Subgroups — #22581395

Order 8896.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984865	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984788	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984787	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984785	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984781	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555786	—	2	1	no	—
9	Q₈×D₁₃₉	—	2	1	no	—
10	Dic₅₅₆	C₂²	1	1	yes	characteristic
11	#14555775	—	2	1	no	—
12	#14555772	—	2	1	no	—
13	D₁₁₁₂	C₂²	1	1	yes	characteristic
14	C₈×D₁₃₉	C₂²	1	1	yes	characteristic
15	C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
17	Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	Dic₂₇₈	—	2	1	no	—
20	Dic₂₇₈	—	4	1	no	—
21	D₅₅₆	—	2	1	no	—
22	D₅₅₆	—	4	1	no	—
23	C₄×D₁₃₉	—	4	1	no	—
24	C₄×D₁₃₉	—	4	1	no	—
25	C₄×D₁₃₉	D₄	1	1	yes	characteristic
26	C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₁₃₉	—	2	1	no	—
28	Q₈×C₁₃₉	—	2	1	no	—
29	C₈×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₂×D₁₃₉	—	4	1	no	—
31	C₂×D₁₃₉	D₈	1	1	yes	characteristic
32	C₄×C₁₃₉	—	4	1	no	—
33	C₄×C₁₃₉	—	4	1	no	—
34	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	Dic₁₃₉	—	4	1	no	—
36	Dic₁₃₉	D₈	1	1	yes	characteristic
37	C₂×C₁₃₉	C₂×D₈	1	1	yes	characteristic
38	D₁₃₉	—	2	1	no	—
39	D₁₃₉	—	8	1	no	—
40	C₁₃₉	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	139	1	no	maximal, Sylow
42	#135	—	139	1	no	—
43	C₂×Q₁₆	—	139	1	no	—
44	Q₃₂	—	139	1	no	—
45	Q₃₂	D₁₃₉	1	1	yes	characteristic, complemented
46	SD₃₂	—	139	1	no	—
47	SD₃₂	—	139	1	no	—
48	M₃₂	—	139	1	no	—
49	#41	—	278	1	no	—
50	C₂×Q₈	—	278	1	no	—
51	Q₁₆	—	278	1	no	—
52	Q₁₆	—	139	1	no	—
53	Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
54	Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
55	SD₁₆	—	278	1	no	—
56	D₈	—	139	1	no	—
57	C₂×C₈	—	139	1	no	—
58	C₁₆	—	139	1	no	—
59	C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	278	1	no	—
63	Q₈	—	556	1	no	—
64	D₄	—	278	1	no	—
65	D₄	—	556	1	no	—
66	C₂×C₄	—	556	1	no	—
67	C₂×C₄	—	556	1	no	—
68	C₂×C₄	—	139	1	no	—
69	C₈	—	139	1	no	—
70	C₈	C₂²×D₁₃₉	1	1	yes	characteristic
71	C₂²	—	556	1	no	—
72	C₂²	—	139	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	556	1	no	—
76	C₄	—	139	1	no	—
77	C₄	D₄×D₁₃₉	1	1	yes	characteristic
78	C₂	—	278	1	no	—
79	C₂	—	1112	1	no	—
80	C₂	D₈×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₃₉
commutator	C₈×C₁₃₉	C₂³
Frattini	C₈	C₂²×D₁₃₉
Fitting	Q₃₂×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×D₈
hypercenter	Q₃₂	D₁₃₉
hypocenter	C₁₃₉	#510
LCS	G ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₁₃₉	C₂	semidirect
3. #17984788	C₂	semidirect
4. #17984787	C₂	semidirect
5. #17984785	C₂	semidirect
6. #17984781	C₂	semidirect
7. Q₃₂×C₁₃₉	C₂	semidirect
40. C₁₃₉	#510	semidirect
45. Q₃₂	D₁₃₉	semidirect