Subgroups — #27159367

Order 12448.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₃₈₉	C₂	1	1	yes	maximal, characteristic
2	#20261945	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261943	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₈₉	C₂²	1	1	yes	characteristic
7	C₂²⋊D₃₈₉	—	2	2	no	—
8	C₂×Dic₃₈₉	C₂²	1	2	yes	—
9	C₂×Dic₃₈₉	C₂²	1	1	yes	characteristic
10	C₂³×C₃₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃₈₉	—	2	2	no	—
12	C₂×C₄×C₃₈₉	C₂²	1	2	yes	—
13	C₂²×C₃₈₉	—	2	2	no	—
14	C₂²×C₃₈₉	D₄	1	2	yes	—
15	C₂²×C₃₈₉	C₂³	1	1	yes	characteristic
16	C₂×D₃₈₉	—	2	1	no	—
17	C₂×D₃₈₉	—	2	2	no	—
18	C₄×C₃₈₉	—	2	2	no	—
19	Dic₃₈₉	—	2	1	no	—
20	Dic₃₈₉	—	2	2	no	—
21	C₂×C₃₈₉	—	2	2	no	—
22	C₂×C₃₈₉	C₂×D₄	1	1	yes	characteristic
23	C₂×C₃₈₉	#41	1	2	yes	—
24	D₃₈₉	—	4	1	no	—
25	C₃₈₉	#123	1	1	yes	characteristic, complemented, Sylow
26	#123	—	389	1	no	maximal, Sylow
27	C₂×D₄	—	389	1	no	—
28	C₂²×C₄	D₃₈₉	1	1	yes	characteristic, complemented
29	C₄⋊C₄	—	389	2	no	—
30	C₂²⋊C₄	—	389	1	no	—
31	C₂²⋊C₄	—	389	2	no	—
32	C₂³	—	389	1	no	—
33	C₂³	C₂×D₃₈₉	1	1	yes	characteristic
34	D₄	—	778	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	389	2	no	—
37	C₂×C₄	—	389	1	no	—
38	C₂×C₄	C₂×D₃₈₉	1	2	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	—	778	1	no	—
41	C₂²	—	778	2	no	—
42	C₂²	C₂²⋊D₃₈₉	1	2	yes	—
43	C₂²	C₂²×D₃₈₉	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	778	1	no	—
46	C₄	—	778	2	no	—
47	C₂	—	2	2	no	—
48	C₂	—	1556	1	no	—
49	C₂	C₂×C₂²⋊D₃₈₉	1	1	yes	characteristic
50	C₂	#20261966	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₈₉
commutator, socle, cosocle	C₂²×C₃₈₉	C₂³
Fitting	C₂²×C₄×C₃₈₉	C₂
prime radical	C₂×C₂²⋊D₃₈₉	C₂
hypercenter	C₂²×C₄	D₃₈₉
hypocenter	C₃₈₉	#123
LCS	G ▷ C₂²×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #20261945	C₂	semidirect
3. C₂²⋊Dic₃₈₉	C₂	semidirect
4. #20261943	C₂	semidirect
5. C₂²×C₄×C₃₈₉	C₂	semidirect
25. C₃₈₉	#123	semidirect
28. C₂²×C₄	D₃₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₃₈₉	2
#20261945	2
C₂²⋊Dic₃₈₉	2
#20261943	2
C₂²×C₄×C₃₈₉	2
#123	389