Subgroups — #20291088

Order 6784.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811546	C₂	1	2	yes	maximal, complemented
2	#16811519	C₂	1	2	yes	maximal
3	#16811515	C₂	1	2	yes	maximal, complemented
4	#16810579	C₂	1	1	yes	maximal, characteristic
5	#13133291	C₂²	1	4	yes	—
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₄	1	1	yes	characteristic
8	#13133128	C₄	1	1	yes	characteristic
9	#13133128	C₂²	1	1	yes	characteristic
10	C₄×#88322	C₄	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂²⋊C₄×C₅₃	—	2	1	no	—
16	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#11719917	—	2	2	no	—
19	C₂×#88322	—	2	2	no	—
20	C₂×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	#11719910	—	2	4	no	—
25	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	#88322	—	2	1	no	—
33	#88322	—	2	1	no	—
34	#88322	M₁₆	1	2	yes	—
35	#88322	#41	1	2	yes	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	4	1	no	—
41	C₄×C₅₃	—	4	1	no	—
42	C₄×C₅₃	#117	1	1	yes	characteristic
43	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	#2166	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	4	1	no	—
47	C₂×C₅₃	#432	1	1	yes	characteristic
48	#405	—	53	1	no	—
49	#364	—	53	2	no	—
50	#349	—	53	2	no	—
51	#347	—	53	2	no	—
52	C₅₃	#2166	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₁₆	—	53	1	no	—
54	C₂²×C₈	—	53	1	no	—
55	#117	#6157	1	1	yes	characteristic
56	M₃₂	—	53	4	no	—
57	#97	—	53	1	no	—
58	#97	—	53	1	no	—
59	C₄×C₈	—	53	1	no	—
60	C₄×C₈	—	53	1	no	—
61	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
62	M₁₆	—	106	2	no	—
63	C₂×C₈	—	53	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	53	1	no	—
67	C₂×C₈	—	106	2	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	—	2	1	no	—
70	C₄²	C₂×#6157	1	1	yes	characteristic
71	C₄²	C₂×#6157	1	1	yes	characteristic
72	C₁₆	—	106	4	no	—
73	C₂³	#11719940	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	#11719940	1	1	yes	characteristic
79	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
80	C₈	—	53	2	no	—
81	C₈	—	53	2	no	—
82	C₈	—	106	1	no	—
83	C₈	—	106	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₂×#11719940	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	4	1	no	—
90	C₄	#13133296	1	1	yes	characteristic
91	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#16811643	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#117×C₅₃	C₄
socle	C₂×C₅₃	#432
hypercenter	#117	#6157
hypocenter	C₅₃	#2166
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₄² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #16811546	C₂	semidirect
3. #16811515	C₂	semidirect
52. C₅₃	#2166	semidirect