Subgroups — C₂×#20289885 #28312672

Order 13568.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289885	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16811462	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811462	C₂²	1	4	yes	complemented
5	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
6	C₂×#13133123	C₂²	1	1	yes	characteristic
7	C₃₂×D₅₃	C₄	1	2	yes	—
8	C₃₂×D₅₃	C₂²	1	2	yes	—
9	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
11	C₂×#11719908	C₈	1	1	yes	characteristic
12	C₁₆×D₅₃	C₈	1	2	yes	—
13	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
14	#13133123	C₂×C₄	1	1	yes	characteristic
15	#13133123	C₂³	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂×#88322	C₁₆	1	1	yes	characteristic
21	C₈×D₅₃	C₁₆	1	2	yes	—
22	C₈×D₅₃	C₂×C₈	1	2	yes	—
23	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	#11719908	C₂×C₈	1	2	yes	—
26	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
29	C₄×D₅₃	C₃₂	1	2	yes	—
30	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
31	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
33	#88322	C₂×C₁₆	1	2	yes	—
34	C₂×M₁₂₈	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
36	C₂×D₅₃	—	2	2	no	—
37	C₂×D₅₃	C₂×C₃₂	1	2	yes	—
38	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
39	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
40	Dic₅₃	C₂×C₃₂	1	2	yes	—
41	C₂²×C₃₂	—	53	1	no	—
42	M₁₂₈	—	53	4	no	—
43	C₂×C₆₄	—	53	2	no	—
44	C₂×C₅₃	M₁₂₈	1	2	yes	complemented
45	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
46	D₅₃	—	2	2	no	—
47	C₂²×C₁₆	—	53	1	no	—
48	C₂×C₃₂	—	53	1	no	—
49	C₂×C₃₂	—	53	2	no	—
50	C₂×C₃₂	—	53	2	no	—
51	C₂×C₃₂	#6157	1	1	yes	characteristic
52	C₆₄	—	53	4	no	—
53	C₅₃	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	—	53	1	no	—
55	C₂×C₁₆	—	53	1	no	—
56	C₂×C₁₆	—	53	2	no	—
57	C₂×C₁₆	—	53	2	no	—
58	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
59	C₃₂	—	53	1	no	—
60	C₃₂	—	53	1	no	—
61	C₃₂	C₂×#6157	1	2	yes	—
62	C₂²×C₄	—	53	1	no	—
63	C₂×C₈	—	53	1	no	—
64	C₂×C₈	—	53	2	no	—
65	C₂×C₈	—	53	2	no	—
66	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
67	C₁₆	—	53	2	no	—
68	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
69	C₁₆	C₂²×#6157	1	1	yes	characteristic
70	C₂³	—	53	1	no	—
71	C₂×C₄	—	53	1	no	—
72	C₂×C₄	—	53	2	no	—
73	C₂×C₄	—	53	2	no	—
74	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
75	C₈	—	53	2	no	—
76	C₈	D₅₃⋊C₁₆	1	1	yes	characteristic
77	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
78	C₂²	—	53	2	no	—
79	C₂²	—	106	2	no	—
80	C₂²	D₅₃⋊C₃₂	1	1	yes	characteristic
81	C₄	—	53	2	no	—
82	C₄	D₅₃⋊C₃₂	1	1	yes	characteristic
83	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
84	C₂	—	106	2	no	—
85	C₂	#20289885	1	2	yes	direct product factor
86	C₂	C₂×D₅₃⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₃₂
Frattini	C₁₆	C₂²×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	#13133123	C₂³
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #20289885	C₂	direct (N × H)
2. C₂×#16811462	C₂	semidirect
4. #16811462	C₂²	semidirect
44. C₂×C₅₃	M₁₂₈	semidirect
53. C₅₃	C₂×M₁₂₈	semidirect
85. C₂	#20289885	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289885	2
C₂×#16811462	2
C₂×C₃₂×D₅₃	2
C₂×M₁₂₈	53

Subgroups — C2×#20289885 #28312672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289885 #28312672