Subgroups — #20274853

Order 6784.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	2	yes	maximal
2	#16810388	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	4	yes	—
5	C₄⋊#88322	—	2	4	no	—
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₅₃	C₄	1	2	yes	—
9	C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄²×C₅₃	D₄	1	2	yes	—
17	#11719917	—	2	8	no	—
18	C₂×#88322	—	2	4	no	—
19	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₅₃	—	2	4	no	—
21	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
22	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#41	1	4	yes	—
26	C₈×C₅₃	—	2	4	no	—
27	#88322	—	4	4	no	—
28	C₂²×C₅₃	—	2	2	no	—
29	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₅₃	#125	1	2	yes	—
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	#108	1	4	yes	—
33	C₄×C₅₃	#117	1	2	yes	—
34	C₄×C₅₃	#124	1	2	yes	—
35	#1837	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	2	2	no	—
37	C₂×C₅₃	#382	1	1	yes	characteristic
38	C₂×C₅₃	C₂×#108	1	2	yes	—
39	#423	—	53	2	no	—
40	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
41	#341	—	53	4	no	—
42	C₅₃	#1837	1	1	yes	characteristic, complemented, Sylow
43	C₂×M₁₆	—	53	4	no	—
44	C₂²×C₈	C₂×D₅₃	1	2	yes	—
45	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
46	C₄⋊C₈	—	106	4	no	—
47	C₄×C₈	Dic₅₃	1	2	yes	—
48	C₄×C₈	Dic₅₃	1	2	yes	—
49	C₂²×C₄	Dic₁₀₆	1	2	yes	—
50	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
51	M₁₆	—	106	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	106	4	no	—
54	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
55	C₄²	D₂₁₂	1	2	yes	—
56	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
57	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
58	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
62	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
63	C₂×C₄	#11719945	1	4	yes	—
64	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
65	C₈	—	2	4	no	—
66	C₈	—	212	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
69	C₂²	#13133209	1	2	yes	—
70	C₄	—	2	2	no	—
71	C₄	#13133153	1	4	yes	—
72	C₄	#13133176	1	2	yes	—
73	C₄	#13133205	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	#16810489	1	1	yes	characteristic
76	C₂	C₂×#13133153	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1837
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
42. C₅₃	#1837	semidirect