Subgroups — #38733852

Order 24896.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15691005	C₂	1	2	yes	maximal
2	C₄×C₈×C₃₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15691005	—	2	2	no	—
4	C₂×C₈×C₃₈₉	C₄	1	1	yes	characteristic
5	C₂×C₈×C₃₈₉	C₄	1	1	yes	characteristic
6	C₄²×C₃₈₉	C₂²	1	1	yes	characteristic
7	#15691005	—	4	2	no	—
8	C₂×C₄×C₃₈₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₃₈₉	D₄	1	1	yes	characteristic
10	C₂×C₄×C₃₈₉	Q₈	1	1	yes	characteristic
11	C₈×C₃₈₉	—	2	1	no	—
12	C₈×C₃₈₉	C₈	1	2	yes	complemented
13	C₂²×C₃₈₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₈₉	—	2	1	no	—
15	C₄×C₃₈₉	C₂×C₈	1	1	yes	characteristic
16	C₄×C₃₈₉	M₁₆	1	1	yes	characteristic
17	C₄×C₃₈₉	D₈	1	1	yes	characteristic
18	C₄×C₃₈₉	Q₁₆	1	1	yes	characteristic
19	C₂×C₃₈₉	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₃₈₉	#107	1	1	yes	characteristic
21	C₂×C₃₈₉	#108	1	1	yes	characteristic
22	C₃₈₉	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	389	1	no	maximal, Sylow
24	C₄⋊C₈	—	389	2	no	—
25	C₄×C₈	D₃₈₉	1	1	yes	characteristic
26	C₂×C₈	—	778	2	no	—
27	C₂×C₈	Dic₃₈₉	1	1	yes	characteristic
28	C₂×C₈	Dic₃₈₉	1	1	yes	characteristic
29	C₄²	C₂×D₃₈₉	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₃₈₉	1	1	yes	characteristic
31	C₂×C₄	D₁₅₅₆	1	1	yes	characteristic
32	C₂×C₄	Dic₇₇₈	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	1556	2	no	—
35	C₈	#15691005	1	2	yes	complemented
36	C₂²	C₄⋊Dic₃₈₉	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×#15691005	1	1	yes	characteristic
39	C₄	#20261950	1	1	yes	characteristic
40	C₄	D₃₁₁₂	1	1	yes	characteristic
41	C₄	Dic₁₅₅₆	1	1	yes	characteristic
42	C₂	C₄⋊#15691005	1	1	yes	characteristic
43	C₂	#27159318	1	1	yes	characteristic
44	C₂	#27159320	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₅₅₆
commutator	C₄×C₃₈₉	C₂×C₈
Frattini	C₄²	C₂×D₃₈₉
Fitting	C₄×C₈×C₃₈₉	C₂
socle, prime radical	C₂²×C₃₈₉	C₄⋊C₄
cosocle	C₄²×C₃₈₉	C₂²
hypercenter	C₄×C₈	D₃₈₉
hypocenter	C₃₈₉	#335
LCS	G ▷ C₄×C₃₈₉ ▷ C₂×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₃₈₉	C₈	semidirect
22. C₃₈₉	#335	semidirect
35. C₈	#15691005	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15691005	2
C₄×C₈×C₃₈₉	2
#335	389