Subgroups — #37556581

Order 23648.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊Dic₇₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×C₇₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₃₉	—	2	1	no	—
5	C₇₃₉⋊C₈	—	2	1	no	—
6	C₂³×C₇₃₉	—	2	1	no	—
7	D₄×C₇₃₉	—	2	1	no	—
8	D₄×C₇₃₉	C₄	1	2	yes	complemented
9	C₂×C₄×C₇₃₉	C₂²	1	1	yes	characteristic
10	C₂²×C₇₃₉	—	2	2	no	—
11	C₂²×C₇₃₉	—	4	1	no	—
12	C₂²×C₇₃₉	—	4	1	no	—
13	C₂²×C₇₃₉	D₄	1	1	yes	characteristic
14	C₄×C₇₃₉	C₂×C₄	1	1	yes	characteristic
15	C₄×C₇₃₉	D₄	1	1	yes	characteristic
16	Dic₇₃₉	—	4	1	no	—
17	C₂×C₇₃₉	—	4	2	no	—
18	C₂×C₇₃₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₇₃₉	D₈	1	1	yes	characteristic
20	C₂×C₇₃₉	SD₁₆	1	1	yes	characteristic
21	C₇₃₉	#102	1	1	yes	characteristic, complemented, Sylow
22	#102	—	739	1	no	maximal, Sylow
23	C₂×D₄	D₇₃₉	1	1	yes	characteristic
24	C₂×C₈	—	739	1	no	—
25	C₄⋊C₄	—	739	1	no	—
26	C₂³	—	2	1	no	—
27	D₄	—	2	1	no	—
28	D₄	Dic₇₃₉	1	2	yes	complemented
29	C₂×C₄	—	1478	1	no	—
30	C₂×C₄	C₂×D₇₃₉	1	1	yes	characteristic
31	C₈	—	1478	1	no	—
32	C₂²	—	2	2	no	—
33	C₂²	—	4	1	no	—
34	C₂²	—	4	1	no	—
35	C₂²	C₂²⋊D₇₃₉	1	1	yes	characteristic
36	C₄	—	2956	1	no	—
37	C₄	C₂×Dic₇₃₉	1	1	yes	characteristic
38	C₄	C₂²⋊D₇₃₉	1	1	yes	characteristic
39	C₂	—	4	2	no	—
40	C₂	C₂²⋊Dic₇₃₉	1	1	yes	characteristic
41	C₂	#26044766	1	1	yes	characteristic
42	C₂	#26044769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊D₇₃₉
commutator	C₄×C₇₃₉	C₂×C₄
Frattini	C₂×C₄	C₂×D₇₃₉
Fitting, prime radical	C₂×D₄×C₇₃₉	C₂
socle	C₂²×C₇₃₉	D₄
cosocle	C₂×C₄×C₇₃₉	C₂²
hypercenter	C₂×D₄	D₇₃₉
hypocenter	C₇₃₉	#102
LCS	G ▷ C₄×C₇₃₉ ▷ C₂×C₇₃₉ ▷ C₇₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₇₃₉⋊C₈	C₂	semidirect
2. C₄⋊Dic₇₃₉	C₂	semidirect
8. D₄×C₇₃₉	C₄	semidirect
21. C₇₃₉	#102	semidirect
28. D₄	Dic₇₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇₃₉⋊C₈	2
C₄⋊Dic₇₃₉	2
C₂×D₄×C₇₃₉	2
#102	739