Subgroups — #16810754

Order 3392.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#88322	C₂	1	2	yes	maximal
2	C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#13133128	C₂	1	2	yes	maximal
4	C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₅₃	C₄	1	2	yes	—
8	C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
9	C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₄²×C₅₃	C₂²	1	2	yes	—
11	C₂×#88322	C₂²	1	1	yes	characteristic
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂×#88322	C₂²	1	1	yes	characteristic
14	Q₈×C₅₃	—	2	2	no	—
15	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	#88322	—	2	1	no	—
21	#88322	—	2	2	no	—
22	#88322	Q₈	1	2	yes	complemented
23	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₃	—	2	2	no	—
25	C₄×C₅₃	—	2	1	no	—
26	C₄×C₅₃	—	2	2	no	—
27	C₄×C₅₃	M₁₆	1	2	yes	—
28	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅₃	#41	1	1	yes	characteristic
30	C₂×C₅₃	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₃	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₅₃	#131	1	1	yes	characteristic
33	#446	—	53	1	no	maximal, Sylow
34	C₅₃	#446	1	1	yes	characteristic, complemented, Sylow
35	C₄×Q₈	D₅₃	1	1	yes	characteristic
36	C₄⋊C₈	—	53	1	no	—
37	C₄⋊C₈	—	53	2	no	—
38	#97	—	53	2	no	—
39	C₄×C₈	—	53	1	no	—
40	C₂×Q₈	Dic₅₃	1	1	yes	characteristic
41	C₂×C₈	—	53	1	no	—
42	C₂×C₈	—	53	1	no	—
43	C₂×C₈	—	53	2	no	—
44	C₄⋊C₄	Dic₅₃	1	2	yes	—
45	C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
46	C₄²	C₂×D₅₃	1	1	yes	characteristic
47	C₄²	C₂×D₅₃	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₅₃	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₅₃	1	2	yes	—
51	C₂×C₄	C₂×Dic₅₃	1	2	yes	—
52	C₂×C₄	C₂×Dic₅₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
54	C₈	—	53	2	no	—
55	C₈	—	106	1	no	—
56	C₈	—	106	2	no	—
57	C₂²	C₂²×Dic₅₃	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#11719917	1	2	yes	—
62	C₄	Q₈×D₅₃	1	1	yes	characteristic
63	C₄	#11719949	1	1	yes	characteristic
64	C₂	Q₈×Dic₅₃	1	1	yes	characteristic
65	C₂	C₂×#11719917	1	1	yes	characteristic
66	C₂	#13133224	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₃
commutator, socle, prime radical	C₂²×C₅₃	C₂²×C₄
Fitting	C₄×Q₈×C₅₃	C₂
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	C₄×Q₈	D₅₃
hypocenter	C₅₃	#446
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
22. #88322	Q₈	semidirect
34. C₅₃	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#88322	2
#13133128	2
C₄×#88322	2
C₄×Q₈×C₅₃	2
#446	53