Subgroups — #20276471

Order 6784.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	2	yes	maximal
2	C₄⋊#11719908	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133157	—	2	4	no	—
5	C₂×#11719908	C₂²	1	4	yes	—
6	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₅₃	C₈	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	—	2	2	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	D₄	1	2	yes	—
17	C₂×C₈×C₅₃	Q₈	1	2	yes	—
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	C₈	1	1	yes	characteristic
20	C₄²×C₅₃	C₈	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	#11719908	—	2	4	no	—
23	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₅₃	—	2	2	no	—
32	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₅₃	—	2	2	no	—
36	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
37	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₅₃	—	2	2	no	—
39	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₄×C₅₃	M₃₂	1	4	yes	complemented
41	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
43	#2150	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	C₂×M₃₂	1	2	yes	—
47	C₂×M₃₂	—	53	2	no	—
48	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
49	C₄⋊C₁₆	—	53	4	no	—
50	C₅₃	#2150	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
53	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
54	M₃₂	—	106	4	no	—
55	C₂×C₁₆	—	53	4	no	—
56	C₄×C₈	Dic₅₃	1	2	yes	—
57	C₄×C₈	C₂×D₅₃	1	2	yes	—
58	C₂²×C₄	#88322	1	2	yes	—
59	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	Dic₁₀₆	1	2	yes	—
62	C₂×C₈	D₂₁₂	1	2	yes	—
63	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
64	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
65	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
66	C₄²	#88322	1	1	yes	characteristic
67	C₄²	#88322	1	1	yes	characteristic
68	C₄²	C₂×Dic₅₃	1	2	yes	—
69	C₁₆	—	106	4	no	—
70	C₂³	C₂×#88322	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	C₂×#88322	1	2	yes	—
73	C₂×C₄	C₂×#88322	1	1	yes	characteristic
74	C₂×C₄	C₂×#88322	1	2	yes	—
75	C₂×C₄	#11719917	1	2	yes	—
76	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
77	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	C₄⋊Dic₅₃	1	2	yes	—
80	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
81	C₈	C₂×D₂₁₂	1	1	yes	characteristic
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊#88322	1	2	yes	—
84	C₂²	C₂²×#88322	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₄⋊#88322	1	2	yes	—
87	C₄	#13133157	1	4	yes	complemented
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	C₂×#13133157	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₂
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#11719908	C₂	semidirect
40. C₄×C₅₃	M₃₂	semidirect
50. C₅₃	#2150	semidirect
87. C₄	#13133157	semidirect