Subgroups — C₂×#20272925 #27305128

Order 13568.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20272925	C₂	1	4	yes	maximal, direct product factor
3	C₂×#16810342	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
6	C₂×#13133123	C₄	1	1	yes	characteristic
7	C₃₂×D₅₃	C₄	1	2	yes	—
8	C₃₂×D₅₃	C₂²	1	2	yes	—
9	#16810342	C₂²	1	2	yes	complemented
10	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×C₆₄	C₂²	1	2	yes	complemented
12	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
13	C₂×#11719908	C₈	1	1	yes	characteristic
14	C₁₆×D₅₃	C₈	1	2	yes	—
15	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
16	#13133123	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₂×#88322	C₁₆	1	1	yes	characteristic
23	C₈×D₅₃	C₁₆	1	2	yes	—
24	C₈×D₅₃	C₂×C₈	1	2	yes	—
25	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	#11719908	C₂×C₈	1	2	yes	—
28	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
30	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
31	C₄×D₅₃	C₃₂	1	2	yes	—
32	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
33	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
34	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
35	#88322	C₂×C₁₆	1	2	yes	—
36	C₂×M₁₂₈	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
38	C₂×D₅₃	—	2	2	no	—
39	C₂×D₅₃	C₂×C₃₂	1	2	yes	—
40	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
42	Dic₅₃	C₂×C₃₂	1	2	yes	—
43	C₂²×C₃₂	—	53	1	no	—
44	M₁₂₈	—	53	4	no	—
45	C₂×C₆₄	—	53	1	no	—
46	C₂×C₆₄	D₅₃	1	1	yes	characteristic, complemented
47	C₂×C₅₃	M₁₂₈	1	2	yes	complemented
48	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
49	D₅₃	—	2	2	no	—
50	C₂²×C₁₆	—	53	1	no	—
51	C₂×C₃₂	—	53	1	no	—
52	C₂×C₃₂	—	53	2	no	—
53	C₂×C₃₂	—	53	2	no	—
54	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
55	C₆₄	—	53	2	no	—
56	C₆₄	C₂×D₅₃	1	2	yes	complemented
57	C₅₃	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	53	1	no	—
59	C₂×C₁₆	—	53	1	no	—
60	C₂×C₁₆	—	53	2	no	—
61	C₂×C₁₆	—	53	2	no	—
62	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
63	C₃₂	—	53	2	no	—
64	C₃₂	C₄×D₅₃	1	1	yes	characteristic
65	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
66	C₂²×C₄	—	53	1	no	—
67	C₂×C₈	—	53	1	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	53	2	no	—
70	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
71	C₁₆	—	53	2	no	—
72	C₁₆	C₈×D₅₃	1	1	yes	characteristic
73	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
74	C₂³	—	53	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	53	2	no	—
77	C₂×C₄	—	53	2	no	—
78	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
79	C₈	—	53	2	no	—
80	C₈	C₁₆×D₅₃	1	1	yes	characteristic
81	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
82	C₂²	—	53	2	no	—
83	C₂²	—	106	2	no	—
84	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
85	C₄	—	53	2	no	—
86	C₄	C₃₂×D₅₃	1	1	yes	characteristic
87	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
88	C₂	—	106	2	no	—
89	C₂	#20272925	1	2	yes	direct product factor
90	C₂	C₂×C₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₃₂×C₅₃	C₂³
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #20272925	C₂	direct (N × H)
3. C₂×#16810342	C₂	semidirect
4. C₂×C₅₃×C₆₄	C₂	semidirect
9. #16810342	C₂²	semidirect
11. C₅₃×C₆₄	C₂²	semidirect
46. C₂×C₆₄	D₅₃	semidirect
47. C₂×C₅₃	M₁₂₈	semidirect
56. C₆₄	C₂×D₅₃	semidirect
57. C₅₃	C₂×M₁₂₈	semidirect
89. C₂	#20272925	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×C₃₂×D₅₃	2
#20272925	2
C₂×#16810342	2
C₂×C₅₃×C₆₄	2
C₂×M₁₂₈	53

Subgroups — C2×#20272925 #27305128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272925 #27305128