Subgroups — #27305090

Order 13568.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₈	C₂	1	4	yes	maximal
2	#20276968	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₄	—	2	2	no	—
5	C₂×Dic₄₂₄	C₂²	1	4	yes	—
6	#16811003	—	2	2	no	—
7	Dic₈₄₈	—	2	8	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁₂	—	2	4	no	—
11	C₂×Dic₂₁₂	—	4	2	no	—
12	#13133217	—	4	2	no	—
13	Dic₄₂₄	—	2	8	no	—
14	Dic₄₂₄	—	4	4	no	—
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	4	yes	—
19	C₂×Dic₁₀₆	—	4	4	no	—
20	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₈	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	8	4	no	—
24	Dic₂₁₂	—	4	4	no	—
25	Dic₂₁₂	—	4	4	no	—
26	Dic₂₁₂	—	8	4	no	—
27	C₁₆×C₅₃	D₈	1	2	yes	—
28	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₅₃	—	8	4	no	—
32	Dic₁₀₆	—	8	4	no	—
33	Dic₁₀₆	—	8	4	no	—
34	C₈×C₅₃	D₁₆	1	2	yes	—
35	C₈×C₅₃	#127	1	1	yes	characteristic
36	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
37	#13726	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
39	C₄×C₅₃	Q₆₄	1	4	yes	complemented
40	C₄×C₅₃	#493	1	1	yes	characteristic
41	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
42	Dic₅₃	—	16	4	no	—
43	C₂×Q₆₄	—	53	4	no	—
44	#2247	—	53	2	no	—
45	C₄×C₃₂	D₅₃	1	1	yes	characteristic
46	C₂×C₅₃	#2246	1	1	yes	characteristic
47	C₂×C₅₃	C₂×Q₆₄	1	2	yes	—
48	C₂×Q₃₂	—	53	4	no	—
49	C₂×Q₃₂	—	106	2	no	—
50	#494	—	106	2	no	—
51	Q₆₄	—	106	8	no	—
52	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
53	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
54	C₅₃	#13726	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	106	4	no	—
56	C₂×Q₁₆	—	212	2	no	—
57	C₄⋊Q₈	—	212	2	no	—
58	Q₃₂	—	106	8	no	—
59	Q₃₂	—	212	4	no	—
60	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
62	C₄×C₈	D₂₁₂	1	1	yes	characteristic
63	C₃₂	D₂₁₂	1	4	yes	—
64	C₂×Q₈	—	212	4	no	—
65	Q₁₆	—	424	4	no	—
66	Q₁₆	—	212	4	no	—
67	Q₁₆	—	212	4	no	—
68	C₂×C₈	D₄₂₄	1	1	yes	characteristic
69	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
70	C₄⋊C₄	—	424	4	no	—
71	C₄²	D₄₂₄	1	1	yes	characteristic
72	C₁₆	D₄₂₄	1	2	yes	—
73	C₁₆	C₂×D₂₁₂	1	2	yes	—
74	Q₈	—	424	4	no	—
75	Q₈	—	424	4	no	—
76	C₂×C₄	—	424	4	no	—
77	C₂×C₄	D₈₄₈	1	2	yes	—
78	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
79	C₈	D₈₄₈	1	2	yes	—
80	C₈	#13133214	1	1	yes	characteristic
81	C₈	C₂×D₄₂₄	1	1	yes	characteristic
82	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
83	C₄	—	848	4	no	—
84	C₄	Dic₈₄₈	1	4	yes	complemented
85	C₄	#16811000	1	1	yes	characteristic
86	C₄	C₂×D₈₄₈	1	1	yes	characteristic
87	C₂	#20276965	1	1	yes	characteristic
88	C₂	C₂×Dic₈₄₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄₈
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#13726
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
39. C₄×C₅₃	Q₆₄	semidirect
54. C₅₃	#13726	semidirect
84. C₄	Dic₈₄₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄₈	2
#20276968	2
C₄×C₃₂×C₅₃	2
#13726	53