Subgroups — #29505501

Order 15296.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439960	C₂	1	3	yes	maximal
2	#21439951	C₂	1	6	yes	maximal
3	Q₈×Dic₂₃₉	C₂	1	2	yes	maximal
4	C₄×Dic₄₇₈	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₈	C₂²	1	3	yes	—
7	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃₉	C₂²	1	6	yes	—
9	C₄⋊Dic₂₃₉	C₂²	1	6	yes	—
10	Dic₂₃₉⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₂₃₉	C₂²	1	6	yes	—
12	C₂×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃₉	C₂²	1	3	yes	—
14	C₄²×C₂₃₉	C₂²	1	3	yes	—
15	Dic₄₇₈	—	2	6	no	—
16	C₂×Dic₂₃₉	C₂³	1	2	yes	—
17	C₂×Dic₂₃₉	C₂³	1	6	yes	—
18	Q₈×C₂₃₉	Q₈	1	4	yes	complemented
19	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
20	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
21	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₃₉	—	2	3	no	—
24	C₄×C₂₃₉	C₂×Q₈	1	6	yes	—
25	C₄×C₂₃₉	#41	1	2	yes	—
26	Dic₂₃₉	—	2	6	no	—
27	Dic₂₃₉	—	2	2	no	—
28	C₂×C₂₃₉	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₂₃₉	C₂×#41	1	1	yes	characteristic
30	C₂×C₂₃₉	2₋¹⁺⁴	1	1	yes	characteristic
31	C₂₃₉	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	239	1	no	maximal, Sylow
33	C₄⋊Q₈	—	239	3	no	—
34	#125	—	239	6	no	—
35	C₄×Q₈	—	239	2	no	—
36	C₄×Q₈	—	239	3	no	—
37	C₄×Q₈	D₂₃₉	1	1	yes	characteristic
38	C₂×Q₈	—	239	3	no	—
39	C₂×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
40	C₄⋊C₄	—	239	1	no	—
41	C₄⋊C₄	—	239	6	no	—
42	C₄⋊C₄	—	239	6	no	—
43	C₄⋊C₄	—	239	6	no	—
44	C₄⋊C₄	C₂×D₂₃₉	1	3	yes	—
45	C₄²	—	239	6	no	—
46	C₄²	C₂×D₂₃₉	1	3	yes	—
47	Q₈	—	478	6	no	—
48	Q₈	Dic₄₇₈	1	4	yes	complemented
49	C₂×C₄	—	239	2	no	—
50	C₂×C₄	—	239	6	no	—
51	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
52	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
53	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
54	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	478	6	no	—
57	C₄	—	478	2	no	—
58	C₄	C₂×Dic₄₇₈	1	6	yes	—
59	C₄	#16838549	1	2	yes	—
60	C₂	C₂²×Dic₄₇₈	1	1	yes	characteristic
61	C₂	C₂×#16838549	1	1	yes	characteristic
62	C₂	#21440007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₉
commutator, socle, prime radical, cosocle	C₂²×C₂₃₉	C₂⁴
Fitting	C₄×Q₈×C₂₃₉	C₂
hypercenter	C₄×Q₈	D₂₃₉
hypocenter	C₂₃₉	#557
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₂₃₉	Q₈	semidirect
31. C₂₃₉	#557	semidirect
48. Q₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#21439960	2
#21439951	2
Q₈×Dic₂₃₉	2
C₄×Dic₄₇₈	2
C₄×Q₈×C₂₃₉	2
#557	239