Subgroups — C₅₃×#12022 #27219534

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	2	yes	maximal
3	C₅₃×#1371	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
5	C₅₃×#423	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#335	—	2	2	no	—
11	C₅₃×#335	C₂²	1	4	yes	—
12	C₂×M₁₆×C₅₃	—	4	2	no	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	—	2	4	no	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₄⋊C₈×C₅₃	—	2	4	no	—
20	C₄⋊C₈×C₅₃	—	4	2	no	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
28	M₁₆×C₅₃	—	8	4	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	4	4	no	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	D₈	1	1	yes	characteristic
38	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₅₃	—	2	2	no	—
41	C₁₆×C₅₃	M₁₆	1	4	yes	complemented
42	C₂³×C₅₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	8	4	no	—
57	C₈×C₅₃	#108	1	2	yes	—
58	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
59	#12022	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	#366	1	2	yes	—
63	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	#366	1	2	yes	—
67	C₄×C₅₃	#368	1	2	yes	—
68	C₄×C₅₃	#423	1	1	yes	characteristic
69	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
70	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
73	#1567	C₂×C₅₃	1	2	yes	—
74	#1371	C₂×C₅₃	1	4	yes	complemented
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	#1567	1	1	yes	characteristic
77	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
79	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
80	#423	—	2	2	no	—
81	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
82	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
83	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
84	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
85	#335	—	2	2	no	—
86	#335	C₂²×C₅₃	1	4	yes	—
87	C₅₃	#12022	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×M₁₆	—	4	2	no	—
89	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
91	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
94	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
98	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
99	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
100	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
101	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
102	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
103	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
104	M₁₆	—	8	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
109	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
112	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
113	C₄²	D₈×C₅₃	1	1	yes	characteristic
114	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
115	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	M₁₆×C₅₃	1	4	yes	complemented
118	C₂³	#107×C₅₃	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
123	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
124	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
125	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
126	C₂×C₄	Q₃₂×C₅₃	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	8	4	no	—
133	C₈	#108×C₅₃	1	2	yes	—
134	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	C₅₃×#366	1	2	yes	—
138	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	C₅₃×#366	1	2	yes	—
142	C₄	C₅₃×#368	1	2	yes	—
143	C₄	C₅₃×#423	1	1	yes	characteristic
144	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
145	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
146	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	C₅₃×#1567	1	1	yes	characteristic
149	C₂	C₂×C₅₃×#366	1	1	yes	characteristic
150	C₂	C₂×C₅₃×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
3. C₅₃×#1371	C₂	semidirect
41. C₁₆×C₅₃	M₁₆	semidirect
59. #12022	C₅₃	direct (N × H)
74. #1371	C₂×C₅₃	semidirect
87. C₅₃	#12022	direct (N × H)
117. 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×C₄×C₁₆×C₅₃	2
C₅₃×#1567	2
C₅₃×#1371	2
#12022	53

Subgroups — C53×#12022 #27219534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12022 #27219534