Subgroups — C₅₃×#7465 #27214977

Order 13568.

202 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	—	2	2	no	—
7	C₂×C₃₂×C₅₃	—	2	2	no	—
8	C₂×C₃₂×C₅₃	C₄	1	4	yes	complemented
9	C₄×C₁₆×C₅₃	—	2	2	no	—
10	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	—	2	1	no	—
14	C₂×C₁₆×C₅₃	—	2	1	no	—
15	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₃₂×C₅₃	—	2	4	no	—
23	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₁₆×C₅₃	—	2	2	no	—
37	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	C₃₂	1	4	yes	complemented
43	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	#95	1	1	yes	characteristic
51	C₈×C₅₃	#97	1	1	yes	characteristic
52	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	#7465	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
59	C₂²×C₅₃	M₆₄	1	2	yes	—
60	C₄×C₅₃	—	2	4	no	—
61	C₄×C₅₃	#336	1	1	yes	characteristic
62	C₄×C₅₃	#346	1	1	yes	characteristic
63	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
64	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₃₂	C₂×C₅₃	1	2	yes	—
66	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
67	C₂×C₅₃	#1374	1	1	yes	characteristic
68	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
69	C₂×C₅₃	#1397	1	1	yes	characteristic
70	C₂×C₅₃	C₂²⋊C₃₂	1	2	yes	—
71	C₂×C₅₃	C₄⋊C₃₂	1	2	yes	—
72	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
73	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
74	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
75	C₂×C₃₂	—	2	2	no	—
76	C₂×C₃₂	—	2	2	no	—
77	C₂×C₃₂	C₄×C₅₃	1	4	yes	complemented
78	C₄×C₁₆	—	2	2	no	—
79	C₅₃	#7465	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
82	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
87	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
88	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
89	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₃₂	—	2	4	no	—
93	C₂²×C₄	C₁₆×C₅₃	1	2	yes	—
94	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₁₆×C₅₃	1	2	yes	—
98	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
102	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
103	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂²⋊C₄×C₅₃	1	2	yes	—
108	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
109	C₂³	C₂×C₁₆×C₅₃	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	C₃₂×C₅₃	1	4	yes	complemented
113	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
115	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
117	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
118	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	#95×C₅₃	1	1	yes	characteristic
121	C₈	#97×C₅₃	1	1	yes	characteristic
122	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
123	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
124	C₂²	C₄×C₁₆×C₅₃	1	1	yes	characteristic
125	C₂²	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
126	C₂²	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
127	C₂²	C₂×C₃₂×C₅₃	1	2	yes	—
128	C₂²	C₅₃×M₆₄	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	C₅₃×#336	1	1	yes	characteristic
131	C₄	C₅₃×#346	1	1	yes	characteristic
132	C₄	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
133	C₄	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
134	C₂	C₅₃×#1374	1	1	yes	characteristic
135	C₂	C₄×C₃₂×C₅₃	1	1	yes	characteristic
136	C₂	C₅₃×#1397	1	1	yes	characteristic
137	C₂	C₅₃×C₂²⋊C₃₂	1	2	yes	—
138	C₂	C₅₃×C₄⋊C₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆×C₅₃	C₂²
commutator	C₂	C₄×C₃₂×C₅₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₃₂×C₅₃	C₄	semidirect
42. C₂×C₄×C₅₃	C₃₂	semidirect
54. #7465	C₅₃	direct (N × H)
77. C₂×C₃₂	C₄×C₅₃	semidirect
79. C₅₃	#7465	direct (N × H)
112. C₂×C₄	C₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₅₃	2
C₂×C₄×C₁₆×C₅₃	2
#7465	53

Subgroups — C53×#7465 #27214977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7465 #27214977