Subgroups — C₅₃×#11399 #27218911

Order 13568.

330 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	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1311	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
8	C₅₃×#431	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#322	C₄	1	2	yes	—
16	C₈²×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	M₃₂×C₅₃	C₈	1	8	yes	complemented
21	C₂×C₁₆×C₅₃	C₈	1	8	yes	complemented
22	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₄⋊C₈×C₅₃	C₈	1	4	yes	—
26	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₄²	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×C₈	1	2	yes	—
43	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
45	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
46	C₁₆×C₅₃	C₂×C₈	1	8	yes	—
47	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
55	C₈×C₅₃	—	2	8	no	—
56	C₈×C₅₃	C₄×C₈	1	2	yes	—
57	C₈×C₅₃	C₄×C₈	1	2	yes	—
58	C₈×C₅₃	C₂²×C₈	1	2	yes	—
59	C₈×C₅₃	C₂²×C₈	1	2	yes	—
60	#11399	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	C₈²	1	2	yes	—
63	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₈²	1	2	yes	—
66	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₅₃	#504	1	4	yes	—
69	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
70	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
71	#1450	C₂×C₅₃	1	1	yes	characteristic
72	#1311	C₂×C₅₃	1	2	yes	complemented
73	C₈×C₁₆	C₂×C₅₃	1	2	yes	complemented
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
76	C₂×C₅₃	#2108	1	2	yes	—
77	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
78	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
79	#431	C₄×C₅₃	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
81	#346	C₄×C₅₃	1	2	yes	—
82	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
83	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
84	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
85	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
86	#322	C₄×C₅₃	1	2	yes	—
87	C₈²	C₂²×C₅₃	1	2	yes	—
88	C₅₃	#11399	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
91	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
92	M₃₂	C₈×C₅₃	1	8	yes	complemented
93	C₂×C₁₆	C₈×C₅₃	1	8	yes	complemented
94	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
95	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
96	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
97	C₄⋊C₈	C₈×C₅₃	1	4	yes	—
98	C₂²⋊C₈	C₈×C₅₃	1	4	yes	—
99	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
101	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
103	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
106	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
107	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
108	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
109	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
110	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
111	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
113	C₄²	C₄²×C₅₃	1	1	yes	characteristic
114	C₄²	C₂×C₈×C₅₃	1	2	yes	—
115	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
116	C₁₆	C₂×C₈×C₅₃	1	4	yes	—
117	C₁₆	C₂×C₈×C₅₃	1	4	yes	—
118	C₁₆	C₂×C₈×C₅₃	1	8	yes	—
119	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
122	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
123	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
124	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
125	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
127	C₈	—	2	8	no	—
128	C₈	C₄×C₈×C₅₃	1	2	yes	—
129	C₈	C₄×C₈×C₅₃	1	2	yes	—
130	C₈	C₂²×C₈×C₅₃	1	2	yes	—
131	C₈	C₂²×C₈×C₅₃	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈²×C₅₃	1	2	yes	—
134	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈²×C₅₃	1	2	yes	—
137	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
138	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
139	C₄	C₅₃×#504	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈²×C₅₃	1	1	yes	characteristic
142	C₂	C₅₃×#2108	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₅₃	C₂²
commutator	C₂	C₂×C₈²×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1311	C₂	semidirect
5. C₈×C₁₆×C₅₃	C₂	semidirect
20. M₃₂×C₅₃	C₈	semidirect
21. C₂×C₁₆×C₅₃	C₈	semidirect
60. #11399	C₅₃	direct (N × H)
72. #1311	C₂×C₅₃	semidirect
73. C₈×C₁₆	C₂×C₅₃	semidirect
88. C₅₃	#11399	direct (N × H)
92. M₃₂	C₈×C₅₃	semidirect
93. C₂×C₁₆	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×#1450	2
C₅₃×#1311	2
C₈×C₁₆×C₅₃	2
#11399	53

Subgroups — C53×#11399 #27218911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11399 #27218911