Subgroups — C₅₃×#2905 #20271842

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#424	C₂	1	8	yes	maximal, complemented
4	C₅₃×#423	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₅₃	—	2	8	no	—
7	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
8	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₈×C₅₃	C₂²	1	4	yes	—
10	#117×C₅₃	C₄	1	4	yes	—
11	#117×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
15	C₄⋊C₈×C₅₃	C₂²	1	16	yes	complemented
16	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	4	yes	—
21	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
24	M₁₆×C₅₃	—	2	8	no	—
25	M₁₆×C₅₃	—	2	8	no	—
26	C₂×C₈×C₅₃	—	2	8	no	—
27	C₂×C₈×C₅₃	C₂³	1	8	yes	—
28	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
30	C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₄²×C₅₃	C₂³	1	4	yes	—
32	C₂³×C₅₃	—	4	2	no	—
33	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	—	4	4	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₈×C₅₃	—	2	8	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	2	4	no	—
50	C₂²×C₅₃	—	4	4	no	—
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	8	no	—
55	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
59	#2905	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	—	2	4	no	—
62	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	2	yes	—
64	C₂²×M₁₆	C₂×C₅₃	1	2	yes	—
65	C₂×#117	C₂×C₅₃	1	1	yes	characteristic
66	#424	C₂×C₅₃	1	8	yes	complemented
67	#423	C₂×C₅₃	1	4	yes	complemented
68	C₅₃	#2905	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
70	C₂×M₁₆	—	2	8	no	—
71	C₂×M₁₆	C₂²×C₅₃	1	4	yes	—
72	C₂×M₁₆	C₂²×C₅₃	1	4	yes	—
73	C₂²×C₈	C₂²×C₅₃	1	4	yes	—
74	#117	C₄×C₅₃	1	4	yes	—
75	#117	C₂²×C₅₃	1	4	yes	—
76	C₂×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
77	C₂×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
78	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
79	C₄⋊C₈	C₂²×C₅₃	1	16	yes	complemented
80	C₂⁴	C₂×C₄×C₅₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄×C₅₃	1	4	yes	—
82	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
84	C₂²×C₄	D₄×C₅₃	1	4	yes	—
85	C₂²×C₄	Q₈×C₅₃	1	4	yes	—
86	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
88	M₁₆	—	2	8	no	—
89	M₁₆	—	2	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	C₂³×C₅₃	1	8	yes	—
92	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
93	C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
94	C₄²	C₂×C₄×C₅₃	1	4	yes	—
95	C₄²	C₂³×C₅₃	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊C₄×C₅₃	1	4	yes	—
98	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
99	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
102	C₂×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
103	C₂×C₄	C₂²×C₄×C₅₃	1	8	yes	—
104	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
106	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
107	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
108	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
109	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
110	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
111	C₈	—	2	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
116	C₂²	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
117	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
120	C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
121	C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
122	C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
126	C₂	C₅₃×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#424	C₂	semidirect
4. C₅₃×#423	C₂	semidirect
15. C₄⋊C₈×C₅₃	C₂²	semidirect
59. #2905	C₅₃	direct (N × H)
66. #424	C₂×C₅₃	semidirect
67. #423	C₂×C₅₃	semidirect
68. C₅₃	#2905	direct (N × H)
79. C₄⋊C₈	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₅₃	2
C₂×#117×C₅₃	2
C₅₃×#424	2
C₅₃×#423	2
#2905	53

Subgroups — C53×#2905 #20271842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2905 #20271842