Subgroups — C₅₃×#7151 #27214663

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₅₃	—	2	2	no	—
4	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₄⋊C₁₆	—	2	2	no	—
8	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₈²×C₅₃	—	2	2	no	—
10	C₂²×C₈×C₅₃	—	2	1	no	—
11	C₂²×C₈×C₅₃	C₈	1	2	yes	—
12	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	—	2	4	no	—
16	C₂×C₁₆×C₅₃	—	4	2	no	—
17	C₂×C₁₆×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₁₆	1	4	yes	complemented
34	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₄²×C₅₃	C₄²	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
43	C₁₆×C₅₃	—	4	4	no	—
44	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
58	C₈×C₅₃	—	2	4	no	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	—	2	4	no	—
61	C₈×C₅₃	#102	1	2	yes	—
62	C₈×C₅₃	#103	1	2	yes	—
63	#7151	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₅₃	#334	1	1	yes	characteristic
65	C₂²×C₅₃	#335	1	1	yes	characteristic
66	C₂²×C₅₃	#336	1	1	yes	characteristic
67	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
68	C₂²×C₅₃	#364	1	2	yes	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	#334	1	1	yes	characteristic
72	C₄×C₅₃	#335	1	1	yes	characteristic
73	C₄×C₅₃	#340	1	1	yes	characteristic
74	C₄×C₅₃	#341	1	1	yes	characteristic
75	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
76	C₄×C₅₃	#346	1	1	yes	characteristic
77	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
78	C₂×C₄⋊C₁₆	C₂×C₅₃	1	2	yes	—
79	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
80	C₂×C₅₃	#1277	1	1	yes	characteristic
81	C₂×C₅₃	#1367	1	2	yes	—
82	C₂×C₅₃	#1370	1	2	yes	—
83	C₂×C₅₃	#1374	1	1	yes	characteristic
84	C₂×C₅₃	#1376	1	1	yes	characteristic
85	C₂²×C₁₆	—	2	2	no	—
86	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	2	no	—
90	C₄⋊C₁₆	C₄×C₅₃	1	4	yes	—
91	C₈²	—	2	2	no	—
92	C₅₃	#7151	1	1	yes	characteristic, direct product factor, Sylow
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₈×C₅₃	1	2	yes	—
95	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
96	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	4	2	no	—
100	C₂×C₁₆	—	4	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	C₈×C₅₃	1	2	yes	—
105	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
106	C₄×C₈	D₄×C₅₃	1	2	yes	—
107	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
109	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₁₆×C₅₃	1	4	yes	complemented
117	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
118	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
119	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
120	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
121	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
122	C₄²	C₄²×C₅₃	1	1	yes	characteristic
123	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
124	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
125	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
126	C₁₆	—	4	4	no	—
127	C₂³	C₄⋊C₈×C₅₃	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
130	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
131	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
135	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
136	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
137	C₂×C₄	#108×C₅₃	1	1	yes	characteristic
138	C₂×C₄	#108×C₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
140	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	#102×C₅₃	1	2	yes	—
145	C₈	#103×C₅₃	1	2	yes	—
146	C₂²	C₅₃×#334	1	1	yes	characteristic
147	C₂²	C₅₃×#335	1	1	yes	characteristic
148	C₂²	C₅₃×#336	1	1	yes	characteristic
149	C₂²	C₅₃×C₄⋊C₁₆	1	2	yes	—
150	C₂²	C₅₃×#364	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	C₅₃×#334	1	1	yes	characteristic
154	C₄	C₅₃×#335	1	1	yes	characteristic
155	C₄	C₅₃×#340	1	1	yes	characteristic
156	C₄	C₅₃×#341	1	1	yes	characteristic
157	C₄	C₄×C₁₆×C₅₃	1	1	yes	characteristic
158	C₄	C₅₃×#346	1	1	yes	characteristic
159	C₄	C₅₃×C₂²⋊C₁₆	1	2	yes	—
160	C₂	C₅₃×#1277	1	1	yes	characteristic
161	C₂	C₅₃×#1367	1	2	yes	—
162	C₂	C₅₃×#1370	1	2	yes	—
163	C₂	C₅₃×#1374	1	1	yes	characteristic
164	C₂	C₅₃×#1376	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₅₃	D₄
commutator	C₄	C₄×C₁₆×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle, prime radical	C₂³×C₅₃	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₂×C₈×C₅₃	C₁₆	semidirect
63. #7151	C₅₃	direct (N × H)
92. C₅₃	#7151	direct (N × H)
116. C₂×C₈	C₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₈²×C₅₃	2
#7151	53

Subgroups — C53×#7151 #27214663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7151 #27214663