Subgroups — #20276474

Order 6784.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	#13133157	—	2	2	no	—
6	C₂×#11719908	—	2	2	no	—
7	C₂×#11719908	C₂²	1	2	yes	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	#117×C₅₃	C₄	1	1	yes	characteristic
12	#97×C₅₃	C₄	1	2	yes	—
13	C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₂×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
22	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
23	C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	#11719908	—	2	2	no	—
25	#11719908	—	2	2	no	—
26	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₅₃	—	2	2	no	—
33	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₅₃	—	2	4	no	—
40	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
43	#2151	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	#504	1	2	yes	—
47	C₂×M₃₂	—	53	1	no	—
48	C₂²×C₁₆	—	53	1	no	—
49	#405	D₅₃	1	1	yes	characteristic
50	C₄⋊C₁₆	—	53	4	no	—
51	C₅₃	#2151	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₁₆	Dic₅₃	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
54	#117	Dic₅₃	1	1	yes	characteristic
55	M₃₂	—	106	2	no	—
56	C₂×C₁₆	—	53	2	no	—
57	C₂×C₁₆	—	53	2	no	—
58	C₂×C₁₆	—	106	2	no	—
59	#97	Dic₅₃	1	2	yes	—
60	C₄×C₈	C₂×D₅₃	1	2	yes	—
61	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	Dic₁₀₆	1	2	yes	—
64	C₂×C₈	D₂₁₂	1	2	yes	—
65	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
66	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
67	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
68	C₄⋊C₄	#88322	1	2	yes	—
69	C₂²⋊C₄	#88322	1	2	yes	—
70	C₄²	C₂×Dic₅₃	1	2	yes	—
71	C₁₆	—	106	2	no	—
72	C₁₆	—	106	2	no	—
73	C₂³	C₂×#88322	1	1	yes	characteristic
74	C₂×C₄	C₂×#88322	1	4	yes	—
75	C₂×C₄	C₂×#88322	1	1	yes	characteristic
76	C₂×C₄	#11719917	1	2	yes	—
77	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
78	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊Dic₅₃	1	2	yes	—
81	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
82	C₈	C₂×D₂₁₂	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#88322	1	2	yes	—
85	C₂²	C₂²×#88322	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#88322	1	2	yes	—
88	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
89	C₄	C₂×#11719917	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic
92	C₂	#16811040	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×#405	C₂
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. C₄⋊#11719908	C₂	semidirect
51. C₅₃	#2151	semidirect