Subgroups — C₂×M₁₆×Dic₅₃ #20283122

Order 6784.

656 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	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₅₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133127	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₅₃	C₄	1	2	yes	—
8	C₂×#11719917	C₄	1	8	yes	complemented
9	C₂×#11719917	C₂²	1	4	yes	—
10	C₂²×#88322	C₂²	1	2	yes	—
11	C₂×C₄×Dic₅₃	C₄	1	4	yes	—
12	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
16	#13133127	C₂²	1	8	yes	complemented
17	C₈×Dic₅₃	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	C₄	1	8	yes	complemented
20	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	8	no	—
30	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₅₃	C₂×C₄	1	8	yes	—
32	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
33	M₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
34	C₂×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
35	C₂×C₈×C₅₃	C₂³	1	4	yes	—
36	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
37	C₄×Dic₅₃	C₂×C₄	1	8	yes	—
38	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₄×Dic₅₃	C₂³	1	2	yes	—
40	C₄×Dic₅₃	C₂³	1	2	yes	—
41	#11719917	C₂×C₄	1	16	yes	complemented
42	C₂×#88322	C₂×C₄	1	8	yes	complemented
43	C₂×#88322	C₂³	1	4	yes	—
44	C₂³×C₅₃	—	2	4	no	—
45	C₂³×C₅₃	C₄²	1	4	yes	—
46	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄²	1	8	yes	—
49	C₂×C₄×C₅₃	C₄²	1	4	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	2	16	no	—
59	C₂×Dic₅₃	M₁₆	1	8	yes	direct product factor
60	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
61	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
62	C₈×C₅₃	C₂²×C₄	1	8	yes	complemented
63	#88322	C₂²×C₄	1	8	yes	complemented
64	C₂²×C₅₃	—	2	4	no	—
65	C₂²×C₅₃	—	2	8	no	—
66	C₂²×C₅₃	C₂×C₄²	1	4	yes	—
67	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
68	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
69	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₅₃	C₂×C₄²	1	2	yes	—
71	C₄×C₅₃	C₂×C₄²	1	4	yes	—
72	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
74	Dic₅₃	—	2	4	no	—
75	Dic₅₃	C₂×M₁₆	1	8	yes	direct product factor
76	C₂×C₄×M₁₆	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	4	no	—
78	C₂×C₅₃	C₄×M₁₆	1	4	yes	complemented
79	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
80	C₂×C₅₃	C₂²×M₁₆	1	2	yes	—
81	C₂²×M₁₆	—	53	1	no	—
82	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
83	C₂²×C₄²	—	53	1	no	—
84	C₄×M₁₆	—	53	8	no	—
85	C₂×#97	—	53	2	no	—
86	C₂×C₄×C₈	—	53	2	no	—
87	C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	—	53	2	no	—
89	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
90	C₂×M₁₆	—	53	8	no	—
91	C₂×M₁₆	—	53	4	no	—
92	C₂×M₁₆	Dic₅₃	1	8	yes	direct product factor
93	C₂×M₁₆	C₂×D₅₃	1	4	yes	—
94	C₂²×C₈	—	53	2	no	—
95	C₂²×C₈	C₂×D₅₃	1	2	yes	—
96	C₂×C₄²	—	53	2	no	—
97	C₂×C₄²	—	53	4	no	—
98	C₂×C₄²	—	53	4	no	—
99	C₂×C₄²	—	53	1	no	—
100	C₂×C₄²	—	53	1	no	—
101	#97	—	53	8	no	—
102	C₄×C₈	—	53	8	no	—
103	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
104	C₂²×C₄	—	53	2	no	—
105	C₂²×C₄	—	53	8	no	—
106	C₂²×C₄	—	53	2	no	—
107	C₂²×C₄	—	106	8	no	—
108	C₂²×C₄	C₄×D₅₃	1	4	yes	—
109	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₅₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	M₁₆	—	53	16	no	—
115	M₁₆	C₂×Dic₅₃	1	16	yes	direct product factor
116	C₂×C₈	—	53	8	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	C₂×Dic₅₃	1	8	yes	complemented
119	C₂×C₈	C₂²×D₅₃	1	4	yes	—
120	C₄²	—	53	2	no	—
121	C₄²	—	53	8	no	—
122	C₄²	—	53	2	no	—
123	C₄²	—	53	2	no	—
124	C₄²	—	53	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	C₄×Dic₅₃	1	4	yes	—
127	C₂³	C₂×C₄×D₅₃	1	2	yes	—
128	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	—	53	8	no	—
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	53	4	no	—
132	C₂×C₄	—	106	16	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	C₄×Dic₅₃	1	8	yes	—
135	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
136	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
140	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
141	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
142	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
143	C₈	—	53	8	no	—
144	C₈	C₂²×Dic₅₃	1	8	yes	complemented
145	C₂²	—	2	8	no	—
146	C₂²	—	2	4	no	—
147	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
148	C₂²	C₂×C₄×Dic₅₃	1	2	yes	—
149	C₂²	M₁₆×D₅₃	1	4	yes	—
150	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
151	C₄	—	53	8	no	—
152	C₄	—	106	4	no	—
153	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
154	C₄	C₂×C₄×Dic₅₃	1	2	yes	—
155	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
156	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
157	C₂	—	2	4	no	—
158	C₂	M₁₆×Dic₅₃	1	4	yes	direct product factor
159	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
160	C₂	C₂×M₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. M₁₆×Dic₅₃	C₂	direct (N × H)
4. C₂×#13133127	C₂	semidirect
5. C₂×C₈×Dic₅₃	C₂	semidirect
8. C₂×#11719917	C₄	semidirect
16. #13133127	C₂²	semidirect
17. C₈×Dic₅₃	C₂²	semidirect
19. C₂×M₁₆×C₅₃	C₄	semidirect
33. M₁₆×C₅₃	C₂×C₄	semidirect
34. C₂×C₈×C₅₃	C₂×C₄	semidirect
41. #11719917	C₂×C₄	semidirect
42. C₂×#88322	C₂×C₄	semidirect
59. C₂×Dic₅₃	M₁₆	direct (N × H)
62. C₈×C₅₃	C₂²×C₄	semidirect
63. #88322	C₂²×C₄	semidirect
75. Dic₅₃	C₂×M₁₆	direct (N × H)
78. C₂×C₅₃	C₄×M₁₆	semidirect
87. C₅₃	C₂×C₄×M₁₆	semidirect
92. C₂×M₁₆	Dic₅₃	direct (N × H)
115. M₁₆	C₂×Dic₅₃	direct (N × H)
118. C₂×C₈	C₂×Dic₅₃	semidirect
144. C₈	C₂²×Dic₅₃	semidirect
158. C₂	M₁₆×Dic₅₃	direct (N × H)

Subgroups — C2×M16×Dic53 #20283122

Conjugacy classes of subgroups

Complements

Subgroups — C₂×M₁₆×Dic₅₃ #20283122