Subgroups — #20289762

Order 6784.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133291	C₂	1	2	yes	maximal
2	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
3	#13133291	—	2	2	no	—
4	#13133291	C₄	1	4	yes	complemented
5	C₂×#11719910	—	2	2	no	—
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	#13133128	—	2	2	no	—
9	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₈	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₅₃	—	2	1	no	—
13	C₄²×C₅₃	—	2	1	no	—
14	C₂×#88322	—	2	1	no	—
15	C₂×#88322	—	2	1	no	—
16	C₂×#88322	C₈	1	2	yes	—
17	C₂×#88322	C₂×C₄	1	2	yes	—
18	C₂×#88322	D₄	1	1	yes	characteristic
19	C₂×#88322	D₄	1	2	yes	—
20	C₂×#88322	Q₈	1	1	yes	characteristic
21	#11719910	—	2	4	no	—
22	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	2	no	—
24	C₂×C₄×C₅₃	—	4	2	no	—
25	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	#88322	—	2	2	no	—
32	#88322	C₂²⋊C₄	1	2	yes	—
33	#88322	C₄⋊C₄	1	2	yes	—
34	C₂²×C₅₃	—	2	2	no	—
35	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₅₃	—	4	2	no	—
39	C₄×C₅₃	#95	1	1	yes	characteristic
40	C₄×C₅₃	#97	1	1	yes	characteristic
41	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
43	#1377	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#336	1	1	yes	characteristic
47	C₂×C₅₃	#349	1	2	yes	—
48	C₂×M₃₂	—	53	2	no	—
49	C₂×#97	—	53	1	no	—
50	C₅₃	#1377	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	—	53	1	no	—
52	C₂²×C₈	—	53	1	no	—
53	C₂×C₄²	#6157	1	1	yes	characteristic
54	M₃₂	—	53	4	no	—
55	M₃₂	—	106	2	no	—
56	C₂×C₁₆	—	106	2	no	—
57	#97	—	106	2	no	—
58	C₂²×C₄	#88324	1	2	yes	—
59	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
60	C₂×C₈	—	53	2	no	—
61	C₂×C₈	—	53	1	no	—
62	C₂×C₈	—	53	1	no	—
63	C₂×C₈	—	53	2	no	—
64	C₂×C₈	—	53	2	no	—
65	C₂×C₈	—	106	1	no	—
66	C₂×C₈	—	106	1	no	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₁₆	—	106	4	no	—
70	C₂³	C₂×#88324	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
74	C₂×C₄	#11719936	1	1	yes	characteristic
75	C₂×C₄	C₄×#6157	1	1	yes	characteristic
76	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
77	C₂×C₄	#11719940	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
79	C₈	—	53	2	no	—
80	C₈	—	53	2	no	—
81	C₈	—	106	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	C₄×#88324	1	1	yes	characteristic
84	C₂²	C₂²⋊#88324	1	1	yes	characteristic
85	C₂²	C₄⋊#88324	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
88	C₄	#13133272	1	1	yes	characteristic
89	C₄	#13133274	1	1	yes	characteristic
90	C₄	#13133284	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#16811489	1	1	yes	characteristic
94	C₂	#16811519	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#6157
commutator	C₂²×C₅₃	C₄×C₈
Frattini	C₂²×C₄	C₂×#6157
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₂×C₄²	#6157
hypocenter	C₅₃	#1377
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #13133291	C₄	semidirect
50. C₅₃	#1377	semidirect