Subgroups — #16837694

Order 3776.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#14273660	C₂	1	2	yes	maximal
3	C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#14273650	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₅₉	—	2	1	no	—
7	C₂×Dic₁₁₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₅₉	—	2	1	no	—
9	C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
10	C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₄	—	2	1	no	—
13	C₄×Dic₅₉	—	2	1	no	—
14	C₂×C₅₉⋊C₈	C₂²	1	2	yes	—
15	Q₈×C₅₉	—	4	2	no	—
16	C₂×C₄×C₅₉	—	2	2	no	—
17	C₂×C₄×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×Dic₅₉	—	2	1	no	—
21	C₂×Dic₅₉	—	2	1	no	—
22	Dic₁₁₈	—	2	1	no	—
23	Dic₁₁₈	Q₈	1	2	yes	complemented
24	C₅₉⋊C₈	—	2	2	no	—
25	C₂²×C₅₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₅₉	—	2	1	no	—
27	C₄×C₅₉	—	4	2	no	—
28	C₄×C₅₉	Q₁₆	1	2	yes	—
29	C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₅₉	#41	1	1	yes	characteristic
31	Dic₅₉	—	2	2	no	—
32	Dic₅₉	—	4	1	no	—
33	C₂×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	#477	—	59	1	no	maximal, Sylow
37	C₅₉	#477	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	D₅₉	1	1	yes	characteristic
39	C₄×Q₈	—	59	1	no	—
40	#107	—	59	2	no	—
41	C₄⋊C₈	—	59	1	no	—
42	#103	—	59	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	59	1	no	—
45	C₂×C₈	—	59	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	59	1	no	—
48	C₄⋊C₄	—	118	1	no	—
49	C₄⋊C₄	C₂×D₅₉	1	2	yes	—
50	C₄²	—	118	1	no	—
51	C₄²	C₂×D₅₉	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	59	2	no	—
54	Q₈	—	118	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	118	1	no	—
57	C₂×C₄	—	118	1	no	—
58	C₂×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
61	C₈	—	118	2	no	—
62	C₂²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	118	2	no	—
66	C₄	—	236	1	no	—
67	C₄	#11745672	1	2	yes	—
68	C₄	Q₈×D₅₉	1	1	yes	characteristic
69	C₄	#11745689	1	1	yes	characteristic
70	C₂	#14273703	1	1	yes	characteristic
71	C₂	C₂×#11745672	1	1	yes	characteristic
72	C₂	#14273753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₉
commutator, cosocle	C₂×C₄×C₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₉
Fitting	C₄⋊Q₈×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#477
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. Dic₁₁₈	Q₈	semidirect
37. C₅₉	#477	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₁₈	2
#14273660	2
C₄⋊(C₅₉⋊C₈)	2
#14273650	2
C₄⋊Q₈×C₅₉	2
#477	59