	1	yes	characteristic
73	C₄×C₅₃	#500	1	1	yes	characteristic
74	Dic₅₃	—	16	4	no	—
75	C₂×#367	—	53	2	no	—
76	C₂×#366	—	53	2	no	—
77	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
78	#1846	—	53	2	no	—
79	C₂×C₅₃	#1845	1	1	yes	characteristic
80	C₂×C₅₃	#2158	1	2	yes	—
81	C₂×C₅₃	#2249	1	2	yes	—
82	C₂×C₅₃	#2253	1	2	yes	—
83	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
85	C₂×#107	—	53	4	no	—
86	C₂×#106	—	106	2	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	#382	—	106	2	no	—
89	#367	—	106	4	no	—
90	#366	—	106	4	no	—
91	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
92	C₅₃	#12907	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	106	4	no	—
96	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
97	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
98	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
99	C₂×C₁₆	D₂₁₂	1	2	yes	—
100	C₂×C₁₆	D₂₁₂	1	2	yes	—
101	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
102	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
103	#107	—	106	4	no	—
104	#107	—	106	4	no	—
105	#106	—	212	4	no	—
106	C₄×C₈	Dic₁₀₆	1	2	yes	—
107	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
108	#95	—	212	4	no	—
109	C₂²×C₄	—	212	4	no	—
110	C₂²×C₄	D₄₂₄	1	2	yes	—
111	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
112	C₂×C₈	D₄₂₄	1	2	yes	—
113	C₂×C₈	Dic₂₁₂	1	2	yes	—
114	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
115	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
117	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
120	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₄⋊C₄	—	212	4	no	—
122	C₄⋊C₄	—	212	4	no	—
123	C₄²	Dic₂₁₂	1	2	yes	—
124	C₄²	C₄⋊Dic₅₃	1	2	yes	—
125	C₁₆	C₄⋊Dic₅₃	1	4	yes	—
126	C₁₆	C₄⋊Dic₅₃	1	4	yes	—
127	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	424	8	no	—
130	C₂×C₄	—	424	4	no	—
131	C₂×C₄	#13133151	1	4	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₂×C₄	#13133214	1	1	yes	characteristic
134	C₂×C₄	#13133217	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
138	C₈	#13133151	1	4	yes	—
139	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
141	C₈	#13133217	1	1	yes	characteristic
142	C₈	#13133217	1	1	yes	characteristic
143	C₂²	C₂×#13133151	1	1	yes	characteristic
144	C₂²	#16811000	1	1	yes	characteristic
145	C₂²	#16811028	1	1	yes	characteristic
146	C₂²	#16811055	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	848	4	no	—
149	C₄	#16810493	1	1	yes	characteristic
150	C₄	C₂×#13133151	1	1	yes	characteristic
151	C₄	#16811003	1	1	yes	characteristic
152	C₄	#16811028	1	1	yes	characteristic
153	C₂	#20274875	1	1	yes	characteristic
154	C₂	#20276499	1	2	yes	—
155	C₂	#20276976	1	2	yes	—
156	C₂	#20276994	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12907
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₆×C₅₃	C₄	semidirect
91. C₄×C₁₆	Dic₅₃	semidirect
92. C₅₃	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810451	2
C₂×#16810449	2
#20274879	2
C₂×C₄×C₁₆×C₅₃	2
#12907	53