Subgroups — #29505180

Order 15296.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439959	C₂	1	1	yes	maximal, characteristic
2	#21439951	C₂	1	1	yes	maximal, characteristic
3	#21439893	C₂	1	1	yes	maximal, characteristic
4	#21439892	C₂	1	1	yes	maximal, characteristic
5	#21439891	C₂	1	1	yes	maximal, characteristic
6	#21439870	C₂	1	1	yes	maximal, characteristic
7	#107×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₈	—	2	1	no	—
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃₉	—	2	1	no	—
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	Dic₂₃₉⋊C₄	—	2	1	no	—
14	Dic₂₃₉⋊C₄	—	2	1	no	—
15	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
19	Dic₄₇₈	—	4	2	no	—
20	C₂×Dic₂₃₉	—	2	1	no	—
21	C₂×Dic₂₃₉	—	2	1	no	—
22	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
23	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
24	C₂₃₉⋊C₈	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₃₉	—	2	1	no	—
26	C₂×C₄×C₂₃₉	—	2	1	no	—
27	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
28	C₈×C₂₃₉	Q₈	1	2	yes	complemented
29	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₃₉	—	4	1	no	—
31	C₄×C₂₃₉	—	4	1	no	—
32	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
34	Dic₂₃₉	—	2	1	no	—
35	Dic₂₃₉	—	2	1	no	—
36	Dic₂₃₉	—	4	1	no	—
37	Dic₂₃₉	—	4	1	no	—
38	C₂×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₂₃₉	#137	1	1	yes	characteristic
41	C₂₃₉	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	239	1	no	maximal, Sylow
43	C₄⋊Q₈	—	239	1	no	—
44	#125	—	239	1	no	—
45	#107	—	239	1	no	—
46	#107	D₂₃₉	1	1	yes	characteristic
47	#106	—	239	1	no	—
48	#106	—	239	1	no	—
49	#97	—	239	1	no	—
50	C₂×Q₈	—	478	1	no	—
51	C₂×C₈	—	239	1	no	—
52	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
53	C₄⋊C₄	—	478	1	no	—
54	C₄⋊C₄	—	478	1	no	—
55	C₄⋊C₄	—	478	1	no	—
56	C₄⋊C₄	—	239	1	no	—
57	C₄⋊C₄	—	239	1	no	—
58	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
60	C₄²	—	239	1	no	—
61	Q₈	—	956	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	478	1	no	—
65	C₂×C₄	—	478	1	no	—
66	C₂×C₄	—	239	1	no	—
67	C₂×C₄	—	239	1	no	—
68	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
69	C₈	—	239	2	no	—
70	C₈	Dic₄₇₈	1	2	yes	complemented
71	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	956	1	no	—
75	C₄	—	956	1	no	—
76	C₄	—	478	1	no	—
77	C₄	—	478	1	no	—
78	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
79	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
80	C₂	#21439959	1	1	yes	characteristic
81	C₂	#21439984	1	1	yes	characteristic
82	C₂	#21439992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#107×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₄
hypercenter	#107	D₂₃₉
hypocenter	C₂₃₉	#501
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
24. C₂₃₉⋊C₈	Q₈	semidirect
28. C₈×C₂₃₉	Q₈	semidirect
41. C₂₃₉	#501	semidirect
70. C₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#21439959	2
#21439951	2
#21439893	2
#21439892	2
#21439891	2
#21439870	2
#107×C₂₃₉	2
#501	239