Subgroups — C₂²×#16811101 #28290677

Order 13568.

11230 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811101	C₂	1	48	yes	maximal, direct product factor
2	C₂²×Q₈×Dic₅₃	C₂	1	8	yes	maximal, complemented
3	C₂²×#13133176	C₂	1	6	yes	maximal
4	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811101	C₂²	1	256	yes	direct product factor
6	C₂×Q₈×Dic₅₃	C₂²	1	192	yes	complemented
7	C₂×#13133176	C₂²	1	144	yes	—
8	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	—
9	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
11	C₂³×Q₈×C₅₃	C₄	1	32	yes	complemented
12	C₂³×Q₈×C₅₃	C₂²	1	4	yes	—
13	C₂³×Q₈×C₅₃	C₂²	1	24	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
15	C₂³×Dic₅₃	C₂³	1	8	yes	—
16	Q₈×Dic₅₃	C₂³	1	512	yes	complemented
17	#13133176	C₂³	1	384	yes	—
18	C₂×C₄⋊Dic₅₃	C₂³	1	144	yes	—
19	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	—
20	C₂×C₄×Dic₅₃	C₂³	1	144	yes	—
21	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Q₈×C₅₃	C₂×C₄	1	64	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	384	yes	complemented
24	C₂²×Q₈×C₅₃	C₂³	1	48	yes	—
25	C₂²×Q₈×C₅₃	C₂³	1	64	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	48	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
29	C₂⁴×C₅₃	—	2	8	no	—
30	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
31	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
32	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	C₂²×C₄	1	384	yes	—
34	C₂×Q₈×C₅₃	C₂²×C₄	1	512	yes	complemented
35	C₂×Q₈×C₅₃	C₂⁴	1	64	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	288	yes	—
38	C₂²×C₄×C₅₃	C₂⁴	1	36	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
40	C₂²×Dic₅₃	—	2	32	no	—
41	C₂²×Dic₅₃	C₂⁴	1	48	yes	—
42	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
43	C₄⋊Dic₅₃	C₂⁴	1	192	yes	—
44	C₄×Dic₅₃	C₂⁴	1	192	yes	—
45	C₂³×C₅₃	—	2	32	no	—
46	C₂³×C₅₃	—	2	24	no	—
47	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
48	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
49	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	—
50	C₂³×C₅₃	C₂⁵	1	4	yes	—
51	C₂³×C₅₃	C₂⁵	1	3	yes	—
52	Q₈×C₅₃	C₂³×C₄	1	256	yes	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	144	yes	—
54	C₂×C₄×C₅₃	C₂³×C₄	1	192	yes	—
55	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
56	C₂×Dic₅₃	—	2	96	no	—
57	C₂×Dic₅₃	C₂⁵	1	32	yes	—
58	C₂²×#519	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	8	no	—
60	C₂²×C₅₃	—	2	48	no	—
61	C₂²×C₅₃	#519	1	16	yes	complemented
62	C₂²×C₅₃	C₂⁴×C₄	1	8	yes	—
63	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
64	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	12	yes	—
65	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
66	C₄×C₅₃	C₂⁴×C₄	1	48	yes	—
67	Dic₅₃	—	2	32	no	—
68	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic
69	C₂×#519	—	53	48	no	—
70	C₂²×C₄×Q₈	—	53	8	no	—
71	C₂²×#117	—	53	6	no	—
72	C₂×C₅₃	—	2	8	no	—
73	C₂×C₅₃	C₂×#519	1	12	yes	complemented
74	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
75	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	2	yes	—
76	C₂³×Q₈	Dic₅₃	1	32	yes	complemented
77	C₂³×Q₈	C₂×D₅₃	1	4	yes	—
78	C₂³×Q₈	C₂×D₅₃	1	24	yes	—
79	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
80	#519	—	53	256	no	—
81	C₂×C₄×Q₈	—	53	192	no	—
82	C₂×#117	—	53	144	no	—
83	C₂²×C₄⋊C₄	—	53	12	no	—
84	C₂²×C₂²⋊C₄	—	53	4	no	—
85	C₂²×C₄²	—	53	12	no	—
86	C₅₃	C₂²×#519	1	1	yes	characteristic, complemented, Sylow
87	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
88	C₂²×Q₈	C₂×Dic₅₃	1	64	yes	—
89	C₂²×Q₈	C₂×Dic₅₃	1	384	yes	complemented
90	C₂²×Q₈	C₂²×D₅₃	1	48	yes	—
91	C₂²×Q₈	C₂²×D₅₃	1	64	yes	—
92	C₂³×C₄	—	53	8	no	—
93	C₂³×C₄	C₂×Dic₅₃	1	48	yes	—
94	C₂³×C₄	C₂²×D₅₃	1	36	yes	—
95	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
96	C₄×Q₈	—	53	512	no	—
97	#117	—	53	384	no	—
98	C₂×C₄⋊C₄	—	53	144	no	—
99	C₂×C₂²⋊C₄	—	53	48	no	—
100	C₂×C₄²	—	53	144	no	—
101	C₂⁴	—	2	8	no	—
102	C₂⁴	C₂²×Dic₅₃	1	8	yes	—
103	C₂⁴	C₂³×D₅₃	1	6	yes	—
104	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
105	C₂×Q₈	C₂²×Dic₅₃	1	384	yes	—
106	C₂×Q₈	C₂²×Dic₅₃	1	512	yes	complemented
107	C₂×Q₈	C₂³×D₅₃	1	64	yes	—
108	C₂²×C₄	—	53	48	no	—
109	C₂²×C₄	—	106	32	no	—
110	C₂²×C₄	C₂²×Dic₅₃	1	48	yes	—
111	C₂²×C₄	C₂²×Dic₅₃	1	288	yes	—
112	C₂²×C₄	C₂³×D₅₃	1	36	yes	—
113	C₂²×C₄	C₂³×D₅₃	1	48	yes	—
114	C₄⋊C₄	—	53	192	no	—
115	C₂²⋊C₄	—	53	64	no	—
116	C₄²	—	53	192	no	—
117	C₂³	—	2	32	no	—
118	C₂³	—	2	24	no	—
119	C₂³	C₂³×Dic₅₃	1	24	yes	—
120	C₂³	C₂³×Dic₅₃	1	4	yes	—
121	C₂³	#13133265	1	8	yes	—
122	C₂³	C₂⁴×D₅₃	1	4	yes	—
123	C₂³	C₂⁴×D₅₃	1	3	yes	—
124	Q₈	C₂³×Dic₅₃	1	256	yes	—
125	C₂×C₄	—	53	32	no	—
126	C₂×C₄	—	106	96	no	—
127	C₂×C₄	C₂³×Dic₅₃	1	144	yes	—
128	C₂×C₄	C₂³×Dic₅₃	1	192	yes	—
129	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
130	C₂²	—	2	48	no	—
131	C₂²	—	2	8	no	—
132	C₂²	#16811101	1	16	yes	direct product factor
133	C₂²	C₂⁴×Dic₅₃	1	8	yes	—
134	C₂²	C₂⁴×Dic₅₃	1	6	yes	—
135	C₂²	C₂×#13133265	1	12	yes	—
136	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
137	C₄	—	106	32	no	—
138	C₄	C₂⁴×Dic₅₃	1	48	yes	—
139	C₂	—	2	8	no	—
140	C₂	C₂×#16811101	1	12	yes	direct product factor
141	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
142	C₂	C₂²×#13133265	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	C₂²×#519
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811101	C₂	direct (N × H)
2. C₂²×Q₈×Dic₅₃	C₂	semidirect
5. #16811101	C₂²	direct (N × H)
6. C₂×Q₈×Dic₅₃	C₂²	semidirect
11. C₂³×Q₈×C₅₃	C₄	semidirect
16. Q₈×Dic₅₃	C₂³	semidirect
23. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
34. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
61. C₂²×C₅₃	#519	semidirect
73. C₂×C₅₃	C₂×#519	semidirect
76. C₂³×Q₈	Dic₅₃	semidirect
86. C₅₃	C₂²×#519	semidirect
89. C₂²×Q₈	C₂×Dic₅₃	semidirect
106. C₂×Q₈	C₂²×Dic₅₃	semidirect
132. C₂²	#16811101	direct (N × H)
140. C₂	C₂×#16811101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811101	2
C₂²×Q₈×Dic₅₃	2
C₂²×#13133176	2
C₂⁴×Q₈×C₅₃	2
C₂²×#519	53

Subgroups — C22×#16811101 #28290677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811101 #28290677