Subgroups — #22580696

Order 8896.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984839	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₅₆	—	2	1	no	—
5	C₂×D₅₅₆	—	2	1	no	—
6	C₂×D₅₅₆	C₄	1	2	yes	—
7	C₂×C₁₃₉⋊C₈	—	2	1	no	—
8	C₂×C₈×C₁₃₉	—	2	1	no	—
9	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂²×D₁₃₉	—	2	2	no	—
11	D₅₅₆	—	4	2	no	—
12	D₅₅₆	—	4	1	no	—
13	D₅₅₆	—	4	1	no	—
14	D₅₅₆	—	4	1	no	—
15	D₅₅₆	—	4	1	no	—
16	C₁₃₉⋊C₈	—	4	1	no	—
17	C₂×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
20	C₈×C₁₃₉	—	4	1	no	—
21	C₂²×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₃₉	—	4	2	no	—
23	C₂×D₁₃₉	—	4	2	no	—
24	C₂×D₁₃₉	—	4	2	no	—
25	C₄×C₁₃₉	—	2	1	no	—
26	C₄×C₁₃₉	D₈	1	1	yes	characteristic
27	C₄×C₁₃₉	D₈	1	1	yes	characteristic
28	C₄×C₁₃₉	SD₁₆	1	1	yes	characteristic
29	C₄×C₁₃₉	SD₁₆	1	1	yes	characteristic
30	C₂×C₁₃₉	#100	1	1	yes	characteristic
31	C₂×C₁₃₉	#102	1	1	yes	characteristic
32	C₂×C₁₃₉	#102	1	1	yes	characteristic
33	D₁₃₉	—	8	2	no	—
34	C₁₃₉	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	139	1	no	maximal, Sylow
36	#127	—	139	1	no	—
37	C₄⋊C₈	—	139	1	no	—
38	C₄⋊C₈	D₁₃₉	1	1	yes	characteristic, complemented
39	C₂×D₄	—	278	1	no	—
40	C₂×D₄	—	278	1	no	—
41	C₂×D₄	—	139	2	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	278	1	no	—
44	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
45	C₂³	—	278	2	no	—
46	D₄	—	556	2	no	—
47	D₄	—	556	1	no	—
48	D₄	—	556	1	no	—
49	D₄	—	556	1	no	—
50	D₄	—	556	1	no	—
51	C₂×C₄	C₄×D₁₃₉	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
53	C₂×C₄	D₅₅₆	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	556	1	no	—
56	C₂²	—	556	2	no	—
57	C₂²	—	556	2	no	—
58	C₂²	—	556	2	no	—
59	C₂²	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#14555770	1	1	yes	characteristic
62	C₄	D₁₁₁₂	1	1	yes	characteristic
63	C₄	#14555772	1	1	yes	characteristic
64	C₄	#14555774	1	1	yes	characteristic
65	C₂	—	1112	2	no	—
66	C₂	#17984758	1	1	yes	characteristic
67	C₂	#17984762	1	1	yes	characteristic
68	C₂	#17984764	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₃₉)⋊C₄
commutator	C₂×C₄×C₁₃₉	C₂×C₄
Frattini	C₄²	C₂×D₁₃₉
Fitting	C₄⋊C₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²⋊C₄
hypercenter	C₄⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#331
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₃₉⋊C₈)	C₂	semidirect
3. C₄⋊C₈×C₁₃₉	C₂	semidirect
34. C₁₃₉	#331	semidirect
38. C₄⋊C₈	D₁₃₉	semidirect