Subgroups — C₅₃×#12887 #27220399

Order 13568.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1383	C₂	1	4	yes	maximal
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×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	M₃₂×C₅₃	—	2	2	no	—
21	M₃₂×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
23	#108×C₅₃	—	2	4	no	—
24	#108×C₅₃	—	4	2	no	—
25	#107×C₅₃	—	4	2	no	—
26	#106×C₅₃	—	2	4	no	—
27	C₄⋊C₈×C₅₃	—	4	2	no	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	#95×C₅₃	—	4	2	no	—
33	C₂²×C₄×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	D₈	1	2	yes	—
42	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₂×C₈×C₅₃	#41	1	2	yes	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	SD₁₆	1	2	yes	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	—	8	4	no	—
56	C₂×C₄×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	#106	1	2	yes	—
58	C₂×C₄×C₅₃	#106	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₅₃	—	8	2	no	—
66	C₈×C₅₃	#107	1	2	yes	—
67	C₈×C₅₃	#107	1	2	yes	—
68	C₈×C₅₃	#117	1	2	yes	—
69	C₈×C₅₃	#125	1	2	yes	—
70	#12887	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
73	C₂²×C₅₃	#486	1	1	yes	characteristic
74	C₂²×C₅₃	#487	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	16	2	no	—
78	C₄×C₅₃	#382	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₄×C₅₃	#486	1	1	yes	characteristic
81	C₄×C₅₃	#487	1	1	yes	characteristic
82	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
83	#1848	C₂×C₅₃	1	1	yes	characteristic
84	#1846	C₂×C₅₃	1	1	yes	characteristic
85	#1383	C₂×C₅₃	1	4	yes	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#1834	1	1	yes	characteristic
89	C₂×C₅₃	#2155	1	2	yes	—
90	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
91	C₂×#108	—	2	1	no	—
92	C₂×#108	C₂²×C₅₃	1	2	yes	—
93	C₂×#107	—	2	1	no	—
94	C₂×#106	C₂²×C₅₃	1	2	yes	—
95	#423	—	2	1	no	—
96	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
97	#382	—	2	1	no	—
98	#346	C₄×C₅₃	1	2	yes	complemented
99	C₄×C₁₆	C₄×C₅₃	1	2	yes	complemented
100	C₅₃	#12887	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
109	#108	—	2	4	no	—
110	#108	—	4	2	no	—
111	#107	—	4	2	no	—
112	#106	—	2	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
118	#95	—	4	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
121	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
122	M₁₆	—	4	2	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
127	C₂×C₈	D₈×C₅₃	1	2	yes	—
128	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
131	C₂×C₈	#41×C₅₃	1	2	yes	—
132	C₂×C₈	#41×C₅₃	1	2	yes	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
136	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
137	C₄²	SD₁₆×C₅₃	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	#106×C₅₃	1	2	yes	—
144	C₂×C₄	#106×C₅₃	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
146	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
147	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
151	C₈	—	8	2	no	—
152	C₈	#107×C₅₃	1	2	yes	—
153	C₈	#107×C₅₃	1	2	yes	—
154	C₈	#117×C₅₃	1	2	yes	—
155	C₈	#125×C₅₃	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
158	C₂²	C₅₃×#486	1	1	yes	characteristic
159	C₂²	C₅₃×#487	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	16	2	no	—
163	C₄	C₅₃×#382	1	1	yes	characteristic
164	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
165	C₄	C₅₃×#486	1	1	yes	characteristic
166	C₄	C₅₃×#487	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	C₅₃×#1834	1	1	yes	characteristic
170	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
13. C₅₃×#346	C₄	semidirect
14. C₄×C₁₆×C₅₃	C₄	semidirect
70. #12887	C₅₃	direct (N × H)
98. #346	C₄×C₅₃	semidirect
99. C₄×C₁₆	C₄×C₅₃	semidirect
100. C₅₃	#12887	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₅₃×#1848	2
C₅₃×#1846	2
C₅₃×#1383	2
#12887	53

Subgroups — C53×#12887 #27220399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12887 #27220399