Subgroups — C₅₃×#7159 #27214671

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 168 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×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
5	C₅₃×#431	—	2	1	no	—
6	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#346	—	2	1	no	—
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×#334	C₄	1	2	yes	—
14	C₂²×C₈×C₅₃	—	4	1	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	M₃₂×C₅₃	—	2	2	no	—
19	M₃₂×C₅₃	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₅₃	—	2	2	no	—
21	C₄⋊C₈×C₅₃	—	2	2	no	—
22	C₄⋊C₈×C₅₃	—	2	2	no	—
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	C₂²⋊C₈×C₅₃	—	4	1	no	—
25	C₄×C₈×C₅₃	—	4	1	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	8	2	no	—
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
43	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	D₈	1	1	yes	characteristic
46	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₁₆×C₅₃	—	2	4	no	—
48	C₂³×C₅₃	#106	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	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
65	C₈×C₅₃	#97	1	1	yes	characteristic
66	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
67	#7159	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	#334	1	2	yes	—
71	C₂²×C₅₃	#340	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#335	1	2	yes	—
76	C₄×C₅₃	#336	1	1	yes	characteristic
77	C₄×C₅₃	#341	1	1	yes	characteristic
78	C₄×C₅₃	#361	1	1	yes	characteristic
79	C₄×C₅₃	#362	1	1	yes	characteristic
80	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
81	#1564	C₂×C₅₃	1	1	yes	characteristic
82	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	#1277	1	1	yes	characteristic
85	C₂×C₅₃	#1387	1	1	yes	characteristic
86	C₂×C₅₃	#1389	1	1	yes	characteristic
87	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
88	#431	—	2	1	no	—
89	C₂×C₄⋊C₈	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
91	#346	—	2	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	#335	—	2	1	no	—
94	#335	—	2	1	no	—
95	#335	C₄×C₅₃	1	2	yes	—
96	#334	C₄×C₅₃	1	2	yes	—
97	C₅₃	#7159	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	C₈×C₅₃	1	4	yes	complemented
104	C₂×C₁₆	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₂²⋊C₈	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
111	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
112	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
113	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
115	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
116	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
126	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
127	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
128	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
129	C₄²	D₈×C₅₃	1	1	yes	characteristic
130	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
131	C₁₆	—	2	4	no	—
132	C₂³	#106×C₅₃	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
137	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
138	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
139	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
140	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
141	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
142	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
143	C₂×C₄	#108×C₅₃	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
149	C₈	#97×C₅₃	1	1	yes	characteristic
150	C₈	C₂²⋊C₈×C₅₃	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	C₅₃×#334	1	2	yes	—
154	C₂²	C₅₃×#340	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	C₅₃×#335	1	2	yes	—
159	C₄	C₅₃×#336	1	1	yes	characteristic
160	C₄	C₅₃×#341	1	1	yes	characteristic
161	C₄	C₅₃×#361	1	1	yes	characteristic
162	C₄	C₅₃×#362	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	C₅₃×#1277	1	1	yes	characteristic
165	C₂	C₅₃×#1387	1	1	yes	characteristic
166	C₂	C₅₃×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₈	C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#106
LCS	G ▷ C₈ ▷ 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
19. M₃₂×C₅₃	C₈	semidirect
67. #7159	C₅₃	direct (N × H)
97. C₅₃	#7159	direct (N × H)
103. M₃₂	C₈×C₅₃	semidirect

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×#1564	2
C₂×C₅₃×#335	2
#7159	53

Subgroups — C53×#7159 #27214671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7159 #27214671