Subgroups — #20274852

Order 6784.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810387	C₂	1	4	yes	maximal
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	C₄×#88322	C₄	1	2	yes	—
9	C₄×#88322	C₄	1	2	yes	—
10	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	2	no	—
13	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	M₁₆×C₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	—	2	2	no	—
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	2	yes	—
20	#11719917	—	2	4	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	—	2	2	no	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	C₂×C₄	1	4	yes	—
25	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	4	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	#41	1	2	yes	—
32	C₂×C₄×C₅₃	#41	1	2	yes	—
33	C₈×C₅₃	—	4	2	no	—
34	#88322	—	2	4	no	—
35	#88322	—	4	2	no	—
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₅₃	#125	1	2	yes	—
39	C₄×C₅₃	—	2	2	no	—
40	C₄×C₅₃	#108	1	4	yes	—
41	C₄×C₅₃	#117	1	2	yes	—
42	C₄×C₅₃	#124	1	1	yes	characteristic
43	C₄×C₅₃	#124	1	1	yes	characteristic
44	#1837	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	2	no	—
46	C₂×C₅₃	#382	1	1	yes	characteristic
47	C₂×C₅₃	C₂×#108	1	2	yes	—
48	#423	—	53	1	no	—
49	#423	D₅₃	1	1	yes	characteristic
50	C₂×C₄×C₈	—	53	1	no	—
51	#341	—	53	4	no	—
52	C₅₃	#1837	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₁₆	—	53	2	no	—
54	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
55	C₂²×C₈	—	53	2	no	—
56	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	106	2	no	—
59	C₄×C₈	—	53	2	no	—
60	C₄×C₈	—	53	2	no	—
61	C₂²×C₄	Dic₁₀₆	1	2	yes	—
62	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	106	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	53	4	no	—
67	C₂×C₈	—	106	2	no	—
68	C₂×C₈	—	106	2	no	—
69	C₂×C₈	—	106	2	no	—
70	C₄²	C₄×D₅₃	1	1	yes	characteristic
71	C₄²	C₄×D₅₃	1	1	yes	characteristic
72	C₄²	C₂²⋊D₅₃	1	2	yes	—
73	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	#11719919	1	2	yes	—
76	C₂×C₄	#11719919	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
78	C₂×C₄	#11719947	1	2	yes	—
79	C₂×C₄	#11719949	1	2	yes	—
80	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	106	4	no	—
83	C₈	—	212	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×#11719919	1	1	yes	characteristic
86	C₂²	#13133208	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#13133152	1	4	yes	—
89	C₄	#13133175	1	2	yes	—
90	C₄	#13133204	1	1	yes	characteristic
91	C₄	#13133206	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#16810488	1	1	yes	characteristic
94	C₂	C₂×#13133152	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1837
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

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