Subgroups — C₂²×C₅₃×#519 #27270158

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

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂	C₂⁵×C₄×C₅₃
Frattini, cosocle	C₂²	C₂⁶×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#519	C₂	direct (N × H)
3. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
5. C₂³×Q₈×C₅₃	C₄	semidirect
9. C₅₃×#519	C₂²	direct (N × H)
10. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
17. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
24. C₄×Q₈×C₅₃	C₂³	semidirect
34. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
58. C₂²×#519	C₅₃	direct (N × H)
61. C₂²×C₅₃	#519	direct (N × H)
69. C₂×#519	C₂×C₅₃	direct (N × H)
70. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
73. C₂×C₅₃	C₂×#519	direct (N × H)
76. C₂³×Q₈	C₄×C₅₃	semidirect
80. #519	C₂²×C₅₃	direct (N × H)
81. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
86. C₅₃	C₂²×#519	direct (N × H)
89. C₂²×Q₈	C₂×C₄×C₅₃	semidirect
96. C₄×Q₈	C₂³×C₅₃	semidirect
106. C₂×Q₈	C₂²×C₄×C₅₃	semidirect
132. C₂²	C₅₃×#519	direct (N × H)
140. C₂	C₂×C₅₃×#519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₅₃	2
C₂×C₅₃×#519	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×#117×C₅₃	2
C₂²×#519	53

Subgroups — C22×C53×#519 #27270158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#519 #27270158