Subgroups — #37564995

Order 24256.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃₇₉⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₃₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇₉⋊C₈	—	2	2	no	—
4	C₂×C₈×C₃₇₉	C₄	1	1	yes	characteristic
5	C₂×C₈×C₃₇₉	C₄	1	1	yes	characteristic
6	C₄²×C₃₇₉	C₂²	1	1	yes	characteristic
7	C₃₇₉⋊C₈	—	4	2	no	—
8	C₂×C₄×C₃₇₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₃₇₉	D₄	1	1	yes	characteristic
10	C₂×C₄×C₃₇₉	Q₈	1	1	yes	characteristic
11	C₈×C₃₇₉	—	2	1	no	—
12	C₈×C₃₇₉	C₈	1	2	yes	complemented
13	C₂²×C₃₇₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₇₉	—	2	1	no	—
15	C₄×C₃₇₉	C₂×C₈	1	1	yes	characteristic
16	C₄×C₃₇₉	M₁₆	1	1	yes	characteristic
17	C₄×C₃₇₉	SD₁₆	1	2	yes	—
18	C₂×C₃₇₉	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₃₇₉	#106	1	1	yes	characteristic
20	C₂×C₃₇₉	#108	1	1	yes	characteristic
21	C₃₇₉	#334	1	1	yes	characteristic, complemented, Sylow
22	#334	—	379	1	no	maximal, Sylow
23	C₄⋊C₈	—	379	2	no	—
24	C₄×C₈	D₃₇₉	1	1	yes	characteristic
25	C₂×C₈	—	758	2	no	—
26	C₂×C₈	Dic₃₇₉	1	1	yes	characteristic
27	C₂×C₈	Dic₃₇₉	1	1	yes	characteristic
28	C₄²	C₂×D₃₇₉	1	1	yes	characteristic
29	C₂×C₄	C₂×Dic₃₇₉	1	1	yes	characteristic
30	C₂×C₄	D₁₅₁₆	1	1	yes	characteristic
31	C₂×C₄	Dic₇₅₈	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	1516	2	no	—
34	C₈	C₃₇₉⋊C₈	1	2	yes	complemented
35	C₂²	C₄⋊Dic₃₇₉	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	C₂×C₃₇₉⋊C₈	1	1	yes	characteristic
38	C₄	#20258590	1	1	yes	characteristic
39	C₄	#20258595	1	2	yes	—
40	C₂	C₄⋊(C₃₇₉⋊C₈)	1	1	yes	characteristic
41	C₂	#27131263	1	1	yes	characteristic
42	C₂	#27131267	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₅₁₆
commutator	C₄×C₃₇₉	C₂×C₈
Frattini	C₄²	C₂×D₃₇₉
Fitting	C₄×C₈×C₃₇₉	C₂
socle, prime radical	C₂²×C₃₇₉	C₄⋊C₄
cosocle	C₄²×C₃₇₉	C₂²
hypercenter	C₄×C₈	D₃₇₉
hypocenter	C₃₇₉	#334
LCS	G ▷ C₄×C₃₇₉ ▷ C₂×C₃₇₉ ▷ C₃₇₉
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×C₃₇₉	C₈	semidirect
21. C₃₇₉	#334	semidirect
34. C₈	C₃₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃₇₉⋊C₈)	2
C₄×C₈×C₃₇₉	2
#334	379