Subgroups — C₄×#16810398 #27298913

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810398	C₂	1	4	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	#16810398	C₄	1	16	yes	direct product factor
11	C₄×#11719908	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₂×#11719908	C₂×C₄	1	4	yes	—
19	C₂×#11719908	C₂×C₄	1	8	yes	—
20	C₄×#88322	C₈	1	4	yes	—
21	C₈×Dic₅₃	C₈	1	8	yes	—
22	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₃	C₄²	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₄²	1	2	yes	—
39	C₄²×C₅₃	C₂×C₈	1	2	yes	—
40	C₄×Dic₅₃	—	2	8	no	—
41	C₄×Dic₅₃	C₄²	1	4	yes	—
42	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
43	C₂×#88322	C₄²	1	4	yes	—
44	C₂×#88322	C₂×C₈	1	4	yes	—
45	C₂×#88322	C₂×C₈	1	4	yes	—
46	C₁₆×C₅₃	C₄²	1	8	yes	complemented
47	#11719908	C₄²	1	8	yes	complemented
48	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₅₃	M₃₂	1	8	yes	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
55	C₂×Dic₅₃	—	2	8	no	—
56	C₂×Dic₅₃	C₄×C₈	1	8	yes	—
57	C₈×C₅₃	C₄×C₈	1	4	yes	—
58	C₈×C₅₃	C₂×C₄²	1	2	yes	—
59	C₈×C₅₃	C₂×C₄²	1	2	yes	—
60	#88322	C₄×C₈	1	8	yes	—
61	C₄×#346	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
63	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
65	C₄×C₅₃	#346	1	8	yes	complemented
66	C₄×C₅₃	C₄³	1	1	yes	characteristic
67	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
68	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	Dic₅₃	—	2	8	no	—
70	C₂×#346	—	53	4	no	—
71	C₂×C₄×C₁₆	—	53	1	no	—
72	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
73	C₄²×C₈	—	53	1	no	—
74	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#346	1	2	yes	—
76	C₂×C₅₃	C₄×M₃₂	1	4	yes	—
77	C₂²×C₁₆	—	53	2	no	—
78	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
79	C₂×C₄×C₈	—	53	2	no	—
80	C₂×C₄×C₈	—	53	2	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₄³	—	53	1	no	—
84	#346	—	53	16	no	—
85	C₄×C₁₆	—	53	4	no	—
86	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
87	C₅₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	2	no	—
89	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
91	C₂×C₄²	—	53	2	no	—
92	C₂×C₄²	—	53	4	no	—
93	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₁₆	—	53	4	no	—
95	C₂×C₁₆	—	53	8	no	—
96	C₂×C₁₆	C₄×D₅₃	1	8	yes	—
97	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
98	C₄×C₈	—	53	8	no	—
99	C₄×C₈	—	53	8	no	—
100	C₄×C₈	—	53	4	no	—
101	C₄×C₈	C₄×D₅₃	1	2	yes	—
102	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
103	C₂²×C₄	—	53	4	no	—
104	C₂²×C₄	C₈×D₅₃	1	2	yes	—
105	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	53	4	no	—
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	C₈×D₅₃	1	4	yes	—
110	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
111	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
113	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₄²	—	53	4	no	—
115	C₄²	—	53	4	no	—
116	C₄²	—	106	8	no	—
117	C₄²	C₈×D₅₃	1	2	yes	—
118	C₄²	C₄×Dic₅₃	1	2	yes	—
119	C₁₆	—	53	8	no	—
120	C₁₆	C₄×Dic₅₃	1	8	yes	complemented
121	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
122	C₂×C₄	—	53	8	no	—
123	C₂×C₄	—	106	8	no	—
124	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
125	C₂×C₄	#13133156	1	8	yes	—
126	C₂×C₄	C₄²×D₅₃	1	2	yes	—
127	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
130	C₈	—	53	8	no	—
131	C₈	C₈×Dic₅₃	1	4	yes	—
132	C₈	C₄²×D₅₃	1	2	yes	—
133	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
134	C₂²	C₄×C₈×D₅₃	1	2	yes	—
135	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
136	C₂²	C₂×#13133156	1	4	yes	—
137	C₄	—	106	8	no	—
138	C₄	#16810398	1	8	yes	direct product factor
139	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
140	C₄	C₄×C₈×D₅₃	1	2	yes	—
141	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
142	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
143	C₂	C₂×#16810398	1	2	yes	—
144	C₂	C₄×#13133156	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₄×#346
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16810398	C₄	direct (N × H)
11. C₄×#11719908	C₄	semidirect
14. C₄×C₁₆×C₅₃	C₄	semidirect
46. C₁₆×C₅₃	C₄²	semidirect
47. #11719908	C₄²	semidirect
65. C₄×C₅₃	#346	semidirect
86. C₄×C₁₆	Dic₅₃	semidirect
87. C₅₃	C₄×#346	semidirect
120. C₁₆	C₄×Dic₅₃	semidirect
138. C₄	#16810398	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×#16810398	2
C₂×C₄×#11719908	2
C₄×C₈×Dic₅₃	2
C₂×C₄×C₁₆×C₅₃	2
C₄×#346	53

Subgroups — C4×#16810398 #27298913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810398 #27298913