Subgroups — C₂×C₅₃×#1312 #27218912

Order 13568.

350 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×C₄×C₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1312	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₅₃	C₄	1	4	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₈	1	2	yes	—
13	C₂²×C₈×C₅₃	C₈	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	—	2	8	no	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
18	C₄×C₈×C₅₃	C₈	1	2	yes	—
19	C₄×C₈×C₅₃	C₈	1	2	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₁₆	1	8	yes	complemented
30	C₂×C₈×C₅₃	C₄²	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
35	C₄²×C₅₃	C₄²	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₅₃	—	2	8	no	—
40	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#97	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₃	—	2	4	no	—
51	C₈×C₅₃	#97	1	4	yes	—
52	C₈×C₅₃	C₂×C₁₆	1	8	yes	complemented
53	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
54	C₂×#1312	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₅₃	#322	1	2	yes	—
56	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
57	C₂²×C₅₃	#346	1	2	yes	—
58	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₄×C₅₃	#322	1	2	yes	—
60	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
61	C₄×C₅₃	#346	1	2	yes	—
62	C₄×C₅₃	C₂×#97	1	2	yes	—
63	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
64	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
65	C₂×C₄×C₁₆	C₂×C₅₃	1	2	yes	—
66	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
67	#1312	C₂×C₅₃	1	4	yes	direct product factor
68	C₂×C₅₃	#1312	1	4	yes	direct product factor
69	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
70	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
72	C₂²×C₁₆	C₄×C₅₃	1	4	yes	—
73	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
74	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
76	C₄×C₁₆	C₄×C₅₃	1	4	yes	—
77	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
78	C₈²	C₄×C₅₃	1	2	yes	—
79	C₈²	C₂²×C₅₃	1	2	yes	—
80	C₅₃	C₂×#1312	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₈	C₈×C₅₃	1	2	yes	—
82	C₂²×C₈	C₈×C₅₃	1	2	yes	—
83	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
84	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
85	C₂×C₁₆	—	2	8	no	—
86	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	—
87	C₄×C₈	C₈×C₅₃	1	2	yes	—
88	C₄×C₈	C₈×C₅₃	1	2	yes	—
89	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
93	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₁₆×C₅₃	1	8	yes	complemented
99	C₂×C₈	C₄²×C₅₃	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
101	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
102	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
103	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
104	C₄²	C₄²×C₅₃	1	1	yes	characteristic
105	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
106	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
107	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
108	C₁₆	—	2	8	no	—
109	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
110	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
111	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
112	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
113	C₂×C₄	#97×C₅₃	1	4	yes	—
114	C₂×C₄	C₂×C₁₆×C₅₃	1	4	yes	—
115	C₂×C₄	M₃₂×C₅₃	1	4	yes	—
116	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	#97×C₅₃	1	4	yes	—
121	C₈	C₂×C₁₆×C₅₃	1	8	yes	complemented
122	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
123	C₂²	C₅₃×#322	1	2	yes	—
124	C₂²	C₄×C₁₆×C₅₃	1	2	yes	—
125	C₂²	C₅₃×#346	1	2	yes	—
126	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
127	C₄	C₅₃×#322	1	2	yes	—
128	C₄	C₄×C₁₆×C₅₃	1	2	yes	—
129	C₄	C₅₃×#346	1	2	yes	—
130	C₄	C₂×#97×C₅₃	1	2	yes	—
131	C₄	C₂²×C₁₆×C₅₃	1	2	yes	—
132	C₄	C₂×M₃₂×C₅₃	1	2	yes	—
133	C₂	C₅₃×#1312	1	4	yes	direct product factor
134	C₂	C₂×C₅₃×#322	1	1	yes	characteristic
135	C₂	C₂×C₄×C₁₆×C₅₃	1	1	yes	characteristic
136	C₂	C₂×C₅₃×#346	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1312	C₂	direct (N × H)
29. C₂×C₈×C₅₃	C₁₆	semidirect
52. C₈×C₅₃	C₂×C₁₆	semidirect
54. C₂×#1312	C₅₃	direct (N × H)
67. #1312	C₂×C₅₃	direct (N × H)
68. C₂×C₅₃	#1312	direct (N × H)
80. C₅₃	C₂×#1312	direct (N × H)
98. C₂×C₈	C₁₆×C₅₃	semidirect
121. C₈	C₂×C₁₆×C₅₃	semidirect
133. C₂	C₅₃×#1312	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×C₄×C₁₆×C₅₃	2
C₂×C₈²×C₅₃	2
C₅₃×#1312	2
C₂×#1312	53

Subgroups — C2×C53×#1312 #27218912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1312 #27218912