Subgroups — #27159416

Order 12448.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261967	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261956	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261955	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261953	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261949	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₇₇₈	—	2	1	no	—
9	Dic₇₇₈	C₂²	1	1	yes	characteristic
10	D₁₅₅₆	—	2	1	no	—
11	D₁₅₅₆	C₂²	1	1	yes	characteristic
12	C₄×D₃₈₉	—	2	1	no	—
13	C₄×D₃₈₉	—	2	1	no	—
14	C₄×D₃₈₉	C₂²	1	1	yes	characteristic
15	#15691005	C₂²	1	1	yes	characteristic
16	Q₈×C₃₈₉	C₂²	1	1	yes	characteristic
17	Q₈×C₃₈₉	C₂²	1	1	yes	characteristic
18	C₈×C₃₈₉	C₂²	1	1	yes	characteristic
19	C₂×D₃₈₉	—	2	1	no	—
20	C₂×D₃₈₉	D₄	1	1	yes	characteristic
21	C₄×C₃₈₉	—	2	1	no	—
22	C₄×C₃₈₉	—	2	1	no	—
23	C₄×C₃₈₉	C₂³	1	1	yes	characteristic
24	Dic₃₈₉	—	2	1	no	—
25	Dic₃₈₉	D₄	1	1	yes	characteristic
26	C₂×C₃₈₉	C₂×D₄	1	1	yes	characteristic
27	D₃₈₉	—	2	1	no	—
28	D₃₈₉	—	4	1	no	—
29	C₃₈₉	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	389	1	no	maximal, Sylow
31	#41	—	389	1	no	—
32	C₂×Q₈	—	389	1	no	—
33	Q₁₆	—	389	1	no	—
34	Q₁₆	D₃₈₉	1	1	yes	characteristic, complemented
35	SD₁₆	—	389	1	no	—
36	SD₁₆	—	389	1	no	—
37	M₁₆	—	389	1	no	—
38	Q₈	—	778	1	no	—
39	Q₈	—	389	1	no	—
40	Q₈	C₂×D₃₈₉	1	1	yes	characteristic
41	Q₈	C₂×D₃₈₉	1	1	yes	characteristic
42	D₄	—	778	1	no	—
43	D₄	—	389	1	no	—
44	C₂×C₄	—	778	1	no	—
45	C₂×C₄	—	778	1	no	—
46	C₂×C₄	—	389	1	no	—
47	C₈	—	389	1	no	—
48	C₈	C₂×D₃₈₉	1	1	yes	characteristic
49	C₂²	—	778	1	no	—
50	C₂²	—	389	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	778	1	no	—
54	C₄	—	389	1	no	—
55	C₄	C₂²×D₃₈₉	1	1	yes	characteristic
56	C₂	—	778	1	no	—
57	C₂	—	1556	1	no	—
58	C₂	D₄×D₃₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₈₉
commutator, cosocle	C₄×C₃₈₉	C₂³
Frattini	C₄	C₂²×D₃₈₉
Fitting	Q₁₆×C₃₈₉	C₂
socle	C₂×C₃₈₉	C₂×D₄
prime radical	#20261967	C₂
hypercenter	Q₁₆	D₃₈₉
hypocenter	C₃₈₉	#137
LCS	G ▷ C₄×C₃₈₉ ▷ C₂×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₃₈₉	C₂	semidirect
3. #20261956	C₂	semidirect
4. #20261955	C₂	semidirect
5. #20261953	C₂	semidirect
6. #20261949	C₂	semidirect
7. Q₁₆×C₃₈₉	C₂	semidirect
29. C₃₈₉	#137	semidirect
34. Q₁₆	D₃₈₉	semidirect

Maximal subgroups

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

subgroup	index
#20261967	2
Q₈×D₃₈₉	2
#20261956	2
#20261955	2
#20261953	2
#20261949	2
Q₁₆×C₃₈₉	2
#137	389