Subgroups — C₂×#20272932 #27305141

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₈	C₂	1	1	yes	maximal, characteristic
2	#20272932	C₂	1	4	yes	maximal, direct product factor
3	C₂×#16810342	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₄	—	2	1	no	—
6	Dic₈₄₈	—	2	1	no	—
7	Dic₈₄₈	C₂²	1	2	yes	—
8	#16810342	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₅₃	—	2	1	no	—
10	C₅₃×Q₆₄	—	2	1	no	—
11	C₅₃×Q₆₄	C₂²	1	2	yes	—
12	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₂	—	4	1	no	—
14	Dic₄₂₄	—	2	2	no	—
15	Dic₄₂₄	—	4	1	no	—
16	C₂×Q₁₆×C₅₃	—	4	1	no	—
17	Q₃₂×C₅₃	—	2	2	no	—
18	Q₃₂×C₅₃	—	4	1	no	—
19	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
20	C₃₂×C₅₃	D₄	1	1	yes	characteristic
21	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	—	8	1	no	—
23	C₂×Dic₁₀₆	—	8	1	no	—
24	Q₁₆×C₅₃	—	4	2	no	—
25	Q₁₆×C₅₃	—	8	1	no	—
26	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
27	Dic₂₁₂	—	4	2	no	—
28	Dic₂₁₂	—	8	1	no	—
29	C₁₆×C₅₃	D₈	1	1	yes	characteristic
30	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₅₃	—	8	2	no	—
32	Q₈×C₅₃	—	16	1	no	—
33	C₂×C₄×C₅₃	—	16	1	no	—
34	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
35	C₂×Dic₅₃	—	16	1	no	—
36	Dic₁₀₆	—	8	2	no	—
37	Dic₁₀₆	—	16	1	no	—
38	C₈×C₅₃	D₁₆	1	1	yes	characteristic
39	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
40	C₂×Q₁₂₈	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
42	C₄×C₅₃	—	16	2	no	—
43	C₄×C₅₃	D₃₂	1	1	yes	characteristic
44	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
45	Dic₅₃	—	16	2	no	—
46	C₂×Q₆₄	—	53	1	no	—
47	C₂×Q₆₄	D₅₃	1	1	yes	characteristic
48	Q₁₂₈	—	53	4	no	—
49	C₂×C₆₄	—	53	1	no	—
50	C₂×C₅₃	Q₁₂₈	1	2	yes	complemented
51	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
52	C₂×Q₃₂	—	2	1	no	—
53	C₂×Q₃₂	—	106	1	no	—
54	Q₆₄	—	2	1	no	—
55	Q₆₄	—	53	2	no	—
56	Q₆₄	—	106	1	no	—
57	Q₆₄	C₂×D₅₃	1	2	yes	—
58	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
59	C₆₄	—	53	2	no	—
60	C₅₃	C₂×Q₁₂₈	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₁₆	—	4	1	no	—
62	C₂×Q₁₆	—	212	1	no	—
63	Q₃₂	—	2	2	no	—
64	Q₃₂	—	4	1	no	—
65	Q₃₂	—	106	2	no	—
66	Q₃₂	—	212	1	no	—
67	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
69	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×Q₈	—	8	1	no	—
71	C₂×Q₈	—	424	1	no	—
72	Q₁₆	—	4	2	no	—
73	Q₁₆	—	8	1	no	—
74	Q₁₆	—	424	1	no	—
75	Q₁₆	—	212	2	no	—
76	C₂×C₈	#11719922	1	1	yes	characteristic
77	C₁₆	#11719922	1	1	yes	characteristic
78	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	Q₈	—	8	2	no	—
80	Q₈	—	16	1	no	—
81	Q₈	—	848	1	no	—
82	Q₈	—	424	2	no	—
83	C₂×C₄	—	16	1	no	—
84	C₂×C₄	—	848	1	no	—
85	C₂×C₄	#13133158	1	1	yes	characteristic
86	C₈	#13133158	1	1	yes	characteristic
87	C₈	C₂×#11719922	1	1	yes	characteristic
88	C₂²	#16810458	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	848	2	no	—
91	C₄	#16810458	1	1	yes	characteristic
92	C₄	C₂×#13133158	1	1	yes	characteristic
93	C₂	#20272932	1	2	yes	direct product factor
94	C₂	C₂×#16810458	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810458
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₆₄	C₂
socle, prime radical	C₂²×C₅₃	D₃₂
hypercenter	C₂×Q₆₄	D₅₃
hypocenter	C₅₃	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
2. #20272932	C₂	direct (N × H)
50. C₂×C₅₃	Q₁₂₈	semidirect
60. C₅₃	C₂×Q₁₂₈	semidirect
93. C₂	#20272932	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×Dic₈₄₈	2
#20272932	2
C₂×#16810342	2
C₂×C₅₃×Q₆₄	2
C₂×Q₁₂₈	53

Subgroups — C2×#20272932 #27305141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272932 #27305141