Subgroups — C₅₃×#12556 #27220068

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	2	yes	maximal
3	C₅₃×#1380	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×#117×C₅₃	—	2	2	no	—
6	C₂²×C₁₆×C₅₃	—	2	4	no	—
7	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	—	2	4	no	—
10	C₅₃×#427	C₄	1	8	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	4	yes	—
12	C₂×#106×C₅₃	—	2	2	no	—
13	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	D₄	1	4	yes	—
17	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	#117×C₅₃	—	2	8	no	—
21	#117×C₅₃	—	4	4	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
23	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
24	C₂×C₄²×C₅₃	—	4	2	no	—
25	C₂×C₁₆×C₅₃	—	2	8	no	—
26	C₂×C₁₆×C₅₃	—	2	4	no	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	#107×C₅₃	—	2	4	no	—
29	#107×C₅₃	C₂×C₄	1	8	yes	—
30	#106×C₅₃	—	2	8	no	—
31	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	4	4	no	—
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
37	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄²	1	4	yes	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄×C₅₃	—	2	8	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₂²⋊C₄×C₅₃	—	4	8	no	—
51	C₄²×C₅₃	—	4	8	no	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	#102	1	4	yes	—
55	C₂³×C₅₃	#107	1	2	yes	—
56	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	4	8	no	—
58	C₂×C₄×C₅₃	—	8	8	no	—
59	C₂×C₄×C₅₃	#95	1	4	yes	—
60	C₂×C₄×C₅₃	#102	1	4	yes	—
61	C₂×C₄×C₅₃	#103	1	8	yes	—
62	C₂×C₄×C₅₃	#106	1	4	yes	—
63	C₂×C₄×C₅₃	#107	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₅₃	#95	1	4	yes	—
69	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	#12556	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	—	2	8	no	—
74	C₂²×C₅₃	—	2	4	no	—
75	C₂²×C₅₃	#340	1	4	yes	—
76	C₂²×C₅₃	C₂×#102	1	2	yes	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#508	1	4	yes	—
79	C₄×C₅₃	—	8	8	no	—
80	C₄×C₅₃	#340	1	4	yes	—
81	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
82	C₄×C₅₃	C₂×#103	1	2	yes	—
83	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
84	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic
85	C₂×#427	C₂×C₅₃	1	2	yes	—
86	#1380	C₂×C₅₃	1	4	yes	complemented
87	C₂×C₅₃	—	2	4	no	—
88	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
89	C₂×C₅₃	#2138	1	4	yes	—
90	C₂×C₅₃	#2158	1	2	yes	—
91	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
92	C₂×#117	—	2	2	no	—
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	C₄×C₅₃	1	4	yes	complemented
95	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
96	#427	—	2	4	no	—
97	#427	C₄×C₅₃	1	8	yes	complemented
98	C₂×#107	C₂²×C₅₃	1	4	yes	—
99	C₂×#106	—	2	2	no	—
100	C₅₃	#12556	1	1	yes	characteristic, direct product factor, Sylow
101	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
104	C₂²×C₈	D₄×C₅₃	1	4	yes	—
105	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
106	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
107	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
108	#117	—	2	8	no	—
109	#117	—	4	4	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₂²⋊C₄	—	4	2	no	—
112	C₂×C₄²	—	4	2	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	—	2	8	no	—
115	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
116	#107	—	2	4	no	—
117	#107	C₂×C₄×C₅₃	1	8	yes	—
118	#106	—	2	8	no	—
119	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
122	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
123	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
124	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
125	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
126	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
127	C₂×C₈	C₄²×C₅₃	1	4	yes	—
128	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
129	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
130	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
131	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
132	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
134	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
135	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	8	no	—
139	C₄²	—	4	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	#102×C₅₃	1	4	yes	—
143	C₂³	#107×C₅₃	1	2	yes	—
144	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	8	8	no	—
147	C₂×C₄	#95×C₅₃	1	4	yes	—
148	C₂×C₄	#102×C₅₃	1	4	yes	—
149	C₂×C₄	#103×C₅₃	1	8	yes	—
150	C₂×C₄	#106×C₅₃	1	4	yes	—
151	C₂×C₄	#107×C₅₃	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
155	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
156	C₈	#95×C₅₃	1	4	yes	—
157	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
159	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
160	C₂²	—	2	8	no	—
161	C₂²	—	2	4	no	—
162	C₂²	C₅₃×#340	1	4	yes	—
163	C₂²	C₂×#102×C₅₃	1	2	yes	—
164	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
165	C₂²	C₅₃×#508	1	4	yes	—
166	C₄	—	8	8	no	—
167	C₄	C₅₃×#340	1	4	yes	—
168	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
169	C₄	C₂×#103×C₅₃	1	2	yes	—
170	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×C₅₃×#340	1	1	yes	characteristic
173	C₂	C₅₃×#2138	1	4	yes	—
174	C₂	C₅₃×#2158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	D₈
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1380	C₂	semidirect
7. C₂²×C₁₆×C₅₃	C₄	semidirect
10. C₅₃×#427	C₄	semidirect
72. #12556	C₅₃	direct (N × H)
86. #1380	C₂×C₅₃	semidirect
94. C₂²×C₁₆	C₄×C₅₃	semidirect
97. #427	C₄×C₅₃	semidirect
100. C₅₃	#12556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₅₃	2
C₂×C₅₃×#427	2
C₅₃×#1380	2
#12556	53

Subgroups — C53×#12556 #27220068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12556 #27220068