Subgroups — C₅₃×#7288 #27214800

Order 13568.

202 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	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#423	—	2	1	no	—
5	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₄⋊C₁₆	—	2	1	no	—
8	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₁₆×C₅₃	—	2	1	no	—
10	C₅₃×#335	—	2	1	no	—
11	C₅₃×#335	—	2	1	no	—
12	C₅₃×#335	C₄	1	2	yes	—
13	C₅₃×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₅₃	—	4	1	no	—
15	C₂²×C₈×C₅₃	—	4	1	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	—	2	2	no	—
20	C₄⋊C₈×C₅₃	—	2	2	no	—
21	C₄⋊C₈×C₅₃	—	2	2	no	—
22	C₄⋊C₈×C₅₃	—	4	1	no	—
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
31	M₁₆×C₅₃	—	8	1	no	—
32	M₁₆×C₅₃	—	8	1	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	8	1	no	—
38	C₂×C₈×C₅₃	—	8	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₅₃	—	4	2	no	—
48	C₂³×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
60	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
61	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	8	2	no	—
66	C₈×C₅₃	—	8	2	no	—
67	C₈×C₅₃	#108	1	2	yes	—
68	#7288	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	#340	1	1	yes	characteristic
72	C₂²×C₅₃	#366	1	1	yes	characteristic
73	C₂²×C₅₃	#367	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	#341	1	1	yes	characteristic
78	C₄×C₅₃	#344	1	1	yes	characteristic
79	C₄×C₅₃	#357	1	1	yes	characteristic
80	C₄×C₅₃	#357	1	1	yes	characteristic
81	C₄×C₅₃	#358	1	1	yes	characteristic
82	C₄×C₅₃	#358	1	1	yes	characteristic
83	#2162	C₂×C₅₃	1	1	yes	characteristic
84	#1567	C₂×C₅₃	1	1	yes	characteristic
85	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#1301	1	1	yes	characteristic
89	C₂×C₅₃	#1380	1	1	yes	characteristic
90	C₂×C₅₃	#1388	1	1	yes	characteristic
91	#423	—	2	1	no	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₂²⋊C₁₆	C₄×C₅₃	1	2	yes	—
96	C₄×C₁₆	—	2	1	no	—
97	#335	—	2	1	no	—
98	#335	—	2	1	no	—
99	#335	C₄×C₅₃	1	2	yes	—
100	#335	C₄×C₅₃	1	2	yes	—
101	C₅₃	#7288	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₁₆	—	4	1	no	—
103	C₂²×C₈	—	4	1	no	—
104	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
106	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
113	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
115	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
117	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
118	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
119	M₁₆	—	8	1	no	—
120	M₁₆	—	8	1	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	—	8	1	no	—
127	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
131	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
132	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
133	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
134	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
135	C₁₆	—	4	2	no	—
136	C₂³	#107×C₅₃	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
141	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
142	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
143	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
144	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
145	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
146	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
147	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
148	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
149	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₈	#108×C₅₃	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₅₃×#340	1	1	yes	characteristic
159	C₂²	C₅₃×#366	1	1	yes	characteristic
160	C₂²	C₅₃×#367	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₅₃×#341	1	1	yes	characteristic
165	C₄	C₅₃×#344	1	1	yes	characteristic
166	C₄	C₅₃×#357	1	1	yes	characteristic
167	C₄	C₅₃×#357	1	1	yes	characteristic
168	C₄	C₅₃×#358	1	1	yes	characteristic
169	C₄	C₅₃×#358	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₅₃×#1301	1	1	yes	characteristic
173	C₂	C₅₃×#1380	1	1	yes	characteristic
174	C₂	C₅₃×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₂×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
68. #7288	C₅₃	direct (N × H)
101. C₅₃	#7288	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2162	2
C₅₃×#1567	2
C₂×C₅₃×#335	2
#7288	53

Subgroups — C53×#7288 #27214800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7288 #27214800