Subgroups — #22581332

Order 8896.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555775	C₂	1	4	yes	maximal
2	#17984842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₇₈	C₂²	1	2	yes	—
6	#14555775	—	2	8	no	—
7	C₂×C₁₃₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₃₉	—	2	2	no	—
9	C₂×Q₈×C₁₃₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₃₉	—	2	2	no	—
11	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	Dic₂₇₈	—	2	4	no	—
13	C₂×Dic₁₃₉	—	2	2	no	—
14	C₁₃₉⋊C₈	D₄	1	4	yes	—
15	Q₈×C₁₃₉	—	2	4	no	—
16	C₂×C₄×C₁₃₉	—	2	2	no	—
17	C₂×C₄×C₁₃₉	D₄	1	2	yes	—
18	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
19	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₃₉	—	4	2	no	—
21	C₄×C₁₃₉	Q₁₆	1	4	yes	complemented
22	C₄×C₁₃₉	C₂×D₄	1	2	yes	—
23	Dic₁₃₉	—	4	2	no	—
24	C₂×C₁₃₉	#127	1	1	yes	characteristic
25	C₂×C₁₃₉	C₂×Q₁₆	1	2	yes	—
26	C₁₃₉	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	139	1	no	maximal, Sylow
28	C₂×Q₁₆	—	139	4	no	—
29	C₄⋊Q₈	—	139	1	no	—
30	C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic
31	C₄×C₈	—	139	1	no	—
32	C₂×Q₈	—	139	2	no	—
33	C₂×Q₈	C₂×D₁₃₉	1	2	yes	—
34	Q₁₆	—	278	8	no	—
35	C₂×C₈	—	139	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	278	2	no	—
38	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	278	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	278	2	no	—
43	C₂×C₄	C₂²⋊D₁₃₉	1	2	yes	—
44	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
45	C₈	—	139	4	no	—
46	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	556	2	no	—
49	C₄	#14555775	1	4	yes	complemented
50	C₄	D₄×D₁₃₉	1	2	yes	—
51	C₂	#17984838	1	1	yes	characteristic
52	C₂	C₂×#14555775	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	C₄⋊Q₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	#494
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₁₃₉	Q₁₆	semidirect
26. C₁₃₉	#494	semidirect
49. C₄	#14555775	semidirect