Subgroups — C₄×#16810457 #27304269

Order 13568.

250 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×#16810457	C₂	1	2	yes	maximal
2	#20272849	C₂	1	2	yes	maximal, complemented
3	C₄×#13133123	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810457	C₄	1	8	yes	direct product factor
6	C₂×#13133123	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	#13133123	C₂×C₄	1	8	yes	complemented
15	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₈	1	4	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₁₆	1	4	yes	—
31	C₂×C₈×C₅₃	C₄²	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₁₆	1	2	yes	—
38	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₅₃	C₄²	1	2	yes	—
41	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
42	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
43	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₅₃	C₄×C₈	1	2	yes	—
53	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
54	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
56	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₄×M₆₄	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
60	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
63	C₄×C₅₃	M₆₄	1	4	yes	complemented
64	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
66	C₂×M₆₄	—	53	2	no	—
67	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
68	#1397	—	53	2	no	—
69	C₄×C₃₂	—	53	2	no	—
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
72	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
73	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
74	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
75	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
76	M₆₄	—	53	8	no	—
77	C₂×C₃₂	—	53	4	no	—
78	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
79	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₅₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	#88322	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂×C₄²	#88322	1	1	yes	characteristic
86	C₂×C₁₆	#88322	1	4	yes	—
87	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
88	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
92	C₄×C₈	#88322	1	1	yes	characteristic
93	C₄×C₈	#88322	1	1	yes	characteristic
94	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
96	C₃₂	—	53	8	no	—
97	C₂²×C₄	#11719908	1	2	yes	—
98	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
99	C₂×C₈	#11719908	1	4	yes	—
100	C₂×C₈	C₂×#88322	1	1	yes	characteristic
101	C₂×C₈	C₂×#88322	1	2	yes	—
102	C₂×C₈	C₂×#88322	1	1	yes	characteristic
103	C₂×C₈	C₂×#88322	1	1	yes	characteristic
104	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
105	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
106	C₄²	#11719908	1	2	yes	—
107	C₄²	C₂×#88322	1	1	yes	characteristic
108	C₄²	C₂×#88322	1	1	yes	characteristic
109	C₁₆	C₂×#88322	1	2	yes	—
110	C₁₆	C₂×#88322	1	2	yes	—
111	C₁₆	C₄×Dic₅₃	1	2	yes	—
112	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
113	C₂³	C₂×#11719908	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×#88322	1	2	yes	—
116	C₂×C₄	C₂×#11719908	1	2	yes	—
117	C₂×C₄	C₂×#11719908	1	2	yes	—
118	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
119	C₂×C₄	C₂×#11719908	1	2	yes	—
120	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
121	C₈	C₄×#88322	1	2	yes	—
122	C₈	C₂×#11719908	1	2	yes	—
123	C₈	C₂×#11719908	1	2	yes	—
124	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
125	C₈	C₂²×#88322	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×#11719908	1	2	yes	—
128	C₂²	C₂²×#11719908	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×#11719908	1	2	yes	—
131	C₄	#16810457	1	4	yes	direct product factor
132	C₄	C₂×C₄×#88322	1	1	yes	characteristic
133	C₄	C₂²×#11719908	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×#11719908	1	1	yes	characteristic
136	C₂	C₂×#16810457	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₄×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20272849	C₂	semidirect
3. C₄×#13133123	C₂	semidirect
5. #16810457	C₄	direct (N × H)
14. #13133123	C₂×C₄	semidirect
63. C₄×C₅₃	M₆₄	semidirect
82. C₅₃	C₄×M₆₄	semidirect
131. C₄	#16810457	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×#16810457	2
#20272849	2
C₄×#13133123	2
C₂×C₄×C₁₆×C₅₃	2
C₄×M₆₄	53

Subgroups — C4×#16810457 #27304269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810457 #27304269