Subgroups — C₅₃×#12597 #27220109

Order 13568.

258 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2110	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1379	C₂	1	2	yes	maximal
5	C₅₃×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#405	C₂²	1	2	yes	—
10	C₅₃×#405	C₂²	1	1	yes	characteristic
11	C₅₃×#347	C₄	1	4	yes	complemented
12	C₅₃×#346	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
14	C₅₃×#338	C₄	1	1	yes	characteristic
15	C₅₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	C₅₃×H(3,ℤ/4)	C₄	1	2	yes	—
17	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	#117×C₅₃	C₂×C₄	1	2	yes	—
21	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₅₃	—	2	2	no	—
23	M₃₂×C₅₃	—	2	2	no	—
24	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	#97×C₅₃	—	2	2	no	—
31	#97×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₅₃	—	2	2	no	—
36	M₁₆×C₅₃	—	2	2	no	—
37	M₁₆×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	C₄²	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₂×C₈×C₅₃	#41	1	2	yes	—
44	C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂²⋊C₄×C₅₃	—	2	1	no	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₄²	1	2	yes	—
50	C₁₆×C₅₃	—	2	1	no	—
51	C₁₆×C₅₃	—	2	1	no	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₁₆×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	#131	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#117	1	2	yes	—
59	C₂×C₄×C₅₃	#131	1	2	yes	—
60	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	—	4	2	no	—
63	C₈×C₅₃	#117	1	1	yes	characteristic
64	C₈×C₅₃	#117	1	2	yes	—
65	C₈×C₅₃	#117	1	1	yes	characteristic
66	#12597	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	#405	1	1	yes	characteristic
69	C₂²×C₅₃	#433	1	2	yes	—
70	C₄×C₅₃	—	4	2	no	—
71	C₄×C₅₃	—	8	2	no	—
72	C₄×C₅₃	#376	1	1	yes	characteristic
73	C₄×C₅₃	#405	1	1	yes	characteristic
74	C₄×C₅₃	#433	1	2	yes	—
75	#2110	C₂×C₅₃	1	1	yes	characteristic
76	#2108	C₂×C₅₃	1	1	yes	characteristic
77	#1747	C₂×C₅₃	1	1	yes	characteristic
78	#1379	C₂×C₅₃	1	2	yes	—
79	#1378	C₂×C₅₃	1	2	yes	—
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	#1746	1	1	yes	characteristic
83	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
84	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
85	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
86	#405	C₂²×C₅₃	1	2	yes	—
87	#405	C₂²×C₅₃	1	1	yes	characteristic
88	#347	C₄×C₅₃	1	4	yes	complemented
89	#346	C₄×C₅₃	1	2	yes	complemented
90	C₄×C₁₆	C₄×C₅₃	1	2	yes	complemented
91	#338	C₄×C₅₃	1	1	yes	characteristic
92	H(3,ℤ/4)	C₄×C₅₃	1	1	yes	characteristic
93	H(3,ℤ/4)	C₄×C₅₃	1	2	yes	—
94	C₅₃	#12597	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
96	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
98	#117	C₂×C₄×C₅₃	1	2	yes	—
99	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
107	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
108	#97	—	2	2	no	—
109	#97	C₂×C₄×C₅₃	1	2	yes	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₄²×C₅₃	1	2	yes	—
118	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
120	C₂×C₈	#41×C₅₃	1	2	yes	—
121	C₂×C₈	#41×C₅₃	1	2	yes	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₄²×C₅₃	1	2	yes	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	2	2	no	—
132	C₂³	#131×C₅₃	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
136	C₂×C₄	#117×C₅₃	1	2	yes	—
137	C₂×C₄	#131×C₅₃	1	2	yes	—
138	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	#117×C₅₃	1	1	yes	characteristic
142	C₈	#117×C₅₃	1	2	yes	—
143	C₈	#117×C₅₃	1	1	yes	characteristic
144	C₂²	—	4	1	no	—
145	C₂²	C₅₃×#405	1	1	yes	characteristic
146	C₂²	C₅₃×#433	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	C₅₃×#376	1	1	yes	characteristic
150	C₄	C₅₃×#405	1	1	yes	characteristic
151	C₄	C₅₃×#433	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	C₅₃×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1746
prime radical	C₂³×C₅₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#347	C₄	semidirect
12. C₅₃×#346	C₄	semidirect
13. C₄×C₁₆×C₅₃	C₄	semidirect
66. #12597	C₅₃	direct (N × H)
88. #347	C₄×C₅₃	semidirect
89. #346	C₄×C₅₃	semidirect
90. C₄×C₁₆	C₄×C₅₃	semidirect
94. C₅₃	#12597	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2110	2
C₅₃×#2108	2
C₅₃×#1747	2
C₅₃×#1379	2
C₅₃×#1378	2
#12597	53

Subgroups — C53×#12597 #27220109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12597 #27220109