Subgroups — #20285239

Order 6784.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811180	C₂	1	2	yes	maximal
2	#16811032	C₂	1	8	yes	maximal, complemented
3	#16810661	C₂	1	4	yes	maximal
4	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133217	C₂²	1	4	yes	—
6	#13133209	C₂²	1	4	yes	—
7	#13133196	—	2	4	no	—
8	C₄×Dic₁₀₆	—	2	4	no	—
9	#13133176	C₂²	1	4	yes	—
10	#13133151	C₂²	1	8	yes	—
11	#13133149	C₂²	1	8	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#97×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₀₆	—	2	4	no	—
18	C₂×C₈×C₅₃	Q₈	1	8	yes	complemented
19	C₂×C₈×C₅₃	C₂³	1	4	yes	—
20	C₄²×C₅₃	D₄	1	2	yes	—
21	C₄²×C₅₃	C₂³	1	2	yes	—
22	C₂²⋊Dic₅₃	—	2	4	no	—
23	C₄⋊Dic₅₃	—	2	4	no	—
24	C₄⋊Dic₅₃	C₂³	1	8	yes	—
25	#11719919	—	2	8	no	—
26	C₄×Dic₅₃	—	2	4	no	—
27	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
32	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	8	no	—
34	Dic₁₀₆	—	4	8	no	—
35	C₈×C₅₃	C₂×Q₈	1	8	yes	—
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₅₃	—	2	4	no	—
40	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
41	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
42	Dic₅₃	—	4	8	no	—
43	#3162	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₅₃	#575	1	2	yes	—
47	#533	—	53	2	no	—
48	#501	—	53	8	no	—
49	#427	—	53	4	no	—
50	C₂×#97	D₅₃	1	1	yes	characteristic
51	C₅₃	#3162	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₈	C₂×D₅₃	1	2	yes	—
53	C₄⋊Q₈	—	53	4	no	—
54	#125	—	53	4	no	—
55	C₂²⋊Q₈	—	106	4	no	—
56	C₄×Q₈	—	106	4	no	—
57	#117	—	53	4	no	—
58	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
59	#107	—	53	8	no	—
60	#106	—	53	8	no	—
61	#97	C₂×D₅₃	1	4	yes	—
62	C₂×Q₈	—	106	4	no	—
63	C₂²×C₄	D₂₁₂	1	2	yes	—
64	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
65	C₂×C₈	Dic₁₀₆	1	8	yes	complemented
66	C₂×C₈	C₂²×D₅₃	1	4	yes	—
67	C₄⋊C₄	—	53	8	no	—
68	C₄⋊C₄	—	106	8	no	—
69	C₄⋊C₄	—	106	4	no	—
70	C₂²⋊C₄	—	106	4	no	—
71	C₄²	—	106	4	no	—
72	C₄²	D₂₁₂	1	2	yes	—
73	C₄²	C₂²×D₅₃	1	2	yes	—
74	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
75	Q₈	—	212	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	106	8	no	—
78	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
79	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
80	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
81	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
82	C₈	C₂×Dic₁₀₆	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#13133217	1	2	yes	—
85	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	212	8	no	—
88	C₄	#13133217	1	2	yes	—
89	C₄	C₂²×Dic₁₀₆	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×#13133217	1	1	yes	characteristic
92	C₂	#16811423	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#3162
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #16811032	C₂	semidirect
18. C₂×C₈×C₅₃	Q₈	semidirect
51. C₅₃	#3162	semidirect
65. C₂×C₈	Dic₁₀₆	semidirect