Subgroups — C₅₃×#11401 #27218913

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 136 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×#1312	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₅₃×#431	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#322	C₄	1	2	yes	—
14	C₈²×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₅₃	—	2	4	no	—
19	C₂×C₁₆×C₅₃	—	2	4	no	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
22	C₄⋊C₈×C₅₃	C₈	1	4	yes	—
23	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
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	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
34	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
35	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₄²	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×C₈	1	2	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₅₃	—	2	4	no	—
42	C₁₆×C₅₃	—	2	4	no	—
43	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#97	1	2	yes	—
48	C₂×C₄×C₅₃	#97	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
51	C₈×C₅₃	—	2	8	no	—
52	C₈×C₅₃	#97	1	2	yes	—
53	C₈×C₅₃	#97	1	2	yes	—
54	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
55	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
56	#11401	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	#322	1	2	yes	—
59	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#322	1	2	yes	—
62	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
64	C₄×C₅₃	#504	1	4	yes	—
65	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
66	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
67	#1450	C₂×C₅₃	1	1	yes	characteristic
68	#1312	C₂×C₅₃	1	4	yes	complemented
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
71	C₂×C₅₃	#2108	1	2	yes	—
72	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
73	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
74	#431	C₄×C₅₃	1	2	yes	—
75	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
76	#346	C₄×C₅₃	1	2	yes	—
77	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
78	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
79	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
80	#322	C₄×C₅₃	1	2	yes	—
81	C₈²	C₂²×C₅₃	1	2	yes	—
82	C₅₃	#11401	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
85	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
89	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
90	C₄⋊C₈	C₈×C₅₃	1	4	yes	—
91	C₂²⋊C₈	C₈×C₅₃	1	4	yes	—
92	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
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	C₂³×C₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
99	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
100	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
102	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
103	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
104	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
106	C₄²	C₄²×C₅₃	1	1	yes	characteristic
107	C₄²	C₂×C₈×C₅₃	1	2	yes	—
108	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
109	C₁₆	—	2	4	no	—
110	C₁₆	—	2	4	no	—
111	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
114	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
115	C₂×C₄	#97×C₅₃	1	2	yes	—
116	C₂×C₄	#97×C₅₃	1	2	yes	—
117	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
119	C₈	—	2	8	no	—
120	C₈	#97×C₅₃	1	2	yes	—
121	C₈	#97×C₅₃	1	2	yes	—
122	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
123	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	C₅₃×#322	1	2	yes	—
126	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	C₅₃×#322	1	2	yes	—
129	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
130	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
131	C₄	C₅₃×#504	1	4	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×C₅₃×#322	1	1	yes	characteristic
134	C₂	C₅₃×#2108	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1312	C₂	semidirect
56. #11401	C₅₃	direct (N × H)
68. #1312	C₂×C₅₃	semidirect
82. C₅₃	#11401	direct (N × H)

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×#1312	2
#11401	53

Subgroups — C53×#11401 #27218913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11401 #27218913