Subgroups — C₅₃×#12019 #27219531

Order 13568.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1368	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
6	C₅₃×#432	—	2	1	no	—
7	C₅₃×#423	—	2	1	no	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₄	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	1	yes	characteristic
11	C₅₃×#346	C₂²	1	2	yes	—
12	C₅₃×#335	—	2	1	no	—
13	C₅₃×#334	—	2	1	no	—
14	C₅₃×#334	C₂²	1	2	yes	—
15	C₅₃×#334	C₂²	1	2	yes	—
16	C₂×M₁₆×C₅₃	—	4	1	no	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	—	2	4	no	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₄⋊C₈×C₅₃	—	2	2	no	—
24	C₄⋊C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₈×C₅₃	—	4	1	no	—
26	C₂²⋊C₈×C₅₃	—	4	1	no	—
27	#97×C₅₃	—	4	1	no	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	M₁₆×C₅₃	—	8	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₅₃	—	2	2	no	—
49	C₁₆×C₅₃	M₁₆	1	4	yes	complemented
50	C₂³×C₅₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	—	8	2	no	—
65	C₈×C₅₃	#108	1	2	yes	—
66	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
67	#12019	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	#365	1	2	yes	—
71	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	#423	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
79	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
80	#1568	C₂×C₅₃	1	1	yes	characteristic
81	#1566	C₂×C₅₃	1	1	yes	characteristic
82	#1368	C₂×C₅₃	1	4	yes	complemented
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	#1566	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
86	C₂×C₅₃	#2155	1	1	yes	characteristic
87	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
88	#432	—	2	1	no	—
89	#423	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
91	#346	C₄×C₅₃	1	1	yes	characteristic
92	#346	C₄×C₅₃	1	1	yes	characteristic
93	#346	C₂²×C₅₃	1	2	yes	—
94	#335	—	2	1	no	—
95	#334	—	2	1	no	—
96	#334	C₂²×C₅₃	1	2	yes	—
97	#334	C₂²×C₅₃	1	2	yes	—
98	C₅₃	#12019	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×M₁₆	—	4	1	no	—
100	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
101	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
102	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
105	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	4	1	no	—
109	C₂²⋊C₈	—	4	1	no	—
110	#97	—	4	1	no	—
111	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
112	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
113	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
114	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
116	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
117	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
118	M₁₆	—	8	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
127	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
128	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
129	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
130	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	M₁₆×C₅₃	1	4	yes	complemented
133	C₂³	#106×C₅₃	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
139	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
140	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₈	#108×C₅₃	1	2	yes	—
149	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₅₃×#365	1	2	yes	—
153	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	C₅₃×#423	1	1	yes	characteristic
160	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₅₃×#1566	1	1	yes	characteristic
163	C₂	C₂×C₅₃×#365	1	1	yes	characteristic
164	C₂	C₅₃×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1368	C₂	semidirect
49. C₁₆×C₅₃	M₁₆	semidirect
67. #12019	C₅₃	direct (N × H)
82. #1368	C₂×C₅₃	semidirect
98. C₅₃	#12019	direct (N × H)
132. C₁₆	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#346	2
C₅₃×#1568	2
C₅₃×#1566	2
C₅₃×#1368	2
#12019	53

Subgroups — C53×#12019 #27219531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12019 #27219531