Subgroups — C₅₃×#12904 #27220416

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#365	C₂	1	4	yes	maximal
2	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
6	C₂×#108×C₅₃	—	2	1	no	—
7	C₂×#108×C₅₃	C₂²	1	2	yes	—
8	C₂×#107×C₅₃	—	2	1	no	—
9	C₂×#106×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#382	—	2	1	no	—
13	C₅₃×#365	—	2	8	no	—
14	C₅₃×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	M₃₂×C₅₃	D₄	1	4	yes	—
22	M₃₂×C₅₃	Q₈	1	4	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
24	#108×C₅₃	—	2	4	no	—
25	#108×C₅₃	—	4	2	no	—
26	#107×C₅₃	—	4	2	no	—
27	#106×C₅₃	—	2	4	no	—
28	C₄⋊C₈×C₅₃	—	4	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	#95×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₅₃	—	4	2	no	—
38	M₁₆×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	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	SD₁₆	1	2	yes	—
53	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
54	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	—	8	4	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	#106	1	2	yes	—
59	C₂×C₄×C₅₃	#106	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₅₃	—	8	2	no	—
67	C₈×C₅₃	#106	1	2	yes	—
68	C₈×C₅₃	#106	1	2	yes	—
69	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
71	#12904	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
74	C₂²×C₅₃	#492	1	1	yes	characteristic
75	C₂²×C₅₃	#498	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	16	2	no	—
79	C₄×C₅₃	#384	1	1	yes	characteristic
80	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
81	C₄×C₅₃	#492	1	1	yes	characteristic
82	C₄×C₅₃	#498	1	1	yes	characteristic
83	C₂×#365	C₂×C₅₃	1	4	yes	—
84	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
85	#1848	C₂×C₅₃	1	1	yes	characteristic
86	#1846	C₂×C₅₃	1	1	yes	characteristic
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1844	1	1	yes	characteristic
90	C₂×C₅₃	#2155	1	2	yes	—
91	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂²×C₅₃	1	2	yes	—
94	C₂×#107	—	2	1	no	—
95	C₂×#106	C₂²×C₅₃	1	2	yes	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
98	#382	—	2	1	no	—
99	#365	—	2	8	no	—
100	#346	C₄×C₅₃	1	2	yes	complemented
101	C₄×C₁₆	C₄×C₅₃	1	2	yes	complemented
102	C₅₃	#12904	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×M₁₆	—	2	2	no	—
104	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
108	M₃₂	D₄×C₅₃	1	4	yes	—
109	M₃₂	Q₈×C₅₃	1	4	yes	—
110	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
111	#108	—	2	4	no	—
112	#108	—	4	2	no	—
113	#107	—	4	2	no	—
114	#106	—	2	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
119	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
120	#95	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
123	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
130	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
131	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
133	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
134	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₄²	SD₁₆×C₅₃	1	2	yes	—
140	C₁₆	C₄⋊C₄×C₅₃	1	8	yes	—
141	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#106×C₅₃	1	2	yes	—
146	C₂×C₄	#106×C₅₃	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#106×C₅₃	1	2	yes	—
155	C₈	#106×C₅₃	1	2	yes	—
156	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
157	C₈	C₄⋊Q₈×C₅₃	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
160	C₂²	C₅₃×#492	1	1	yes	characteristic
161	C₂²	C₅₃×#498	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	C₅₃×#384	1	1	yes	characteristic
166	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
167	C₄	C₅₃×#492	1	1	yes	characteristic
168	C₄	C₅₃×#498	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₅₃×#1844	1	1	yes	characteristic
172	C₂	C₅₃×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×SD₁₆
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
14. C₅₃×#346	C₄	semidirect
15. C₄×C₁₆×C₅₃	C₄	semidirect
71. #12904	C₅₃	direct (N × H)
100. #346	C₄×C₅₃	semidirect
101. C₄×C₁₆	C₄×C₅₃	semidirect
102. C₅₃	#12904	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×#365	2
C₄×M₃₂×C₅₃	2
C₅₃×#1848	2
C₅₃×#1846	2
#12904	53

Subgroups — C53×#12904 #27220416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12904 #27220416