Subgroups — #16810901

Order 3392.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133216	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133151	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133150	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133146	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133138	C₂	1	1	yes	maximal, characteristic
7	#102×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₃	—	2	1	no	—
10	C₂×Dic₁₀₆	—	2	1	no	—
11	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅₃	—	2	1	no	—
14	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	#11719919	—	2	1	no	—
17	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#88322	C₂²	1	1	yes	characteristic
19	C₂³×C₅₃	—	2	1	no	—
20	D₄×C₅₃	—	2	1	no	—
21	D₄×C₅₃	Q₈	1	2	yes	complemented
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₅₃	—	2	1	no	—
25	C₂×Dic₅₃	—	2	1	no	—
26	C₂×Dic₅₃	—	4	2	no	—
27	C₂×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	D₄	1	1	yes	characteristic
29	Dic₁₀₆	—	4	2	no	—
30	C₈×C₅₃	—	2	1	no	—
31	#88322	—	2	1	no	—
32	C₂²×C₅₃	—	2	2	no	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	—	4	1	no	—
35	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅₃	—	4	1	no	—
37	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₅₃	#41	1	1	yes	characteristic
39	Dic₅₃	—	2	1	no	—
40	Dic₅₃	—	4	1	no	—
41	Dic₅₃	—	4	1	no	—
42	Dic₅₃	D₈	1	2	yes	—
43	C₂×C₅₃	—	4	2	no	—
44	C₂×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
45	C₂×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₅₃	#137	1	1	yes	characteristic
47	#474	—	53	1	no	maximal, Sylow
48	C₅₃	#474	1	1	yes	characteristic, complemented, Sylow
49	C₄⋊Q₈	—	53	1	no	—
50	C₄×D₄	—	53	1	no	—
51	#107	—	53	1	no	—
52	#107	—	53	1	no	—
53	C₄⋊C₈	—	53	1	no	—
54	#102	—	53	1	no	—
55	#102	D₅₃	1	1	yes	characteristic
56	C₂×Q₈	—	106	1	no	—
57	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
58	C₂²×C₄	—	106	1	no	—
59	C₂×C₈	—	53	1	no	—
60	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₄⋊C₄	—	53	1	no	—
62	C₄⋊C₄	—	53	1	no	—
63	C₄⋊C₄	—	106	1	no	—
64	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
65	C₂²⋊C₄	—	106	1	no	—
66	C₄²	—	53	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	212	2	no	—
69	D₄	—	2	1	no	—
70	D₄	Dic₁₀₆	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	53	1	no	—
73	C₂×C₄	—	53	1	no	—
74	C₂×C₄	—	106	1	no	—
75	C₂×C₄	—	106	1	no	—
76	C₂×C₄	—	212	2	no	—
77	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	106	1	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₅₃	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	53	2	no	—
86	C₄	—	106	1	no	—
87	C₄	—	212	1	no	—
88	C₄	—	212	1	no	—
89	C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
90	C₄	#11719947	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	#13133194	1	1	yes	characteristic
93	C₂	D₈×D₅₃	1	1	yes	characteristic
94	C₂	#13133246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#102×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	C₂×D₄×C₅₃	C₂²
hypercenter	#102	D₅₃
hypocenter	C₅₃	#474
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
3. #13133151	C₂	semidirect
4. #13133150	C₂	semidirect
5. #13133146	C₂	semidirect
21. D₄×C₅₃	Q₈	semidirect
48. C₅₃	#474	semidirect
70. D₄	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#13133216	2
D₄×Dic₅₃	2
#13133151	2
#13133150	2
#13133146	2
#13133138	2
#102×C₅₃	2
#474	53