Subgroups — C₂×#20283110 #27590186

Order 13568.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133214	C₂	1	14	yes	maximal, complemented
2	#20283110	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₁₂	C₂²	1	28	yes	—
5	C₂×#13133214	—	2	112	no	—
6	C₂×#13133214	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₅₃	C₂²	1	7	yes	—
8	C₄³×C₅₃	C₂²	1	8	yes	complemented
9	C₂⁴×D₅₃	C₂³	1	8	yes	—
10	C₂²×D₂₁₂	—	2	336	no	—
11	C₂²×D₂₁₂	C₂³	1	112	yes	—
12	#13133214	—	2	448	no	—
13	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
14	C₂×C₄²×C₅₃	D₄	1	56	yes	complemented
15	C₂×C₄²×C₅₃	C₂³	1	28	yes	—
16	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
17	C₂³×D₅₃	—	2	112	no	—
18	C₂³×D₅₃	C₂⁴	1	16	yes	—
19	C₂²×C₄×C₅₃	C₂×D₄	1	84	yes	—
20	C₂²×C₄×C₅₃	C₂⁴	1	14	yes	—
21	C₂×D₂₁₂	—	2	672	no	—
22	C₂×D₂₁₂	—	4	448	no	—
23	C₄²×C₅₃	C₂×D₄	1	112	yes	complemented
24	C₂³×C₅₃	C₂²×D₄	1	14	yes	—
25	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
26	C₂²×D₅₃	—	2	112	no	—
27	C₂²×D₅₃	—	4	224	no	—
28	C₂×C₄×C₅₃	#127	1	112	yes	complemented
29	C₂×C₄×C₅₃	C₂²×D₄	1	84	yes	—
30	D₂₁₂	—	4	448	no	—
31	C₂×#2867	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	C₂×#127	1	28	yes	—
33	C₂²×C₅₃	C₂³×D₄	1	7	yes	—
34	C₂×D₅₃	—	4	112	no	—
35	C₂×D₅₃	—	8	64	no	—
36	C₄×C₅₃	C₂×#127	1	56	yes	complemented
37	C₂²×#127	—	53	14	no	—
38	#2867	—	53	16	no	—
39	C₂×C₄³	D₅₃	1	1	yes	characteristic, complemented
40	C₂×C₅₃	#2867	1	8	yes	complemented
41	C₂×C₅₃	C₂²×#127	1	7	yes	—
42	D₅₃	—	8	16	no	—
43	C₂³×D₄	—	53	28	no	—
44	C₂×#127	—	53	112	no	—
45	C₂×#127	—	106	112	no	—
46	C₂²×C₄²	C₂×D₅₃	1	7	yes	—
47	C₄³	C₂×D₅₃	1	8	yes	complemented
48	C₅₃	C₂×#2867	1	1	yes	characteristic, complemented, Sylow
49	C₂⁵	—	53	8	no	—
50	C₂²×D₄	—	53	112	no	—
51	C₂²×D₄	—	106	336	no	—
52	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
53	#127	—	106	448	no	—
54	C₂×C₄²	D₂₁₂	1	56	yes	complemented
55	C₂×C₄²	C₂²×D₅₃	1	28	yes	—
56	C₂⁴	—	53	16	no	—
57	C₂⁴	—	106	112	no	—
58	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
59	C₂×D₄	—	106	672	no	—
60	C₂×D₄	—	212	448	no	—
61	C₂²×C₄	C₂×D₂₁₂	1	84	yes	—
62	C₂²×C₄	C₂³×D₅₃	1	14	yes	—
63	C₄²	C₂×D₂₁₂	1	112	yes	complemented
64	C₂³	—	106	112	no	—
65	C₂³	—	212	224	no	—
66	C₂³	C₂²×D₂₁₂	1	14	yes	—
67	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
68	D₄	—	212	448	no	—
69	C₂×C₄	#13133214	1	112	yes	complemented
70	C₂×C₄	C₂²×D₂₁₂	1	84	yes	—
71	C₂²	—	424	64	no	—
72	C₂²	—	212	112	no	—
73	C₂²	C₂×#13133214	1	28	yes	—
74	C₂²	C₂³×D₂₁₂	1	7	yes	—
75	C₄	C₂×#13133214	1	56	yes	complemented
76	C₂	—	424	16	no	—
77	C₂	#20283110	1	8	yes	direct product factor
78	C₂	C₂²×#13133214	1	7	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₂×#2867
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133214	C₂	semidirect
2. #20283110	C₂	direct (N × H)
3. C₂×C₄³×C₅₃	C₂	semidirect
6. C₂×#13133214	C₂²	semidirect
8. C₄³×C₅₃	C₂²	semidirect
14. C₂×C₄²×C₅₃	D₄	semidirect
23. C₄²×C₅₃	C₂×D₄	semidirect
28. C₂×C₄×C₅₃	#127	semidirect
36. C₄×C₅₃	C₂×#127	semidirect
39. C₂×C₄³	D₅₃	semidirect
40. C₂×C₅₃	#2867	semidirect
47. C₄³	C₂×D₅₃	semidirect
48. C₅₃	C₂×#2867	semidirect
54. C₂×C₄²	D₂₁₂	semidirect
63. C₄²	C₂×D₂₁₂	semidirect
69. C₂×C₄	#13133214	semidirect
75. C₄	C₂×#13133214	semidirect
77. C₂	#20283110	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×#13133214	2
#20283110	2
C₂×C₄³×C₅₃	2
C₂×#2867	53

Subgroups — C2×#20283110 #27590186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283110 #27590186