Subgroups — #29504879

Order 15296.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₃₉	C₂	1	2	yes	maximal
2	#21439892	C₂	1	1	yes	maximal, characteristic
3	#21439884	C₂	1	2	yes	maximal
4	#21439870	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₃₉	—	2	2	no	—
8	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	C₄×Dic₂₃₉	—	2	2	no	—
10	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₂₃₉	C₂²	1	2	yes	—
12	Q₁₆×C₂₃₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₃₉	—	2	2	no	—
15	C₂×Dic₂₃₉	D₄	1	2	yes	—
16	C₂₃₉⋊C₈	—	2	1	no	—
17	Q₈×C₂₃₉	—	2	2	no	—
18	Q₈×C₂₃₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₂₃₉	—	2	2	no	—
20	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
21	C₈×C₂₃₉	C₂×C₄	1	2	yes	—
22	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₃₉	—	2	4	no	—
24	C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₃₉	#41	1	1	yes	characteristic
26	Dic₂₃₉	—	2	2	no	—
27	Dic₂₃₉	—	4	2	no	—
28	C₂×C₂₃₉	C₄×D₄	1	1	yes	characteristic
29	C₂×C₂₃₉	#137	1	2	yes	—
30	C₂₃₉	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	239	1	no	maximal, Sylow
32	C₂×Q₁₆	D₂₃₉	1	1	yes	characteristic
33	C₄×Q₈	—	239	2	no	—
34	#106	—	239	1	no	—
35	#103	—	239	2	no	—
36	#97	—	239	1	no	—
37	C₂×Q₈	C₂×D₂₃₉	1	2	yes	—
38	Q₁₆	Dic₂₃₉	1	4	yes	complemented
39	C₂×C₈	—	239	1	no	—
40	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
41	C₄⋊C₄	—	478	2	no	—
42	C₄⋊C₄	—	239	2	no	—
43	C₄²	—	478	2	no	—
44	C₄²	—	239	1	no	—
45	Q₈	—	2	2	no	—
46	Q₈	C₂×Dic₂₃₉	1	4	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	478	2	no	—
49	C₂×C₄	—	239	2	no	—
50	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
51	C₈	—	478	1	no	—
52	C₈	C₂×Dic₂₃₉	1	2	yes	—
53	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	956	2	no	—
56	C₄	—	478	2	no	—
57	C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
58	C₄	#16838547	1	1	yes	characteristic
59	C₂	D₄×Dic₂₃₉	1	1	yes	characteristic
60	C₂	#21439992	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator	C₄×C₂₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₂×Q₁₆×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₄
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	C₂×Q₁₆	D₂₃₉
hypocenter	C₂₃₉	#441
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
12. Q₁₆×C₂₃₉	C₄	semidirect
30. C₂₃₉	#441	semidirect
38. Q₁₆	Dic₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₃₉	2
#21439892	2
#21439884	2
#21439870	2
C₂×Q₁₆×C₂₃₉	2
#441	239