Subgroups — C₂×C₄×#13133123 #27304262

Order 13568.

270 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	2	yes	maximal
2	C₄×#13133123	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	8	yes	direct product factor
5	C₂×#13133123	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	#13133123	C₂×C₄	1	8	yes	direct product factor
12	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	C₈	1	4	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₈	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₁₆	1	4	yes	—
25	C₂×C₈×C₅₃	C₄²	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₁₆	1	2	yes	—
31	C₄²×C₅₃	C₂×C₈	1	2	yes	—
32	C₁₆×C₅₃	C₄²	1	2	yes	—
33	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
34	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₃₂	1	8	yes	complemented
37	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
41	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
42	C₈×C₅₃	C₄×C₈	1	2	yes	—
43	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
44	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
45	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₃₂	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
48	C₂²×C₅₃	C₂×C₃₂	1	4	yes	—
49	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
51	C₄×C₅₃	C₂×C₃₂	1	8	yes	complemented
52	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₂²×C₃₂	—	53	2	no	—
55	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
56	C₄×C₃₂	—	53	4	no	—
57	C₂×C₅₃	C₄×C₃₂	1	4	yes	complemented
58	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₅₃	C₂²×C₃₂	1	2	yes	—
60	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
61	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
62	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
63	C₂×C₃₂	—	53	8	no	—
64	C₂×C₃₂	—	53	4	no	—
65	C₄×C₁₆	Dic₅₃	1	2	yes	—
66	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
67	C₅₃	C₂×C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	#88322	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
70	C₂×C₄²	#88322	1	1	yes	characteristic
71	C₂×C₁₆	#88322	1	4	yes	—
72	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
73	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
74	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
76	C₄×C₈	#88322	1	2	yes	—
77	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
78	C₃₂	—	53	8	no	—
79	C₂²×C₄	#11719908	1	2	yes	—
80	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
81	C₂×C₈	#11719908	1	4	yes	—
82	C₂×C₈	C₂×#88322	1	1	yes	characteristic
83	C₂×C₈	C₂×#88322	1	2	yes	—
84	C₂×C₈	C₂×#88322	1	2	yes	—
85	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
86	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₄²	#11719908	1	2	yes	—
88	C₄²	C₂×#88322	1	2	yes	—
89	C₁₆	C₂×#88322	1	4	yes	—
90	C₁₆	C₄×Dic₅₃	1	2	yes	—
91	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
92	C₂³	C₂×#11719908	1	1	yes	characteristic
93	C₂×C₄	#13133123	1	8	yes	direct product factor
94	C₂×C₄	C₄×#88322	1	2	yes	—
95	C₂×C₄	C₂×#11719908	1	2	yes	—
96	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
97	C₂×C₄	C₂×#11719908	1	4	yes	—
98	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
99	C₈	C₄×#88322	1	2	yes	—
100	C₈	C₂×#11719908	1	4	yes	—
101	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
102	C₈	C₂²×#88322	1	1	yes	characteristic
103	C₂²	C₄×#11719908	1	2	yes	—
104	C₂²	C₂×#13133123	1	4	yes	—
105	C₂²	C₂²×#11719908	1	1	yes	characteristic
106	C₄	C₄×#11719908	1	2	yes	—
107	C₄	C₂×#13133123	1	8	yes	direct product factor
108	C₄	C₂×C₄×#88322	1	1	yes	characteristic
109	C₄	C₂²×#11719908	1	1	yes	characteristic
110	C₂	C₄×#13133123	1	4	yes	direct product factor
111	C₂	C₂×C₄×#11719908	1	1	yes	characteristic
112	C₂	C₂²×#13133123	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₅₃
commutator, hypocenter	C₅₃	C₂×C₄×C₃₂
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#13133123	C₂	direct (N × H)
4. C₂×#13133123	C₄	direct (N × H)
11. #13133123	C₂×C₄	direct (N × H)
36. C₂×C₄×C₅₃	C₃₂	semidirect
51. C₄×C₅₃	C₂×C₃₂	semidirect
57. C₂×C₅₃	C₄×C₃₂	semidirect
67. C₅₃	C₂×C₄×C₃₂	semidirect
93. C₂×C₄	#13133123	direct (N × H)
107. C₄	C₂×#13133123	direct (N × H)
110. C₂	C₄×#13133123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133123	2
C₄×#13133123	2
C₂×C₄×C₁₆×C₅₃	2
C₂×C₄×C₃₂	53

Subgroups — C2×C4×#13133123 #27304262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133123 #27304262