Subgroups — C₂×C₅₃×#1398 #27221062

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1398	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	C₄	1	4	yes	complemented
8	C₅₃×M₆₄	C₂²	1	4	yes	—
9	C₂×C₃₂×C₅₃	C₄	1	4	yes	complemented
10	C₅₃×#346	C₄	1	2	yes	—
11	C₅₃×#346	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
16	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₈	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	—	2	1	no	—
37	C₄²×C₅₃	C₂×C₈	1	2	yes	—
38	C₁₆×C₅₃	C₄²	1	2	yes	—
39	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
40	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×#1398	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	#346	1	1	yes	characteristic
58	C₂²×C₅₃	#346	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	#346	1	1	yes	characteristic
62	C₄×C₅₃	#346	1	1	yes	characteristic
63	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
65	C₂×M₆₄	C₂×C₅₃	1	2	yes	—
66	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
67	#1398	C₂×C₅₃	1	4	yes	direct product factor
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#1398	1	2	yes	direct product factor
71	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
72	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
73	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
74	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
75	M₆₄	C₄×C₅₃	1	4	yes	complemented
76	M₆₄	C₂²×C₅₃	1	4	yes	—
77	C₂×C₃₂	C₄×C₅₃	1	4	yes	complemented
78	#346	C₄×C₅₃	1	2	yes	—
79	#346	C₂²×C₅₃	1	2	yes	—
80	C₅₃	C₂×#1398	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
83	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
84	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
85	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
90	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
91	C₄×C₈	C₈×C₅₃	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
93	C₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
99	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
101	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
103	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂×C₈×C₅₃	1	2	yes	—
107	C₁₆	C₄²×C₅₃	1	2	yes	—
108	C₁₆	C₂×C₈×C₅₃	1	4	yes	—
109	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
110	C₂³	M₃₂×C₅₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
114	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
115	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
116	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
117	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
121	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
122	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
123	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	C₅₃×#346	1	1	yes	characteristic
126	C₂²	C₅₃×#346	1	1	yes	characteristic
127	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	C₅₃×#346	1	1	yes	characteristic
130	C₄	C₅₃×#346	1	1	yes	characteristic
131	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
132	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	C₅₃×#1398	1	2	yes	direct product factor
136	C₂	C₂×C₅₃×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₅₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂²×C₅₃	#346
prime radical	C₂³×C₅₃	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1398	C₂	direct (N × H)
7. C₅₃×M₆₄	C₄	semidirect
9. C₂×C₃₂×C₅₃	C₄	semidirect
24. C₃₂×C₅₃	C₂×C₄	semidirect
55. C₂×#1398	C₅₃	direct (N × H)
67. #1398	C₂×C₅₃	direct (N × H)
70. C₂×C₅₃	#1398	direct (N × H)
75. M₆₄	C₄×C₅₃	semidirect
77. C₂×C₃₂	C₄×C₅₃	semidirect
80. C₅₃	C₂×#1398	direct (N × H)
93. C₃₂	C₂×C₄×C₅₃	semidirect
135. C₂	C₅₃×#1398	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×M₆₄	2
C₂×C₅₃×#346	2
C₅₃×#1398	2
C₂×#1398	53

Subgroups — C2×C53×#1398 #27221062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1398 #27221062