Subgroups — C₅₃×#13638 #27221150

Order 13568.

182 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1426	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#108×C₅₃	—	2	1	no	—
7	C₅₃×#427	—	2	1	no	—
8	C₅₃×M₆₄	C₄	1	2	yes	complemented
9	C₅₃×M₆₄	C₂²	1	2	yes	—
10	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
11	C₅₃×#368	—	2	1	no	—
12	C₅₃×#368	C₂²	1	2	yes	—
13	C₅₃×#367	C₂²	1	2	yes	—
14	C₅₃×#366	—	2	1	no	—
15	C₂×M₁₆×C₅₃	—	4	1	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	#117×C₅₃	—	4	1	no	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
24	#108×C₅₃	—	2	2	no	—
25	#108×C₅₃	—	4	1	no	—
26	#107×C₅₃	—	2	2	no	—
27	#106×C₅₃	—	4	1	no	—
28	C₃₂×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
30	M₁₆×C₅₃	—	4	2	no	—
31	M₁₆×C₅₃	—	8	1	no	—
32	C₂×C₈×C₅₃	—	8	1	no	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₂²⋊C₄×C₅₃	—	8	1	no	—
41	C₄²×C₅₃	—	8	1	no	—
42	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₅₃	SD₃₂	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	8	2	no	—
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
50	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
51	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
52	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	C₈×C₅₃	—	8	2	no	—
55	C₈×C₅₃	#107	1	1	yes	characteristic
56	C₈×C₅₃	#107	1	1	yes	characteristic
57	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	#13638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	#367	1	1	yes	characteristic
62	C₂²×C₅₃	#367	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
64	C₄×C₅₃	—	16	2	no	—
65	C₄×C₅₃	#367	1	1	yes	characteristic
66	C₄×C₅₃	#367	1	1	yes	characteristic
67	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
68	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
69	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
70	C₂×#368	C₂×C₅₃	1	1	yes	characteristic
71	#2158	C₂×C₅₃	1	1	yes	characteristic
72	#1426	C₂×C₅₃	1	4	yes	complemented
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	—	2	1	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
77	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
78	C₂×#108	—	2	1	no	—
79	#427	—	2	1	no	—
80	M₆₄	C₄×C₅₃	1	2	yes	complemented
81	M₆₄	C₂²×C₅₃	1	2	yes	—
82	C₂×C₃₂	C₄×C₅₃	1	2	yes	complemented
83	#368	—	2	1	no	—
84	#368	C₂²×C₅₃	1	2	yes	—
85	#367	C₂²×C₅₃	1	2	yes	—
86	#366	—	2	1	no	—
87	C₅₃	#13638	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×M₁₆	—	4	1	no	—
89	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
90	#117	—	4	1	no	—
91	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
92	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
93	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
94	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
95	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
96	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
97	#108	—	2	2	no	—
98	#108	—	4	1	no	—
99	#107	—	2	2	no	—
100	#106	—	4	1	no	—
101	C₃₂	C₂×C₄×C₅₃	1	4	yes	—
102	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
103	M₁₆	—	4	2	no	—
104	M₁₆	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
108	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
109	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
110	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
112	C₄⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	8	1	no	—
114	C₄²	—	8	1	no	—
115	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
116	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
117	C₁₆	C₂²×C₄×C₅₃	1	1	yes	characteristic
118	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
119	C₂³	SD₃₂×C₅₃	1	1	yes	characteristic
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
122	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
123	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
124	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
125	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
127	C₈	—	8	2	no	—
128	C₈	#107×C₅₃	1	1	yes	characteristic
129	C₈	#107×C₅₃	1	1	yes	characteristic
130	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
131	C₈	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
132	C₂²	—	4	1	no	—
133	C₂²	C₅₃×#367	1	1	yes	characteristic
134	C₂²	C₅₃×#367	1	1	yes	characteristic
135	C₂²	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
136	C₄	—	16	2	no	—
137	C₄	C₅₃×#367	1	1	yes	characteristic
138	C₄	C₅₃×#367	1	1	yes	characteristic
139	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
140	C₄	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	C₂×C₅₃×#367	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#367
commutator	C₁₆	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₂×#367
prime radical	C₂³×C₅₃	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1426	C₂	semidirect
8. C₅₃×M₆₄	C₄	semidirect
10. C₂×C₃₂×C₅₃	C₄	semidirect
59. #13638	C₅₃	direct (N × H)
72. #1426	C₂×C₅₃	semidirect
80. M₆₄	C₄×C₅₃	semidirect
82. C₂×C₃₂	C₄×C₅₃	semidirect
87. C₅₃	#13638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×M₆₄	2
C₂×C₅₃×#368	2
C₅₃×#2158	2
C₅₃×#1426	2
#13638	53

Subgroups — C53×#13638 #27221150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13638 #27221150