Subgroups — C₂²×C₅₃×#367 #27240918

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₅₃×#367	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
6	C₂²×C₁₆×C₅₃	C₄	1	8	yes	complemented
7	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
8	C₂×#107×C₅₃	—	2	6	no	—
9	C₂×#107×C₅₃	C₂²	1	12	yes	—
10	C₅₃×#367	C₂²	1	16	yes	direct product factor
11	C₂³×C₄×C₅₃	—	4	2	no	—
12	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂²×C₈×C₅₃	D₄	1	3	yes	—
15	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
16	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
17	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
18	C₂×C₄⋊C₄×C₅₃	—	4	6	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	24	yes	complemented
20	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
21	#107×C₅₃	—	2	12	no	—
22	#107×C₅₃	C₂³	1	8	yes	—
23	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	4	12	no	—
25	C₂²×C₄×C₅₃	—	8	8	no	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
27	C₂²×C₄×C₅₃	D₈	1	3	yes	—
28	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
32	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
33	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
34	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	8	no	—
36	C₄⋊C₄×C₅₃	—	4	12	no	—
37	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
38	C₂³×C₅₃	#107	1	4	yes	—
39	C₂³×C₅₃	SD₃₂	1	8	yes	—
40	C₂³×C₅₃	C₂×D₈	1	3	yes	—
41	C₂×C₄×C₅₃	—	4	8	no	—
42	C₂×C₄×C₅₃	—	8	24	no	—
43	C₂×C₄×C₅₃	#107	1	12	yes	—
44	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
45	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
46	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
47	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
48	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
49	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×#367	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	#367	1	16	yes	direct product factor
53	C₂²×C₅₃	C₂×#107	1	6	yes	—
54	C₂²×C₅₃	C₂×SD₃₂	1	12	yes	—
55	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
56	C₄×C₅₃	—	8	8	no	—
57	C₄×C₅₃	C₂×#107	1	6	yes	—
58	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
60	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic
61	C₂²×#107	C₂×C₅₃	1	2	yes	—
62	C₂×#367	C₂×C₅₃	1	12	yes	direct product factor
63	C₂×C₅₃	C₂×#367	1	12	yes	direct product factor
64	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
65	C₂×C₅₃	C₂²×SD₃₂	1	2	yes	—
66	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
67	C₂²×C₄⋊C₄	—	2	2	no	—
68	C₂²×C₁₆	C₄×C₅₃	1	8	yes	complemented
69	C₂²×C₁₆	C₂²×C₅₃	1	6	yes	—
70	C₂×#107	—	2	6	no	—
71	C₂×#107	C₂²×C₅₃	1	12	yes	—
72	#367	C₂²×C₅₃	1	16	yes	direct product factor
73	C₅₃	C₂²×#367	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₄	—	4	2	no	—
75	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
76	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
77	C₂²×C₈	D₄×C₅₃	1	3	yes	—
78	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
79	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
80	C₂×C₄⋊C₄	—	2	12	no	—
81	C₂×C₄⋊C₄	—	4	6	no	—
82	C₂×C₁₆	C₂×C₄×C₅₃	1	24	yes	complemented
83	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
84	#107	—	2	12	no	—
85	#107	C₂³×C₅₃	1	8	yes	—
86	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	—	4	12	no	—
88	C₂²×C₄	—	8	8	no	—
89	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
90	C₂²×C₄	D₈×C₅₃	1	3	yes	—
91	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
92	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
93	C₂×C₈	C₄⋊C₄×C₅₃	1	12	yes	—
94	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
95	C₂×C₈	C₂×D₄×C₅₃	1	3	yes	—
96	C₂×C₈	C₂×Q₈×C₅₃	1	6	yes	—
97	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	8	no	—
99	C₄⋊C₄	—	4	12	no	—
100	C₁₆	C₂²×C₄×C₅₃	1	8	yes	complemented
101	C₂³	#107×C₅₃	1	4	yes	—
102	C₂³	SD₃₂×C₅₃	1	8	yes	—
103	C₂³	C₂×D₈×C₅₃	1	3	yes	—
104	C₂×C₄	—	4	8	no	—
105	C₂×C₄	—	8	24	no	—
106	C₂×C₄	#107×C₅₃	1	12	yes	—
107	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
108	C₂×C₄	C₂×D₈×C₅₃	1	3	yes	—
109	C₂×C₄	C₂×Q₁₆×C₅₃	1	6	yes	—
110	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
111	C₈	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
112	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
113	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
114	C₂²	C₅₃×#367	1	16	yes	direct product factor
115	C₂²	C₂×#107×C₅₃	1	6	yes	—
116	C₂²	C₂×SD₃₂×C₅₃	1	12	yes	—
117	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
118	C₄	—	8	8	no	—
119	C₄	C₂×#107×C₅₃	1	6	yes	—
120	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
121	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
122	C₂	C₂×C₅₃×#367	1	12	yes	direct product factor
123	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic
124	C₂	C₂²×SD₃₂×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#367	C₂	direct (N × H)
6. C₂²×C₁₆×C₅₃	C₄	semidirect
10. C₅₃×#367	C₂²	direct (N × H)
19. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
37. C₁₆×C₅₃	C₂²×C₄	semidirect
51. C₂²×#367	C₅₃	direct (N × H)
52. C₂²×C₅₃	#367	direct (N × H)
62. C₂×#367	C₂×C₅₃	direct (N × H)
63. C₂×C₅₃	C₂×#367	direct (N × H)
68. C₂²×C₁₆	C₄×C₅₃	semidirect
72. #367	C₂²×C₅₃	direct (N × H)
73. C₅₃	C₂²×#367	direct (N × H)
82. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
100. C₁₆	C₂²×C₄×C₅₃	semidirect
114. C₂²	C₅₃×#367	direct (N × H)
122. C₂	C₂×C₅₃×#367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₅₃	2
C₂²×#107×C₅₃	2
C₂×C₅₃×#367	2
C₂²×#367	53

Subgroups — C22×C53×#367 #27240918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#367 #27240918