Subgroups — #16810751

Order 3392.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#13133146	C₂	1	1	yes	maximal, characteristic
4	#13133146	C₂	1	1	yes	maximal, characteristic
5	#13133127	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	#97×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₆	C₄	1	1	yes	characteristic
9	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
13	#11719919	C₄	1	1	yes	characteristic
14	#11719919	C₄	1	1	yes	characteristic
15	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×#88322	C₂²	1	1	yes	characteristic
18	C₂×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	Dic₁₀₆	—	2	2	no	—
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	Q₈	1	2	yes	complemented
29	#88322	—	2	1	no	—
30	#88322	—	2	1	no	—
31	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₅₃	#41	1	1	yes	characteristic
36	Dic₅₃	—	2	1	no	—
37	Dic₅₃	—	2	1	no	—
38	Dic₅₃	—	2	1	no	—
39	Dic₅₃	M₁₆	1	2	yes	—
40	C₂×C₅₃	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₅₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₅₃	#131	1	1	yes	characteristic
43	#446	—	53	1	no	maximal, Sylow
44	C₅₃	#446	1	1	yes	characteristic, complemented, Sylow
45	C₄×Q₈	—	53	1	no	—
46	C₄⋊C₈	—	53	1	no	—
47	C₄⋊C₈	—	53	1	no	—
48	C₄⋊C₈	—	53	1	no	—
49	#97	—	53	1	no	—
50	#97	D₅₃	1	1	yes	characteristic
51	C₄×C₈	—	53	1	no	—
52	C₂×Q₈	—	53	1	no	—
53	C₂×C₈	—	53	1	no	—
54	C₂×C₈	—	53	1	no	—
55	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
57	C₄⋊C₄	—	53	1	no	—
58	C₄⋊C₄	—	53	1	no	—
59	C₄⋊C₄	—	53	1	no	—
60	C₄²	—	53	1	no	—
61	C₄²	—	53	1	no	—
62	C₄²	C₂×D₅₃	1	1	yes	characteristic
63	Q₈	—	106	2	no	—
64	C₂×C₄	—	53	1	no	—
65	C₂×C₄	—	53	1	no	—
66	C₂×C₄	—	53	1	no	—
67	C₂×C₄	—	53	1	no	—
68	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
69	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	106	1	no	—
73	C₈	—	106	1	no	—
74	C₈	Dic₁₀₆	1	2	yes	complemented
75	C₂²	C₂×C₄×D₅₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	53	2	no	—
79	C₄	—	106	1	no	—
80	C₄	—	106	1	no	—
81	C₄	—	106	1	no	—
82	C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
83	C₄	#11719945	1	1	yes	characteristic
84	C₂	C₄×Dic₁₀₆	1	1	yes	characteristic
85	C₂	M₁₆×D₅₃	1	1	yes	characteristic
86	C₂	#13133225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₃
commutator, socle, prime radical	C₂²×C₅₃	C₂²×C₄
Fitting	#97×C₅₃	C₂
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	#97	D₅₃
hypocenter	C₅₃	#446
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
28. C₈×C₅₃	Q₈	semidirect
44. C₅₃	#446	semidirect
74. C₈	Dic₁₀₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×Dic₁₀₆	2
C₄⋊#88322	2
#13133146	2
#13133127	2
C₈×Dic₅₃	2
#97×C₅₃	2
#446	53