Subgroups — #29505577

Order 15296.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440006	C₂	1	2	yes	maximal, complemented
2	#21439988	C₂	1	6	yes	maximal, complemented
3	#21439982	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₉₁₂	C₂	1	3	yes	maximal, complemented
5	#131×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16838549	—	2	2	no	—
7	#16838548	C₂²	1	6	yes	complemented
8	C₂×D₉₅₆	C₂²	1	6	yes	complemented
9	D₄×D₂₃₉	—	2	6	no	—
10	Dic₉₅₆	C₂²	1	1	yes	characteristic, complemented
11	#16838537	C₂²	1	6	yes	complemented
12	D₁₉₁₂	C₂²	1	3	yes	complemented
13	D₁₉₁₂	C₂²	1	6	yes	complemented
14	#41×C₂₃₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₃₉	C₂²	1	3	yes	complemented
16	C₂×C₈×C₂₃₉	C₂²	1	3	yes	complemented
17	C₂²×D₂₃₉	—	2	6	no	—
18	Dic₄₇₈	C₂³	1	2	yes	—
19	D₉₅₆	—	2	6	no	—
20	D₉₅₆	C₂³	1	6	yes	—
21	C₂²⋊D₂₃₉	—	2	6	no	—
22	C₄×D₂₃₉	—	2	6	no	—
23	Q₈×C₂₃₉	D₄	1	1	yes	characteristic
24	D₄×C₂₃₉	D₄	1	3	yes	—
25	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
26	C₈×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₈×C₂₃₉	C₂³	1	3	yes	—
28	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
29	C₂×D₂₃₉	—	2	6	no	—
30	C₂×D₂₃₉	—	8	6	no	—
31	C₄×C₂₃₉	C₂×D₄	1	3	yes	—
32	C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
33	Dic₂₃₉	—	2	2	no	—
34	C₂×C₂₃₉	—	2	3	no	—
35	C₂×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
36	D₂₃₉	—	4	6	no	—
37	C₂₃₉	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	239	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	239	2	no	—
40	AGL(1,ℤ/8)	—	239	6	no	—
41	#135	—	239	3	no	—
42	C₂×D₈	—	239	3	no	—
43	#131	D₂₃₉	1	1	yes	characteristic, complemented
44	#41	—	478	2	no	—
45	#41	—	239	6	no	—
46	#41	C₂×D₂₃₉	1	1	yes	characteristic
47	C₂×D₄	—	478	6	no	—
48	C₂×D₄	—	239	6	no	—
49	Q₁₆	—	239	1	no	—
50	SD₁₆	—	239	6	no	—
51	D₈	—	239	3	no	—
52	D₈	—	239	6	no	—
53	M₁₆	C₂×D₂₃₉	1	3	yes	complemented
54	C₂×C₈	C₂×D₂₃₉	1	3	yes	complemented
55	C₂³	—	478	6	no	—
56	Q₈	—	239	2	no	—
57	Q₈	D₉₅₆	1	1	yes	characteristic
58	D₄	—	478	6	no	—
59	D₄	—	478	6	no	—
60	D₄	—	239	6	no	—
61	D₄	D₉₅₆	1	3	yes	—
62	C₂×C₄	—	478	6	no	—
63	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
64	C₈	C₂²×D₂₃₉	1	1	yes	characteristic
65	C₈	C₂²×D₂₃₉	1	3	yes	—
66	C₂²	—	1912	6	no	—
67	C₂²	—	478	6	no	—
68	C₂²	C₂×D₉₅₆	1	3	yes	—
69	C₄	—	478	2	no	—
70	C₄	C₂×D₉₅₆	1	3	yes	—
71	C₄	C₂³×D₂₃₉	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	956	6	no	—
74	C₂	C₂²×D₉₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₉₅₆
commutator, cosocle	C₄×C₂₃₉	C₂⁴
Frattini	C₄	C₂³×D₂₃₉
Fitting	#131×C₂₃₉	C₂
socle	C₂×C₂₃₉	C₂²×D₄
hypercenter	#131	D₂₃₉
hypocenter	C₂₃₉	#576
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #21440006	C₂	semidirect
2. #21439988	C₂	semidirect
3. #21439982	C₂	semidirect
4. C₂×D₁₉₁₂	C₂	semidirect
5. #131×C₂₃₉	C₂	semidirect
7. #16838548	C₂²	semidirect
8. C₂×D₉₅₆	C₂²	semidirect
10. Dic₉₅₆	C₂²	semidirect
11. #16838537	C₂²	semidirect
12. D₁₉₁₂	C₂²	semidirect
13. D₁₉₁₂	C₂²	semidirect
15. M₁₆×C₂₃₉	C₂²	semidirect
16. C₂×C₈×C₂₃₉	C₂²	semidirect
37. C₂₃₉	#576	semidirect
43. #131	D₂₃₉	semidirect
53. M₁₆	C₂×D₂₃₉	semidirect
54. C₂×C₈	C₂×D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21440006	2
#21439988	2
#21439982	2
C₂×D₁₉₁₂	2
#131×C₂₃₉	2
#576	239