Subgroups — #22581236

Order 8896.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984841	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#17984774	C₂	1	1	yes	maximal, characteristic
4	#17984773	C₂	1	1	yes	maximal, characteristic
5	Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#17984766	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₇₈	—	2	1	no	—
9	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₉	—	2	1	no	—
11	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
13	Dic₁₃₉⋊C₄	—	2	1	no	—
14	C₄×Dic₁₃₉	—	2	1	no	—
15	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	Dic₂₇₈	—	4	2	no	—
20	C₂×Dic₁₃₉	—	2	1	no	—
21	C₂×Dic₁₃₉	—	2	1	no	—
22	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
24	C₁₃₉⋊C₈	—	2	1	no	—
25	Q₈×C₁₃₉	—	2	1	no	—
26	Q₈×C₁₃₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₁₃₉	—	2	1	no	—
28	C₂×C₄×C₁₃₉	—	2	1	no	—
29	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₈×C₁₃₉	—	2	1	no	—
31	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₃₉	—	2	2	no	—
33	C₄×C₁₃₉	—	4	1	no	—
34	C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₁₃₉	#41	1	1	yes	characteristic
36	Dic₁₃₉	—	2	1	no	—
37	Dic₁₃₉	—	4	1	no	—
38	Dic₁₃₉	—	4	1	no	—
39	Dic₁₃₉	SD₁₆	1	2	yes	—
40	C₂×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₁₃₉	#137	1	1	yes	characteristic
43	C₁₃₉	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	139	1	no	maximal, Sylow
45	C₄⋊Q₈	—	139	1	no	—
46	C₄×Q₈	—	139	1	no	—
47	#106	—	139	1	no	—
48	#106	—	139	1	no	—
49	C₄⋊C₈	—	139	1	no	—
50	#103	—	139	1	no	—
51	#103	D₁₃₉	1	1	yes	characteristic
52	C₂×Q₈	—	278	1	no	—
53	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
54	C₂×C₈	—	139	1	no	—
55	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
56	C₄⋊C₄	—	278	1	no	—
57	C₄⋊C₄	—	278	1	no	—
58	C₄⋊C₄	—	139	1	no	—
59	C₄⋊C₄	—	139	1	no	—
60	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
61	C₄²	—	278	1	no	—
62	C₄²	—	139	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	556	2	no	—
65	Q₈	Dic₂₇₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	278	1	no	—
69	C₂×C₄	—	278	1	no	—
70	C₂×C₄	—	139	1	no	—
71	C₂×C₄	—	139	1	no	—
72	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	278	1	no	—
75	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	278	1	no	—
79	C₄	—	556	1	no	—
80	C₄	—	556	1	no	—
81	C₄	—	139	2	no	—
82	C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
83	C₄	#14555784	1	1	yes	characteristic
84	C₂	#17984819	1	1	yes	characteristic
85	C₂	SD₁₆×D₁₃₉	1	1	yes	characteristic
86	C₂	#17984874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#103×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#103	D₁₃₉
hypocenter	C₁₃₉	#475
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₁₃₉	Q₈	semidirect
43. C₁₃₉	#475	semidirect
65. Q₈	Dic₂₇₈	semidirect