Subgroups — C₅₃×#13556 #27221068

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1400	C₂	1	4	yes	maximal, complemented
4	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
5	C₅₃×#504	—	2	2	no	—
6	C₅₃×#504	—	2	2	no	—
7	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×M₆₄	—	2	2	no	—
12	C₅₃×M₆₄	C₂²	1	4	yes	—
13	C₂×C₃₂×C₅₃	—	2	2	no	—
14	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
15	#131×C₅₃	—	2	2	no	—
16	#131×C₅₃	—	2	2	no	—
17	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	2	no	—
22	M₃₂×C₅₃	—	2	4	no	—
23	C₂×C₁₆×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
28	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
30	C₃₂×C₅₃	—	2	4	no	—
31	#41×C₅₃	—	2	2	no	—
32	#41×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
34	C₂×D₄×C₅₃	C₁₆	1	2	yes	—
35	C₂×D₄×C₅₃	C₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₅₃	—	2	2	no	—
39	M₁₆×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	—	2	2	no	—
48	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
51	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
52	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
53	Q₈×C₅₃	—	2	2	no	—
54	D₄×C₅₃	—	2	4	no	—
55	D₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
65	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	#13556	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	—	4	2	no	—
71	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
72	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
75	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
77	C₂×#504	C₂×C₅₃	1	1	yes	characteristic
78	C₂×M₆₄	C₂×C₅₃	1	2	yes	complemented
79	#1400	C₂×C₅₃	1	4	yes	complemented
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	—	4	2	no	—
83	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
84	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
85	#504	—	2	2	no	—
86	#504	—	2	2	no	—
87	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
88	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
89	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
90	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
91	M₆₄	—	2	2	no	—
92	M₆₄	C₂²×C₅₃	1	4	yes	—
93	C₂×C₃₂	—	2	2	no	—
94	C₅₃	#13556	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×#41	C₈×C₅₃	1	1	yes	characteristic
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	C₂×M₁₆	C₈×C₅₃	1	1	yes	characteristic
99	C₂×M₁₆	C₈×C₅₃	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
101	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
107	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
109	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
110	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
111	C₃₂	—	2	4	no	—
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	C₂×Q₈	C₁₆×C₅₃	1	1	yes	characteristic
115	C₂×D₄	C₁₆×C₅₃	1	2	yes	—
116	C₂×D₄	C₁₆×C₅₃	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
118	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
124	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
125	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
126	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
127	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
130	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
131	C₁₆	C₂×D₄×C₅₃	1	2	yes	—
132	C₂³	C₂×C₁₆×C₅₃	1	2	yes	—
133	C₂³	C₂×C₁₆×C₅₃	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
140	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
141	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
142	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
143	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₂²⋊C₈×C₅₃	1	2	yes	—
146	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
147	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₅₃×C₂²⋊C₁₆	1	2	yes	—
152	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₅₃×C₂²⋊C₁₆	1	2	yes	—
155	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
156	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₅₃	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×M₆₄	C₂	semidirect
3. C₅₃×#1400	C₂	semidirect
67. #13556	C₅₃	direct (N × H)
78. C₂×M₆₄	C₂×C₅₃	semidirect
79. #1400	C₂×C₅₃	semidirect
94. C₅₃	#13556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#504	2
C₂×C₅₃×M₆₄	2
C₅₃×#1400	2
#13556	53

Subgroups — C53×#13556 #27221068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13556 #27221068