Subgroups — C₅₃×#11404 #27218916

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 138 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	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1313	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
6	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₂²	1	2	yes	—
10	C₅₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
13	C₅₃×#322	C₄	1	2	yes	—
14	C₅₃×#322	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₅₃	—	2	4	no	—
20	M₃₂×C₅₃	C₈	1	8	yes	complemented
21	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₅₃	—	2	2	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₄²	1	1	yes	characteristic
40	C₄²×C₅₃	M₁₆	1	2	yes	—
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₅₃	—	2	4	no	—
43	C₁₆×C₅₃	C₂×C₈	1	8	yes	complemented
44	C₂³×C₅₃	#97	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	2	yes	—
49	C₂×C₄×C₅₃	#97	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
52	C₈×C₅₃	—	4	4	no	—
53	C₈×C₅₃	C₄×C₈	1	2	yes	—
54	C₈×C₅₃	#97	1	2	yes	—
55	C₈×C₅₃	C₂²×C₈	1	2	yes	—
56	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
57	#11404	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	#322	1	2	yes	—
60	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	#322	1	2	yes	—
63	C₄×C₅₃	#347	1	4	yes	—
64	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
66	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
67	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
68	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
69	#1313	C₂×C₅₃	1	4	yes	complemented
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#347	1	2	yes	—
73	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
74	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
75	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
76	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
77	#346	C₂²×C₅₃	1	2	yes	—
78	#346	C₂²×C₅₃	1	2	yes	—
79	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
80	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
81	#322	C₄×C₅₃	1	2	yes	—
82	#322	C₂²×C₅₃	1	2	yes	—
83	C₅₃	#11404	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
88	M₃₂	—	2	4	no	—
89	M₃₂	C₈×C₅₃	1	8	yes	complemented
90	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
91	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
94	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
99	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
103	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
105	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
106	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
108	C₄²	C₄²×C₅₃	1	1	yes	characteristic
109	C₄²	M₁₆×C₅₃	1	2	yes	—
110	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
111	C₁₆	—	2	4	no	—
112	C₁₆	C₂×C₈×C₅₃	1	8	yes	complemented
113	C₂³	#97×C₅₃	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
116	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
117	C₂×C₄	#97×C₅₃	1	2	yes	—
118	C₂×C₄	#97×C₅₃	1	2	yes	—
119	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₈	C₄×C₈×C₅₃	1	2	yes	—
123	C₈	#97×C₅₃	1	2	yes	—
124	C₈	C₂²×C₈×C₅₃	1	2	yes	—
125	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₅₃×#322	1	2	yes	—
128	C₂²	C₂×#97×C₅₃	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₅₃×#322	1	2	yes	—
131	C₄	C₅₃×#347	1	4	yes	—
132	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
133	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₅₃×#322	1	1	yes	characteristic
136	C₂	C₂×C₅₃×#347	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1313	C₂	semidirect
20. M₃₂×C₅₃	C₈	semidirect
43. C₁₆×C₅₃	C₂×C₈	semidirect
57. #11404	C₅₃	direct (N × H)
69. #1313	C₂×C₅₃	semidirect
83. C₅₃	#11404	direct (N × H)
89. M₃₂	C₈×C₅₃	semidirect
112. C₁₆	C₂×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×C₅₃×#322	2
C₅₃×#1313	2
#11404	53

Subgroups — C53×#11404 #27218916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11404 #27218916