Subgroups — #31806022

Order 17792.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581358	C₂	1	4	yes	maximal
2	#22581333	C₂	1	2	yes	maximal
3	C₈²×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₅₆	—	2	4	no	—
5	#17984842	C₂²	1	4	yes	—
6	#17984776	—	2	8	no	—
7	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
9	C₂×Dic₂₇₈	—	2	4	no	—
10	Dic₅₅₆	—	4	8	no	—
11	C₄⋊Dic₁₃₉	—	2	8	no	—
12	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
13	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
14	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
15	Dic₂₇₈	—	4	8	no	—
16	C₂×Dic₁₃₉	—	4	4	no	—
17	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
19	C₈×C₁₃₉	D₈	1	4	yes	—
20	C₈×C₁₃₉	Q₁₆	1	8	yes	complemented
21	C₂²×C₁₃₉	#127	1	1	yes	characteristic
22	C₄×C₁₃₉	C₂×D₈	1	2	yes	—
23	C₄×C₁₃₉	C₂×Q₁₆	1	4	yes	—
24	Dic₁₃₉	—	8	4	no	—
25	C₂×C₁₃₉	#493	1	1	yes	characteristic
26	C₂×C₁₃₉	#494	1	2	yes	—
27	C₁₃₉	#1713	1	1	yes	characteristic, complemented, Sylow
28	#1713	—	139	1	no	maximal, Sylow
29	#500	—	139	4	no	—
30	#494	—	139	2	no	—
31	C₈²	D₁₃₉	1	1	yes	characteristic
32	C₂×Q₁₆	—	278	4	no	—
33	C₄⋊Q₈	—	139	4	no	—
34	#107	—	278	8	no	—
35	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
36	C₄×C₈	C₂×D₁₃₉	1	2	yes	—
37	C₂×Q₈	—	278	4	no	—
38	Q₁₆	—	556	8	no	—
39	C₂×C₈	D₅₅₆	1	2	yes	—
40	C₂×C₈	D₅₅₆	1	4	yes	—
41	C₄⋊C₄	—	278	8	no	—
42	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
43	Q₈	—	556	8	no	—
44	C₂×C₄	—	556	4	no	—
45	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
46	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
47	C₈	D₁₁₁₂	1	4	yes	—
48	C₈	Dic₅₅₆	1	8	yes	complemented
49	C₂²	#17984839	1	1	yes	characteristic
50	C₄	—	1112	4	no	—
51	C₄	C₂×D₁₁₁₂	1	2	yes	—
52	C₄	C₂×Dic₅₅₆	1	4	yes	—
53	C₂	#22581330	1	1	yes	characteristic
54	C₂	#22581333	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984839
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	C₈²×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#127
hypercenter	C₈²	D₁₃₉
hypocenter	C₁₃₉	#1713
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₃₉	Q₁₆	semidirect
27. C₁₃₉	#1713	semidirect
48. C₈	Dic₅₅₆	semidirect

Maximal subgroups

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

subgroup	index
#22581358	2
#22581333	2
C₈²×C₁₃₉	2
#1713	139